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System optimization for micron and sub-micron particle identification using spectroscopy-based techniques

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System optimization for micron and sub-micron particle identification using spectroscopy-based techniques
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Zurek, Eduardo
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Modeling
Signal processing
Compensation
Optical effects
Simulation
Dissertations, Academic -- Electrical Engineering -- Doctoral -- USF
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Abstract:
ABSTRACT: This dissertation describes an approach and a model for the analysis of critical parameters related to the optical and electronic components of spectroscopy systems. The model described herein enables a systematic study of the impact of these parameters on the total performance of the system; therefore, it is a tool for the design and optimization of spectrometers.Although the physics of the optical and electronic components in spectroscopy systems are known and well established, the systemic approach to the understanding of their interactions is recent and it is an area of active research. The results from this study are at several levels: from an engineering perspective, the method developed based on an integrated spectroscopy model enables not only the study of the interactions between the components of the spectrometer, but also the design and optimization of spectrometers for specific applications. From the signal analysis point of view, the understanding of the inter actions between components enables a better identification and filtering of the noise. From the applications point of view, the resulting integrated model enables the translation of data between different spectrometer systems through appropriate compensation algorithms.The approach followed in this dissertation is based on the integration of the models of each one of the components of a spectro-photometer: slit, grating, collimating elements, photo-detectors and analog-to-digital converters. An important contribution of this research has been the simplification of the diffraction grating model. The simplification of the diffraction grating model enables the implementation of a general spectrometer model with two important characteristics: first, it facilitates the analysis of the effect of the parameters of the spectrometer on the spectra readings; second, it allows a computational efficient simulation of the complete model of the spectrometer.The simplified spectrometer model present ed in this dissertation predicts the instrumental effects detected in observed spectra. The results obtained with the model are validated against measured spectra of polystyrene particles suspended in de-ionized water. It is demonstrated that the integrated spectrometer model is capable of representing all the instrumental effects identified.
Thesis:
Dissertation (Ph.D.)--University of South Florida, 2006.
Bibliography:
Includes bibliographical references.
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by Eduardo Zurek.
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Document formatted into pages; contains 118 pages.
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Includes vita.

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ABSTRACT: This dissertation describes an approach and a model for the analysis of critical parameters related to the optical and electronic components of spectroscopy systems. The model described herein enables a systematic study of the impact of these parameters on the total performance of the system; therefore, it is a tool for the design and optimization of spectrometers.Although the physics of the optical and electronic components in spectroscopy systems are known and well established, the systemic approach to the understanding of their interactions is recent and it is an area of active research. The results from this study are at several levels: from an engineering perspective, the method developed based on an integrated spectroscopy model enables not only the study of the interactions between the components of the spectrometer, but also the design and optimization of spectrometers for specific applications. From the signal analysis point of view, the understanding of the inter actions between components enables a better identification and filtering of the noise. From the applications point of view, the resulting integrated model enables the translation of data between different spectrometer systems through appropriate compensation algorithms.The approach followed in this dissertation is based on the integration of the models of each one of the components of a spectro-photometer: slit, grating, collimating elements, photo-detectors and analog-to-digital converters. An important contribution of this research has been the simplification of the diffraction grating model. The simplification of the diffraction grating model enables the implementation of a general spectrometer model with two important characteristics: first, it facilitates the analysis of the effect of the parameters of the spectrometer on the spectra readings; second, it allows a computational efficient simulation of the complete model of the spectrometer.The simplified spectrometer model present ed in this dissertation predicts the instrumental effects detected in observed spectra. The results obtained with the model are validated against measured spectra of polystyrene particles suspended in de-ionized water. It is demonstrated that the integrated spectrometer model is capable of representing all the instrumental effects identified.
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PAGE 1

System Optimization for Micron and Sub-Micron Parti cle Identification Using Spectroscopy-Based Techniques by Eduardo Zurek A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Electrical Engineering College of Engineering University of South Florida Co-Major Professor: Luis Garca-Rubio, Ph.D. Co-Major Professor: Wilfrido Moreno, Ph.D. Grisselle Centeno, Ph.D. Miguel Labrador, Ph.D. James Leffew, Ph.D. Jos Olivares, Ph.D. Date of Approval: May 5, 2006 Keywords: Modeling, Signal Processing, Compensation Optical Effects, Simulation Copyright 2006, Eduardo Zurek

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DEDICATION Esta disertacin esta dedicada a mis padres Eduardo y Adriana, a mi hija Daniela, a mi mami-abuela Emilce, a mi abuelita Julia (Q.E.P .D.), a mi hermano Leandro, y todos mis familiares: Gloria, Raquel, Orlando el viejo, G ustavo el viejo, Fortunato, Gustavo el joven, Zahide, Valentina, Orlando el joven, Yelenis Andrea, Hernando, Luis Felipe, Lucy y por supuesto a Dios. A mi amigo Homero San J uan, quien me recibi en su casa y me dio apoyo incondicional durante mis primeros ao s en Tampa. A mi amiga Cristina Jan, quien me ha respaldado durante la etapa final de este proceso y sin su apoyo no hubiese podido superar las dificultades que se han presentado.

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ACKNOWLEDGMENTS I would like to thank Dr. Luis Garca-Rubio, for hi s support, patience and guidance throughout the development of this dissert ation. I want to express my deepest gratitude to Dr. Wilfrido Moreno for giving me the opportunity to come to the University of South Florida. I also would like to thank the fo llowing institutions for sponsoring me during the process to obtain my doctorate: Universi dad del Norte (Uninorte, Barranquilla, Colombia), University of South Florida (USF), Los A lamos National Laboratory (LANL), Particle Engineering Research Center (PERC) Center for Disaster Management and Humanitarian Action (CDMHA), Ibero American Sci ence & Technology Educational Consortium (ISTEC).

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i TABLE OF CONTENTS LIST OF TABLES..................................... ................................................... .....................iii LIST OF FIGURES.................................... ................................................... ....................iv ABSTRACT........................................... ................................................... ........................xii 1. INTRODUCTION.................................... ................................................... .................1 1.1. Introduction.................................. ................................................... .....................1 1.2. Overview of Chapters.......................... ................................................... .............4 2. MATERIALS, METHODS AND EXPERIMENTAL RESULTS..... ..........................6 2.1. Overview of Spectrometers Used for Obtaining t he Experimental Results............................................ ................................................... ...................7 2.2. Theoretical Expected Spectra of Mono-Disperse Polystyrene Particles..............8 2.3. Experimental Spectra of Mono-Disperse Polystyr ene Particles..........................9 3. THE SPECTROMETER MODEL.......................... ................................................... 14 3.1. General Model of a Spectroscopy System........ .................................................15 3.2. Slit.......................................... ................................................... .........................16 3.3. Collimating Elements.......................... ................................................... ............17 3.4. Diffraction Grating........................... ................................................... ...............19 3.5. Photo-Detector Array and Analog-to-Digital-Con verter...................................21 3.6. Noise Analysis and Description................ ................................................... ......23 4. SIMPLIFICATION OF THE SPECTROMETER MODEL........ ...............................29 4.1. Simplification of the Spectrophotometer Model. ...............................................29 4.1.1. Slit........................................ ................................................... ...............31 4.1.2. Grating..................................... ................................................... ...........32 4.1.3. Photo-Detector Array and Analog-to-Digital-C onverter.......................33 4.2. Simplification of the Grating Model........... ................................................... ....35 5. COMPUTATIONAL IMPLEMENTATION OF THE SPECTROMETER MODEL.............................................. ................................................... .....................39 5.1. Representation of the Spectrometer Model Using Discrete Variables..............40 5.2. Computational Implementation.................. ................................................... ....42

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ii 6. EFFECTS OF MINIATURIZATION IN SPECTROSCOPY SYSTE MS.................44 6.1. Optical and Electronic Configuration.......... ................................................... ...45 6.2. Overview of the Effects Caused by the Optical Components............................46 6.3. Effect Caused by the Slit Size................ ................................................... .........52 6.4. Effect Caused by the Distance between the Slit and the Grating......................53 6.5. Effect Caused by the Distance between the Grat ing and the PhotoDetector Array..................................... ................................................... ...........55 6.6. Effect Caused by the Grating Configuration.... .................................................56 7. COMPENSATION OF INSTRUMENTAL EFFECTS IN SPECTROS COPY SYSTEMS............................................ ................................................... ....................61 7.1. Identify the Parameters of the Simplified Spec trometer Model........................63 7.2. Identify the Wavelength(s) of the Beam(s) Caus ing the Peaks.........................65 7.3. Calculate the Simulated Diffraction of the Bea m(s) Causing the Peaks...........66 7.4. Estimate the Effect of the Simulated Diffracti on on the Spectrum Being Compensated........................................ ................................................... ...........69 7.5. Subtract the Effect from the Spectrum Being Co mpensated.............................72 8. CONCLUSIONS..................................... ................................................... .................75 8.1. Conclusions................................... ................................................... ..................75 8.2. Contributions................................. ................................................... ..................76 8.3. Recommendations and Future Work............... ..................................................7 6 REFERENCES......................................... ................................................... .....................77 APPENDICES......................................... ................................................... ......................79 Appendix A: Spectra of Mono-Disperse Polystyrene Pa rticles Used for Calibration........................................ ................................................... 80 Appendix B: Simulation of the Model of the Spectrop hotometer..........................105 ABOUT THE AUTHOR................................... ................................................... .End Page

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iii LIST OF TABLES Table 2.1: De-Ionized Water Dilution for Each Polys tyrene Particle Size....................13 Table 6.1: Most Important Features of the Effect of the Diffraction Grating over a Light Beam with a Wavelength of 250 Nanometers... ...............................49 Table 6.2: Relation between the Grating to Photo-de tector Distance and the Photo-Detector Array Length........................ ...............................................55 Table 6.3: Relation between the Slope of the Reflec tive Surface of the Grating and the Size of the Photo-detector Array........... ...........................................57 Table 6.4: Relation between the Size of Each One of the Grooves in the Grating and the Size of the Photo-Detector Array........... ..........................................58 Table B.1: Theoretical Spectrum Used for the Simula tion...........................................115 Table B.2: Spectrum Obtained with the Ocean Optics HR2000..................................116

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iv LIST OF FIGURES Figure 1.1: Observed Spectra of 1.3 Micron Polysty rene Particles Suspended in De-Ionized Water – Spectrometers Agilent 8453 and O cean Optics HR2000............................................. ................................................... ...........3 Figure 2.1: Theoretical Spectra of Samples Mono-Di sperse Polystyrene Particles..........9 Figure 2.2: Spectra of Different Concentrations of Samples of 1.3 Micron Mono-Disperse Polystyrene Particles Suspended in De -Ionized Water – Spectrometers Ocean Optics HR2000 and Agile nt 8453................10 Figure 2.3: Spectra of Different Concentrations of Samples of 1.3 Micron Mono-Disperse Polystyrene Particles Suspended in De -Ionized Water – Spectrometers Ocean Optics USB2000 and Agil ent 8453..............11 Figure 2.4: Spectra of Different Concentrations of Samples of 1.3 Micron Mono-Disperse Polystyrene Particles Suspended in De -Ionized Water – Spectrometers Ocean Optics USB2000 and Ocea n Optics HR2000............................................. ................................................... .........11 Figure 2.5: Spectra of Different Concentrations of Samples of 1.3 Micron Mono-Disperse Polystyrene Particles Suspended in De -Ionized Water – Spectrometers Agilent 8453 and Perkin Elmer Lambda 900..........12 Figure 2.6: Spectra of 2 Micron Polystyrene Partic les Suspended in De-Ionized Water – Theoretical Spectrum and Spectrum Observed with the Spectrometer Agilent 8453.......................... .................................................13 Figure 3.1: General Structure of a Diffraction Gra ting Spectrometer.............................15 Figure 3.2: Diffraction Effect Caused by a Slit... ................................................... .........17

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v Figure 3.3: Collimating Effect Caused by a Converg ing Lens.......................................18 Figure 3.4: Collimating Effect Caused by a Concave Mirror.........................................19 Figure 3.5: Effect of the Diffraction Grating..... ................................................... ...........21 Figure 3.6: Spectral Response of Some Commercial P hoto-Detector Arrays.................23 Figure 3.7: Log10 of (Variance of Intensity) vs. L og10 of (Intensity)...........................25 Figure 3.8: Variance of Intensity vs. Intensity... ................................................... ..........25 Figure 3.9: Probability Density Function of the Li ght Intensity Measurement..............26 Figure 3.10: Relation between Variance and Number of Samples Averaged...................27 Figure 4.1: Simplified Model of a Spectrophotomete r.................................................. ..30 Figure 4.2: Effect of the Diffraction Grating Inte grated to the Simplified Model..........31 Figure 6.1: Theoretical Spectral Response of the P hotodiodes Used for the Simulations Presented in Chapter Six............... ............................................46 Figure 6.2: Diffraction of a Light Beam with a Wav elength of 250 Nanometers – Intensity Measured in a Linear Scale versus Positio n of the Diffracted Beams Measured in Centimeters........... ......................................47 Figure 6.3: Diffraction of a Light Beam with a Wav elength of 250 Nanometers – Intensity Measured in a Logarithmic Scale versus Po sition of the Diffracted Beams Measured in Centimeters........... ......................................47 Figure 6.4: Diffraction of a Light Beam with a Wav elength of 250 Nanometers – Intensity Measured in a Linear Scale versus Wavelen gth............................48 Figure 6.5: Diffraction of a Light Beam with a Wav elength of 250 Nanometers – Intensity Measured in a Logarithmic Scale versus Wa velength...................49 Figure 6.6: Theoretical and Experimentally Obtaine d Spectra of 1.3 Micron Mono-Disperse Polystyrene Particles................ ...........................................50

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vi Figure 6.7: Spectrum of the Reference Used For Sim ulation Presented in Chapter Six........................................ ................................................... .........51 Figure 6.8: Theoretical Spectrum of 1.3 Micron Mon o-Disperse Polystyrene Particles.......................................... ................................................... ............51 Figure 6.9: Simulated and Experimentally Obtained Spectra of 1.3 Micron Mono-Disperse Polystyrene Particles................ ...........................................52 Figure 6.10: Effect Caused by Changes in the Slit Width.............................................. ...53 Figure 6.11: Detail of the Effect Caused By Change s in the Slit Width...........................53 Figure 6.12: Effect Caused by Changes in the Distan ce between the Slit and the Grating............................................ ................................................... ...........54 Figure 6.13: Detail of the Effect Caused by Change s in the Distance between the Slit and the Grating............................... ................................................... .....54 Figure 6.14: Effect Caused by Changes in the Dista nce between the Grating and the Photo-Detector Array........................... ................................................... 56 Figure 6.15: Detail of the Effect Caused by Change s in the Distance between the Grating and the Photo-Detector Array............... ...........................................56 Figure 6.16: Effect Caused by Changes in the Angle the Slope of the Reflective Surface of Each Groove in the Grating.............. ...........................................57 Figure 6.17: Detail of the Effect Caused by Change s in the Angle of the Slope of the Reflective Surface of Each One of the Grooves i n the Grating..............58 Figure 6.18: Effect Caused by Changes in the Size of Each One of the Grooves in the Grating........................................ ................................................... .........59 Figure 6.19: Detail of the Effect Caused by Change s in the Size of Each One of the Grooves in the Grating......................... ................................................... 59 Figure 6.20: Effect Caused by Changes in the Size of the Reflective Surface of Each One of the Grooves in the Grating............. ..........................................60

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vii Figure 6.21: Detail of the Effect Caused by Change s in the Size of the Reflective Surface of Each One of the Grooves in the Grating.. ...................................60 Figure 7.1: Current Approach for Spectral Compensa tion..............................................6 2 Figure 7.2: New Approach for Compensation of Spect ral Features...............................62 Figure 7.3: Description of the Process Applied to Calculated the Spectra of 1.3 Micron Mono-Disperse Polystyrene Particles......... .....................................63 Figure 7.4: Theoretical Spectral Response of the P hotodiodes Used for the Simulation Presented in Chapter Seven.............. ..........................................65 Figure 7.5: Spectrum of the Reference Used for Sim ulation Presented in Chapter Seven.............................................. ................................................... ............66 Figure 7.6: Diffraction Generated by a Beam with a Wavelength of 210 Nanometers......................................... ................................................... .......67 Figure 7.7: Logarithm of the Diffraction Generated by a Beam with a Wavelength of 210 Nanometers....................... .............................................67 Figure 7.8: Diffraction Generated by a Beam with a Wavelength of 325 Nanometers......................................... ................................................... .......68 Figure 7.9: Logarithm of the Diffraction Generated by a Beam with a Wavelength of 325 Nanometers....................... .............................................68 Figure 7.10: Selection of Points Used for Interpol ation.............................................. ......69 Figure 7.11: Threshold Applied to the Spectrum of the Reference Used for the Simulation......................................... ................................................... .........70 Figure 7.12: Threshold Applied to the Spectrum of the Sample Used for the Simulation......................................... ................................................... .........70 Figure 7.13: Spectra in Logarithmic and Linear Sca le................................................. .....71 Figure 7.14: Estimation of the Diffraction Generat ed by a Beam with a Wavelength of 325 Nanometers: Subtraction Result... .................................72

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viii Figure 7.15: Estimation of the Diffraction Generat ed by a Beam with a Wavelength of 325 Nanometers: Estimated Peaks...... .................................73 Figure 7.16: Spectrum Compensated Subtracting the Estimated Peaks............................73 Figure 7.17: Absorption Spectrum after Correction of the Spectral Features...................74 Figure A.1: Spectra of 1 Micron Mono-Disperse Poly styrene Particles – Spectrometers Ocean Optics HR2000 and Agilent 8453. .............................80 Figure A.2: Spectra of 1 Micron Mono-Disperse Poly styrene Particles – Spectrometers Ocean Optics USB2000 and Ocean Optics HR2000............81 Figure A.3: Spectra of 1 Micron Mono-Disperse Poly styrene Particles – Spectrometers Agilent 8453 and Perkin-Elmer Lambda 900.......................81 Figure A.4: Spectra of 1 Micron Mono-Disperse Poly styrene Particles – Spectrum Observed with the Spectrometers Agilent 84 53 and Theoretical Spectrum............................... ................................................... ..82 Figure A.5: Spectra of 40 Nanometers Mono-Disperse Polystyrene Particles – Spectrometers Ocean Optics HR2000 and Agilent 8453. .............................83 Figure A.6: Spectra of 40 Nanometers Mono-Disperse Polystyrene Particles – Spectrometers Ocean Optics USB2000 and Ocean Optics HR2000............84 Figure A.7: Spectra of 40 Nanometers Mono-Disperse Polystyrene Particles – Spectrum Observed with the Spectrometers Agilent 84 53 and Theoretical Spectrum............................... ................................................... ..84 Figure A.8: Spectra of 150 Nanometers Mono-Dispers e Polystyrene Particles – Spectrometers Ocean Optics HR2000 and Agilent 8453. .............................85 Figure A.9: Spectra of 150 Nanometers Mono-Dispers e Polystyrene Particles – Spectrometers Ocean Optics USB2000 and Ocean Optics HR2000............86 Figure A.10: Spectra of 150 Nanometers Mono-Dispers e Polystyrene Particles – Spectrum Observed with the Spectrometers Agilent 84 53 and Theoretical Spectrum............................... ................................................... ..86

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ix Figure A.11: Spectra of 500 Nanometers Mono-Dispers e Polystyrene Particles – Spectrometers Ocean Optics HR2000 and Agilent 8453. .............................87 Figure A.12: Spectra of 500 Nanometers Mono-Dispers e Polystyrene Particles – Spectrometers Ocean Optics USB2000 and Ocean Optics HR2000............88 Figure A.13: Spectra of 500 Nanometers Mono-Dispers e Polystyrene Particles – Spectrum Observed with the Spectrometers Agilent 84 53 and Theoretical Spectrum............................... ................................................... ..88 Figure A.14: Spectra of 700 Nanometers Mono-Dispers e Polystyrene Particles – Spectrometers Ocean Optics HR2000 and Agilent 8453. .............................89 Figure A.15: Spectra of 700 Nanometers Mono-Dispers e Polystyrene Particles – Spectrometers Ocean Optics USB2000 and Ocean Optics HR2000............90 Figure A.16: Spectra of 700 Nanometers Mono-Dispers e Polystyrene Particles – Spectrum Observed with the Spectrometers Agilent 84 53 and Theoretical Spectrum............................... ................................................... ..90 Figure A.17: Spectra of 1.3 Micron Mono-Disperse Po lystyrene Particles – Spectrometers Ocean Optics HR2000 and Agilent 8453. .............................91 Figure A.18: Spectra of 1.3 Micron Mono-Disperse Po lystyrene Particles – Spectrometers Ocean Optics USB2000 and Ocean Optics HR2000............92 Figure A.19: Spectra of 1.3 Micron Mono-Disperse Po lystyrene Particles – Spectrometers Agilent 8453 and Perkin-Elmer Lambda 900.......................92 Figure A.20: Spectra of 1.3 Micron Mono-Disperse Po lystyrene Particles – Spectrum Observed with the Spectrometers Agilent 84 53 and Theoretical Spectrum............................... ................................................... ..93 Figure A.21: Spectra of 2 Micron Mono-Disperse Poly styrene Particles – Spectrometers Ocean Optics HR2000 and Agilent 8453. .............................94 Figure A.22: Spectra of 2 Micron Mono-Disperse Poly styrene Particles – Spectrometers Ocean Optics USB2000 and Ocean Optics HR2000............95

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x Figure A.23: Spectra of 2 Micron Mono-Disperse Poly styrene Particles – Spectrometers Agilent 8453 and Perkin-Elmer Lambda 900.......................95 Figure A.24: Spectra of 2 Micron Mono-Disperse Poly styrene Particles – Spectrum Observed with the Spectrometers Agilent 84 53 and Theoretical Spectrum............................... ................................................... ..96 Figure A.25: Spectra of 4 Micron Mono-Disperse Poly styrene Particles – Spectrometers Ocean Optics HR2000 and Agilent 8453. .............................97 Figure A.26: Spectra of 4 Micron Mono-Disperse Poly styrene Particles – Spectrometers Ocean Optics USB2000 and Ocean Optics HR2000............98 Figure A.27: Spectra of 4 Micron Mono-Disperse Poly styrene Particles – Spectrometers Agilent 8453 and Perkin-Elmer Lambda 900.......................98 Figure A.28: Spectra of 4 Micron Mono-Disperse Poly styrene Particles – Spectrum Observed with the Spectrometers Agilent 84 53 and Theoretical Spectrum............................... ................................................... ..99 Figure A.29: Spectra of 9 Micron Mono-Disperse Poly styrene Particles – Spectrometers Ocean Optics HR2000 and Agilent 8453. ...........................100 Figure A.30: Spectra of 9 Micron Mono-Disperse Poly styrene Particles – Spectrometers Ocean Optics USB2000 and Ocean Optics HR2000..........101 Figure A.31: Spectra of 9 Micron Mono-Disperse Poly styrene Particles – Spectrometers Agilent 8453 and Perkin-Elmer Lambda 900.....................101 Figure A.32: Spectra of 9 Micron Mono-Disperse Poly styrene Particles – Spectrum Observed with the Spectrometers Agilent 84 53 and Theoretical Spectrum............................... ................................................... 102 Figure A.33: Spectra of 15 Micron Mono-Disperse Pol ystyrene Particles – Spectrometers Ocean Optics HR2000 and Agilent 8453. ...........................103 Figure A.34: Spectra of 15 Micron Mono-Disperse Pol ystyrene Particles – Spectrometers Ocean Optics USB2000 and Ocean Optics HR2000..........104

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xi Figure A.35: Spectra of 15 Micron Mono-Disperse Pol ystyrene Particles – Spectrum Observed with the Spectrometers Agilent 84 53 and Theoretical Spectrum............................... ................................................... 104

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xii SYSTEM OPTIMIZATION FOR MICRON AND SUB-MICRON PARTI CLE IDENTIFICATION USING SPECTROSCOPY-BASED TECHNIQUES EDUARDO ZUREK ABSTRACT This dissertation describes an approach and a model for the analysis of critical parameters related to the optical and electronic co mponents of spectroscopy systems. The model described herein enables a systematic study o f the impact of these parameters on the total performance of the system; therefore, it is a tool for the design and optimization of spectrometers. Although the physics of the optical and electronic components in spectroscopy systems are known and well established, the systemi c approach to the understanding of their interactions is recent and it is an area of a ctive research. The results from this study are at several levels: from an engineering perspect ive, the method developed based on an integrated spectroscopy model enables not only the study of the interactions between the components of the spectrometer, but also the design and optimization of spectrometers for specific applications. From the signal analysis point of view, the understanding of the

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xiii interactions between components enables a better id entification and filtering of the noise. From the applications point of view, the resulting integrated model enables the translation of data between different spectrometer systems thro ugh appropriate compensation algorithms. The approach followed in this dissertation is based on the integration of the models of each one of the components of a spectro-p hotometer: slit, grating, collimating elements, photo-detectors and analog-to-digital con verters. An important contribution of this research has been the simplification of the di ffraction grating model. The simplification of the diffraction grating model ena bles the implementation of a general spectrometer model with two important characteristi cs: first, it facilitates the analysis of the effect of the parameters of the spectrometer on the spectra readings; second, it allows a computational efficient simulation of the complet e model of the spectrometer. The simplified spectrometer model presented in this dissertation predicts the instrumental effects detected in observed spectra. The results obtained with the model are validated against measured spectra of polystyrene p articles suspended in de-ionized water. It is demonstrated that the integrated spect rometer model is capable of representing all the instrumental effects identifie d.

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1 1. INTRODUCTION 1.1. Introduction Analyzing a spectrometer from systemic approach imp lies identifying its most important components, modeling each one of these co mponents, and integrating these models into a general system. The fundamental eleme nts of a spectrometer are: a light entrance slit, a diffraction element, and a device to measure the intensity of the diffracted light. A spectrometer has to be designed for a spec ific set of spectroscopy requirements considering the kind of particles in suspension tha t has to be analyzed. These requirements are defined by the wavelength range of the spectra used for the analysis, and the ability to distinguish a feature associated to a specific wavelength. This ability to discern a feature is related to the optical resolut ion of a spectrometer; and, the wavelength range is related to the nature of the light source used for the spectral analysis, which can be VIS (visible) light, UV (ultra-violet) light, UV -VIS light, or IR (infra-red) light. When the requirements cannot be fulfilled due to the phy sical limitations of the fundamental elements, additional optical elements (i.e. lens, m irrors, and filters) are required to collimate the light. An important contribution of this research is to st ate a systemic approach to analyze the critical parameters related to the opti cal and electronic components of

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2 spectroscopy systems, and to study the impact of th ese parameters in the total performance of the system. Although the physics of the optical and electronic components in spectroscopy systems are known and we ll established, the application of a systemic approach to the understanding of their int eractions is recent [1, 2] and it is an area of active research [3]. The spectra shown in F igure 1.1 1 (in page 3) depict the differences between the readings obtained using two different spectrometers; the curves shown in that figure are the observed spectra of 1. 3 microns mono-disperse polystyrene particles suspended in de-ionized water; the concen tration of particles used for the sample is 1:100 in volume units. Figure 1.1 shows both the spectrum obtained using a spectrometer Ocean Optics HR2000 and the spectrum o btained with a spectrometer Agilent 8453. As result of this research, software has been devel oped based on an integrated spectroscopy model. This software enables not only the study of the interactions between the components of the spectrometer, but also the de sign and optimization of spectrometry systems for specific applications. A clear understa nding of the interactions between components facilitates the identification and filte ring of the noise inherent to the system. Other important contribution of this research is al gorithm that determines the instrumental effects of a given spectrometer. This algorithm accounts for the apparently spurious peaks remarkably present in some observed spectra; such peaks are often attributed to the light source. 1 The results presented in this dissertation are re lated to measurements of extinction of light, but t he scale of Optical Density in absorbance units allows a bet ter perception of some features commonly seen in spectroscopy. Chapter two provides further explanat ion about the optical density units.

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3 250 300 350 400 450 500 550 600 650 700 750 800 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Wavelength in nanometersOD in absorbance units Agilent 8453HR2000 Figure 1.1: Observed Spectra of 1.3 Micron Polystyr ene Particles Suspended in DeIonized Water – Spectrometers Agilent 8453 and Ocea n Optics HR2000 The spectrum of a given sample of particles in susp ension is affected by two physical phenomena: the instrument used to acquire the spectra, and the optical properties of the suspended particles that are being analyzed [4]. Having the ability to determine the instrumental effects of a given spectrometer has a sensitive impact on the spectroscopy related areas because new technologies on the devel opment of spectroscopy systems are aimed to the reduction of size and cost of this kin d of instruments [5]. Smaller and cheaper instruments are limited in their accuracy a nd applications; but having the ability to remove instrumental effects lead to increase the possibilities for applications of affordable spectroscopy tools.

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4 1.2. Overview of Chapters The contents of the chapters of this dissertation c an be summarized as follows. Chapter two describes the materials and methods use d to validate the model described in chapters three and four. The results obtained with the model are validated against measured spectra of polystyrene particles suspended in de-ionized water. The selection of suspended particles for validation is based on the fact that particles in suspension accentuate the optical effects of the spectrometer components on the observed spectra; and, the use of well-characterized standards facili tates the identification of these effects. Three commercial UV-VIS spectrophotometers were use d to measure the spectra of the mixtures: Agilent 8453, Ocean Optics HR2000, Ocean Optics USB2000, and a PerkinElmer Lambda 900. Chapter three describes a general model of a spectr oscopy system. Each one of the fundamental elements required to build a spectr ometer is discussed: slit, diffraction grating, photodetector array, and collimating eleme nts. Chapter four describes the simplification of the general model of a spectromet er. The first simplification considers the waveguide concave mirrors as gain functions; th is reduces the number of variables required for the analysis. The second simplificatio n deals with the grating model and the fact that when the number of grating grooves is lar ge, a pulse train can approximate the Dirichlet function used to model the grating. Chapter five discusses the computational implementa tion and requirements for the simulation of the spectroscopy model. Chapter six d eals with the application of the simulation developed in chapter five to study the e ffects of miniaturization on

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5 spectroscopy systems; this study focuses on the ana lysis of the interaction of the lightsource peaks and the grating; this study explains h ow this interaction affects the observed absorption spectra of the particles. Chapter seven describes a linearization methodology for modeling the instrumental effects, and it also presents an algor ithm for compensating such effects. Chapter eight presents the conclusions, contributio ns, recommendations and future work derived from this research.

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6 2. MATERIALS, METHODS AND EXPERIMENTAL RESULTS As stated in chapter one, a spectrometer has to be designed for a specific set of spectroscopy requirements considering the kind of p articles in suspension that has to be analyzed. The results obtained with the model descr ibed in chapters three and four are validated against measured spectra of polystyrene p articles suspended in de-ionized water. The selection of suspended particles for val idation is based on the fact that particles in suspension accentuate the optical effe cts of the spectrometer components on the observed spectra; and, the use of well-characte rized standards facilitates the identification of these effects. Four different spe ctrometers were used to obtain the experimental results: an Ocean Optics HR2000, an Oc ean Optics USB2000, an Agilent 8453, and Perkin-Elmer Lambda 900. Section 2.1 over views these spectrometers. Section 2.2 describes the theoretical expected spectra of s ome mono-disperse polystyrene particles. Section 2.3 discusses and compares the r esult obtained with each one of the spectrometers when measuring the spectra of 1.3 mic ron mono-disperse polystyrene particles in de-ionized water. The description of spectra of particles in suspensi on usually requires two measurements: transmission and scattering. Scatteri ng occurs when light interacts with a particle; after the interaction, scattering measure ments indicate how much of the light

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7 leave the particle at any direction. Transmission m easurements, on the other hand, indicate how much of the light leaves the particle in the forward direction. Transmission measurements do not discriminate betwe en the light absorbed by the particle and the light scattered-forward by the particle; it provides a total result of the interaction of this two factors. In order to separa te the measurements of forward scattering and absorption, two measurements have be en used in literature: turbidity or optical depth, and optical density. Turbidity has b een usually applied to describe the attenuation caused by forward scattering; and, opti cal density has been usually applied to describe the attenuation caused by absorption. As suggested in [6], two assumptions can be made in order to clarify the notation used in this dissertation: first, turbidity measure s the total attenuation caused by forwardscattering and absorption; second, turbidity will b e measured in absorption units per path length, which are the units used to describe optica l density. 2.1. Overview of Spectrometers Used for Obtaining the Ex perimental Results The experimental spectra presented in section 2.3 a nd used through this dissertation have been obtained with four UV-VIS sp ectrophotometers: an Ocean Optics HR2000, an Ocean Optics USB2000, an Agilent 8453, a nd a Perkin-Elmer Lambda 900. The Ocean Optics spectrometers use flat blazed diff raction grating. These spectrometers are equipped with photo-detector arra ys with 2048 elements. Some elements2 of these spectrometers can be customized: the slit size, the collimating 2 Further explanation about the fundamental elements of a spectrometer is presented in Chapter Three.

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8 elements, and the number of grating groove elements per millimeter. Some characteristics of the optical bench of these spectrometers cannot be customized: the distance between the slit and the grating, and the distance between the grating and the photo-detector array. The wavelength range of these spectrometers depends on the customization, but it is usually between 200 nanometers and 1100 nanometers [7, 8]. An important fact is that the Ocean Optics’ spectrophotometers are portable a nd inexpensive when compared to the Agilent 8453 or Perkin-Elmer Lambda 900. The Agilent 8453 uses a concave holographic grating and a photo-detector array that has 1024 elements. The wavelength range of thi s spectrometer is from 190 nanometers to 1100 nanometers. This spectrometer is equipped with slit width of 1 nanometer [9]. The Perkin-Elmer Lambda 900 is a double-beam and do uble-monochromator UV/VIS/NIR spectrophotometer based on holographic g ratings. The wavelength range of this spectrometer is from 175 nanometers to 3300 na nometers [10]. 2.2. Theoretical Expected Spectra of Mono-Disperse Polys tyrene Particles The theoretical expected spectra used herein has be en generated using a software implementation of the model proposed in [11]. This model is based on the assumption that the light extinction caused by spherical parti cles can be described using Mie theory3. This theory has no restrictions in regard of the si ze of the particles and the values of the refractive index. Particles whose size is in the ra nge from 0.5 micron to 10 micron 3 Appendix A describes the fundamentals of Mie theor y.

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9 usually forward scatter light; turbidity measuremen ts applied to this kind of particles will count for both: attenuation due to absorption and s cattered light. Figure 2.1 shows the theoretical expected spectra o f some samples of monodisperse polystyrene particles. In the ideal case, in a sample containing mono-disperse particles of a given size, each one of the particle s should be of the same size. In an experimental situation, an average size and an inte rval bounding4 describe a sample of mono-disperse particles. Usually, the size and the interval describe 99% of the particles in the sample. 200 300 400 500 600 700 800 900 0.5 1 1.5 2 2.5 3 3.5 4 Theoretical Expected Spectra Of Polystyrene Particl es Wavelength in nanometersOD in absorbance units 700 nm 1.3 mm 2 mm Figure 2.1: Theoretical Spectra of Samples Mono-Dis perse Polystyrene Particles 2.3. Experimental Spectra of Mono-Disperse Polystyrene P articles Different concentrations of mono-disperse 1.3 micro n polystyrene particles generate the same spectral pattern, but, with the a mplitude being proportional to the 4 For example in a sample of mono-disperse particles described by 700 nanometers 0.5%, the average size of the particles is 700 nanometers and the siz e of 99% of the particles is in the interval from 6 96.5 nanometers to 703.5 nanometers.

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10 concentration. Three important differences are noti ceable in the curves shown in Figure 2.2; first, the spectra obtained with the Ocean Opt ics HR2000 spectrophotometer contains peaks 5 that are not present in the spectra obtained with the Agilent spectrophotometer; second, the values of the spectra obtained with the Agilent spectrophotometer are slightly smaller than the values obtained with the HR2000 fo r the same suspension; and, third, at short wavelengths (wavelength < 230 nanometers), th e differences between the spectra are accentuated. 250 300 350 400 450 500 550 600 650 700 750 800 0.4 0.6 0.8 1 1.2 1.4 1.6 HR2000 (lighter lines) and Agilent8453 (darker line s) Wavelength in nanometersOD in absorbance units1:100 1:175 1:250 peaks 250 300 350 400 450 500 550 600 650 700 750 800 0.4 0.6 0.8 1 1.2 1.4 1.6 HR2000 (lighter lines) and Agilent8453 (darker line s) Wavelength in nanometersOD in absorbance units 250 300 350 400 450 500 550 600 650 700 750 800 0.4 0.6 0.8 1 1.2 1.4 1.6 HR2000 (lighter lines) and Agilent8453 (darker line s) Wavelength in nanometersOD in absorbance units1:100 1:175 1:250 peaks Figure 2.2: Spectra of Different Concentrations of Samples of 1.3 Micron Mono-Disperse Polystyrene Particles Suspended in De-Ionized Water – Spectrometers Ocean Optics HR2000 and Agilent 8453 Figure 2.3 shows the spectra obtained with the Agil ent 8453 and Ocean Optics USB2000. Figure 2.4 shows the spectra obtained with the Ocean Optics USB2000 and the Ocean Optics HR2000. Figure 2.5 shows the spect ra obtained with the Agilent 8453 and the Perkin-Elmer Lambda 900. 5 Chapter Six describes a detailed explanation about how these spurious peaks and other effects are generated.

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11 250 300 350 400 450 500 550 600 650 700 750 800 0.4 0.6 0.8 1 1.2 1.4 1.6 USB2000 (lighter lines) and Agilent8453 (darker lin es) Wavelength in nanometersOD in absorbance units1:100 1:175 1:250 peaks 250 300 350 400 450 500 550 600 650 700 750 800 0.4 0.6 0.8 1 1.2 1.4 1.6 USB2000 (lighter lines) and Agilent8453 (darker lin es) Wavelength in nanometersOD in absorbance units 250 300 350 400 450 500 550 600 650 700 750 800 0.4 0.6 0.8 1 1.2 1.4 1.6 USB2000 (lighter lines) and Agilent8453 (darker lin es) Wavelength in nanometersOD in absorbance units1:100 1:175 1:250 peaks Figure 2.3: Spectra of Different Concentrations of Samples of 1.3 Micron Mono-Disperse Polystyrene Particles Suspended in De-Ionized Water – Spectrometers Ocean Optics USB2000 and Agilent 8453 Figure 2.4: Spectra of Different Concentrations of Samples of 1.3 Micron Mono-Disperse Polystyrene Particles Suspended in De-Ionized Water – Spectrometers Ocean Optics USB2000 and Ocean Optics HR2000

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12 200 300 400 500 600 700 800 900 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Spectra Of 1.3 Micron Mono-disperse Polystyrene Par ticles Wavelength in nanometersOD in absorbance units Agilent, 1:175 Perkin-Elmer, 1:175Agilent, 1:250 Perkin-Elmer, 1:250 Figure 2.5: Spectra of Different Concentrations of Samples of 1.3 Micron Mono-Disperse Polystyrene Particles Suspended in De-Ionized Water – Spectrometers Agilent 8453 and Perkin Elmer Lambda 900 Figure 2.6 shows the theoretical spectrum of 2 micr on mono-disperse polystyrene (1:50) particles compared to the spectrum obtained with the Agilent 8453 spectrophotometer. The spectra obtained with the sp ectrophotometer have undulations and peaks that are not present in the theoretical s pectra, and there is a difference between the wavelength location of the theoretical spectrum and the experimental spectrum. Therefore, the data should be analyzed having in ac count the spectrometer configuration and the optical effects of the particles.

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13 200 300 400 500 600 700 800 900 0.4 0.5 0.6 0.7 0.8 0.9 1 Spectra Of 2 Micron Mono-disperse Polystyrene Parti cles (1:50) Wavelength in nanometersOD in absorbance units Agilent Theoretical Figure 2.6: Spectra of 2 Micron Polystyrene Particl es Suspended in De-Ionized Water – Theoretical Spectrum and Spectrum Observed with the Spectrometer Agilent 8453 The particle sizes and suspension concentrations us ed to obtain the validation spectra are summarized in Table 2.1. The measured a nd theoretical spectra for all the particles suspensions are presented in Appendix B. Table 2.1: De-Ionized Water Dilution for Each Polys tyrene Particle Size 1 ml of Polystyrene particles is diluted in Di wate r with a proportion of: Polystyrene Particle Size 1:2 1:3 1:5 1:10 1:25 1:50 1:75 1:100 1:175 1:250 1 :500 1:1000 40 nm X X X X 150 nm X X 500 nm X X 700 nm X X X 1 m X X 1.3 m X X X 2 m X X X 4 m X X X 9 m X X 15 m X X X

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14 3. THE SPECTROMETER MODEL As indicated in chapter one, analyzing a spectromet er from systemic approach implies identifying its most important components, modeling each one of these components, and integrating these models into a gen eral system. This chapter describes a general model of a spectrometer, analyzes each one of its elements, and describes and analyzes the noise inherent to these elements. The elements analyzed here are: slit, collimating elements, diffraction grating, photo-de tector array and analog-to-digital converter. Section 3.1 overviews the spectrometer m odel. Section 3.2 presents the model of the Fraunhofer diffraction pattern generated by a slit; this pattern depends on the slit width and the wavelength of the light passing throu gh the slit. Two types of collimating elements are discussed in section 3.3: converging l ens and concave mirrors. Section 3.4 presents the model of the diffraction generated by a blazed grating; this type of grating can be described with four parameters: the number o f grating-grooves per millimeter, the total number of grooves, the width of the reflectin g part of each groove, and the angle between the reflecting part and the grating normal. Section 3.5 describes the model of a general photo-detector array; this model indicates that the transduction function of the photo-detector is not linear and that the photo-det ector response depends on the wavelength of the incident light.

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15 3.1. General Model of a Spectroscopy System The general structure of a diffraction grating base d spectrometer can be described as per Figure 3.1. Figure 3.1: General Structure of a Diffraction Grat ing Spectrometer The interaction between the elements indicated in F igure 3.1 can be described as follows: The light enters to the spectrometer through the Sl it, indicated with number (1) in Figure 3.1. A collimating lens and a filter usually accompany the slit entrance. A SMA (Sub-Miniature v ersion A) connector is required if the light is transported to the slit entrance via fiber optics. The light from the slit is collimated using a conca ve mirror that projects the light into the grating. This concave mirror is denoted with number (2) in Figure 3.1.

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16 The grating element diffracts the light collimated by the concave mirror. The grating is marked with number (3) in Figure 3.1 A focusing concave mirror, numbered (4) in Figure 3 .1, receives the light diffracted by the grating. This mirror projects the light into the photodetector array. Each element of the photo-detector array transduces the received light into electrical current. The electrical current is trans duced into a discrete numerical representation by an analog-to-digital co nverter. The discrete data are sent to a computer for further processing. The photo-detector array is indicated with number (5) in Figure 3.1. 3.2. Slit The effect of the slit over the light passing throu gh it is depicted by Figure 3.2. The slit diffracts the light and this effect can be described by equation 3.1 [1, 4, 12]: ( ) + = s L s G s Linput slit outputfor ) ( ) ( ) (l l l (3.1) Where: () + + = s z s s a Sinc z a s Gslitfor 2 ) (2 2 2l l lp (3.2) ) (linputL is the spectrum of the input light intensity, a is the slit size, l is the light wavelength that is being analyzed,

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17 z is the distance from the slit output to the surfac e is being where the light is being dispersed, And s as per Figure 3.2, is the variable indicating the reference axis for measuring the dispersion of the light. Equations (3.1) and (3.2) can also be represented a s follows: ( ) 2/ 2/ for ) ( ) ( ) (p p q l l q l q+ = L G Linput slit output (3.3) Where: ()2/ 2/ for ) ( 2 ) (2 2 2p p q q l l l qp+ + = Sine z s a Sinc z a Gslit (3.4) Equations (3.1), (3.2), (3.3) and (3.4) are related by the following equation[4, 12]: + =2 2z s s ArcSineq (3.5) Figure 3.2: Diffraction Effect Caused by a Slit 3.3. Collimating Elements The collimating lens attached to the slit is a conv erging lens, and this lens is meant to reduce the diffraction angle of the light diffracted by the slit. There are three

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18 possible cases for the position of the slit relativ e to the focal point of the lens. For the case shown in Figure 3.3, when the light rays trave l through the focal point before reaching the lens, they will exit the lens travelin g parallel to the principal axis of the lens. When the incident light rays travel through a point that is slightly beyond the focal point of the lens, i.e. p1 in Figure 3.3, they will exit the lens traveling toward the principal axis of the lens. When the incident light rays travel th rough a point near to the focal point and between the focal point and the lens, i.e. p2 in Fi gure 3.3, they will exit the lens traveling away of the principal axis of the lens. Figure 3.3: Collimating Effect Caused by a Convergi ng Lens The effect of the collimating mirrors, indicated wi th numbers (2) and (4) in Figure 3.1, is depicted in Figure 3.4. The incident rays t raveling parallel to the principal axis on the way to the mirror will pass through the focal p oint after reflection. When the incident rays come from a direction that points away of the principal axis of the mirror, the reflected rays will pass through a point beyond the point, i.e. p1 in Figure 3.4. Focal Point Light Rays Entering Into The Lens Light Rays Leaving The Lens Principal Axis p1 p2

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19 Figure 3.4: Collimating Effect Caused by a Concave Mirror 3.4. Diffraction Grating The grating, indicated with number (3) in Figure 3. 1, can be modeled as per the following equations [1, 4, 12]. ] ,0[ for ) ,' ( ) ,' )] ( ) ( [ ( ) (max mins s ds s L s z s G s Lincident g grating diffracted =l l l b l b l (3.6) Where: a l l b+ = n r) ( 2 2 ) ( C Sine d m ArcCos, (3.7) n r n r+ + = l s z d Diric s z w Sinc ) (s, Gg g grating, ) / ( 1 2 ) / ( 12 2 2 2l p lp, (3.8) ) ,'(ls Lincident describes the intensity of the incident light on t he grating surface. The parameters used in the grating model are:

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20 l is the wavelength that is being analyzed s is the variable indicating the reference axis over the surface where the diffracted light hit s is the variable indicating the reference axis perp endicular to the direction of the light incidence angle m is the diffraction order d is the groove spacing w is usually equal to 0.9 d l is the total number of grating grooves is the light incidence angle zg is the distance between the grating and the surfac e hit by the diffracted light C is the slope angle of the grating groove reflectiv e surface The Sinus Cardinalis function used for the slit and grating models is d efined as follows [13]: = = 0 for ) ( ) ( 0 for ,1 ) (2x x x Sin x x Sincp pp (3.9)

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21 The Dirichlet or Periodic Sinus Cardinalis function used for the grating model is defined as follows [13]: = = =Otherwise )2/ ( )2/ ( ,... 2 ,1 2 for )1 ( ) ()1 (x Sin n nx Sin k k x n x Diricn kp (3.10) Figure 3.5 illustrates the model defined by equatio ns (3.6) to (3.10). Figure 3.5: Effect of the Diffraction Grating 3.5. Photo-Detector Array and Analog-to-Digital-Converte r The photo-detector array, indicated with number (5) in Figure 3.1, can be modeled by the following equations [1]: ( ) D D - =max min)1 ( _) ( ) ( ) (l l gl l ls k s k ph incident ph output phd ds G s L k C (3.11)

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22 N k k Coutput ph,..., 1 for ), (_= D + Where: ) (_k Coutput ph is the output current of the kth photodetector, ) (_ls Lincident ph represents the intensity of the light of waveleng th l incident at the point s of the photodetector array surface, ) (lphG represents the photodetector spectral response for the wavelength l N is the number of detectors in the photodetector a rray, s indicates the reference axis over the photodetecto r array surface, s is the leght of each photodetector, g indicates the photodetector non-linearity and 97.0 > g And, ) (_k Coutput phD is the noise, which is Gaussian and is a function of ) (_k Coutput ph The photo-detector response function, ) (lphG, depends on the properties of the photo-detector. Figure 3.6 shows the ) (lphG function of some commercial photodetector arrays.

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23 Figure 3.6: Spectral Response of Some Commercial Ph oto-Detector Arrays The Analog-to-Digital quantization process is perfo rmed as per the following equation using a quantization step size of q [14]: q q k C k Coutput ph output ph n r = ) ( of part Integer )} ( {_ (3.12) For k = 1, ... N 3.6. Noise Analysis and Description The spectrum of a sample is affected by three impor tant factors: the light source characteristics, the photo-detector array propertie s, and the sample itself. The particles contained in the sample are affected by gravity and their own chemical properties; given these factors, the parti cles tend to stick together and to precipitate to the bottom of the sample-holder. In order to avoid this phenomenon the

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24 sample has to be stirred and its spectrum has to be measured when the particles are homogeneously distributed. The total noise generated by the light source and p hotodetector effects can be modeled as a random variable with normal distributi on and its variance can be calculated from a set of spectra samples [1]. The noise depend s on the measured intensity. For example, for a set of spectra measured with the Oce an Optics HR2000 with tungsten light-source, the variance of the noise is related to the intensity by the following equation: 0.90851 ) ( 0.86949 )) ( (10 10 = Intensity Log Intensity Var Log (3.13) Equation 3.13 can be written as follows: 0.12345 ) (0.86949Intensity Intensity Var = (3.14) Figure 3.7 illustrates equation (3.13), the thick l ine remarks the equation, and the thin lines describe the upper and lower limits cont aining the 95% of the analyzed data. Figure 3.8 illustrates equation (3.14).

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25 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 Log10 of (Intensity)Log10 of (Variance of Intensity)Logarithmic Analysis of the Variance of Intensity Figure 3.7: Log10 of (Variance of Intensity) vs. Lo g10 of (Intensity) 0 500 1000 1500 2000 2500 3000 3500 0 20 40 60 80 100 120 140 160 180 200 IntensityVariance of IntensityAnalysis of the Variance of Intensity Figure 3.8: Variance of Intensity vs. Intensity Figure 3.9 shows the histogram of the distribution of the light intensity around the mean and normalized to the standard deviation.

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26 -5 -4 -3 -2 -1 0 1 2 3 4 5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Probability Density Function of the Light Intensity Measurement Normalized IntensityPDF Histogram Normal PDF Figure 3.9: Probability Density Function of the Lig ht Intensity Measurement The curves shown in Figure 3.7, Figure 3.8 and Figu re 3.9 were made with signals obtained configuring the spectrometer to take one s pectrum every time. The Central Limit Theorem indicates [15] that the variance is r educed as the sample size is increased. The following equation explains the relation: Let X be a random variable with mean m and variance 2s Let Y be a random variable defined as: Y is equal to the mean of n samples of X Then: n n X Y2 of Variance of Variances= = (3.15) Figure 3.10 illustrates this relation.

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27 0 5 10 15 20 25 30 35 40 45 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Variance versus number of data averaged Number of data averagedVariance Figure 3.10: Relation between Variance and Number o f Samples Averaged The noise described in this section is caused by th e interaction of a set of noise sources that can be individually described as follo ws [4, 16, 17]: The number of photons per time-interval received by the photo-detector behaves in a stochastic manner that can be describe d using statistical models. The presence of these random variations in the measured signals is known as photon noise Electrical resistance is inherent to electrical det ectors. The effect of the thermal fluctuations of electrons in a resistance i s known as Johnson Noise Defect states at surfaces and interfaces at contact s lead the tangling and reemission of carriers. The noise related to this phe nomenon is named flicker or 1/f noise

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28 The noise associated with the creation and annihila tion of electron-hole pairs across the band gap is called generation-recombination noise Even on the absence of light, electrons are created inside the detector; the signal generated as the photodiode counts these ele ctrons is named dark current noise Dark current is generated by crystal defects and imperfections, impurities in the depletion region, and diffusion of carriers out of the high-resistivity sidewalls of the deplet ion region. Dark current is thermally activated [17].

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29 4. SIMPLIFICATION OF THE SPECTROMETER MODEL Chapter three overviews the spectrometer model and details each one of the model elements. This chapter describes an approach to simplify the spectrometer model. This approach can be divided in two parts. The firs t part deals with the reduction of the number of elements of the model. The second part de als with the simplification of the grating model and the integration of this simplifie d version into the total system. Section 4.1 describes the process of reducing the number el ements of the spectrometer model; this reduction is based on the assumption that the effect of the collimating elements can be neglected under certain conditions; this section also describes the interaction between the elements of the simplified model. Section 4.2 d escribes the simplification of the diffraction grating model; this simplification is b ased on an approximation of the Dirichlet function. 4.1. Simplification of the Spectrophotometer Model The model presented in chapter two has three elemen ts whose main function is to keep the light rays aligned to the principal axis o f the optical elements of the spectrophotometer. These three elements are: the le ns attached to the slit, the concave mirror that reflects the light coming from the slit and the concave mirror that reflects the

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30 diffracted light coming from the grating. The effec t of these elements can be neglected when two conditions are verified: first, the slit a perture is placed in front of the diffraction grating; and, second, the photo-detecto r array is placed in front of the diffraction grating normal to the direction of the refraction of the mean of the wavelengths incident to the grating. Figure 4.1 sho ws the simplified spectrophotometer model. Figure 4.1: Simplified Model of a Spectrophotometer In Figure 4.1, z1 represents the distance between the slit and the g rating, and z2 represents the distance between the grating and the photo-detector array. Figure 4.2 details the interaction between the optical element s, and the equations in subsections 4.1.1 through 4.1.3 describe the details.

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31 Figure 4.2: Effect of the Diffraction Grating Integ rated to the Simplified Model 4.1.1. Slit As indicated in section 3.2, the slit diffracts the light. When the spectrophotometer model is simplified as shown in F igure 4.1, the light diffracted by the slit reaches the diffraction grating. The slit outp ut curves shown in Figure 4.2 depict the shape of the intensity of then output of a light be am as it leaves the slit; a detailed description of the slit output curved is represente d by equation 4.1 and subsequent equations [1, 4, 12]: ( ) + = s L s G s Linput slit slit' for ) ( ) ,' ( ) ,' (l l l (4.1) Where: () + + = s z s s a Sinc z a s Gslit' for ' 2 ) ,' (2 1 2 2 1l l lp (4.2)

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32 The function ) (linputL denotes the spectrum of the input light intensity, The variable a represents the slit size, The variable l indicates the light’s wavelength that is being ana lyzed, The variable1z represents the distance from the slit output to th e grating’s surface, And s is the variable indicating the reference axis over the grating’s surface. 4.1.2. Grating The light diffracted by the slit reaches a blazed g rating. The blazed grating diffracts the light again with a diffraction angle that depends on the wavelength of the incident beam. The intensity of the light leaving t he grating is also affected and this effect can be described as the result of a convolution bet ween the incident beam and the grating function. These two phenomena are represented by eq uations 4.3, 4.4 and 4.5 [1, 4, 12]. ] ,0[ for ) ,' ( ) ,' )] ( ) ( [ ( ) (max min 2s s ds s L s z s G s Lslit grating diffracted =l l l b l b l (4.3) Where: a l l b+ = n r) ( 2 2 ) ( C Sine d m ArcCos, (4.4) n r n r+ + = l s z d Diric s z w Sinc ) (s,Ggrating, ) / ( 1 2 ) / ( 12 2 2 2 2 2l p lp, (4.5) ) (l b represents the diffraction angle and ) (s, Ggrating represents the grating function. The parameters used in the grating model are:

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33 The variable l represents the wavelength of the incident beam tha t is being analyzed s is the variable indicating the reference axis over the surface of the photo-detector array m is the variable indicating the diffraction order The variable d indicates the groove spacing The variable w indicates the size of the reflective surface and i s usually equal to 0.9 d The variable l represents the total number of grating grooves The variable indicates the angle of the incident beam z2 is the variable representing the distance between the grating and the photo-detector array The variable C is the angle of the slope of the reflective surfac e of each groove in the grating 4.1.3. Photo-Detector Array and Analog-to-Digital-Converte r As indicated by Figure 4.1 and Figure 4.2, the ligh t diffracted by the grating reaches a photo-detector array. The photo-detector array is coupled to an analog-todigital-converter. The light received by each eleme nt of the photo-detector array is transduced into electric voltage; this voltage is t aken by the analog-to-digital-converter

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34 and it transduced into a numeric representation use ful for computational purposes. The transduction process of the photo-detector array ca n be modeled by equation 4.6 [1]. ( ) ) ( ) ( ) ( ) (_ )1 ( _max mink C d ds G s L k Coutput ph s k s k ph diffracted output phD + =D D -l l gl l l (4.6) For k = 1, ... N Where: ) (_k Coutput ph is the output current of the kth photo-detector ) (lphG represents the photo-detector spectral response fo r each wavelength l N is the number of detectors in the photo-detector array s is the leght of each photo-detector g indicates the photo-detector non-linearity and 97. 0 > g ) (_k Coutput phD is the noise, which is Gaussian [1], as indicated in section 3.6, and is a function of ) (_k Coutput ph The photo-detector response function, ) (lphG depends on the properties of the photo-detector.

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35 The Analog-to-Digital quantization process is descr ibed by the following equation using a quantization step size of q : q q k C k Coutput ph output ph n r = ) ( of part Integer )} ( {_ (4.7) For k = 1, ... N 4.2. Simplification of the Grating Model Equation 4.3 can be written: ] ,0[ for ) ,'( ) ,' ( ) (maxs s ds s L s s G s Lslit grating diffracted =l l ll (4.8) Where: )] ( ) ( [min 2l b l bl= z s s (4.9) )] ( ) ( [min 2l b l bl+ = z s s (4.10) Let be constant for the equations above, and let Ldiffracted_(s) be defined as follows: )]] ( ) ( [ )], ( ) ( [ [ for )'( )' ( ) (min 2 max min 2 _ _l b l b l b l bl l l l l l =z s z s ds s L s s G s Lslit grating diffracted (4.11) Where: n r + n r + = ,l /s) (z d Diric /s) (z w Sinc ) (s Ggrating_ 2 2 2 2 2 21 2 1p (4.12) The convolution function is defined as:

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36 t t t t t td t f g d t g f g f ) ( ) ( ) ( ) ( = (4.13) Then: )]] ( ) ( [ )], ( ) ( [ [ for ) ( ) ( ) (min 2 max min 2 _ _l b l b l b l bl l l l l l l = z s z s s L s G s Lslit grating diffracted (4.14) In equations 4.5 and 4.12, if l then ) (2l x Diric can be approximated by: = =k apprk x x D ) 2 ( ) (p d (4.15) Then ) (_l ls Ggrating becomes: n r + n r + = =k ) /s (z d ) /s (z w Sinc s Gk gratingp dl l p l l2 1 2 1 ) (2 2 2 2 2 (4.16) Equation 4.16 has values different to zero only whe n: k d ) /s (z = +2 21l (4.17) Then: 2 2 2 2 2 ,) ( 1 k d k z k d z sk = =l l ll (4.18) Let: n r l = 5.0 of part integer d n (4.19) Then:

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37 n n n n k,1 ,..., 1 + = (4.20) Using equations 4.16 through 4.20, ) (_l ls Ggrating can be expressed in a discrete form as follows [4, 12]: n r =d k w Sinc k Gsimple grating2 _) (p (4.21) Equation 4.21 greatly simplifies the modeling of gr atings and facilitates the computational simulation of the model of a spectrop hotometer. Applying the definition of convolution, equation 4.8 can be written as follows : ] ,0[ for ) ,' ( ) ,'( ) (maxs s ds s s L s G s Lslit grating diffracted =l l ll (4.22) Where: maxs is the photo-detector array’s length, And, )] ( ) ( [min 2l b l bl=z s s Then, applying the simplification of the grating mo del to equation 4.22: s z s s s s a Sinc d wk Sinc L z a s Lk k n n k input diffractedD n r + n r @= 2 2 , 2 2 1) ( ) ( ) ( 2 ) (l l l l p pl l l l (4.23) Where: s s D = D (4.24) And,

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38 array tor photodetec the of elements of Number array tor photodetec the of Length = D s (4.25)

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39 5. COMPUTATIONAL IMPLEMENTATION OF THE SPECTROMETER MODEL This chapter deals with the computational implement ation of the model described in chapter four. The model in chapter four has to b e solved in two steps: first, calculating the diffraction generated by the grating for each w avelength using equation 4.23; and, second, calculating the output of the photo-detecto r array using a numerical implementation of the integral represented by equat ion 4.6. Section 5.1 describes a representation of the spectrometer model using disc rete variables; this section describes the application of a two-dimensional numerical inte gration algorithm based on NewtonCotes rules; the integration along the wavelength a xis is implemented using the Euler's rule, and the integration along the photo-detector array axis is implemented using the Boole's rule. Section 5.2 discuss a computational i mplementation of the discrete model presented in section 5.1; this discussion includes a description of the matrices, arrays and scalars used for the implementation, an explanation on the implementation algorithm, and, an analysis of the computational resources and execution time required by the program.

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40 5.1. Representation of the Spectrometer Model Using Disc rete Variables Equation 5.1 represents the photo-detector array mo del [1]: ( ) D D - =max min)1 ( _) ( ) ( ) (l l gl l ls k s k ph diffracted output phd ds G s L k C (5.1) ) (_k Coutput phD + For k = 1, ... N Where, N is the number of elements of the photo-det ector array. Let ) ( k sph represent the position of the center of the kth photo-detector: ) ( ) (2 1 D = k s k sph (5.2) The diffraction angle for each wavelength is descri bed by equation 5.3, and the location of the diffracted beam on the surface of t he photo-detector array is described by equation 5.4. a l l b+ = n r) ( 2 2 ) (C Sine d m ArcCos (5.3) )] ( ) ( [min 2l b l bl+ = z s s (5.4) Then, from equations 5.2, 5.3 and 5.4 with s = sk and s=0: ) ( )] ( ) ( [2 1 min 2 D = k s zl b l b (5.5) Then: ) ( ) ( ) (min 2 2 1l b l b+ D = z k s (5.6)

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41 And, from equation 5.3: = 2 ) ( ) ( 2l b a lCos m C Sine d (5.7) Then, from equations 5.6 and 5.7, let ) ( kphl represent the mean value of the wavelength of the light incident on the kth photo-detector element: m C Sine d kph) ( 2 ) ( =l (5.8) + D 2 )) ( /) 2 1 ( ( min 2l b az k s Cos Let ) ( lwavl be defined as follows: ) ( ) (2 1 min D + = l lwavl l l (5.9) For l = 1, …, M Where: M is the number of wavelengths to be analyzed, M /) (min maxl l l= D max is the maximum wavelength to be analyzed Equation 5.1 can be approximated using numerical in tegration techniques [18]. Equation 5.10 shows a numerical solution based on E uler’s rule. l l lgD D ==s l G l k s L k CM l wav ph wav ph diffracted approx ph1 _))] ( ( )) ( ), ( ( [ ) ( (5.10) ) (_k Capprox phD +

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42 The ) (lphG function depends on the properties of the photo-de tector. Equation 5.11 implements a more accurate numerical integration based on Newton-Cotes rules [18]. This implementation uses a n Euler’s rule for the integration of l d component in equation 4.6 in chapter four. The int egration of the ds component in equation 4.6 in chapter four is implemented using 5 -point Newton-Cotes rule, also known as Boole’s rule. ==M l approx phl k A k C1 _) ( ) ( (5.11) ) (_k Capprox phD + Where: l k l k l kf f f s l k A,1 4 ,1 1 4 ,2 1 412 32 7[ 90 4 ) (-+ + D D = l (5.12) ] 7 32,2 1 4 ,1 1 4l k l kf f+ + -+ + And, )) ( ), 1 4( ( [, 1 4l i k s L fwav ph diffracted li kl+ =+ (5.13) 2,1,0,1 ,2 for ))] ( ( = l gi l Gwav ph 5.2. Computational Implementation The computational implementation of the system mode led by equations 5.11 through 5.13 can be approached as follows: N is the number of elements of the photo-detector ar ray. Nph is the number of points on the photo-detector arra y’s surface.

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43 M is the number of wavelengths used for the simulati on. W is an array containing the wavelengths used for th e simulation, the size of W is 1 x M and its values are calculated as per equation 5.9. S is an array containing the position of points on t he surface of the photodetector, the size of S is 1 x Nph and its values are calculated as per equation 5.2. Gph is an array containing the spectral response of th e photo-detector array, the size of Gph is 1 x M and its values are taken from tabulations of characteristics of the photo-detector array. Linput is an array containing the spectrum of the input l ight, the size of Linput is 1 x M Ldiffracted is a matrix calculated using equation 4.23 in chap ter four, the size of Ldiffracted is Nph x M Cphout is an array containing the photo-detector array out put calculated as per equation 5.11, the size of Cphout is 1 x N The noise associated to this output can be estimated as per equations 3.14 or 3.15 in c hapter three. Wout is an array containing the wavelengths represented by each element of the photo-detector array, the size of Wout is 1 x N and its values are calculated as per equation 5.8. A Matlab program according to this implementation i s posted in the Appendix B.

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44 6. EFFECTS OF MINIATURIZATION IN SPECTROSCOPY SYSTEMS As stated in chapter one, the software developed as result of this research enables not only the study of the interactions between the components of the spectrometer, but also the design and optimization of spectrometry sy stems for specific applications. Chapter five discuss a software implementation of t he model presented in chapter four. This chapter describes an approach for analyzing an d designing the fundamental features of the optical bench of a spectrometer; this approa ch is based on the computational tools developed in chapter five. Section 6.1 describes th e optical configuration used for the simulations presented through this chapter; the opt ical configuration is described setting the slit width, the distance from the slit to the g rating, the number of grating grooves per millimeter, the size of each reflecting surface of the grating, the angle between each reflecting surface normal and the grating normal, t he distance front the grating to the photo-detector array, the photo-detector array leng th, and the number of elements of the photo-detector array. Section 6.2 presents an overv iew of the effects caused by the optical components of the spectrometer. Section 6.3 discuss es the effect cause by the slit size. Section 6.4 discusses the effect caused by the dist ance between the slit and the grating. The effect caused by the distance between the grati ng and the photo-detector array is

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45 discussed in section 6.5. Section 6.6 discusses the effect caused by the grating configuration. 6.1. Optical and Electronic Configuration The optical and electronic configuration used for t he simulations presented in this chapter is the following: Slit width: a = 100 microns Angle of the light incident into the grating: (5/36 0)* 2* radians Distance from the slit to the grating: z1 = 3.75 centimeters Grating configuration: d = 1/300 millimeters w = 0.9d C = (15/200)* radians Distance from the grating to the photo-detector arr ay: z2 = 2.5 centimeters Wavelength range: from 200 nanometers to 900 nanome ters Photo-detector array length: smax = 2.54 centimeters The Number of elements of the photo-detector array is 1024, and, ) (lphG the theoretical expected spectral response6 of the photo-detectors is presented in Figure 6.1 [19] 6 This theoretical spectral response was obtained fr om the technical information for the photodiode arr ay Hamamatsu S3903-1024Q.

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46 200 300 400 500 600 700 800 900 1000 1100 1200 0 0.05 0.1 0.15 0.2 0.25 0.3 Photosensitivity (A/W)Wavelength in nanometers Figure 6.1: Theoretical Spectral Response of the Ph otodiodes Used for the Simulations Presented in Chapter Six 6.2. Overview of the Effects Caused by the Optical Compo nents As indicated by equation 4.23 in chapter four, for each wavelength, the light leaving the grating is composed by main bean and se t of secondary beams which are symmetrically distributed around the main. In order to illustrate this effect, the diffraction of a light beam with a wavelength of 250 nanometers is presented in Figure 6.2 and Figure 6.3. This diffraction was calculated using t he optical configuration established in section 6.1, the model presented in chapter four an d the computational implementation proposed in chapter five.

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47 0 0.5 1 1.5 2 2.5 0 500 1000 1500 2000 2500 3000 3500 Position on the Photodetector Array Plane (cm)Intensity Figure 6.2: Diffraction of a Light Beam with a Wave length of 250 Nanometers – Intensity Measured in a Linear Scale versus Positio n of the Diffracted Beams Measured in Centimeters 0 0.5 1 1.5 2 2.5 -4 -3 -2 -1 0 1 2 3 4 Position on the Photodetector Array Plane (cm)Log10(Intensity) Figure 6.3: Diffraction of a Light Beam with a Wave length of 250 Nanometers – Intensity Measured in a Logarithmic Scale versus Po sition of the Diffracted Beams Measured in Centimeters

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48 The zero of the reference axis was set at the posit ion of the photo-detector reached by the shortest wavelength that could be analyzed b y the spectrometer. Figure 6.2 shows that, in a linear scale, the size of the secondary components are small compared to the main component. But, given that the calculations re quired for the absorption and transmission spectra are based on the logarithmic f unction7, the real impact of the secondary components can be appreciated analyzing F igure 6.3. Table 6.1 summarizes the most important features verifiable in figures f rom Figure 6.2 through Figure 6.5. Figure 6.4 and Figure 6.5 show the impact of the di ffraction effect when the positions of the photo-detector array elements are mapped into t heir corresponding wavelengths. The corresponding wavelengths were calculated using equ ations 4.4 and 4.9 in chapter four. 200 300 400 500 600 700 800 900 0 500 1000 1500 2000 2500 3000 3500 Wavelength (nm)Intensity Figure 6.4: Diffraction of a Light Beam with a Wave length of 250 Nanometers – Intensity Measured in a Linear Scale versus Wavelen gth 7 Chapter two describes why herein absorbance units are being used to measure transmission.

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49 200 300 400 500 600 700 800 900 -4 -3 -2 -1 0 1 2 3 4 Wavelength (nm)Log10(Intensity) Figure 6.5: Diffraction of a Light Beam with a Wave length of 250 Nanometers – Intensity Measured in a Logarithmic Scale versus Wa velength Table 6.1: Most Important Features of the Effect of the Diffraction Grating over a Light Beam with a Wavelength of 250 Nanometers Wavelength In Nanometers Position In Centimeters Diffraction Angle in Degrees Intensity in Linear Scale Intensity in Logarithmic Scale 250.061 307.295 365.215 424.634 486.486 552.338 623.581 702.716 793.183 0.170 0.357 0.549 0.747 0.955 1.181 1.429 1.711 2.045 66.440 62.140 57.755 53.214 48.432 43.266 37.576 31.105 23.470 3500.000 45.679 41.546 35.052 27.303 19.508 12.379 6.654 2.740 3.544 1.660 1.619 1.545 1.436 1.290 1.093 0.823 0.438 The effect of the diffraction features described ab ove can be readily appreciated on the spectra measured with the Ocean Optics HR200 0 spectrometer. Figure 6.6 shows the theoretical expected spectrum of 1.3 micron mon o-disperse polystyrene particles compared to the spectrum obtained with the HR2000; the spectral features observed in the experimentally obtained spectrum that are not p resent in the theoretical spectrum can

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50 be due to the light source, the suspending media or the sample itself. Herein those features are simulated using the set up presented i n section 6.1, the reference shown in Figure 6.7 (which has a peak at 198 nanometers) and the theoretical spectrum shown in Figure 6.8. The results presented here are measured as explained in chapter two: Sample) the of Spectrum ( Log Reference) the of Spectrum ( Log Units Absorbance in Spectrum Measured10 10= (6.1) 200 300 400 500 600 700 800 900 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Wavelength in nanometersOD in absorbance units HR2000 Theoretical Figure 6.6: Theoretical and Experimentally Obtained Spectra of 1.3 Micron MonoDisperse Polystyrene Particles

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51 100 200 300 400 500 600 700 800 900 1000 0 50 100 150 200 250 Wavelength in nanometersIntensity Figure 6.7: Spectrum of the Reference Used For Simu lation Presented in Chapter Six 200 300 400 500 600 700 800 900 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Wavelength in nanometersOD in absorbance units Figure 6.8: Theoretical Spectrum of 1.3 Micron Mono -Disperse Polystyrene Particles

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52 200 300 400 500 600 700 800 900 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Wavelength in nanometersOD in absorbance units HR2000 Simulated Figure 6.9: Simulated and Experimentally Obtained S pectra of 1.3 Micron MonoDisperse Polystyrene Particles Figure 6.9 shows the effect of applying the model t o the theoretical expected spectrum. Notice that the perturbations are located at the same positions that the spectral features of the spectrum obtained with the Ocean Op tics HR2000. The following sections deal with the details of the effect caused bay each optical component. 6.3. Effect Caused by the Slit Size Figure 6.10 shows the absorption spectra generated by the simulation system for three different sizes of slits and keeping the othe r parameters constant at the values stated in section 6.1. Figure 6.11 shows the details for t he interval from 300 nanometers to 600 nanometers where the spectral features are discerni ble. Notice that smaller the slit, wider the shape of the effect.

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53 200 300 400 500 600 700 800 900 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Wavelength in nanometersOD in absorbance units Slit size: 100 micronSlit size: 20 micron Figure 6.10: Effect Caused by Changes in the Slit W idth 300 350 400 450 500 550 600 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Wavelength in nanometersOD in absorbance units Slit size: 100 micronSlit size: 20 micron Figure 6.11: Detail of the Effect Caused By Changes in the Slit Width 6.4. Effect Caused by the Distance between the Slit and the Grating With the optical parameters constant at the values indicated in section 6.1; and z1 equal to 2.5 cm and 15 cm, the shape of the spectra l features becomes wider and

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54 shallower as z1 becomes larger. Figure 6.12 depicts the simulation results superimposed, and Figure 6.13 shows the detail for the interval f rom 300 nanometers to 600 nanometers. 200 300 400 500 600 700 800 900 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Wavelength in nanometersOD in absorbance units Slit to Grating distance: 3.75 cmSlit to Grating distance: 15 cm Figure 6.12: Effect Caused by Changes in the Distan ce between the Slit and the Grating 300 350 400 450 500 550 600 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Wavelength in nanometersOD in absorbance units Slit to Grating distance: 3.75 cmSlit to Grating distance: 15 cm Figure 6.13: Detail of the Effect Caused by Changes in the Distance between the Slit and the Grating

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55 6.5. Effect Caused by the Distance between the Grating a nd the Photo-Detector Array When z2 changes, two effects are generated in the spectrom eter: first, the relation between the range of the wavelengths that can be an alyzed by the spectrometer and the length of the photo-detector array changes; second, the width and depth of the shape of the distortion changes. Table 6.2 contains the leng th of the photo-detector array required for various grating-to-photo-detector distances; th e grating used for these calculations is configured as per section 6.1. The changes in the s ize of the shape can be explained as per section 6.5, similarly to the effect of changin g z1. The overall effect generated by the grating-to-photo-detector distance is shown in Figu re 6.14, details of the effect are shown in Figure 6.15. Table 6.2: Relation between the Grating to Photo-de tector Distance and the PhotoDetector Array Length Grating to Photo-Detector Distance in Millimeters Photo-Detector Array Length in Millimeters 2 5 10 20 30 40 60 100 150 250 1.959542 4.898854 9.797709 19.595420 29.393130 39.190840 58.786250 97.977090 146.965600 244.942700

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56 200 300 400 500 600 700 800 900 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Wavelength in nanometersOD in absorbance units Grating to Photodetector distance: 2.5 cmGrating to Photodetector distance: 0.5 cm Figure 6.14: Effect Caused by Changes in the Distan ce between the Grating and the Photo-Detector Array 300 350 400 450 500 550 600 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Wavelength in nanometersOD in absorbance units Grating to Photodetector distance: 2.5 cmGrating to Photodetector distance: 0.5 cm Figure 6.15: Detail of the Effect Caused by Changes in the Distance between the Grating and the Photo-Detector Array 6.6. Effect Caused by the Grating Configuration The effect of the grating has been approached consi dering each one of the parameters of the grating: the slope of the reflect ive surface of each groove, indicated by

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57 the variable C in the model, affects the size of the photo-detect or array required to analyze a given wavelength interval, as indicated i n Table 6.3; C also affects the diffraction pattern as shown in Figure 6.16 and Fig ure 6.17. Table 6.3: Relation between the Slope of the Reflec tive Surface of the Grating and the Size of the Photo-detector Array Angle of the Slope of the Reflective Surface (degrees) Size of the Photodetector Array (cm) 10.00 13.45 33.00 45.00 4.30165 2.93931 1.17223 0.897954 200 300 400 500 600 700 800 900 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Wavelength in nanometersOD in absorbance units Angle of the slope: 3/40 p radians Angle of the slope: 1/4 p radians Figure 6.16: Effect Caused by Changes in the Angle the Slope of the Reflective Surface of Each Groove in the Grating

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58 300 350 400 450 500 550 600 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Wavelength in nanometersOD in absorbance units Angle of the slope: 3/40 p radians Angle of the slope: 1/4 p radians Figure 6.17: Detail of the Effect Caused by Changes in the Angle of the Slope of the Reflective Surface of Each One of the Grooves in th e Grating The size of the each one of the grooves in the grat ing, indicated by d in the model, affects the location of the secondary components of the diffraction; the effect cause by d is shown in Figure 6.18 and Figure 6.19; for illust rative purposes, the parameter C was set to 45 degrees to generate the results shown in those figures. The parameter d also affects the length of the photo-detector array requ ired to analyze a given wavelength interval, this relation is indicated in Table 6.4. Table 6.4: Relation between the Size of Each One of the Grooves in the Grating and the Size of the Photo-Detector Array Size of the Groove Size of the Photodetector Array 1/300 mm = 3.33333 microns 1/600 mm = 1.66667 microns 1/1200 mm = 0.83333 microns 0.89795 cm 1.84100 cm 4.19070 cm

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59 200 300 400 500 600 700 800 900 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Wavelength in nanometersOD in absorbance units 300 grooves per mm600 grooves per mm Figure 6.18: Effect Caused by Changes in the Size o f Each One of the Grooves in the Grating 300 350 400 450 500 550 600 650 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Wavelength in nanometersOD in absorbance units 300 grooves per mm600 grooves per mm Figure 6.19: Detail of the Effect Caused by Changes in the Size of Each One of the Grooves in the Grating The size of the reflective surface of each one of t he grooves in the grating, indicate by w in the model, affects the intensity of the main an d secondary components of

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60 the diffraction; which generates major spectral fea tures in the absorption spectra as w / d becomes smaller; Figure 6.20 and Figure 6.21, and i llustrate the effect. 200 300 400 500 600 700 800 900 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Wavelength in nanometersOD in absorbance units w/d = 0.9 w/d = 0.999 Figure 6.20: Effect Caused by Changes in the Size o f the Reflective Surface of Each One of the Grooves in the Grating 300 350 400 450 500 550 600 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Wavelength in nanometersOD in absorbance units w/d = 0.9 w/d = 0.999 Figure 6.21: Detail of the Effect Caused by Changes in the Size of the Reflective Surface of Each One of the Grooves in the Grating

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61 7. COMPENSATION OF INSTRUMENTAL EFFECTS IN SPECTROSCOP Y SYSTEMS Chapter six describes an approach for analyzing and designing the fundamental features of the optical bench of a spectrometer. Th is chapter describes an algorithm for compensating the effects caused by a spectroscopy i nstrument. This algorithm is based on the assumption, made after the results presented in chapter six, that the effects generated by a given spectrometer can be replicated. This rep lication ability is applied to remove some specific features, as are the spurious peaks r emarkably present in some observed spectra, as stated in the introductory chapter. Current spectral correction methods are approached based on deconvolution methods, as indicated in Figure 7.1. The approach p resented in this chapter is summarized in Figure 7.2.

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62 Figure 7.1: Current Approach for Spectral Compensat ion Figure 7.2: New Approach for Compensation of Spectr al Features The algorithm summarized in Figure 7.2 is explained in detail in the following sections. The example is illustrated using the theo retical spectra of polystyrene particles with an average size of 1.3 microns. The theoretica l expected output and the simulation result considering the perturbation caused by the p eaks that will be use to illustrate the

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63 algorithm presented in this chapter are shown in Fi gure 7.3. The absorption spectrum is calculated using the following equation: ) ( ) (10 10 sample referenceS Log S Log A = (7.1) The algorithm proposed in this chapter deals with t he reduction of the effects present in Sreference and Ssample; the variables containing the processed spectra wi ll be named SC_reference and SC_sample, respectively. These new spectra will be used to c alculate a new version of the absorption, AC, for which the effect of the light peaks has been reduced. Figure 7.3: Description of the Process Applied to C alculated the Spectra of 1.3 Micron Mono-Disperse Polystyrene Particles 7.1. Identify the Parameters of the Simplified Spectrome ter Model The parameters of the theoretical system used for t his simulation are the following:

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64 Slit aperture width ( a ): 100 microns Focal distance from the slit to the grating ( z1): 3.75 centimeters Grating groove size ( d ): (1/300) millimeters Grating angle ( C ): 13.45 degrees Grating reflecting surface size ( w ): 0.95 d Focal distance from the grating to the photo-detect or array ( z2): 2.5 centimeters Photo-detector array length: 2.54 centimeters; numb er of photo-detector array elements: 1024 The spectral response of the photo-detector array i s shown in Figure 7.4 The total noise of the system can be modeled as per the equations presented in chapter two. In order to reduce the noise, the Cent ral Limit Theorem [15] has been applied using 10000 samples

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65 200 300 400 500 600 700 800 900 1000 1100 1200 0 0.05 0.1 0.15 0.2 0.25 0.3 Photosensitivity (A/W)Wavelength in nanometers Figure 7.4: Theoretical Spectral Response of the Ph otodiodes Used for the Simulation Presented in Chapter Seven 7.2. Identify the Wavelength(s) of the Beam(s) Causing t he Peaks For illustrative purposes the results presented in this chapter has been calculated using the theoretical reference whose spectrum is s hown in Figure 7.5, this spectrum has two peaks, one located at 210 nanometers, and other located at 325 nanometers.

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66 200 300 400 500 600 700 800 900 1 1.5 2 2.5 3 3.5 4 4.5 Wavelength in nanometersLog10(Intensity) Figure 7.5: Spectrum of the Reference Used for Simu lation Presented in Chapter Seven 7.3. Calculate the Simulated Diffraction of the Beam(s) Causing the Peaks Figure 7.6 and Figure 7.7 show the diffraction gene rated by a beam with a wavelength of 210 nanometers. Figure 7.6 shows the diffraction in linear scale and Figure 7.7 shows the diffraction in logarithmic scale. Fig ure 7.8 and Figure 7.9 show the diffraction generated by a beam with a wavelength o f 325 nanometers.

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67 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 3.5 x 10-7 Photodetector Array Plane Distance in centimetersIntensity Figure 7.6: Diffraction Generated by a Beam with a Wavelength of 210 Nanometers 0 0.5 1 1.5 2 2.5 3 -14 -13 -12 -11 -10 -9 -8 -7 -6 Photodetector Array Plane Distance in centimetersLog10(Intensity) Figure 7.7: Logarithm of the Diffraction Generated by a Beam with a Wavelength of 210 Nanometers

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68 0 0.5 1 1.5 2 2.5 3 0 1 2 3 4 5 x 10-7 Photodetector Array Plane Distance in centimetersIntensity Figure 7.8: Diffraction Generated by a Beam with a Wavelength of 325 Nanometers 0 0.5 1 1.5 2 2.5 3 -14 -13 -12 -11 -10 -9 -8 -7 -6 Photodetector Array Plane Distance in centimetersLog10(Intensity) Figure 7.9: Logarithm of the Diffraction Generated by a Beam with a Wavelength of 325 Nanometers

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69 7.4. Estimate the Effect of the Simulated Diffraction on the Spectrum Being Compensated In order to estimate the effect of the simulated di ffraction, the moving average of the logarithm of the output generated by each peak is calculated and it is used as a threshold. A window size of 21 data was used to cal culate the threshold depicted by the black curve in the figure on the left side of Figur e 7.10. The threshold values are subtracted from the spectrum. The spectral points a ssociated to a negative subtraction result are included in a set that is used for inter polation. The points selected for interpolation are represented with black dots in th e figure on the right side of Figure 7.10. Figure 7.10: Selection of Points Used for Interpola tion The curves in black in Figure 7.11 and Figure 7.12 represent the result of interpolating the reference and sample spectra usin g the wavelengths selected as indicated in the paragraph above, and the gray curv es represent the result of a simulation of the spectra measured by the photo-detector array Figure 7.11 shows the spectra of the

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70 reference used for simulation. Figure 7.12 shows th e spectra of the sample used for simulation. 0 0.5 1 1.5 2 2.5 3 -9 -8.5 -8 -7.5 -7 -6.5 -6 Photodetector Array Plane Distance in centimetersLog10(Intensity) All Values Threshold: diffraction 210 nanometers Figure 7.11: Threshold Applied to the Spectrum of t he Reference Used for the Simulation 0 0.5 1 1.5 2 2.5 3 -10 -9.5 -9 -8.5 -8 -7.5 -7 -6.5 Photodetector Array Plane Distance in centimetersLog10(Intensity) All Values Threshold: diffraction 210 nanometers Figure 7.12: Threshold Applied to the Spectrum of t he Sample Used for the Simulation

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71 Figure 7.13: Spectra in Logarithmic and Linear Scal e Figure 7.13 shows the spectrum of the reference in logarithmic and linear scale. An initial approximation to the peaks to be removed is obtained after subtracting the interpolated spectrum in linear scale, in black on the right side in Figure 7.13, from the simulated photo-detector measurement in linear scal e, in gray on the right side in Figure 7.13. Figure 7.14 depicts the procedure and shows t he result of the subtraction for the spectrum of the reference.

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72 Figure 7.14: Estimation of the Diffraction Generate d by a Beam with a Wavelength of 325 Nanometers: Subtraction Result 7.5. Subtract the Effect from the Spectrum Being Compens ated The spectral features are subtracted from the refer ence spectrum using a scaled version of the diffraction shown in Figure 7.6. Thi s diffraction is scaled using the estimations shown in Figure 7.14. Figure 7.15 depic ts the procedure and shows the result of estimating the peaks that have to be removed fro m the spectrum of the reference. Figure 7.16 depicts how the estimated peaks are rem oved from the spectrum of the reference. The spectrum of the sample is compensate d using the same procedure. Figure 7.17 shows the absorption spectrum affected by the peaks and the absorption spectrum obtained after the algorithm was applied.

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73 Figure 7.15: Estimation of the Diffraction Generate d by a Beam with a Wavelength of 325 Nanometers: Estimated Peaks Figure 7.16: Spectrum Compensated Subtracting the E stimated Peaks

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74 200 300 400 500 600 700 800 900 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 Photodetector Array Plane (centimeters)OD (absorbance units)Spectrum of the Reference Affected Spectrum Corrected Spectrum Figure 7.17: Absorption Spectrum after Correction o f the Spectral Features

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75 8. CONCLUSIONS 8.1. Conclusions The simplification of the diffraction grating model enables an elaboration of the general spectrometer model with two important chara cteristics: first, facilitates the analysis of the effect of the optical parameters of the spectrometer on the spectra readings; second, allows a computational efficient implementation of the model of the spectrometer for simulation purposes. The simplified model of the spectrometer presented in this dissertation predicts the instrumental effects detected in the spectra of polystyrene particles used for calibration. Considering the parameters included in the model, the instrumental effects can be replicated meaning that the model and its co mputational implementation can be used for design purposes. The spectral features due to the peaks generated by the diffraction of a light beam can be reduced applying an algorithm based on the s implified model of the spectrometer. The model allows calculating the effect of a given light beam, which is non-linear, and, once the effect has been calculated, the compensati on can be made using a linear approach.

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76 8.2. Contributions This dissertation proposed a method to analyze the critical parameters related to the optical and electronic components of spectrosco py systems, and to study the impact of these parameters in the total performance of the system. A tool modeling the interaction of the elements of the spectroscopy sys tem is presented. The methods developed in this dissertation enable the study of the interactions between the components of the spectrometer and the design and o ptimization of spectrometer systems for specific applications. A methodology for applyi ng the information from the instrumental effects to correct the spectrum of a g iven solution is presented. 8.3. Recommendations and Future Work Future work that can be developed based on the outc omes of this research regarding the possibility to generate databases con taining instrument-independent spectra. These types of spectra, which are affected only by the optical effects due to the particles, can be used to develop correlation-based particle identification algorithms. The simplified model can be easily adapted to fit sever al optical configurations and to test spectrometer components such as types of photo-dete ctors, and diffraction technologies (i.e. prisms, holographic gratings, curved gratings ). Further work can be done on the solution of the inverse problem based on a matrix i mplementation of the simplified model.

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77 REFERENCES [1] M. P. Wisniewski, R. Z. Morawski, and A. Barwic z, "Modeling the spectrometric microtransducer," Instrumentation and Measurement, IEEE Transactions on vol. 48, pp. 747-752, 1999. [2] M. P. Wisniewski, R. Z. Morawski, and A. Barwic z, "Using rational filters for digital correction of a spectrometric microtransduc er," Instrumentation and Measurement, IEEE Transactions on vol. 49, pp. 43-48, 2000. [3] R. Z. Morawski, "Digital signal processing in m easurement microsystems," Instrumentation & Measurement Magazine, IEEE vol. 7, pp. 43-50, 2004. [4] E. G. Steward, Fourier optics : an introduction Chichester, West Sussex, New York: E. Horwood; Halsted Press, 1983. [5] Ocean Optics Inc., at http://www.oceanoptics.com 2006. [6] A. Garcia-Lopez, "Hybrid model for characteriza tion of submicron particles using multiwavelength spectroscopy," in Department of Electrical Engineering Thesis (Ph.D.) ed. Tampa, Florida: University of South Flo rida, 2005. [7] Ocean Optics Inc., "HR2000+ High-resolution Spe ctrometer." at http://www.oceanoptics.com 2006. [8] Ocean Optics Inc., "USB2000-UV-VIS Spectrometer ." at http://www.oceanoptics.com 2006. [9] Agilent Technologies, "8453 UV-Visible Spectrop hotometer." at http://www.chem.agilent.com 2006. [10] PerkinElmer Inc., "Lambda 800/900 Systems." at http://las.perkinelmer.com 2006. [11] C. E. Alupoaei, J. A. Olivares, and L. H. Garc ia-Rubio, "Quantitative spectroscopy analysis of prokaryotic cells: vegetat ive cells and spores," Biosensors and Bioelectronics vol. 19, pp. 893-903, 2004.

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78 [12] J. F. James and R. S. Sternberg, The design of optical spectrometers London,: Chapman & Hall, 1969. [13] A. V. Oppenheim, A. S. Willsky, and S. H. Nawa b, Signals and Systems Upper Saddle River, N.J.: Prentice-Hall, 1997. [14] A. V. Oppenheim, R. W. Schafer, and J. R. Buck Discrete-Time Signal Processing 2nd Edition ed: Prentice-Hall, 1999. [15] H. P. Hsu and NetLibrary Inc., Schaum's outline of theory and problems of probability, random variables, and random processes New York: McGraw-Hill, 1997. [16] K. A. Jones, Introduction to optical electronics New York: Harper & Row, 1987. [17] W. B. Leigh, Devices for optoelectronics New York: Marcel Dekker, 1996. [18] M. N. O. Sadiku, Numerical techniques in electromagnetics 2nd ed. Boca Raton: CRC Press, 2001. [19] Hamamatsu Corporation. at http://sales.hamamatsu.com/ 2006.

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79 APPENDICES

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80 Appendix A: Spectra of Mono-Disperse Polystyrene Pa rticles Used for Calibration Figure A.1 shows the spectra of samples of 1 micron mono-disperse polystyrene particles suspended in de-ionized water. These spec tra were obtained with the Agilent 8453 and the Ocean Optics HR2000. A higher concentr ation of polystyrene particles suspended in the de-ionized water implies a larger maximum value of the spectrum. The concentrations used to obtain the spectra shown in Figure A.1 are: higher concentration 1:250, medium concentration 1:500, and smaller conc entration 1:1000. Figure A.2 shows the spectra obtained with the Ocean Optics HR2000 a nd the Ocean Optics USB2000. Figure A.3 shows the spectra obtained with the Agil ent 8453 and the Perkin-Elmer Lambda 900. Figure A.4 shows the theoretical expect ed spectrum compared to the spectra obtained with the spectrophotometer Agilent 8453. 200 300 400 500 600 700 800 900 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Wavelength in nanometersOD in absorbance units HR2000 Agilent 8453 Figure A.1: Spectra of 1 Micron Mono-Disperse Polys tyrene Particles – Spectrometers Ocean Optics HR2000 and Agilent 8453

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81 Appendix A: (Continued) 200 300 400 500 600 700 800 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Wavelength in nanometersOD in absorbance units USB2000HR2000 Figure A.2: Spectra of 1 Micron Mono-Disperse Polys tyrene Particles – Spectrometers Ocean Optics USB2000 and Ocean Optics HR2000 200 300 400 500 600 700 800 900 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Wavelength in nanometersOD in absorbance units Agilent Perkin-Elmer Figure A.3: Spectra of 1 Micron Mono-Disperse Polys tyrene Particles – Spectrometers Agilent 8453 and Perkin-Elmer Lambda 900

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82 Appendix A: (Continued) 200 300 400 500 600 700 800 900 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Wavelength in nanometersOD in absorbance units Agilent, 1:500Theoretical Figure A.4: Spectra of 1 Micron Mono-Disperse Polys tyrene Particles – Spectrum Observed with the Spectrometers Agilent 8453 and Th eoretical Spectrum

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83 Appendix A: (Continued) Figure A.5 shows the spectra of samples of 40 nanom eters mono-disperse polystyrene particles suspended in de-ionized water These spectra were obtained with the Agilent 8453 and the Ocean Optics HR2000. A hig her concentration of polystyrene particles suspended in the de-ionized water implies a larger maximum value of the spectrum. The concentrations used to obtain the spe ctra shown in Figure A.5 are: 1:50, 1:100, 1:500 and 1:1000. Figure A.6 shows the spect ra obtained with the Ocean Optics HR2000 and the Ocean Optics USB2000. Figure A.7 sho ws the theoretical expected spectrum compared to the spectra obtained with the spectrophotometer Agilent 8453. 200 300 400 500 600 700 800 900 0 0.5 1 1.5 2 2.5 3 3.5 4 Wavelength in nanometersOD in absorbance units HR2000 Agilent 8453 Figure A.5: Spectra of 40 Nanometers Mono-Disperse Polystyrene Particles – Spectrometers Ocean Optics HR2000 and Agilent 8453

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84 Appendix A: (Continued) 200 300 400 500 600 700 800 -0.5 0 0.5 1 1.5 2 2.5 Wavelength in nanometersOD in absorbance units USB2000HR2000 Figure A.6: Spectra of 40 Nanometers Mono-Disperse Polystyrene Particles – Spectrometers Ocean Optics USB2000 and Ocean Optics HR2000 200 300 400 500 600 700 800 900 0 0.5 1 1.5 2 2.5 3 3.5 4 Wavelength in nanometersOD in absorbance units Agilent, 1:500Theoretical Figure A.7: Spectra of 40 Nanometers Mono-Disperse Polystyrene Particles – Spectrum Observed with the Spectrometers Agilent 8453 and Th eoretical Spectrum

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85 Appendix A: (Continued) Figure A.8 shows the spectra of samples of 150 nano meters mono-disperse polystyrene particles suspended in de-ionized water These spectra were obtained with the Agilent 8453 and the Ocean Optics HR2000. A hig her concentration of polystyrene particles suspended in the de-ionized water implies a larger maximum value of the spectrum. The concentrations used to obtain the spe ctra shown in Figure A.8 are: 1:500 and 1:1000. Figure A.9 shows the spectra obtained w ith the Ocean Optics HR2000 and the Ocean Optics USB2000. Figure A.10 shows the the oretical expected spectrum compared to the spectra obtained with the spectroph otometer Agilent 8453. 200 300 400 500 600 700 800 900 0 0.5 1 1.5 2 2.5 3 Wavelength in nanometersOD in absorbance units HR2000 Agilent 8453 Figure A.8: Spectra of 150 Nanometers Mono-Disperse Polystyrene Particles – Spectrometers Ocean Optics HR2000 and Agilent 8453

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86 Appendix A: (Continued) 200 300 400 500 600 700 800 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Wavelength in nanometersOD in absorbance units USB2000HR2000 Figure A.9: Spectra of 150 Nanometers Mono-Disperse Polystyrene Particles – Spectrometers Ocean Optics USB2000 and Ocean Optics HR2000 200 300 400 500 600 700 800 900 0 0.5 1 1.5 2 2.5 3 Wavelength in nanometersOD in absorbance units Agilent, 1:500Theoretical Figure A.10: Spectra of 150 Nanometers Mono-Dispers e Polystyrene Particles – Spectrum Observed with the Spectrometers Agilent 84 53 and Theoretical Spectrum

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87 Appendix A: (Continued) Figure A.11 shows the spectra of samples of 500 nan ometers mono-disperse polystyrene particles suspended in de-ionized water These spectra were obtained with the Agilent 8453 and the Ocean Optics HR2000. A hig her concentration of polystyrene particles suspended in the de-ionized water implies a larger maximum value of the spectrum. The concentrations used to obtain the spe ctra shown in Figure A.11 are: 1:500 and 1:1000. Figure A.12 shows the spectra obtained with the Ocean Optics HR2000 and the Ocean Optics USB2000. Figure A.13 shows the the oretical expected spectrum compared to the spectra obtained with the spectroph otometer Agilent 8453. 200 300 400 500 600 700 800 900 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Wavelength in nanometersOD in absorbance units HR2000 Agilent 8453 Figure A.11: Spectra of 500 Nanometers Mono-Dispers e Polystyrene Particles – Spectrometers Ocean Optics HR2000 and Agilent 8453

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88 Appendix A: (Continued) 200 300 400 500 600 700 800 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Wavelength in nanometersOD in absorbance units USB2000HR2000 Figure A.12: Spectra of 500 Nanometers Mono-Dispers e Polystyrene Particles – Spectrometers Ocean Optics USB2000 and Ocean Optics HR2000 200 300 400 500 600 700 800 900 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Wavelength in nanometersOD in absorbance units Agilent, 1:500Theoretical Figure A.13: Spectra of 500 Nanometers Mono-Dispers e Polystyrene Particles – Spectrum Observed with the Spectrometers Agilent 84 53 and Theoretical Spectrum

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89 Appendix A: (Continued) Figure A.14 shows the spectra of samples of 700 nan ometers mono-disperse polystyrene particles suspended in de-ionized water These spectra were obtained with the Agilent 8453 and the Ocean Optics HR2000. A hig her concentration of polystyrene particles suspended in the de-ionized water implies a larger maximum value of the spectrum. The concentrations used to obtain the spe ctra shown in Figure A.14 are: 1:500 and 1:1000. Figure A.15 shows the spectra obtained with the Ocean Optics HR2000 and the Ocean Optics USB2000. Figure A.16 shows the the oretical expected spectrum compared to the spectra obtained with the spectroph otometer Agilent 8453. 200 300 400 500 600 700 800 900 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Wavelength in nanometersOD in absorbance units HR2000 Agilent 8453 Figure A.14: Spectra of 700 Nanometers Mono-Dispers e Polystyrene Particles – Spectrometers Ocean Optics HR2000 and Agilent 8453

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90 Appendix A: (Continued) 200 300 400 500 600 700 800 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Wavelength in nanometersOD in absorbance units USB2000HR2000 Figure A.15: Spectra of 700 Nanometers Mono-Dispers e Polystyrene Particles – Spectrometers Ocean Optics USB2000 and Ocean Optics HR2000 200 300 400 500 600 700 800 900 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Wavelength in nanometersOD in absorbance units Agilent, 1:500Theoretical Figure A.16: Spectra of 700 Nanometers Mono-Dispers e Polystyrene Particles – Spectrum Observed with the Spectrometers Agilent 84 53 and Theoretical Spectrum

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91 Appendix A: (Continued) Figure A.17 shows the spectra of samples of 1.3 mic ron mono-disperse polystyrene particles suspended in de-ionized water These spectra were obtained with the Agilent 8453 and the Ocean Optics HR2000. A hig her concentration of polystyrene particles suspended in the de-ionized water implies a larger maximum value of the spectrum. The concentrations used to obtain the spe ctra shown in Figure A.17 are: 1:100, 1:175 and 1:250. Figure A.18 shows the spectra obta ined with the Ocean Optics HR2000 and the Ocean Optics USB2000. Figure A.19 shows the spectra obtained with the Agilent 8453 and the Perkin-Elmer Lambda 900. Figure A.20 s hows the theoretical expected spectrum compared to the spectra obtained with the spectrophotometer Agilent 8453. 200 300 400 500 600 700 800 900 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Wavelength in nanometersOD in absorbance units HR2000 Agilent 8453 Figure A.17: Spectra of 1.3 Micron Mono-Disperse Po lystyrene Particles – Spectrometers Ocean Optics HR2000 and Agilent 8453

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92 Appendix A: (Continued) 200 300 400 500 600 700 800 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Wavelength in nanometersOD in absorbance units USB2000HR2000 Figure A.18: Spectra of 1.3 Micron Mono-Disperse Po lystyrene Particles – Spectrometers Ocean Optics USB2000 and Ocean Optics HR2000 200 300 400 500 600 700 800 900 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Wavelength in nanometersOD in absorbance units Agilent Perkin-Elmer Figure A.19: Spectra of 1.3 Micron Mono-Disperse Po lystyrene Particles – Spectrometers Agilent 8453 and Perkin-Elmer Lambda 900

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93 Appendix A: (Continued) 200 300 400 500 600 700 800 900 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 Wavelength in nanometersOD in absorbance units Agilent, 1:250Theoretical Figure A.20: Spectra of 1.3 Micron Mono-Disperse Po lystyrene Particles – Spectrum Observed with the Spectrometers Agilent 8453 and Th eoretical Spectrum

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94 Appendix A: (Continued) Figure A.21 shows the spectra of samples of 2 micro n mono-disperse polystyrene particles suspended in de-ionized water. These spec tra were obtained with the Agilent 8453 and the Ocean Optics HR2000. A higher concentr ation of polystyrene particles suspended in the de-ionized water implies a larger maximum value of the spectrum. The concentrations used to obtain the spectra shown in Figure A.21 are: 1:50, 1:75 and 1:100. Figure A.22 shows the spectra obtained with the Oce an Optics HR2000 and the Ocean Optics USB2000. Figure A.23 shows the spectra obtai ned with the Agilent 8453 and the Perkin-Elmer Lambda 900. Figure A.24 shows the theo retical expected spectrum compared to the spectra obtained with the spectroph otometer Agilent 8453. 200 300 400 500 600 700 800 900 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Wavelength in nanometersOD in absorbance units HR2000 Agilent 8453 Figure A.21: Spectra of 2 Micron Mono-Disperse Poly styrene Particles – Spectrometers Ocean Optics HR2000 and Agilent 8453

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95 Appendix A: (Continued) 200 300 400 500 600 700 800 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Wavelength in nanometersOD in absorbance units USB2000HR2000 Figure A.22: Spectra of 2 Micron Mono-Disperse Poly styrene Particles – Spectrometers Ocean Optics USB2000 and Ocean Optics HR2000 200 300 400 500 600 700 800 900 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Wavelength in nanometersOD in absorbance units Agilent Perkin-Elmer Figure A.23: Spectra of 2 Micron Mono-Disperse Poly styrene Particles – Spectrometers Agilent 8453 and Perkin-Elmer Lambda 900

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96 Appendix A: (Continued) 200 300 400 500 600 700 800 900 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 Wavelength in nanometersOD in absorbance units Agilent, 1:100Theoretical Figure A.24: Spectra of 2 Micron Mono-Disperse Poly styrene Particles – Spectrum Observed with the Spectrometers Agilent 8453 and Th eoretical Spectrum

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97 Appendix A: (Continued) Figure A.25 shows the spectra of samples of 4 micro n mono-disperse polystyrene particles suspended in de-ionized water. These spec tra were obtained with the Agilent 8453 and the Ocean Optics HR2000. A higher concentr ation of polystyrene particles suspended in the de-ionized water implies a larger maximum value of the spectrum. The concentrations used to obtain the spectra shown in Figure A.25 are: 1:10, 1:25 and 1:50. Figure A.26 shows the spectra obtained with the Oce an Optics HR2000 and the Ocean Optics USB2000. Figure A.27 shows the spectra obtai ned with the Agilent 8453 and the Perkin-Elmer Lambda 900. Figure A.28 shows the theo retical expected spectrum compared to the spectra obtained with the spectroph otometer Agilent 8453. 200 300 400 500 600 700 800 900 0 0.5 1 1.5 Wavelength in nanometersOD in absorbance units HR2000 Agilent 8453 Figure A.25: Spectra of 4 Micron Mono-Disperse Poly styrene Particles – Spectrometers Ocean Optics HR2000 and Agilent 8453

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98 Appendix A: (Continued) 200 300 400 500 600 700 800 0 0.5 1 1.5 Wavelength in nanometersOD in absorbance units USB2000HR2000 Figure A.26: Spectra of 4 Micron Mono-Disperse Poly styrene Particles – Spectrometers Ocean Optics USB2000 and Ocean Optics HR2000 200 300 400 500 600 700 800 900 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Wavelength in nanometersOD in absorbance units Agilent Perkin-Elmer Figure A.27: Spectra of 4 Micron Mono-Disperse Poly styrene Particles – Spectrometers Agilent 8453 and Perkin-Elmer Lambda 900

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99 Appendix A: (Continued) 200 300 400 500 600 700 800 900 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 Wavelength in nanometersOD in absorbance units Agilent, 1:50Theoretical Figure A.28: Spectra of 4 Micron Mono-Disperse Poly styrene Particles – Spectrum Observed with the Spectrometers Agilent 8453 and Th eoretical Spectrum

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100 Appendix A: (Continued) Figure A.29 shows the spectra of samples of 9 micro n mono-disperse polystyrene particles suspended in de-ionized water. These spec tra were obtained with the Agilent 8453 and the Ocean Optics HR2000. A higher concentr ation of polystyrene particles suspended in the de-ionized water implies a larger maximum value of the spectrum. The concentrations used to obtain the spectra shown in Figure A.29 are: 1:5 and 1:10. Figure A.30 shows the spectra obtained with the Ocean Opti cs HR2000 and the Ocean Optics USB2000. Figure A.31 shows the spectra obtained wit h the Agilent 8453 and the PerkinElmer Lambda 900. Figure A.32 shows the theoretical expected spectrum compared to the spectra obtained with the spectrophotometer Agi lent 8453. 200 300 400 500 600 700 800 900 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Wavelength in nanometersOD in absorbance units HR2000 Agilent 8453 Figure A.29: Spectra of 9 Micron Mono-Disperse Poly styrene Particles – Spectrometers Ocean Optics HR2000 and Agilent 8453

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101 Appendix A: (Continued) 200 300 400 500 600 700 800 0 0.5 1 1.5 2 2.5 3 3.5 4 Wavelength in nanometersOD in absorbance units USB2000HR2000 Figure A.30: Spectra of 9 Micron Mono-Disperse Poly styrene Particles – Spectrometers Ocean Optics USB2000 and Ocean Optics HR2000 200 300 400 500 600 700 800 900 0 0.5 1 1.5 2 2.5 3 Wavelength in nanometersOD in absorbance units Agilent Perkin-Elmer Figure A.31: Spectra of 9 Micron Mono-Disperse Poly styrene Particles – Spectrometers Agilent 8453 and Perkin-Elmer Lambda 900

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102 Appendix A: (Continued) 200 300 400 500 600 700 800 900 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Wavelength in nanometersOD in absorbance units Agilent, 1:10Theoretical Figure A.32: Spectra of 9 Micron Mono-Disperse Poly styrene Particles – Spectrum Observed with the Spectrometers Agilent 8453 and Th eoretical Spectrum

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103 Appendix A: (Continued) Figure A.33 shows the spectra of samples of 15 micr on mono-disperse polystyrene particles suspended in de-ionized water These spectra were obtained with the Agilent 8453 and the Ocean Optics HR2000. A hig her concentration of polystyrene particles suspended in the de-ionized water implies a larger maximum value of the spectrum. The concentrations used to obtain the spe ctra shown in Figure A.33 are: 1:2, 1:3 and 1:5. Figure A.34 shows the spectra obtained with the Ocean Optics HR2000 and the Ocean Optics USB2000. Figure A.35 shows the the oretical expected spectrum compared to the spectra obtained with the spectroph otometer Agilent 8453. 200 300 400 500 600 700 800 900 0 0.5 1 1.5 2 2.5 3 Wavelength in nanometersOD in absorbance units HR2000 Agilent 8453 Figure A.33: Spectra of 15 Micron Mono-Disperse Pol ystyrene Particles – Spectrometers Ocean Optics HR2000 and Agilent 8453

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104 Appendix A: (Continued) 200 300 400 500 600 700 800 0 0.5 1 1.5 2 2.5 3 Wavelength in nanometersOD in absorbance units USB2000HR2000 Figure A.34: Spectra of 15 Micron Mono-Disperse Pol ystyrene Particles – Spectrometers Ocean Optics USB2000 and Ocean Optics HR2000 200 300 400 500 600 700 800 900 0 0.5 1 1.5 2 2.5 Wavelength in nanometersOD in absorbance units Agilent, 1:5Theoretical Figure A.35: Spectra of 15 Micron Mono-Disperse Pol ystyrene Particles – Spectrum Observed with the Spectrometers Agilent 8453 and Th eoretical Spectrum

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105 Appendix B: Simulation of the Model of the Spectrop hotometer %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Simulation of a Miniaturized Spectrometer % by Eduardo Zurek % Department of Electrical Engineering % University of South Florida % Summer 2006 % %%%%%%%%%%%%%%%%%%%% % This program is based on the model and computatio nal % implementation presented on chapters 5 and 6 of t he % Ph.D. Dissertation: % "System Optimization of Micron and Sub-Micron % Particle Identification Using % Spectroscopy Based Techniques" % Author: Eduardo Zurek %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% close all;clear all;clc;pause(0.1);disp('Working .. .');tic; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Files used for calibration fnames = {'p40nm50','p40nm100','p40nm500','p40nm100 0',... 'p150nm500','p150nm1000',... 'p500nm500','p500nm1000',... 'p700nm250','p700nm500','p700nm1000',... 'p1um250','p1um500',... 'p1_3um100','p1_3um175','p1_3um250',... 'p2um50','p2um75','p2um100',... 'p4um10','p4um25','p4um50',... 'p9um5','p9um10',... 'p15um2','p15um3','p15um5'}; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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106 Appendix B: (Continued) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Loading the files containg the theoretical expect ed % spectrum used for the simulation file_ezv = 2; if file_ezv == 1 load p700; th_wav = p700(:,1)'; th_data = p700(:,2)'; index_fnames = 11; elseif file_ezv == 2 load p1300; % p1300.mat has to be previously cr eated % and it has to contain the data presented in table 6 th_wav = p1300(:,1)'; th_data = p1300(:,2)'; index_fnames = 14; else load p2000; th_wav = p2000(:,1)'; th_data = p2000(:,2)'; index_fnames = 19; end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Loading the files containing the spectra obtained with % the Ocean Optics HR2000 HR_fname = [char(fnames(index_fnames)),'hr.txt'] ; % The file used for this example is: 'p1_3um100hr.txt', % its cont ents are presented in table 7 HR_temp = load(HR_fname); HR_wav = HR_temp(:,1)'; HR_data = HR_temp(:,2)'; HR_raw_index = HR_wav>=190 & HR_wav<=900; HR1_wav = HR_wav(HR_raw_index); HR1_data = HR_data(HR_raw_index); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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107 Appendix B: (Continued) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Loading the files containing the spectra obtained with % the Ocean Optics USB2000 USB_fname = [char(fnames(index_fnames)),'usb.txt'] ; USB_temp = load(USB_fname); USB_wav = USB_temp(:,1)'; USB_data = USB_temp(:,2)'; USB_raw_index = USB_wav>=190 & USB_wav<=900; USB1_wav = USB_wav(USB_raw_index); USB1_data = USB_data(USB_raw_index); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Theoretical spectrum of the light source used for the % simulation % light source: d2000spectra %pw = [375, 150, 175, 375, 700, 1100, 1450, 1600]; % light source: dtminispectra pw = [140, 50, 20, 20, 40, 75, 120, 180]; pw = pw/max(pw); wav1 = [2:9]*100; th_light = spline(wav1,pw,th_wav); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Light source's peaks wav2 = [200]; th_light = th_light/max(th_light); th_light(wav2-199)=50; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Theoretical reference spectrum nth = length(th_light)-1; kth = 0.9; th_reference = th_light.*exp(-[0:nth]/nth*kth); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Theoretical sample spectrum th_sample = th_reference.*(10.^(-th_data)); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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108 Appendix B: (Continued) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Spectral response of photodiode wav3 = (th_wav-200)/700; y = (wav3-0.7).^2; y = y/max(y)*0.75; y = 1-y; y2 = y-min(y); y2 = y2/max(y2); y2 = y2.*sin(1./(1+7*wav3)*15*pi); th_ph = 5*y+y2; th_ph = th_ph/max(th_ph); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Optical configuration % Grating's configuration grooves_mm = 300; % number of grroves per millimete r m = 1; % diffraction order L = 3e-2; % Grating's length l = L*1000*grooves_mm; % total number of grating's grooves % As per the simplification proposed in Chapter 5 % the total number of grating's grooves has been se t to % infinity d = 1e-3/grooves_mm; % size of the grating's groove w = 0.9*d; % size of the reflective surface of each groove C = 13.45/360*2*pi; % slope of the reflective surfa ce of % groove alpha = 5*pi/180; % light's incidence angle wav_min = 200*1e-9; % minimun wavelength incident o n the % grating's su rface beta0 = -2*acos(m*wav_min/(2*d*sin(C)))+alpha;% min imum % diffraction angle a = 100e-6; % slit size % Other possible values for a are: % 5e-6, 10e-6, 25e-6, 50e-6, 100e-6, 200e-6 z1 = 2e-2; % distance from the slit to the grating z2 = 3e-2; % distance from the grating to the % photodetector array N = 1024; % Number of elements of the photodetector array ph_nonlin = 0.98;% Photodetector non-linearity %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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109 Appendix B: (Continued) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Calculating the grating's diffraction angles % and the minimun required photodetector array leng th wav_arr = th_wav*1e-9; beta1 = -2*acos(m*wav_arr/(2*d*sin(C)))+alpha; s_shift_angle = beta1-beta0; s_max = z2*max(s_shift_angle); disp('Photodetector array length (in centimeters)') disp(s_max*100) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Central Limit Theorem % the following variable is required to estimate th e noise related % to the spectra obtained with this model % Number of samples to average n_samples_avg = 10000; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % The array s represents the photodetector array s urface s_min = 0; n = 4*N+1; s = s_min+[0:n-1]/(n-1)*(s_max-s_min); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Simplified Grating's model nmax = floor(d/wav_arr(1)); sinc2k = sinc(w*[-nmax:nmax]/d).^2; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Parameters for the numerical integration % using the Boole's rule delta_s = s(2)-s(1); delta_wav = wav_arr(2) wav_arr(1); i2 = [0:4]; boole1 = (4/90)*delta_s*[7;32;12;32;7]; boole2 = zeros(1,5); i_boole = [0:4]; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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110 Appendix B: (Continued) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Process to calculate the spectrometer's output L_input = th_reference; ph_t = zeros(1,N); C_ph_approx = zeros(1,N); n_wav = length(wav_arr); for i1 = 1:n_wav, wav1 = wav_arr(i1); n_grating = floor(d/wav1); s_sht = z2*s_shift_angle(i1); s_shift = s-s_sht; % s,s_shift: 1x(4N+1) y = zeros(size(s)); % y: 1x(4N+1) for k = -n_grating:n_grating, s_k = z2*wav1*k/sqrt(d*d-(wav1*k)*(wav1*k)) ; s_grating = s_shift-s_k;% s_grating,s_shift : 1x(4N+1) i2 = nmax k +1; y = y +... sinc2k(i2)*... sinc(a*s_grating./(wav1*sqrt(s_grating. *s_grating+z1*z1))).^2; end y = a*y*L_input(i1)/(2*wav1*z1); y = (y.*th_ph(i1)).^ph_nonlin; for k =0:N-1, boole2 = y(4*k+1+i_boole); ph_t(k+1) = boole2*boole1; end C_ph_approx = C_ph_approx + ph_t*delta_wav; end % Adding noise max_out = max(C_ph_approx); C_ph_approx = C_ph_approx/max_out*3500; var1 = 0.12345*C_ph_approx.^0.86949/n_samples_avg; delta_C_ph = sqrt(var1).*randn(size(var1)); C_ph_approx = C_ph_approx + delta_C_ph; C_ph_approx(C_ph_approx<=0)=realmin; C_ph_approx = C_ph_approx*max_out/3500; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% C_ph_reference = C_ph_approx;

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111 Appendix B: (Continued) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Process to calculate the spectrometer's output L_input = th_sample; ph_t = zeros(1,N); C_ph_approx = zeros(1,N); n_wav = length(wav_arr); for i1 = 1:n_wav, wav1 = wav_arr(i1); n_grating = floor(d/wav1); s_sht = z2*s_shift_angle(i1); s_shift = s-s_sht; % s,s_shift: 1x(4N+1) y = zeros(size(s)); % y: 1x(4N+1) for k = -n_grating:n_grating, s_k = z2*wav1*k/sqrt(d*d-(wav1*k)*(wav1*k)) ; s_grating = s_shift-s_k;% s_grating,s_shift : 1x(4N+1) i2 = nmax k +1; y = y +... sinc2k(i2)*... sinc(a*s_grating./(wav1*sqrt(s_grating. *s_grating+z1*z1))).^2; end y = a*y*L_input(i1)/(2*wav1*z1); y = (y.*th_ph(i1)).^ph_nonlin; for k =0:N-1, boole2 = y(4*k+1+i_boole); ph_t(k+1) = boole2*boole1; end C_ph_approx = C_ph_approx + ph_t*delta_wav; end % Adding noise max_out = max(C_ph_approx); C_ph_approx = C_ph_approx/max_out*3500; var1 = 0.12345*C_ph_approx.^0.86949/n_samples_avg; delta_C_ph = sqrt(var1).*randn(size(var1)); C_ph_approx = C_ph_approx + delta_C_ph; C_ph_approx(C_ph_approx<=0)=realmin; C_ph_approx = C_ph_approx*max_out/3500; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% C_ph_sample = C_ph_approx;

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112 Appendix B: (Continued) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Calculating the Absorption spectrum of the sample % based on the spectrometer output: abs_out = log10(C_ph_reference./C_ph_sample); na = 3; abs_out = conv(ones(1,na)/na,abs_out); abs_out =abs_out([2:N+1]); % Relating the photodetector array positions % to the equivalent wavelengths: wavs = 2*d*sin(C)/m*cos((alpha-(s([3:4:n])/z2+beta0 ))/2); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% disp('Done!') disp('') disp('Execution time (in seconds):') disp(toc) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Plotting the results %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% m1 = colormap(gray); % m1 size is 64x3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Figure 1 shows the simulation output % compared to the spectrum obtained with the HR2000 ind1 = HR1_wav>=210 & HR1_wav<=900; ind2 = wavs*1e9>=210 & wavs*1e9<=900; me1 = mean(HR1_data(HR1_wav>=210 & HR1_wav<=800)); me2 = mean(abs_out(wavs*1e9>=210 & wavs*1e9<=800)); figure(1) plot(HR1_wav(ind1),HR1_data(ind1),... 'LineWidth',1,'Color',m1(1,:)) hold on plot(wavs(ind2)*1e9,abs_out(ind2)/me2*me1,... 'LineWidth',1,'Color',m1(40,:)) legend('HR2000','Simulated') xlabel('Wavelength (nanometers)') ylabel('OD (Absorption units)') title('HR2000 and Simulated Spectra') %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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113 Appendix B: (Continued) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Figure 2 shows the simulation output % compared to the spectrum obtained with the USB200 0 ind1 = USB1_wav>=210 & USB1_wav<=900; ind2 = wavs*1e9>=210 & wavs*1e9<=900; me1 = mean(USB1_data(USB1_wav>=210 & USB1_wav<=800) ); me2 = mean(abs_out(wavs*1e9>=210 & wavs*1e9<=800)); figure(2) plot(USB1_wav(ind1),USB1_data(ind1),... 'LineWidth',1,'Color',m1(1,:)) hold on plot(wavs(ind2)*1e9,abs_out(ind2)/me2*me1,... 'LineWidth',1,'Color',m1(40,:)) legend('USB2000','Simulated') xlabel('Wavelength (nanometers)') ylabel('OD (Absorption units)') title('USB2000 and Simulated Spectra') %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Figure 3 shows the simulation output % compared to the theoretical expected spectrum ind1 = th_wav>=210 & th_wav<=900; ind2 = wavs*1e9>=210 & wavs*1e9<=900; me1 = mean(th_data(th_wav>=210 & th_wav<=800)); me2 = mean(abs_out(wavs*1e9>=210 & wavs*1e9<=800)); figure(3) plot(th_wav(ind1),th_data(ind1),... 'LineWidth',1,'Color',m1(1,:)) hold on plot(wavs(ind2)*1e9,abs_out(ind2)/me2*me1,... 'LineWidth',1,'Color',m1(40,:)) legend('Theoretical','Simulated') xlabel('Wavelength (nanometers)') ylabel('OD (Absorption units)') title('Theoretical and Simulated Spectra') %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

PAGE 130

114 Appendix B: (Continued) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Figure 4 shows the theoretical spectrum of the li ght source % with a peaks at 200 nanometers figure(4) plot(th_wav,th_light,'LineWidth',2,'Color',m1(1,:)) axis([190 910 0 1.1]) xlabel('Wavelength (nm)') ylabel('Intensity') title('Theoretical Spectrum of the Light Source') %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% return

PAGE 131

115 Appendix B: (Continued) Table B.1: Theoretical Spectrum Used for the Simula tion l OD l OD l OD l OD l OD l OD l OD l OD l OD 207,0.77433 208,0.77451 209,0.77468 210,0.77486 211,0.77504 212,0.77523 213,0.77543 214,0.77564 215,0.7759 216,0.7763 217,0.77675 218,0.7772 219,0.77767 220,0.77816 221,0.77867 222,0.77917 223,0.77967 224,0.78015 225,0.78065 226,0.78129 227,0.78219 228,0.78356 229,0.78562 230,0.78853 231,0.79151 232,0.79204 233,0.78414 234,0.76662 235,0.74629 236,0.7345 237,0.73783 238,0.76596 239,0.80701 240,0.84109 241,0.86604 242,0.8824 243,0.88746 244,0.87727 245,0.85818 246,0.83685 247,0.81477 248,0.79013 249,0.76593 250,0.74386 251,0.7248 252,0.71074 253,0.70233 254,0.69746 255,0.69464 256,0.69348 257,0.6949 258,0.69991 259,0.70892 260,0.72128 261,0.73329 262,0.74468 263,0.7555 264,0.76613 265,0.77721 266,0.78896 267,0.80116 268,0.81407 269,0.82741 270,0.83983 271,0.85116 272,0.86078 273,0.86846 274,0.87499 275,0.88172 276,0.88892 277,0.89738 278,0.90671 279,0.91599 287,0.95854 288,0.96002 289,0.96038 290,0.9581 291,0.95181 292,0.94077 293,0.92694 294,0.91537 295,0.90499 296,0.89588 297,0.88844 298,0.88258 299,0.87772 300,0.87317 301,0.86767 302,0.86047 303,0.85132 304,0.84022 305,0.82757 306,0.81411 307,0.80077 308,0.78863 309,0.77821 310,0.76957 311,0.76244 312,0.7563 313,0.7505 314,0.7444 315,0.73736 316,0.72863 317,0.7186 318,0.70754 319,0.69597 320,0.68456 321,0.67398 322,0.66475 323,0.65728 324,0.65178 325,0.64767 326,0.64451 327,0.64182 328,0.63911 329,0.63591 330,0.63198 331,0.62712 332,0.62143 333,0.61546 334,0.60963 335,0.60438 336,0.60015 337,0.59723 338,0.59579 339,0.59598 340,0.59745 341,0.59981 342,0.60271 343,0.6058 344,0.60866 345,0.61091 346,0.6125 347,0.61346 348,0.614 349,0.61427 350,0.61448 351,0.61489 352,0.6157 353,0.61709 354,0.61926 355,0.62233 356,0.6262 357,0.63076 358,0.6359 359,0.64145 367,0.68226 368,0.68609 369,0.68986 370,0.69374 371,0.69791 372,0.7025 373,0.70758 374,0.71315 375,0.71922 376,0.72576 377,0.73274 378,0.7401 379,0.7477 380,0.75535 381,0.763 382,0.77053 383,0.77784 384,0.78491 385,0.79172 386,0.79814 387,0.80432 388,0.81009 389,0.81521 390,0.81996 391,0.82474 392,0.82955 393,0.83456 394,0.83977 395,0.84527 396,0.85101 397,0.85703 398,0.86331 399,0.86974 400,0.87637 401,0.88308 402,0.88986 403,0.89658 404,0.90324 405,0.90979 406,0.91617 407,0.92235 408,0.92825 409,0.93393 410,0.93934 411,0.9446 412,0.94971 413,0.95467 414,0.95955 415,0.96437 416,0.96923 417,0.97413 418,0.9791 419,0.98421 420,0.98948 421,0.99489 422,1.0004 423,1.0061 424,1.0118 425,1.0177 426,1.0236 427,1.0296 428,1.0355 429,1.0413 430,1.0471 431,1.0527 432,1.0582 433,1.0635 434,1.0686 435,1.0735 436,1.0783 437,1.0828 438,1.0872 439,1.0914 447,1.1211 448,1.1246 449,1.1281 450,1.1317 451,1.1354 452,1.1391 453,1.1428 454,1.1466 455,1.1504 456,1.1542 457,1.1579 458,1.1617 459,1.1654 460,1.169 461,1.1725 462,1.1759 463,1.1793 464,1.1825 465,1.1855 466,1.1885 467,1.1913 468,1.1939 469,1.1965 470,1.199 471,1.2013 472,1.2036 473,1.2058 474,1.2079 475,1.21 476,1.2121 477,1.2141 478,1.2162 479,1.2182 480,1.2203 481,1.2223 482,1.2244 483,1.2265 484,1.2286 485,1.2307 486,1.2329 487,1.235 488,1.2372 489,1.2393 490,1.2414 491,1.2435 492,1.2456 493,1.2477 494,1.2496 495,1.2515 496,1.2534 497,1.2551 498,1.2568 499,1.2584 500,1.2599 501,1.2614 502,1.2627 503,1.2639 504,1.2651 505,1.2662 506,1.2672 507,1.2681 508,1.269 509,1.2699 510,1.2706 511,1.2713 512,1.272 513,1.2727 514,1.2734 515,1.274 516,1.2746 517,1.2752 518,1.2758 519,1.2763 527,1.281 528,1.2816 529,1.2821 530,1.2826 531,1.2832 532,1.2837 533,1.2841 534,1.2845 535,1.2849 536,1.2852 537,1.2855 538,1.2858 539,1.286 540,1.2861 541,1.2862 542,1.2862 543,1.2862 544,1.2861 545,1.286 546,1.2858 547,1.2856 548,1.2854 549,1.2851 550,1.2847 551,1.2843 552,1.2839 553,1.2835 554,1.283 555,1.2826 556,1.2821 557,1.2815 558,1.281 559,1.2805 560,1.28 561,1.2795 562,1.2789 563,1.2784 564,1.2779 565,1.2774 566,1.2769 567,1.2764 568,1.2759 569,1.2754 570,1.2749 571,1.2744 572,1.2739 573,1.2734 574,1.2729 575,1.2724 576,1.2718 577,1.2713 578,1.2708 579,1.2702 580,1.2696 581,1.269 582,1.2684 583,1.2677 584,1.267 585,1.2663 586,1.2655 587,1.2647 588,1.2639 589,1.2631 590,1.2622 591,1.2612 592,1.2603 593,1.2593 594,1.2583 595,1.2572 596,1.2561 597,1.255 598,1.2538 599,1.2526 607,1.2424 608,1.241 609,1.2397 610,1.2383 611,1.237 612,1.2356 613,1.2342 614,1.2329 615,1.2315 616,1.2301 617,1.2288 618,1.2276 619,1.2264 620,1.2252 621,1.224 622,1.2228 623,1.2216 624,1.2204 625,1.2192 626,1.218 627,1.2169 628,1.2157 629,1.2145 630,1.2133 631,1.2122 632,1.211 633,1.2098 634,1.2086 635,1.2075 636,1.2063 637,1.2051 638,1.2039 639,1.2027 640,1.2014 641,1.2002 642,1.199 643,1.1977 644,1.1964 645,1.1951 646,1.1938 647,1.1925 648,1.1912 649,1.1899 650,1.1885 651,1.1871 652,1.1857 653,1.1843 654,1.1829 655,1.1814 656,1.1799 657,1.1784 658,1.177 659,1.1755 660,1.174 661,1.1724 662,1.1709 663,1.1693 664,1.1678 665,1.1662 666,1.1646 667,1.163 668,1.1614 669,1.1598 670,1.1582 671,1.1566 672,1.155 673,1.1533 674,1.1517 675,1.1501 676,1.1484 677,1.1468 678,1.1452 679,1.1436 687,1.1308 688,1.1291 689,1.1276 690,1.126 691,1.1244 692,1.1228 693,1.1213 694,1.1197 695,1.1181 696,1.1165 697,1.115 698,1.1135 699,1.1119 700,1.1104 701,1.1088 702,1.1073 703,1.1057 704,1.1042 705,1.1026 706,1.1011 707,1.0995 708,1.098 709,1.0964 710,1.0948 711,1.0933 712,1.0917 713,1.0902 714,1.0886 715,1.087 716,1.0854 717,1.0838 718,1.0822 719,1.0806 720,1.079 721,1.0774 722,1.0757 723,1.0741 724,1.0725 725,1.0708 726,1.0692 727,1.0675 728,1.0659 729,1.0642 730,1.0625 731,1.0609 732,1.0592 733,1.0575 734,1.0558 735,1.0541 736,1.0524 737,1.0507 738,1.049 739,1.0473 740,1.0456 741,1.0438 742,1.0422 743,1.0404 744,1.0387 745,1.037 746,1.0353 747,1.0336 748,1.0318 749,1.0301 750,1.0284 751,1.0267 752,1.025 753,1.0233 754,1.0215 755,1.0198 756,1.0181 757,1.0164 758,1.0147 759,1.013 767,1.0005 768,0.99892 769,0.99736 770,0.99585 771,0.99436 772,0.99282 773,0.99131 774,0.98976 775,0.98826 776,0.98678 777,0.98527 778,0.98379 779,0.98231 780,0.98079 781,0.97927 782,0.9778 783,0.97631 784,0.97486 785,0.97337 786,0.97185 787,0.97037 788,0.96896 789,0.96745 790,0.96594 791,0.96451 792,0.96303 793,0.96155 794,0.96007 795,0.95858 796,0.95713 797,0.95567 798,0.95417 799,0.9527 800,0.95125 801,0.94976 802,0.94827 803,0.94681 804,0.94533 805,0.94385 806,0.94235 807,0.94084 808,0.93938 809,0.9379 810,0.9364 811,0.93491 812,0.9334 813,0.93191 814,0.93043 815,0.92891 816,0.92744 817,0.92596 818,0.92442 819,0.92289 820,0.92139 821,0.91989 822,0.91836 823,0.91685 824,0.91532 825,0.91381 826,0.91232 827,0.91078 828,0.90923 829,0.90773 830,0.90624 831,0.90474 832,0.90321 833,0.90168 834,0.90018 835,0.89867 836,0.89712 837,0.89562 838,0.89413 839,0.8926 847,0.88057 848,0.87904 849,0.87753 850,0.87606 851,0.87458 852,0.87313 853,0.87163 854,0.87012 855,0.86869 856,0.86725 857,0.86576 858,0.86432 859,0.86287 860,0.86139 861,0.85996 862,0.8585 863,0.85708 864,0.85567 865,0.8542 866,0.85276 867,0.85136 868,0.84996 869,0.84858 870,0.84715 871,0.84572 872,0.84433 873,0.84298 874,0.84156 875,0.84021 876,0.83884 877,0.83745 878,0.83608 879,0.83471 880,0.83333 881,0.83198 882,0.83065 883,0.82927 884,0.82795 885,0.82662 886,0.82527 887,0.82393 888,0.82263 889,0.82132 890,0.82001 891,0.81867 892,0.81732 893,0.81604 894,0.81474 895,0.81342 896,0.81212 897,0.8108 898,0.80952 899,0.80823 900,0.80692

PAGE 132

116 Appendix B: (Continued) Table B.2: Spectrum Obtained with the Ocean Optics HR2000 l OD l OD l OD l OD l OD l OD l OD l OD l OD 188.590,0 189.059,0 189.528,-1.447 189.996,-0.072 190.465,-0.319 190.933,-0.171 191.401,-0.473 191.870,-0.391 192.338,-0.336 192.807,-0.211 193.275,-0.238 193.743,-0.184 194.212,-0.207 194.680,-0.27 195.148,-0.242 195.617,-0.194 196.085,-0.052 196.553,-0.187 197.021,-0.13 197.489,-0.235 197.958,-0.275 198.426,-0.293 198.894,-0.335 199.362,0 199.830,0 200.298,0 200.766,3.339 201.234,1.701 201.702,1.556 202.170,1.449 202.638,1.407 203.106,1.387 203.574,1.374 204.042,1.341 204.510,1.327 204.978,1.306 205.446,1.295 205.913,1.276 206.381,1.263 206.849,1.24 207.317,1.239 207.785,1.229 208.252,1.229 208.720,1.221 209.188,1.215 209.655,1.207 210.123,1.206 210.591,1.201 211.058,1.198 211.526,1.187 211.993,1.183 212.461,1.183 212.928,1.187 213.396,1.187 213.863,1.185 214.331,1.18 214.798,1.178 215.266,1.172 215.733,1.164 216.201,1.152 216.668,1.148 217.135,1.139 217.603,1.136 218.070,1.132 218.537,1.132 219.004,1.13 219.472,1.13 219.939,1.124 220.406,1.127 220.873,1.124 221.340,1.121 221.808,1.116 222.275,1.112 222.742,1.108 223.209,1.106 223.676,1.102 224.143,1.1 224.610,1.097 225.077,1.095 225.544,1.094 226.011,1.092 226.478,1.089 226.945,1.087 227.412,1.087 227.879,1.09 228.345,1.095 228.812,1.103 229.279,1.111 229.746,1.117 230.213,1.123 230.679,1.122 231.146,1.117 231.613,1.106 232.080,1.086 232.546,1.065 233.013,1.046 233.479,1.031 233.946,1.018 234.413,1.007 234.879,1 235.346,0.999 235.812,1.002 236.279,1.01 236.745,1.021 237.212,1.036 237.678,1.053 238.145,1.07 238.611,1.09 239.077,1.108 239.544,1.124 240.010,1.138 240.476,1.15 240.943,1.163 241.409,1.174 241.875,1.183 242.342,1.187 242.808,1.19 243.274,1.189 243.740,1.186 244.206,1.181 244.672,1.178 245.138,1.172 245.605,1.168 246.071,1.161 246.537,1.152 247.003,1.142 247.469,1.133 247.935,1.121 248.401,1.111 248.867,1.098 249.333,1.09 249.798,1.082 250.264,1.077 250.730,1.07 251.196,1.063 251.662,1.056 252.128,1.05 252.593,1.045 253.059,1.039 253.525,1.032 253.991,1.028 254.456,1.024 254.922,1.022 255.388,1.021 255.853,1.02 256.319,1.021 256.785,1.022 257.250,1.023 257.716,1.026 258.181,1.028 258.647,1.029 259.112,1.031 259.578,1.032 260.043,1.033 260.509,1.034 260.974,1.035 261.439,1.037 261.905,1.038 262.370,1.04 262.835,1.042 263.301,1.046 263.766,1.051 264.231,1.056 264.696,1.06 265.162,1.066 265.627,1.07 266.092,1.075 266.557,1.079 267.022,1.083 267.487,1.086 267.952,1.09 268.417,1.093 268.883,1.095 269.348,1.097 269.813,1.099 270.278,1.1 270.742,1.103 271.207,1.107 271.672,1.114 272.137,1.123 272.602,1.135 273.067,1.147 273.532,1.159 273.997,1.169 274.461,1.181 274.926,1.19 275.391,1.198 275.856,1.203 276.320,1.206 276.785,1.208 277.250,1.211 277.714,1.211 278.179,1.213 278.644,1.216 279.108,1.221 279.573,1.227 280.037,1.237 280.502,1.246 280.966,1.255 281.431,1.263 281.895,1.27 282.360,1.275 282.824,1.281 283.288,1.283 283.753,1.283 284.217,1.282 284.681,1.28 285.146,1.275 285.610,1.271 286.074,1.262 286.539,1.257 287.003,1.251 287.467,1.248 287.931,1.245 288.395,1.245 288.859,1.245 289.324,1.249 289.788,1.251 290.252,1.256 290.716,1.257 291.180,1.259 291.644,1.258 292.108,1.259 292.572,1.256 293.036,1.255 293.500,1.249 293.963,1.244 294.427,1.235 294.891,1.228 295.355,1.217 295.819,1.208 296.283,1.197 296.746,1.188 297.210,1.179 297.674,1.174 298.138,1.169 298.601,1.167 299.065,1.163 299.528,1.161 299.992,1.16 300.456,1.16 300.919,1.158 301.383,1.156 301.846,1.154 302.310,1.153 302.773,1.149 303.237,1.146 303.700,1.14 304.164,1.135 304.627,1.127 305.090,1.117 305.554,1.107 306.017,1.096 306.480,1.085 306.944,1.074 307.407,1.062 307.870,1.052 308.333,1.043 308.797,1.036 309.260,1.029 309.723,1.024 310.186,1.019 310.649,1.017 311.112,1.014 311.575,1.012 312.038,1.01 312.501,1.009 312.965,1.008 313.427,1.007 313.890,1.004 314.353,1.003 314.816,1 315.279,0.996 315.742,0.992 316.205,0.985 316.668,0.978 317.131,0.972 317.593,0.963 318.056,0.954 318.519,0.945 318.982,0.936 319.444,0.926 319.907,0.919 320.370,0.91 320.832,0.903 321.295,0.896 321.757,0.89 322.220,0.886 322.683,0.884 323.145,0.881 323.608,0.88 324.070,0.88 324.533,0.88 324.995,0.88 325.457,0.88 325.920,0.879 326.382,0.88 326.845,0.88 327.307,0.88 327.769,0.879 328.231,0.879 328.694,0.877 329.156,0.875 329.618,0.873 330.080,0.869 330.543,0.864 331.005,0.86 331.467,0.854 331.929,0.848 332.391,0.842 332.853,0.836 333.315,0.83 333.777,0.825 334.239,0.82 334.701,0.816 335.163,0.812 335.625,0.81 336.087,0.808 336.549,0.807 337.011,0.806 337.473,0.807 337.934,0.808 338.396,0.809 338.858,0.811 339.320,0.812 339.781,0.815 340.243,0.818 340.705,0.82 341.166,0.822 341.628,0.824 342.090,0.827 342.551,0.829 343.013,0.831 343.474,0.831 343.936,0.833 344.397,0.833 344.859,0.833 345.320,0.831 345.782,0.831 346.243,0.829 346.705,0.829 347.166,0.826 347.627,0.825 348.089,0.822 348.550,0.821 349.011,0.818 349.473,0.817 349.934,0.814 350.395,0.813 350.856,0.813 351.317,0.814 351.779,0.815 352.240,0.816 352.701,0.818 353.162,0.82 353.623,0.823 354.084,0.825 354.545,0.828 355.006,0.831 355.467,0.834 355.928,0.838 356.389,0.841 356.850,0.845 357.311,0.849 357.771,0.853 358.232,0.857 358.693,0.861 359.154,0.865 359.615,0.869 360.075,0.872 360.536,0.876 360.997,0.879 361.458,0.881 361.918,0.883 362.379,0.886 362.839,0.887 363.300,0.889 363.761,0.89 364.221,0.89 364.682,0.891 365.142,0.892 365.603,0.892 366.063,0.893 366.524,0.893 366.984,0.895 367.444,0.898 367.905,0.899 368.365,0.901 368.825,0.904 369.286,0.906 369.746,0.909 370.206,0.912 370.666,0.916 371.127,0.92 371.587,0.923 372.047,0.927 372.507,0.932 372.967,0.936 373.427,0.94 373.887,0.944 374.347,0.948 374.807,0.953 375.267,0.959 375.727,0.963 376.187,0.969 376.647,0.974 377.107,0.979 377.567,0.984 378.027,0.989 378.487,0.991 378.947,0.997 379.406,1 379.866,1.004 380.326,1.007 380.786,1.011 381.245,1.014 381.705,1.019 382.165,1.021 382.624,1.024 383.084,1.026 383.543,1.029 384.003,1.03 384.463,1.034 384.922,1.035 385.382,1.038 385.841,1.04 386.301,1.044 386.760,1.046 387.219,1.051 387.679,1.052 388.138,1.056 388.597,1.058 389.057,1.061 389.516,1.064 389.975,1.067 390.435,1.07 390.894,1.075 391.353,1.077 391.812,1.082 392.271,1.085 392.730,1.089 393.190,1.094 393.649,1.1 394.108,1.103 394.567,1.109 395.026,1.112 395.485,1.118 395.944,1.122 396.403,1.127 396.862,1.13 397.320,1.136 397.779,1.141 398.238,1.147 398.697,1.151 399.156,1.157 399.615,1.159 400.073,1.163 400.532,1.165 400.991,1.17 401.449,1.173 401.908,1.178 402.367,1.181 402.825,1.188 403.284,1.189 403.743,1.189 404.201,1.193 404.660,1.196 405.118,1.198 405.577,1.202 406.035,1.204 406.494,1.211 406.952,1.216 407.410,1.219 407.869,1.221 408.327,1.225 408.785,1.226 409.244,1.228 409.702,1.228 410.160,1.232 410.618,1.234 411.077,1.237 411.535,1.239 411.993,1.243 412.451,1.245 412.909,1.249 413.367,1.252 413.825,1.257 414.283,1.261 414.741,1.267 415.199,1.27 415.657,1.275 416.115,1.279 416.573,1.284 417.031,1.287 417.489,1.293 417.947,1.292 418.405,1.299 418.862,1.302 419.320,1.306 419.778,1.308 420.236,1.313 420.693,1.315 421.151,1.324 421.609,1.326 422.066,1.332 422.524,1.336 422.982,1.342 423.439,1.346 423.897,1.352 424.354,1.354 424.812,1.358 425.269,1.36 425.727,1.363 426.184,1.367 426.642,1.37 427.099,1.372 427.556,1.376 428.014,1.377 428.471,1.38 428.928,1.381 429.385,1.383 429.843,1.381 430.300,1.383 430.757,1.383 431.214,1.388 431.671,1.388 432.129,1.39 432.586,1.392 433.043,1.398 433.500,1.397 433.957,1.393 434.414,1.372 434.871,1.343 435.328,1.338 435.785,1.34 436.242,1.342 436.698,1.348 437.155,1.358 437.612,1.382 438.069,1.419 438.526,1.429 438.983,1.434 439.439,1.438 439.896,1.439 440.353,1.439 440.809,1.439 441.266,1.443 441.723,1.445 442.179,1.446 442.636,1.447 443.092,1.451 443.549,1.456 444.005,1.461 444.462,1.46 444.918,1.466 445.375,1.47 445.831,1.476 446.288,1.475 446.744,1.475 447.200,1.477 447.657,1.483 448.113,1.484 448.569,1.492 449.025,1.492 449.482,1.497 449.938,1.499 450.394,1.504 450.850,1.507 451.306,1.508 451.762,1.503 452.218,1.51 452.674,1.512 453.130,1.517 453.586,1.519 454.042,1.523 454.498,1.526 454.954,1.53 455.410,1.529 455.866,1.535 456.322,1.537 456.778,1.539 457.234,1.534 457.689,1.538 458.145,1.541 458.601,1.541 459.057,1.541 459.512,1.544 459.968,1.545 460.424,1.553 460.879,1.555 461.335,1.557 461.790,1.56 462.246,1.565 462.701,1.562 463.157,1.564 463.612,1.564 464.068,1.562 464.523,1.564 464.979,1.563 465.434,1.56 465.889,1.564 466.345,1.565 466.800,1.568 467.255,1.568 467.710,1.571 468.166,1.574 468.621,1.578 469.076,1.578 469.531,1.581 469.986,1.581 470.441,1.588 470.897,1.589 471.352,1.592 471.807,1.592 472.262,1.596 472.717,1.596 473.172,1.596 473.627,1.592 474.081,1.593 474.536,1.594 474.991,1.599 475.446,1.599 475.901,1.599 476.356,1.601 476.810,1.606 477.265,1.607 477.720,1.61 478.175,1.608 478.629,1.612 479.084,1.612 479.539,1.616 479.993,1.62 480.448,1.623 480.902,1.621 481.357,1.623 481.811,1.621 482.266,1.624 482.720,1.622 483.175,1.619 483.629,1.616 484.083,1.616 484.538,1.612 484.992,1.612 485.446,1.61 485.901,1.609 486.355,1.609 486.809,1.609 487.263,1.607 487.718,1.611 488.172,1.61 488.626,1.611 489.080,1.612 489.534,1.613 489.988,1.615 490.442,1.618 490.896,1.619 491.350,1.623 491.804,1.625 492.258,1.627 492.712,1.628 493.166,1.632 493.620,1.633 494.073,1.635 494.527,1.634 494.981,1.636 495.435,1.638 495.889,1.642 496.342,1.641 496.796,1.643 497.250,1.645 497.703,1.649 498.157,1.65 498.611,1.651 499.064,1.65 499.518,1.653 499.971,1.654 500.425,1.656 500.878,1.658 501.332,1.662 501.785,1.665 502.238,1.668 502.692,1.669 503.145,1.671 503.598,1.67 504.052,1.671 504.505,1.669 504.958,1.669 505.411,1.668 505.865,1.669 506.318,1.667 506.771,1.67 507.224,1.669 507.677,1.67 508.130,1.67 508.583,1.671 509.036,1.672 509.489,1.676 509.942,1.675 510.395,1.678 510.848,1.678 511.301,1.68 511.754,1.681 512.207,1.683 512.659,1.683 513.112,1.687 513.565,1.687 514.018,1.689 514.471,1.689 514.923,1.692 515.376,1.692 515.828,1.692 516.281,1.69 516.734,1.691 517.186,1.69 517.639,1.691 518.091,1.69 518.544,1.691 518.996,1.691 519.449,1.69 519.901,1.692

PAGE 133

117 Appendix B: (Continued) Table B.2: (Continued) l OD l OD l OD l OD l OD l OD l OD l OD l OD 520.354,1.695 520.806,1.695 521.258,1.696 521.711,1.696 522.163,1.698 522.615,1.699 523.067,1.699 523.520,1.699 523.972,1.702 524.424,1.701 524.876,1.704 525.328,1.704 525.780,1.705 526.232,1.705 526.685,1.706 527.137,1.702 527.589,1.703 528.040,1.698 528.492,1.701 528.944,1.699 529.396,1.7 529.848,1.699 530.300,1.702 530.752,1.702 531.204,1.705 531.655,1.702 532.107,1.704 532.559,1.702 533.010,1.704 533.462,1.702 533.914,1.702 534.365,1.702 534.817,1.702 535.269,1.699 535.720,1.698 536.172,1.694 536.623,1.696 537.075,1.692 537.526,1.689 537.977,1.684 538.429,1.682 538.880,1.677 539.331,1.674 539.783,1.666 540.234,1.658 540.685,1.647 541.137,1.633 541.588,1.615 542.039,1.601 542.490,1.587 542.941,1.577 543.392,1.568 543.843,1.564 544.294,1.562 544.746,1.558 545.197,1.546 545.648,1.554 546.098,1.565 546.549,1.579 547.000,1.59 547.451,1.603 547.902,1.623 548.353,1.651 548.804,1.656 549.254,1.659 549.705,1.658 550.156,1.66 550.607,1.661 551.057,1.663 551.508,1.665 551.959,1.668 552.409,1.668 552.860,1.67 553.310,1.673 553.761,1.678 554.211,1.68 554.662,1.682 555.112,1.683 555.563,1.686 556.013,1.689 556.463,1.691 556.914,1.691 557.364,1.693 557.814,1.693 558.265,1.695 558.715,1.695 559.165,1.694 559.615,1.693 560.065,1.694 560.516,1.692 560.966,1.693 561.416,1.691 561.866,1.69 562.316,1.69 562.766,1.69 563.216,1.69 563.666,1.69 564.116,1.689 564.566,1.69 565.016,1.691 565.465,1.691 565.915,1.691 566.365,1.693 566.815,1.691 567.264,1.691 567.714,1.689 568.164,1.688 568.614,1.686 569.063,1.686 569.513,1.684 569.962,1.686 570.412,1.686 570.862,1.685 571.311,1.685 571.761,1.686 572.210,1.684 572.659,1.683 573.109,1.68 573.558,1.678 574.008,1.678 574.457,1.678 574.906,1.675 575.356,1.676 575.805,1.673 576.254,1.671 576.703,1.67 577.152,1.668 577.602,1.666 578.051,1.663 578.500,1.66 578.949,1.661 579.398,1.66 579.847,1.658 580.296,1.657 580.745,1.656 581.194,1.658 581.643,1.658 582.092,1.656 582.541,1.657 582.989,1.655 583.438,1.654 583.887,1.653 584.336,1.653 584.785,1.651 585.233,1.652 585.682,1.651 586.131,1.65 586.579,1.646 587.028,1.645 587.476,1.644 587.925,1.643 588.373,1.644 588.822,1.644 589.270,1.645 589.719,1.65 590.167,1.651 590.616,1.653 591.064,1.652 591.512,1.652 591.961,1.651 592.409,1.651 592.857,1.648 593.306,1.648 593.754,1.646 594.202,1.647 594.650,1.647 595.098,1.65 595.546,1.65 595.995,1.653 596.443,1.652 596.891,1.653 597.339,1.652 597.787,1.652 598.234,1.649 598.682,1.648 599.130,1.647 599.578,1.646 600.026,1.646 600.474,1.646 600.922,1.644 601.369,1.645 601.817,1.646 602.265,1.647 602.713,1.648 603.160,1.648 603.608,1.649 604.055,1.651 604.503,1.651 604.951,1.65 605.398,1.648 605.846,1.648 606.293,1.646 606.741,1.645 607.188,1.641 607.635,1.64 608.083,1.638 608.530,1.636 608.977,1.631 609.425,1.623 609.872,1.605 610.319,1.583 610.766,1.567 611.214,1.559 611.661,1.555 612.108,1.553 612.555,1.556 613.002,1.568 613.449,1.586 613.896,1.599 614.343,1.604 614.790,1.608 615.237,1.611 615.684,1.614 616.131,1.615 616.578,1.618 617.025,1.621 617.471,1.623 617.918,1.624 618.365,1.624 618.812,1.624 619.258,1.625 619.705,1.623 620.152,1.621 620.598,1.622 621.045,1.622 621.492,1.623 621.938,1.624 622.385,1.623 622.831,1.624 623.278,1.624 623.724,1.623 624.170,1.622 624.617,1.621 625.063,1.62 625.510,1.621 625.956,1.62 626.402,1.621 626.848,1.621 627.295,1.621 627.741,1.621 628.187,1.621 628.633,1.619 629.079,1.619 629.525,1.616 629.971,1.614 630.417,1.611 630.863,1.609 631.309,1.608 631.755,1.608 632.201,1.608 632.647,1.609 633.093,1.611 633.539,1.613 633.985,1.615 634.431,1.615 634.876,1.613 635.322,1.613 635.768,1.612 636.214,1.611 636.659,1.609 637.105,1.608 637.550,1.606 637.996,1.607 638.442,1.605 638.887,1.606 639.333,1.605 639.778,1.604 640.224,1.603 640.669,1.603 641.114,1.601 641.560,1.601 642.005,1.6 642.450,1.599 642.896,1.598 643.341,1.598 643.786,1.596 644.231,1.596 644.677,1.594 645.122,1.593 645.567,1.592 646.012,1.591 646.457,1.589 646.902,1.588 647.347,1.586 647.792,1.586 648.237,1.584 648.682,1.583 649.127,1.581 649.572,1.581 650.017,1.58 650.461,1.58 650.906,1.58 651.351,1.581 651.796,1.581 652.240,1.582 652.685,1.581 653.130,1.581 653.574,1.579 654.019,1.578 654.464,1.575 654.908,1.561 655.353,1.532 655.797,1.543 656.242,1.549 656.686,1.549 657.130,1.548 657.575,1.547 658.019,1.556 658.464,1.587 658.908,1.574 659.352,1.566 659.796,1.566 660.241,1.566 660.685,1.565 661.129,1.564 661.573,1.562 662.017,1.561 662.461,1.559 662.905,1.557 663.349,1.554 663.793,1.554 664.237,1.553 664.681,1.552 665.125,1.551 665.569,1.551 666.013,1.551 666.457,1.551 666.901,1.551 667.344,1.55 667.788,1.549 668.232,1.549 668.676,1.546 669.119,1.545 669.563,1.544 670.007,1.543 670.450,1.542 670.894,1.542 671.337,1.54 671.781,1.541 672.224,1.539 672.668,1.539 673.111,1.538 673.555,1.538 673.998,1.536 674.441,1.536 674.885,1.535 675.328,1.536 675.771,1.534 676.214,1.533 676.658,1.533 677.101,1.533 677.544,1.533 677.987,1.532 678.430,1.531 678.873,1.531 679.316,1.529 679.759,1.527 680.202,1.525 680.645,1.523 681.088,1.521 681.531,1.521 681.974,1.52 682.417,1.52 682.860,1.52 683.303,1.521 683.745,1.522 684.188,1.521 684.631,1.519 685.073,1.52 685.516,1.519 685.959,1.519 686.401,1.518 686.844,1.516 687.286,1.514 687.729,1.513 688.172,1.51 688.614,1.509 689.056,1.507 689.499,1.506 689.941,1.506 690.384,1.506 690.826,1.506 691.268,1.506 691.710,1.505 692.153,1.505 692.595,1.504 693.037,1.504 693.479,1.503 693.921,1.503 694.363,1.503 694.806,1.502 695.248,1.5 695.690,1.499 696.132,1.496 696.574,1.494 697.015,1.492 697.457,1.491 697.899,1.489 698.341,1.489 698.783,1.488 699.225,1.488 699.667,1.487 700.108,1.488 700.550,1.488 700.992,1.488 701.433,1.487 701.875,1.488 702.317,1.486 702.758,1.485 703.200,1.483 703.641,1.481 704.083,1.478 704.524,1.476 704.966,1.474 705.407,1.473 705.848,1.471 706.290,1.47 706.731,1.467 707.172,1.467 707.614,1.466 708.055,1.466 708.496,1.465 708.937,1.464 709.378,1.463 709.819,1.463 710.260,1.463 710.702,1.462 711.143,1.46 711.584,1.461 712.025,1.46 712.466,1.459 712.906,1.458 713.347,1.458 713.788,1.457 714.229,1.459 714.670,1.457 715.111,1.456 715.551,1.455 715.992,1.455 716.433,1.453 716.874,1.453 717.314,1.45 717.755,1.449 718.195,1.449 718.636,1.449 719.076,1.447 719.517,1.446 719.957,1.445 720.398,1.445 720.838,1.444 721.279,1.443 721.719,1.441 722.159,1.441 722.600,1.439 723.040,1.438 723.480,1.437 723.920,1.435 724.361,1.433 724.801,1.433 725.241,1.431 725.681,1.431 726.121,1.43 726.561,1.43 727.001,1.43 727.441,1.43 727.881,1.428 728.321,1.428 728.761,1.426 729.201,1.425 729.641,1.423 730.081,1.423 730.520,1.422 730.960,1.422 731.400,1.421 731.839,1.42 732.279,1.42 732.719,1.419 733.158,1.418 733.598,1.417 734.038,1.416 734.477,1.416 734.917,1.414 735.356,1.413 735.796,1.412 736.235,1.412 736.674,1.41 737.114,1.41 737.553,1.409 737.992,1.41 738.432,1.409 738.871,1.409 739.310,1.408 739.749,1.407 740.189,1.405 740.628,1.404 741.067,1.402 741.506,1.401 741.945,1.399 742.384,1.399 742.823,1.399 743.262,1.398 743.701,1.397 744.140,1.396 744.579,1.394 745.017,1.394 745.456,1.393 745.895,1.391 746.334,1.389 746.773,1.388 747.211,1.387 747.650,1.386 748.089,1.385 748.527,1.384 748.966,1.384 749.404,1.384 749.843,1.382 750.281,1.382 750.720,1.38 751.158,1.379 751.597,1.378 752.035,1.377 752.474,1.376 752.912,1.377 753.350,1.376 753.788,1.375 754.227,1.375 754.665,1.374 755.103,1.373 755.541,1.372 755.979,1.371 756.417,1.37 756.856,1.369 757.294,1.368 757.732,1.366 758.170,1.365 758.608,1.363 759.045,1.361 759.483,1.36 759.921,1.359 760.359,1.358 760.797,1.358 761.235,1.357 761.672,1.357 762.110,1.356 762.548,1.354 762.986,1.352 763.423,1.351 763.861,1.35 764.298,1.35 764.736,1.35 765.173,1.35 765.611,1.35 766.048,1.35 766.486,1.349 766.923,1.348 767.361,1.346 767.798,1.344 768.235,1.341 768.673,1.34 769.110,1.339 769.547,1.338 769.984,1.337 770.421,1.337 770.859,1.338 771.296,1.338 771.733,1.338 772.170,1.338 772.607,1.337 773.044,1.336 773.481,1.335 773.918,1.334 774.355,1.332 774.792,1.331 775.228,1.33 775.665,1.329 776.102,1.328 776.539,1.328 776.976,1.327 777.412,1.328 777.849,1.327 778.286,1.327 778.722,1.326 779.159,1.326 779.595,1.323 780.032,1.322 780.468,1.32 780.905,1.32 781.341,1.319 781.778,1.318 782.214,1.316 782.650,1.317 783.087,1.316 783.523,1.317 783.959,1.316 784.396,1.314 784.832,1.313 785.268,1.313 785.704,1.311 786.140,1.31 786.576,1.308 787.012,1.307 787.448,1.308 787.884,1.307 788.320,1.307 788.756,1.306 789.192,1.305 789.628,1.304 790.064,1.3 790.500,1.297 790.936,1.294 791.371,1.291 791.807,1.29 792.243,1.289 792.679,1.289 793.114,1.29 793.550,1.289 793.985,1.289 794.421,1.289 794.856,1.29 795.292,1.289 795.727,1.289 796.163,1.287 796.598,1.288 797.034,1.286 797.469,1.287 797.904,1.284 798.340,1.284 798.775,1.283 799.210,1.283 799.645,1.281 800.081,1.281 800.516,1.279 800.951,1.279 801.386,1.277 801.821,1.277 802.256,1.275 802.691,1.275 803.126,1.274 803.561,1.274 803.996,1.272 804.431,1.271 804.866,1.269 805.300,1.267 805.735,1.264 806.170,1.262 806.605,1.259 807.039,1.259 807.474,1.258 807.909,1.258 808.343,1.257 808.778,1.258 809.212,1.258 809.647,1.257 810.081,1.256 810.516,1.255 810.950,1.254 811.385,1.253 811.819,1.252 812.253,1.251 812.688,1.251 813.122,1.25 813.556,1.249 813.991,1.248 814.425,1.246 814.859,1.244 815.293,1.242 815.727,1.24 816.161,1.239 816.595,1.238 817.029,1.237 817.463,1.237 817.897,1.236 818.331,1.237 818.765,1.237 819.199,1.237 819.633,1.237 820.067,1.237 820.500,1.236 820.934,1.236 821.368,1.236 821.802,1.236 822.235,1.235 822.669,1.234 823.102,1.233 823.536,1.233 823.970,1.233 824.403,1.232 824.836,1.231 825.270,1.23 825.703,1.229 826.137,1.227 826.570,1.225 827.003,1.223 827.437,1.221 827.870,1.219 828.303,1.218 828.736,1.218 829.170,1.216 829.603,1.215 830.036,1.215 830.469,1.216 830.902,1.216 831.335,1.215 831.768,1.212 832.201,1.211 832.634,1.209 833.067,1.207 833.500,1.205 833.933,1.204 834.365,1.202 834.798,1.202 835.231,1.201 835.664,1.201 836.096,1.201 836.529,1.201 836.962,1.199 837.394,1.199 837.827,1.197 838.259,1.196 838.692,1.195

PAGE 134

118 Appendix B: (Continued) Table B.2: (Continued) l OD l OD l OD l OD l OD l OD l OD l OD 839.124,1.196 839.557,1.196 839.989,1.196 840.422,1.195 840.854,1.197 841.286,1.197 841.719,1.198 842.151,1.195 842.583,1.194 843.015,1.193 843.448,1.192 843.880,1.19 844.312,1.189 844.744,1.187 845.176,1.185 845.608,1.183 846.040,1.182 846.472,1.179 846.904,1.177 847.336,1.176 847.768,1.175 848.200,1.173 848.631,1.173 849.063,1.171 849.495,1.171 849.927,1.169 850.358,1.168 850.790,1.167 851.222,1.167 851.653,1.167 852.085,1.167 852.516,1.167 852.948,1.167 853.379,1.166 853.811,1.165 854.242,1.163 854.674,1.161 855.105,1.16 855.536,1.16 855.968,1.158 856.399,1.158 856.830,1.158 857.261,1.159 857.693,1.159 858.124,1.16 858.555,1.16 858.986,1.162 859.417,1.161 859.848,1.162 860.279,1.162 860.710,1.161 861.141,1.159 861.572,1.156 862.003,1.153 862.433,1.152 862.864,1.15 863.295,1.148 863.726,1.146 864.156,1.147 864.587,1.148 865.018,1.148 865.448,1.146 865.879,1.146 866.310,1.145 866.740,1.146 867.171,1.145 867.601,1.145 868.032,1.146 868.462,1.146 868.892,1.145 869.323,1.145 869.753,1.143 870.183,1.141 870.614,1.137 871.044,1.135 871.474,1.132 871.904,1.129 872.334,1.126 872.764,1.126 873.195,1.124 873.625,1.125 874.055,1.124 874.484,1.125 874.914,1.124 875.344,1.124 875.774,1.124 876.204,1.126 876.634,1.125 877.064,1.125 877.494,1.124 877.923,1.125 878.353,1.125 878.783,1.124 879.212,1.122 879.642,1.122 880.071,1.119 880.501,1.12 880.931,1.12 881.360,1.12 881.789,1.121 882.219,1.121 882.648,1.121 883.078,1.122 883.507,1.121 883.936,1.121 884.366,1.12 884.795,1.118 885.224,1.116 885.653,1.114 886.082,1.112 886.511,1.111 886.940,1.108 887.370,1.106 887.799,1.104 888.228,1.103 888.656,1.104 889.085,1.104 889.514,1.102 889.943,1.102 890.372,1.101 890.801,1.1 891.230,1.099 891.658,1.097 892.087,1.095 892.516,1.094 892.944,1.093 893.373,1.094 893.802,1.094 894.230,1.094 894.659,1.093 895.087,1.095 895.516,1.095 895.944,1.096 896.372,1.095 896.801,1.095 897.229,1.093 897.657,1.093 898.086,1.09 898.514,1.089 898.942,1.086 899.370,1.086 899.798,1.085 900.227,1.084 900.655,1.083 901.083,1.083 901.511,1.08 901.939,1.079 902.367,1.079 902.795,1.078 903.223,1.077 903.650,1.075 904.078,1.074 904.506,1.075 904.934,1.075 905.362,1.074 905.789,1.073 906.217,1.073 906.645,1.072 907.072,1.07 907.500,1.068 907.927,1.068 908.355,1.065 908.782,1.064 909.210,1.063 909.637,1.063 910.065,1.063 910.492,1.064 910.919,1.063 911.347,1.064 911.774,1.064 912.201,1.064 912.628,1.064 913.056,1.064 913.483,1.063 913.910,1.063 914.337,1.062 914.764,1.062 915.191,1.061 915.618,1.062 916.045,1.06 916.472,1.057 916.899,1.053 917.326,1.05 917.752,1.047 918.179,1.045 918.606,1.042 919.033,1.042 919.459,1.041 919.886,1.042 920.313,1.041 920.739,1.041 921.166,1.04 921.592,1.039 922.019,1.036 922.445,1.036 922.872,1.034 923.298,1.035 923.725,1.036 924.151,1.036 924.577,1.036 925.004,1.038 925.430,1.036 925.856,1.037 926.282,1.035 926.708,1.035 927.135,1.034 927.561,1.032 927.987,1.028 928.413,1.029 928.839,1.026 929.265,1.026 929.691,1.026 930.117,1.027 930.542,1.029 930.968,1.029 931.394,1.026 931.820,1.025 932.246,1.022 932.671,1.019 933.097,1.015 933.523,1.012 933.948,1.012 934.374,1.013 934.799,1.014 935.225,1.014 935.651,1.013 936.076,1.014 936.501,1.013 936.927,1.012 937.352,1.01 937.778,1.01 938.203,1.011 938.628,1.013 939.053,1.012 939.479,1.015 939.904,1.015 940.329,1.018 940.754,1.015 941.179,1.016 941.604,1.015 942.029,1.016 942.454,1.011 942.879,1.013 943.304,1.011 943.729,1.013 944.154,1.011 944.579,1.013 945.003,1.011 945.428,1.015 945.853,1.013 946.278,1.014 946.702,1.01 947.127,1.012 947.551,1.008 947.976,1.007 948.401,1.001 948.825,1 949.250,0.999 949.674,0.999 950.098,0.994 950.523,0.993 950.947,0.991 951.371,0.992 951.796,0.99 952.220,0.99 952.644,0.99 953.068,0.991 953.492,0.992 953.917,0.993 954.341,0.992 954.765,0.993 955.189,0.991 955.613,0.991 956.037,0.99 956.461,0.988 956.885,0.985 957.308,0.985 957.732,0.982 958.156,0.981 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ABOUT THE AUTHOR Eduardo E. Zurek was born in Gamarra (Departamento del Cesar, Colombia). He completed his Bachelor’s degree in Computer Science (Ingeniera de Sistemas) from the Universidad del Norte in Barranquilla (Departamento del Atlntico, Colombia) in 1994, and his Master of Science in Electrical Engineering from the University of South Florida (Tampa, Florida) in 2002. He is currently professor at the Universidad del Norte in Barranquilla.