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Modeling Holographic Grating Imaging Systems Using the Angular Spectrum Propagation Method

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Modeling Holographic Grating Imaging Systems Using the Angular Spectrum Propagation Method Rochester Institute of Technology RIT Scholar Works Theses 8-15-2006 Modeling holographic grating imaging systems using the angular spectrum propagation method Thomas Blasiak Follow this and additional works at: https://scholarworks.rit.edu/theses Recommended Citation Blasiak, Thomas, "Modeling holographic grating imaging systems using the angular spectrum propagation method" (2006). Thesis. Rochester Institute of Technology. Accessed from This Thesis is brought to you for free and open access by RIT Scholar Works. It has been accepted for inclusion in Theses by an authorized administrator of RIT Scholar Works. For more information, please contact [email protected]. CHESTER F. CARLSON CENTER FOR IMAGING SCIENCE ROCHESTER INSTITUTE OF TECHNOLOGY ROCHESTER, NEW YORK CERTIFICATE OF APPROVAL ______________________________________________________________ M.S. DEGREE THESIS ______________________________________________________________ The M.S. Degree Thesis of Thomas C. Blasiak has been examined and approved by the thesis committee as satisfactory for the thesis requirement of the Master of Science degree __________________________________Dr. Roger Easton __________________________________Dr. Zoran Ninkov __________________________________Dr. Michael Kotlarchyk Date ii Modeling Holographic Grating Imaging Systems Using The Angular Spectrum Propagation Method by Thomas Blasiak Submitted to the Chester F. Carlson Center for Imaging Science in partial fulfillment of the requirements for the Master of Science Degree at the Rochester Institute of Technology Abstract The goal of this research was to describe the angular spectrum propagation method for the numerical calculation of scalar optical propagation phenomena. The angular spectrum propagation method has some advantages over the Fresnel propagation method for modeling low F# optical systems and non-paraxial systems. An example of one such system was modeled, namely the diffraction and propagation from a holographic diffraction grating. To accomplish this goal MATLAB® code was developed to implement the angular spectrum propagation method. Some classical imaging problems such as diffraction from a rectangular aperture, Talbot imaging, focal shift for converging beam illumination, and two beam interference were described in detail in order to demonstrate the capabilities of this method. Results from modeling the image formation of a holographic diffraction grating were compared to a ZEMAX® ray trace model. iii Acknowledgements I would like to acknowledge my wife Tina Bray for encouraging me at all of the critical junctures, and my daughter Riley Bray Blasiak who inspires me. I must also acknowledge Sue Chan in the Chester F. Carlson Center for Imaging Science for cutting red tape, and my advisor Professor Roger Easton who taught me everything I know about linear systems and who was extremely patient with me along the way. I want to acknowledge my committee, Roger Easton, Zoran Ninkov and Michael Kotlarchyk who were always knowledgeable and personable. I would also like to acknowledge my employer Newport Corporation and its predecessors who have supported me in my education and the various managers I have had along the way who allowed me the freedom to pursue my interests. Lastly, I want to dedicate this work to my father Leon Blasiak who worked as an electrical engineer for his entire career and has now forgotten more than I know, and to my mother Adele Blasiak who has always been there for him. iv Contents 1 Introduction ................................................................................................................... 1 1.1 Background ........................................................................................................................6 1.2 Physical optics propagation models ...................................................................................7 1.2.1 Fresnel propagation...................................................................................................8 1.2.2 Angular spectrum propagation................................................................................14 1.3 Diffraction gratings ..........................................................................................................21 1.4 Imaging properties of holographic gratings......................................................................24 1.5 Geometrical ray trace models...........................................................................................29 2 Approach ..................................................................................................................... 39 2.1 Plane-wave propagation modeling with the Angular Spectrum .......................................41 2.2 Comparison between the Fresnel method and the ASPM for aperture diffraction modeling ..........................................................................................................................................54 2.3 The application of the Angular Spectrum Propagation Method to Talbot imaging..........64 2.4 Two-beam interference with polarized light described using a modification to the ASPM. ..........................................................................................................................................67 2.5 Axial focal shift for converging beam illumination prediction using the ASPM .............73 2.6 ASPM applied to modeling off-axis converging beam illumination with high NA .........76 3 Results ......................................................................................................................... 78 3.1 Moiré fringe phenomenon description and modeling using the Angular Spectrum Propagation Method ..........................................................................................................................................78 3.2 Holographic diffraction grating image formation predicted using the ASPM..................84 4 Conclusions and discussion......................................................................................... 99 5 Appendix I ................................................................................................................. 101 5.1 Zemax Lens Data entry for modeling a holographic grating..........................................101 6 Appendix II................................................................................................................ 102 6.1 Matlab® code for calculating phase from a complex number array...............................102 6.2 Matlab® code for unwrapping phase .............................................................................103 7 Appendix III .............................................................................................................. 104 7.1 Matlab® code for calculating the image from a two-dimensional square aperture........104 7.2 Matlab® code for plotting two-dimensional gray scale image.......................................108 8 References ................................................................................................................. 109 v List of Figures Figure 1.1. Typical optical system layout showing the traveling wavefront incident on the exit pupil, the exit pupil, and the image plane. The wavefront at the exit pupil is sampled and propagated to the image plane...........................................................................................................3 Figure 1.2. Sampling of the exit pupil wavefront. Nyquist sampling would require two samples per minimum spatial period. Practical sampling may require from five to ten samples per minimum spatial period to reduce numerical artifacts..........................................................................4 Figure 1.3. Geometry for Fresnel propagation. The input plane is on the left, and the image plane is on the right. Z is the distance between the two parallel planes, and r is the distance between a point on plane 1 and a point on plane 2. ......................................................................................10 Figure 1.4. Relative accuracy for the binomial expansion of the propagation distance r. The value “a” is the sum of the maximum dimensions in the two planes. When this value is half of the propagation distance, the estimate for r can be off by ≈ 0.6%. This explains why the Fresnel approximation is limited to systems with large F#’s, generally near the axial zone...........................................................................................12 Figure 1.5. Geometry for the direction cosines. k is the propagation vector. The direction cosines (α,β,γ) are the cosines of the angles between the propagation vector and the x,y, and z axes respectively. (The angles are the inverse cosines of the direction cosines)......................................16 Figure 1.6. Classical ruled diffraction grating construction on a concave surface. During the ruling process, the diamond tool travels very smoothly in the direction perpendicular to the plane of the drawing, while the grating carriage moves very precisely to the right, traveling the distance of one groove for each stroke of the diamond tool. For this type of grating, the grooves are straight and parallel in a plane that is tangent to the grating surface...................................................................22 Figure 1.7. Holographic diffraction grating construction. A fringe field is formed by the interference of two coherent light sources; the intensity pattern at the grating surface
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