ASAP . Technical Guide DIFFRACTION GRATINGS AND DOES Breault Research Organization, Inc. This Technical Guide is for use with ASAP®. Comments on this manual are welcome at: [email protected] For technical support or information about other BRO products, contact: US/Canada:1-800-882-5085 Outside US/Canada:+1-520-721-0500 Fax:+1-520-721-9630 E-Mail: Technical Customer Service:[email protected] General Information:[email protected] Web Site:http://www.breault.com Breault Research Organization, Inc., (BRO) provides this document as is without warranty of any kind, either express or implied, including, but not limited to, the implied warranty of merchantability or fitness for a particular purpose. Some states do not allow a disclaimer of express or implied warranties in certain transactions; therefore, this statement may not apply to you. Information in this document is subject to change without notice. Copyright © 2000-2015 Breault Research Corporation, Inc. All rights reserved. This product and related documentation are protected by copyright and are distributed under licenses restricting their use, copying, distribution, and decompilation. No part of this product or related documentation may be reproduced in any form by any means without prior written authorization of Breault Research Organization, Inc., and its licensors, if any. Diversion contrary to United States law is prohibited. ASAP is a registered trademark of Breault Research Organization, Inc. Breault Research Organization, Inc. 6400 East Grant Road, Suite 350 Tucson, AZ 85715 brotg0915_diffraction (April 9, 2007) ASAP Technical Guide 3 Contents . Fundamentals of Diffractive Optical Elements 9 Huygens-Fresnel principle 9 Diffraction gratings 11 Blazed gratings 14 Phase function 16 Binary optics 23 Transition Points 27 Etch Depths 28 Holographic optical elements 28 Multiple exposure holograms 32 Volume holograms 32 Simulating Diffractive Optical Elements in ASAP 34 MULTIPLE command 35 INTERFACE command for DOEs 40 DOE examples 46 Linear Phase Grating 46 Sinusoidal Phase Grating 53 Circular diffraction grating 59 DOE Lens 69 References 78 ASAP Technical Guide 5 DIFFRACTION GRATINGS AND DOES . In this technical guide, we discuss how to model diffractive optical elements in the Advanced Systems Analysis Program (ASAP®) from Breault Research Organization (BRO). These elements diffract light by producing periodic changes in the phase of the incident wave. The ASAP Primer discusses how the laws of reflection and refraction are used to transform a ray at a specular interface into reflected and refracted rays. In this technical guide, we discuss a theory for diffracting light from phase gratings according to the grating equation law. Diffractive optical elements (DOEs) are a general class of optical elements that include diffraction gratings, binary optics, and holographic optical elements. These optical elements diffract light by producing periodic changes in the amplitude, phase, or both the amplitude and phase of an incident wave. Before you learn how to perform DOE simulations, we must first introduce certain diffractive optical element definitions, nomenclature, and concepts. For some readers, this will be a review and for others it may be the first time that you have seen these definitions and concepts. In either case, you must first have a basic understanding of the physical behavior of diffractive optical elements to associate those definitions and concepts with ASAP procedures and commands for performing DOE simulations. The remainder of this technical guide is divided into two primary sections: • “Fundamentals of Diffractive Optical Elements” on page 9 introduces definitions and concepts. You may choose to skip this section if you already have a background in the subject. Introductory topics include the grating equation, transmission and reflection phase diffraction gratings, binary optics, and holographic optical elements. • “Simulating Diffractive Optical Elements in ASAP” on page 34 presents DOE definitions and concepts with the equivalent ASAP procedures and commands for simulating DOE systems. We discuss how to set up a variety of phase gratings, binary optics, and holograms. 7 DIFFRACTION GRATINGS AND DOES Many of the scripts included in this document are available as INR files on the Quick Start toolbar in ASAP: Example Files> Scripts by Keyword> Diffraction. 8 ASAP Technical Guide DIFFRACTION GRATINGS AND DOES Fundamentals of Diffractive Optical Elements . FUNDAMENTALS OF DIFFRACTIVE OPTICAL .ELEMENTS . Two common approaches exist for calculating the diffraction patterns from DOEs: vector calculation and scalar calculation. Using vector diffraction theory, we can calculate the directions of propagation of the various reflected and transmitted orders of a beam after interacting with the DOE as well as the fractional power contained in each order. The fractional power diffracted into each order is called the diffraction efficiency, and is determined by the diffractive surface profile. This complicated calculation involves not only the DOE structural data, but also the physical and electromagnetic properties of the DOE. ASAP does not perform this type of calculation. Using scalar diffraction theory, we can calculate the idealized directions of propagation of the various reflected and transmitted orders of a beam after interacting with a diffractive element. The scalar diffraction theory technique is valid if the grating period is larger than the wavelength of the diffracting light. Swanson notes that the scalar approximation can be used in Maxwell’s equations if the grating period is approximately five times the wavelength of the diffracting light. The diffraction orders are selected by adjusting the direction cosines of the exiting beams according to the grating equation. The diffraction efficiencies cannot be generally or easily calculated with this method since this is a scalar technique. Most optical design and analysis software programs, including ASAP, use this technique since it is adaptable to ray tracing. The grating equation can also be derived from the Huygens-Fresnel principle. This is, perhaps, a more physically intuitive derivation than one from scalar diffraction theory, yet mathematically less rigorous. Therefore, we use Huygens-Fresnel principle to derive the grating equation. Huygens-Fresnel principle Christian Huygens proposed a theory of diffraction based upon primary and secondary wavefronts. A wavefront is a mathematical surface of constant phase. In a contemporary, geometrical sense, wavefronts are mathematical surfaces over which the optical path lengths from rays of a point source have the same length. Huygens proposed that every point on a primary wavefront can be considered a source of secondary wavelets (or sources), each with the same frequency and velocity, such that at a later time, the primary wavefront is the envelope of the ASAP Technical Guide 9 DIFFRACTION GRATINGS AND DOES Fundamentals of Diffractive Optical Elements wavefronts from the secondary wavelets (or sources). Huygens’ principle is illustrated in Figure 1. Primary Wavefront Envelope of Secondary Wavelets Secondary Wavelets Figure 1 Huygens’ wavelet construction Augustin Jean Fresnel added to Huygens’ intuitive ideas by including Thomas Young’s principle of interference. In doing so, Fresnel assumed under certain conditions that the amplitudes and phases of Huygens’ secondary wavefronts could interfere which each other. The amplitude and phase of a source are a manifest part of its oscillatory behavior. The amplitude is a scalar number or function describing the maximum extent of the electric field vibration. The squared modulus of the amplitude is a measure of the energy density or power in the electric field. The phase is the optical path length of a ray, or the fraction of an oscillatory cycle measured from a specific reference or fixed origin, such as a point source. 10 ASAP Technical Guide DIFFRACTION GRATINGS AND DOES Fundamentals of Diffractive Optical Elements . NOTE For more information regarding amplitudes and phases, see the introductory material in the ASAP technical guide, Wave Optics. The Huygens-Fresnel principle is a superposition principle using spherical wavefronts as the basis functions in the superposition. The principle of wavefront superposition states that the resultant perturbations on a wavefront are the algebraic summations of the individual waves comprising the wavefront. Superposition is also used to algebraically sum amplitudes. The Gaussian beam superposition algorithm in ASAP is similar to Huygens- Fresnel. However, ASAP uses quadratic phase fronts and Gaussian beam amplitudes as the basis functions to superpose individual Gaussian beams to simulate arbitrary optical fields. Gustav Kirchhoff developed a mathematical theory of the Huygens-Fresnel principle, which eventually became the Fresnel-Kirchhoff scalar wave diffraction theory. Arnold Johannes Wilhelm Sommerfeld later developed the Rayleigh- Sommerfeld scalar diffraction theory, which did away with certain nonphysical assumptions of the Fresnel-Kirchhoff scalar wave diffraction theory. Joseph W. Goodman’s book, An Introduction to Fourier Optics, is an excellent introductory resource for learning about the Huygens-Fresnel principal and scalar wave diffraction theory. Several papers describing the Gaussian beam superposition algorithm in ASAP are available in the Knowledge Base at http://www.breault.com/k-base.php. Diffraction gratings Diffraction gratings are regular arrays