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Spie Press Book

Thin-Film Design: Modulated Thickness and Other Stopband Design Methods
Author(s): Bruce E. Perilloux
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Book Description

This text presents several new thin-film design methods that can produce multiple stopbands as well as passbands. It is written for thin-film designers and students with advanced knowledge of multilayer, optical thin-film coatings. The text focuses on coatings that have high reflectance performance requirements in more than one spectral wavelength band or region.

Book Details

Date Published: 15 October 2002
Pages: 134
ISBN: 9780819445254
Volume: TT57

Table of Contents
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Chapter 1 Introduction
1.1 Review of mathematics for thin-film design
0.2 Analytical, discrete-layer, thin-film designs
1.3 Thin-film design methods
1.4 Inhomogeneous coating designs and synthesis
0.3.1 Rugate and synthesized rugate designs
0.3.2 Fourier transform method of inhomogeneous coating design
0.4 Summary
0.5 Exercises
Chapter 1 figures
Chapter 2 Sinusoidal Thickness Modulated Designs
1.1 Review of stopbands and passbands
1.1.1 Periodic thin-film structures
1.1.2 Stopbands and passbands
1.2 Sinusoidal modulation
1.2.1 Modulation frequency range for all possible TMDs
1.3 Degenerate cases of TMDs
1.3.1 Quarter-wave stacks
1.3.2 Progressive series stacks (approximate)
1.4 Sinusoidal thickness modulated designs
1.4.1 Integer modulation periods
1.4.2 Spacing or position of TMD stopbands
1.4.3 Noninteger modulation periods
2.5 Determination of all possible TMD stopbands
1.4.1 Dispersion correction for spectral position of TMD
1.4.2 Effect of nonzero phase term on TMD stopband positions
2.6 Electric field analysis of a TMD
2.7 Reflectance phase shift of a TMD
2.8 Applications
1.7.1 Dual-band high reflector
1.7.2 Triple-band high reflector
1.7.3 Rugate TMD
1.8 Exercises
Chapter 2 figures
Chapter 3 Discrete Apodization of TMDs
2.1 Introduction
2.2 Amplitude modulation
2.2.1 AM frequencies that approach TMD modulation
3.3 Gaussian envelope functions
3.4 Application
2.3.1 Multiband reflective and transmissive filter
2.4 Exercises
Chapter 3 figures
Chapter 4 Other Complex TMDs
3.1 Chirped TMD modulation (C-TMD)
3.1.1 Degenerate cases of C-TMD
3.1.2 Selected cases of C-TMD
3.1.3 Example C-TMD (bandwidth of stopband and GD
3.1.4 Limitations and applications of C-TMD to design problems
3.2 Half-modulation TMDs (H-TMD)
3.2.1 Example H-TMD (bandwidth of stopband and GD
3.2.2 Limitations and applications of H-TMD to design problems
3.3 Exercises
Chapter 4 figures
Chapter 5 Quarter-wave Stack Transformation Method
5.1 Stack transformation method
4.2 Quarter-wave stack transforms
4.2.1 Example of a 3:1 quarter-wave stack transform
4.3 Linear quarter-wave stack transformation method
4.3.1 Example of stack transform method
5.4 High-order harmonic stopbands of transformed quarter-wave stacks
5.5 Applications
5.5.1 Dual-wavelength high-reflector example no. 1
5.5.2 Dual-wavelength high-reflector example no. 2
4.6 Exercises
Chapter 5 figures
Appendix A Some useful equations for discrete-layer thin-film calculations
Appendix B Chebyshev polynomials (0 6th order of the second kind)
Appendix C Fortran 90 source code for the determination of all possible TMD stopbands
Appendix D Stopband positions for selected TMDs
Appendix E Summary of TMD and LOST Equations


This text is written for thin-film designers and students with advanced knowledge of multilayer, optical thin-film coatings. It focuses specifically on coatings that have high reflectance performance requirements in more than one spectral wavelength band or region. Many advanced optical systems that employ optical thin-film coatings rely on the performance attributes of multiple spectral bands. Several new analytical design methods are presented in this text that produce multiple stopbands as well as passbands. These analytical design methods produce discrete thin-film designs that will, in most cases, achieve specifications without any re- finement of the design. If needed, the spectral performance of these analytical designs could be improved via common computer refinement algorithms. The theory of each design method in this text is presented along with design examples. Relatively basic exercises are provided for students as well as challenging ones for researchers.

This text does not attempt to cover the vast amount of material already published on thin films. The reader is expected to have a general knowledge of classical thin-film physics and designs. Several other texts cover the gamut of classical thin- film theory and designs, and additional information is readily available in several conference proceedings and many journal articles. Detailed derivations of Fresnel equations, complex refractive index, effective index, p- and s-polarization, phase, classical thin-film coating designs, etc., were intentionally omitted from this text. However, a brief introductory chapter on thin-film theory and design is included for completeness.

The coating designs that readers produce using the methods described in the text can be readily manufactured using common coating materials and process methods. In general, the layer thicknesses of these designs all vary from a quarter- wave stack. For some designs, process adjustments may be required for thin layers. However, thin-film manufacturing processes are not covered in this text since other texts and technical papers cover them.

The designs produced using the methods in this text can also be used as initial designs for computer refinement and synthesis algorithms. With some commercially available thin-film design software, some designs are produced with no starting design or specified layers; the algorithm adds layers of selected materials until the desired spectral performance is achieved. Still, a good starting design helps to reduce the time or effort required to produce a final design that (1) achieves desired performance specifications; (2) is insensitive to layer thickness errors; and (3) can be manufactured. The design methods presented here are also expected to accomplish these tasks for some applications that require stopbands.

In general, the analytical design methods presented were developed using the following methodology. First, layer thicknesses of an arbitrary quarter-wave stack were modulated using various mathematical functions (e.g., sinusoidal). Next, a computer program was written to determine the existence of all stopbands produced from the modulated design. The resulting patterns of stopbands were evaluated graphically as functions of modulation parameters and spectral frequency. Then, based on these graphical patterns of stopbands, analytical (linear) equations were tested by direct calculation of spectral performance to see if the stopbands could be reproduced analytically.

Several variations on the modulation of layer thickness are presented in this text, including an inhomogeneous rugate design. The last chapter presents a related, novel design method where one quarter-wave stack is linearly transformed into another. Here, empirical testing of layer thickness was used to develop general transform equations. A summary of each chapter and the appendixes follows:

Chapter 1 reviews the fundamental mathematics for thin-film design that applies to the proposed methods. Again, the objective here was to keep this chapter brief since this information can be found in many texts.

Chapter 2 introduces sinusoidal thickness modulation of quarter-wave stacks. First, stopbands and passbands are defined. Next, modulation parameters are assessed and many designs are evaluated. Then linear equations are determined that predict all possible stopbands. The last section evaluates the electric fields and the reflected differential phase shift of some modulated designs.

Chapter 3 introduces discrete apodization of the modulated designs from Chapter 2. Two specific apodization functions are evaluated: amplitude modulation functions and Gaussian envelop functions. Linear equations are again determined that predict all possible stopbands.

Chapter 4 describes two variations of the modulation scheme from Chapter 3. First, chirped-modulation designs are evaluated for spectral performance. Next, a half-modulation is discussed where every other layer of a quarter-wave stack is modulated. Both of these methods are applied to dispersion-controlled mirrors used to produce femtosecond laser pulses. Two design examples and the limitations of these modulation schemes are covered.

Chapter 5 presents a novel, linear transform method that can be used to partially transform a given quarter-wave stack into a second quarter-wave stack. This transformation is accomplished by adjusting the individual layer thickness while the total thickness of the original quarter-wave stack remains constant. General transform equations are developed by direct numerical testing of the transform method. The purpose of this transform is to obtain, or achieve, some of the spectral properties of both quarter-wave stacks (i.e., stopbands).

The five appendixes provide some useful thin-film equations, the Chebychev polynomials used in Chapter 2, the FORTRAN source code used to determine all possible stopbands of modulated designs, several graphs of stopband positions, and a summary of the linear equations that predict stopband positions and the general linear transform equations.

Hopefully this text will provide readers with some new thin-film design tools, further insight into design methods, and inspiration for further research on thin-film design.

I would like to thank my family for their support of my research and writing of this text. I greatly appreciate several helpful discussions with Dr. Philip Baumeister, Dr. Angus Macleod, and Dennis Fischer on these modulation design methods. I also acknowledge the collaborative investigation of rugate versions of these modulated designs, and the rugate designs and calculations provided by Dr. Pierre Verly at the National Research Council Canada. I would also like to thank the SPIE reviewers for suggesting several improvements to this text. I would like to thank Coherent, Inc. for supporting this work. Lastly, I would like to acknowledge the inspiration from my graduate professor, Dr. Rasheed M.A. Azzam, to continue my research on thin films.

Bruce E. Perilloux
May 1988

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