diffraction gratings and does - Breault Research Organization, Inc.

. . . . .This Technical Guide is for use with ASAP®.Comments on this manual are welcome at: support@breault.comFor technical support or information about other BRO products, contact:US/Canada:1-800-882-5085Outside US/Canada:+1-520-721-0500Fax:+1-520-721-9630E-Mail:Technical Customer Service:support@breault.comGeneral Information:info@breault.comWeb Site:http://www.breault.comBreault Research Organization, Inc., (BRO) provides this document as is without warranty of any kind, eitherexpress or implied, including, but not limited to, the implied warranty of merchantability or fitness for a particularpurpose. 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 © 2000-2014 Breault Research Corporation, Inc. All rights reserved.This product and related documentation are protected by copyright and are distributed under licenses restrictingtheir use, copying, distribution, and decompilation. No part of this product or related documentation may bereproduced 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 350Tucson, AZ 85715brotg0915_diffraction (April 9, 2007)ASAP Technical Guide 3

Contents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . .Fundamentals of Diffractive Optical Elements 9Huygens-Fresnel principle 9Diffraction gratings 11Blazed gratings 14Phase function 16Binary optics 23Transition Points 27Etch Depths 28Holographic optical elements 28Multiple exposure holograms 32Volume holograms 32Simulating Diffractive Optical Elements in ASAP 34MULTIPLE command 35INTERFACE command for DOEs 40DOE examples 46Linear Phase Grating 46Sinusoidal Phase Grating 53Circular diffraction grating 59DOE Lens 69References 78ASAP Technical Guide 5

DIFFRACTION GRATINGS AND DOES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . .In this technical guide, we discuss how to model diffractive optical elements in theAdvanced Systems Analysis Program (ASAP®) from Breault ResearchOrganization (BRO). These elements diffract light by producing periodic changesin the phase of the incident wave.The ASAP Primer discusses how the laws of reflection and refraction are used totransform a ray at a specular interface into reflected and refracted rays. In thistechnical guide, we discuss a theory for diffracting light from phase gratingsaccording to the grating equation law.Diffractive optical elements (DOEs) are a general class of optical elements thatinclude diffraction gratings, binary optics, and holographic optical elements. Theseoptical 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 certaindiffractive optical element definitions, nomenclature, and concepts. For somereaders, this will be a review and for others it may be the first time that you haveseen these definitions and concepts. In either case, you must first have a basicunderstanding of the physical behavior of diffractive optical elements to associatethose definitions and concepts with ASAP procedures and commands forperforming DOE simulations.The remainder of this technical guide is divided into two primary sections:• “Fundamentals of Diffractive Optical Elements” on page 9 introducesdefinitions and concepts. You may choose to skip this section if you alreadyhave a background in the subject. Introductory topics include the gratingequation, transmission and reflection phase diffraction gratings, binary optics,and holographic optical elements.• “Simulating Diffractive Optical Elements in ASAP” on page 34 presents DOEdefinitions and concepts with the equivalent ASAP procedures and commandsfor simulating DOE systems. We discuss how to set up a variety of phasegratings, binary optics, and holograms.7

DIFFRACTION GRATINGS AND DOESMany of the scripts included in this document are available as INR files on theQuick Start toolbar in ASAP: Example Files> Scripts by Keyword> Diffraction.8 ASAP Technical Guide

DIFFRACTION GRATINGS AND DOESFundamentals of Diffractive Optical Elements. . . . .FUNDAMENTALS OF DIFFRACTIVE OPTICALELEMENTS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .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 ofthe various reflected and transmitted orders of a beam after interacting with theDOE as well as the fractional power contained in each order. The fractional powerdiffracted into each order is called the diffraction efficiency, and is determined bythe diffractive surface profile. This complicated calculation involves not only theDOE structural data, but also the physical and electromagnetic properties of theDOE. ASAP does not perform this type of calculation.Using scalar diffraction theory, we can calculate the idealized directions ofpropagation of the various reflected and transmitted orders of a beam afterinteracting with a diffractive element. The scalar diffraction theory technique isvalid 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 ifthe grating period is approximately five times the wavelength of the diffractinglight. The diffraction orders are selected by adjusting the direction cosines of theexiting beams according to the grating equation. The diffraction efficienciescannot be generally or easily calculated with this method since this is a scalartechnique. 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. Thisis, perhaps, a more physically intuitive derivation than one from scalar diffractiontheory, yet mathematically less rigorous. Therefore, we use Huygens-Fresnelprinciple to derive the grating equation.Huygens-Fresnel principleChristian Huygens proposed a theory of diffraction based upon primary andsecondary wavefronts. A wavefront is a mathematical surface of constant phase. Ina contemporary, geometrical sense, wavefronts are mathematical surfaces overwhich 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 asource of secondary wavelets (or sources), each with the same frequency andvelocity, such that at a later time, the primary wavefront is the envelope of theASAP Technical Guide 9

DIFFRACTION GRATINGS AND DOESFundamentals of Diffractive Optical Elementswavefronts from the secondary wavelets (or sources). Huygens’ principle isillustrated in Figure 1.Primary WavefrontEnvelope ofSecondaryWaveletsSecondary WaveletsFigure 1 Huygens’ wavelet constructionAugustin Jean Fresnel added to Huygens’ intuitive ideas by including ThomasYoung’s principle of interference. In doing so, Fresnel assumed under certainconditions that the amplitudes and phases of Huygens’ secondary wavefrontscould 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 ofthe electric field vibration. The squared modulus of the amplitude is a measure ofthe energy density or power in the electric field. The phase is the optical pathlength of a ray, or the fraction of an oscillatory cycle measured from a specificreference or fixed origin, such as a point source.10 ASAP Technical Guide

DIFFRACTION GRATINGS AND DOESFundamentals of Diffractive Optical Elements. . . . .NOTE For more information regarding amplitudes and phases, seethe introductory material in the ASAP technical guide, WaveOptics.The Huygens-Fresnel principle is a superposition principle using sphericalwavefronts as the basis functions in the superposition. The principle of wavefrontsuperposition states that the resultant perturbations on a wavefront are thealgebraic 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 beamamplitudes as the basis functions to superpose individual Gaussian beams tosimulate arbitrary optical fields.Gustav Kirchhoff developed a mathematical theory of the Huygens-Fresnelprinciple, which eventually became the Fresnel-Kirchhoff scalar wave diffractiontheory. Arnold Johannes Wilhelm Sommerfeld later developed the Rayleigh-Sommerfeld scalar diffraction theory, which did away with certain nonphysicalassumptions of the Fresnel-Kirchhoff scalar wave diffraction theory.Joseph W. Goodman’s book, An Introduction to Fourier Optics, is an excellentintroductory resource for learning about the Huygens-Fresnel principal and scalarwave diffraction theory.Several papers describing the Gaussian beam superposition algorithm in ASAP areavailable in the Knowledge Base at http://www.breault.com/k-base.php.Diffraction gratingsDiffraction gratings are regular arrays of apertures or phase structures that, bynature of their periodic structures, produce periodic perturbations in the amplitude,phase, or both amplitude and phase of a wavefront incident on the structure. Ifapertures are used, the diffraction grating is an amplitude grating and, specifically,an amplitude transmission grating if the light passes through and is not reflected bythe grating. Young’s double slit is an example of a simple amplitude transmissiongrating.If some type of phase altering material is used, the diffraction grating is a phasegrating. If light transmits through the diffraction grating, the grating is called aphase transmission grating. If light reflects off the grating, it is called a phasereflection grating.ASAP Technical Guide 11

DIFFRACTION GRATINGS AND DOESFundamentals of Diffractive Optical ElementsGenerally, phase gratings are more efficient at diffracting monochromatic lightthan amplitude gratings, and surface relief phase gratings are more efficient atdiffracting broadband light than volume phase gratings. Theoretically, a phasegrating can diffract 100% of incident monochromatic light into a single diffractionorder. A surface relief phase grating has higher diffraction efficiencies than avolume phase grating operating over the same bandwidth. An amplitude gratingcannot diffract 100% of the incident monochromatic light into a single diffractionorder, because some of the incident light is reflected, absorbed, and so on, by thegrating mask.The simplest type of surface relief—phase transmission diffraction grating—is alinear phase transmission grating embossed on a flat, slab substrate. We can gainan intuitive understanding of more complicated diffraction gratings fromexamining the linear grating. A linear grating is a series of regularly arrayed linesor scratches embossed in a transmissive material known as the substrate. Phasetransmission gratings are usually replicated from a master on a plastic substrate.The master is quite often a series of regularly arrayed lines or scratches carved oretched onto a glass substrate.As a simple physical model, the regularly arrayed lines can be consideredscratches, troughs, or humps in the substrate. In the case of scratches, each lineacts like a linear scattering center. In the case of troughs or humps, each line actslike a very short radius of curvature, highly divergent cylindrical lens. In bothcases, the effect is to produce equivalent linear or line sources.Each effective line source is illuminated by the same, or more correctly, differentparts of the same wavefront. Therefore, each line source resembles the secondarysources or wavelets of the Huygens-Fresnel principle. See Figure 1Their superposition at a later time or, equivalently, a later position, will determinethe optical field at that point in time or position.A constructive interference of all the wavelets in a particular direction leads todiffraction orders. Some of the light is not scattered or diverged in transmittingthrough the diffraction grating. This occurs in between the grating lines. This light,basically unperturbed from the specular direction, is called the zeroth order. Inother words, the zeroth order of the diffraction grating is specular direction of theincident light.We have constructed a simple graphical model of the Huygens-Fresnel principle asapplied to linear phase transmission and reflection gratings in Figure 2.12 ASAP Technical Guide

DIFFRACTION GRATINGS AND DOESFundamentals of Diffractive Optical Elements. . . . .P 4 mDiffraction Orders mPhase Transmission GratingPhase Reflection GratingDiffraction Orders mSpecular - Zeroth OrderP 2P i m3 d iP 1 mP 3P 2P 4 idSpecular - Zeroth Order iP 1Figure 2 Transmission and reflection gratingsFor the linear phase transmission illustration, note that the trigonometric sine ofthe incident angle is,Equation 1(EQ 1)Here d is the grating line spacing. Similarly, note that the trigonometric sine of thediffracted order angle is,Equation 2(EQ 2)ASAP Technical Guide 13

DIFFRACTION GRATINGS AND DOESFundamentals of Diffractive Optical ElementsIf the phase difference between lines P 1 P 2 and P 3 P 4 is an integer number ofmultiple wavelengths , all the waves from the secondary wavelets are in phase inthe direction corresponding to m . Operationally, this is,Equation 3The above equation is called the grating equation, and m is called the grating ordiffraction order. Note that the grating equation is independent of the refractiveindex of the slab substrate. This can be easily validated with Snell’s law. Thegrating equation is the same, even in the case of the phase reflection grating.(EQ 3)The grating equation is the same as the equation that describes the maximumorders for Young’s double slit experiment. However, there are some fundamentalphysical differences in the diffraction patterns for each situation:• The diffraction or interference pattern from Young’s double-slit experiment isdue to an amplitude transmission grating. It is also more diffuse or less sharpthan that from the phase diffraction grating. This is primarily due to the largersize of the apertures in Young’s double slit experiment as compared to therelatively small line sources in the diffraction grating. In Young’s experiment,different points across the two apertures, other than the center points, can be inphase with each other and constructively interfere at slightly different locationsthan the center points causing a broadening of the diffraction pattern in thatorder.• The linear phase transmission grating behaves more like a Fabry-Perot etalon orfilter.Blazed gratingsIn many applications that use linear diffraction gratings, we are not interested inthe light in the zeroth order. This light is often wasted. It is useful to transfer lightout of the zeroth order and into another order or arbitrary direction other than thespecular direction. Many modern diffraction gratings, especially phase reflectiongratings are blazed to accomplish this task.John William Strutt Rayleigh, or Lord Rayleigh suggested a clever way to transferenergy out of the zeroth order into other orders. This involved changing the angle14 ASAP Technical Guide

DIFFRACTION GRATINGS AND DOESFundamentals of Diffractive Optical Elements. . . . .of or tilting the surfaces that the specular light sees in between the grating lines.Such a grating is called a blazed grating.The key to a blazed grating is that the angular direction of the non-zero diffractionorders are a function of the grating spacing d, the wavelength of the incident light, and the angle of incidence i . However, m and i are measured from thegrating plane and not the blazed (tilted) surface. If we tilt the specular surfaces inbetween the grating lines, we change the direction of the specular output, but notthe higher diffraction grating orders. Figure 3.Blazed GratingdGratingPeriodBlaze AngleGrating has RefractiveIndex nFigure 3 Blazed transmission gratingThe diffraction efficiency of a given order for regular and blazed linear phasegratings can be calculated using scalar diffraction theory. Swanson, for example,gives the result for diffraction efficiencies as,ASAP Technical Guide 15

DIFFRACTION GRATINGS AND DOESFundamentals of Diffractive Optical ElementsEquation 4Here, m is the diffraction order, T is the grating period, and :(EQ 4)Equation 5(EQ 5)where n is the refractive index of the grating, d is the blaze thickness, and is thewavelength of light. From the grating equation and these equations, we determinethat the diffraction angles and efficiencies are highly dependent on wavelength andblaze thickness.Light from large bandwidth sources will be diffracted into many orders. This is atype of scatter, but it is deterministic scatter since its direction and diffraction orscatter efficiency can be exactly computed with the above equations.NOTE Deterministic scatter is different from the surface scatter dueto random surface variations that are covered in the ASAPtechnical guide, Scattering.The concept of the linear grating in this context can be used to develop a simplebut useful physical model of random scatter. You can consider a random scatteringsurface as being composed of many randomly oriented linear gratings, whosecombined grating spacing and orientation yield the surface scatter pattern.Phase functionBlazed gratings—and, in fact, other types of optical elements such as apertures,prisms, lenses and mirrors—can be described in terms of a transmittance function.The transmittance function describes how the optical element changes the16 ASAP Technical Guide

DIFFRACTION GRATINGS AND DOESFundamentals of Diffractive Optical Elements. . . . .amplitude and phase of a wavefront propagating through the component. We learnlater that we must know the phase function of a diffractive optical element tomodel it in ASAP.Generally, the transmittance function is defined as,Equation 6A(x,y) is the amplitude function and (x,y) is the phase function of the element.(EQ 6)The simplest transmittance function is perhaps that of a one-dimensional aperturewhose transmittance function is one inside the aperture and zero outside of theaperture. Mathematically,Equation 7A prism has a transmittance function equal to,(EQ 7)Equation 8(EQ 8)Here, we assumed that phase is changed, but the amplitude was unalteredpropagating through the prism. k is the wave number whose physical effect is toconvert the linear dimension of the phase function into a fraction of an oscillatorycycle measured in radians. o is a constant term of the phase function.A blazed grating has a transmittance function equal to,Equation 9(EQ 9)ASAP Technical Guide 17

DIFFRACTION GRATINGS AND DOESFundamentals of Diffractive Optical ElementsHereis a modulo 2 operation on the phase function of the blazed grating.Mathematically, a modulo operation keeps the remainder after a division.Physically, it produces a kinoform whose incremental depth is 2 radians or,equivalently, one wavelength. A kinoform is a surface-relief profile of themodulated phase. The common, constant-depth Fresnel lens is an example of atype of kinoform whose modulo operation is much larger than a wavelength oflight. Its kinoform is a surface relief profile of the actual physical surface orsurface sag, and not the phase change (transmittance function) through theelement. See Figure 4.Phase Function of a Prism(x)x(x) Modulo orxPhase Function of a Blazed GratingFigure 4 Phase functions of a prism and blazed grating18 ASAP Technical Guide

DIFFRACTION GRATINGS AND DOESFundamentals of Diffractive Optical Elements. . . . .The transmittance function of the prism looks similar to the blazed grating.Mathematically, the difference is manifest in the modulo operation. Physically, theblazed grating is the diffractive counterpart of the refractive prism. In general, amodulo 2 operation on the phase function of a refractive element yields itsdiffractive counterpart’s phase function.We are now in a position to examine a more complicated phase behavior; namely,that of a simple thin lens. A thin lens is one whose thickness is very smallcompared to its diameter. A ray intersecting one side of a thin lens essentially exitsat the same point on the other side of the lens.A thin lens is a phase transformer in the sense that it delays, in time, the phase ofthe wavefront in proportion to the thickness of the lens. A positive lens, such as abiconvex lens whose outer edge thickness is less than its center thickness, delayscollimated light, for example, more at the center of the lens than at the edge. Theedge of the wavefront has propagated further in the same amount of time than thecenter of the wavefront when it exits the center of the lens, because it is not“slowed” down by the additional refractive lens material. Since the wavefront iscontinuous and is a surface of constant phase, the wavefront must converge towarda point in space, which in this particular case, is the focal point of the lens.We can equivalently understand this phenomena by examining a ray of light. Sincethe wavefront is delayed more at the center of the lens than at its edge, the ray isalso delayed by the same amount because it is related to the wavefront and in factis perpendicular to the wavefront. The magnitude of the optical path length of theray at the edge of the lens is less than at the center. The optical path length of a rayis the distance that light travels in a vacuum in the same time it travels a givendistance in a material. Therefore, the edge ray is bent toward the focal point of thelens.The geometry for a radially symmetric convex-plano lens is illustrated in Equation10.Figure 5 The wavefront experiences a total phase delay due to the lens and theremaining region around the lens. Mathematically, this is,ASAP Technical Guide 19

DIFFRACTION GRATINGS AND DOESFundamentals of Diffractive Optical ElementsEquation 10(EQ 10)20 ASAP Technical Guide

DIFFRACTION GRATINGS AND DOESFundamentals of Diffractive Optical Elements. . . . .R()Plano-Convex Lens with a Spherical SurfacedQuadratic (Parabolic) Phase Function from LensModuloPhase of the Quadratic Phase FunctionFigure 5 Phase-transforming lens, resulting quadratic phase function, and diffractivemodulo 2 phase functionBy invoking the paraxial approximation with the binomial theorem, we can replacethe spherical surface with a quadratic approximation. This leads to,Equation 11ASAP Technical Guide 21

DIFFRACTION GRATINGS AND DOESFundamentals of Diffractive Optical ElementsOur phase function then becomes,Equation 12We recognize the last term of the previous equations as a form of the lens maker’sequation for a convex-plano lens with focal length f.Therefore, the phase function becomes,Equation 13(EQ 13)Equation 14(EQ 14)The first term is a constant phase term of little interest, and the second term is aquadratic approximation to a spherical wavefront. It is a quadratic phase functionof a focused beam. The lens transforms the planar phase (wavefront) of theincident collimated beam into a converging quadratic phase (wavefront) focusingat the focal point of the lens. If modulo 2 operation is carried out on the phasefunction, we obtain the diffractive counterpart to the refractive lens.Arbitrary phase functions and their diffractive counterparts are obtained in similarmanners, usually with the aid of lens design software. However, the wavefrontsand, subsequently, the phase functions of these elements can be much morecomplicated than the plane waves and quadratic waves we examined for the lineargrating and a diffractive lens.22 ASAP Technical Guide

DIFFRACTION GRATINGS AND DOESFundamentals of Diffractive Optical Elements. . . . .Even with more complicated phase functions, the grating equation can still be usedto determine, in a local sense, the direction of the orders of the diffracted light. Thegrating spacing used in the grating equation is the local grating spacing where asmall pencil beam of light, or perhaps a ray, intersects the diffractive element. Thediffraction efficiencies are still governed by the wavelength and depthrelationships developed for the simple linear grating. However, the fundamentalproblem is that with more complicated phase surfaces, the overall depth andgrating spacing can vary radically across the element, making it impractical andusually impossible to properly apply these equations. We recommend resorting tovector diffraction theory to properly account for the canonical diffractionefficiencies.Binary opticsThe diffractive phase kinoforms discussed in the previous section are continuousin profile over the 2 phase depth. However, these profiles cannot bemanufactured in a continuous profile over such small intervals. Even most Fresnellenses, whose kinoform depths and intervals are many orders of magnitudes largerthan those of diffractive optical elements, typically use linear approximations tothe continuous surface profiles. Fortunately, in the last decade the techniques of theintegrated circuit (IC) industry were adapted to produce approximations to adiffractive optical element’s kinoform. The manufacturing technique has led to theterm and field called binary optics.The IC manufacturing process used to generate binary optics is similar to chiselingor carving stair-step structures into a substrate that, on a microscopic scale,approximate the continuous profile of the modulo 2 phase function.This concept is illustrated in Figure 6 The diffraction efficiency for a given order isa function of the number of levels. The more levels, the higher the diffractionefficiency. Two levels produce diffraction efficiencies on the order of 40%. Eightlevels and above produce diffraction efficiencies of 95% and higher.ASAP Technical Guide 23

DIFFRACTION GRATINGS AND DOESFundamentals of Diffractive Optical ElementsPhase function of a blazed gratingMulti-level phase structure or binary optic equivalentA more complicated phase functionMulti-level phase structure or binary optic equivalentFigure 6 Four-level binary optic representation of some phase functionsA series of lithographic masks are made whose total number is determined by thenumber of etch levels and, therefore, the desired diffraction efficiency. The stepsare listed below.The masks are like zero order amplitude gratings, which transmit and reflect acertain percentage and pattern of the light incident on the substrate.1 The masks are placed one at a time over a substrate coated with a photoresist.2 The transmitting portion of the mask allows light to expose the photoresist. If thephotoresist is exposed to light, it is developed and washed away from thesubstrate.24 ASAP Technical Guide

DIFFRACTION GRATINGS AND DOESFundamentals of Diffractive Optical Elements. . . . .3 The substrate areas without photoresist are then etched to a special depthleaving a binary step. The unexposed photoresist remaining on the substrate isimpervious to the etching process.4 The rest of the photoresist is removed and the entire substrate is again coatedwith photoresist.5 The next mask is positioned above the coated substrate and the combination, inturn, is illuminated.6 The exposed photoresist is removed and that portion of the substrate is etchedto a special depth.7 This process is repeated until you have etched the desired number of levels intothe substrate. See Figure 7.ASAP Technical Guide 25

DIFFRACTION GRATINGS AND DOESFundamentals of Diffractive Optical ElementsAmplitude mask for second levelSubstrate recoated with photoresistRemove exposed photoresistEtch bare substrate to required depthRemove remaining photoresistResulting Binary Optic (only 4 levels)Figure 7 Binary optic fabrication process26 ASAP Technical Guide

DIFFRACTION GRATINGS AND DOESFundamentals of Diffractive Optical Elements. . . . .The width of the masks for each step is a function of the type of phase surface. Forthe blazed grating, the width of the mask on the first etch is half of the modulo 2phase interval. The width in subsequent mask intervals is half that of the previousmask. For a circularly symmetric case, like the thin lens, the width of the mask onthe first etch, step, or transition begins at the radial value corresponding to amodulo phase depth. Subsequent radial transition points occur at multiples of themodulo phase depth. On the second mask, the transitions occur at integermultiple modulo /2 phase depths, and so on.The entire binary optics process as well as required transition points and etchdepths are discussed in great detail in Swanson’s report from the MassachusettsInstitute of Technology (MIT) titled “Binary Optics Technology: The Theory andDesign of Multi-level Diffractive Optical Elements”. The transition points andetch depths are shown in Equation 15 and Equation 15.TRANSITION POINTSEquation 15(EQ 15)ASAP Technical Guide 27

DIFFRACTION GRATINGS AND DOESFundamentals of Diffractive Optical ElementsETCH DEPTHSEquation 16Even though binary optics are manufactured in discrete steps, their diffractivebehavior in ASAP is still simulated by defining the grating lines and using thegrating equation. The actual stair-step phase structures are not modeled. Theresulting diffraction efficiencies are adjusted according to the number of etchlevels or steps.Holographic optical elementsHolographic optical elements are actually the result of a diffractive application ofholography. Holography is a process where an interference pattern of a threedimensionalobject is produced that contains not only amplitude information aboutthe object but also the phase information. The two-dimensional pictures that weview, including this page and these words, are irradiance maps of objects. Anirradiance map contains information only about the amplitude of the object and notits phase. In this sense, we get only half the picture, as the phase information aboutthe object is missing. Perhaps this lead Dennis Gabor to name the phenomena hediscovered in 1947 “holography”, and the pictures created with this process:“holograms”, after the Greek word “holos” meaning whole.The interference pattern of a hologram is produced by interfering a referencewavefront with a split portion of itself reflected from an object. The hologram istypically recorded on a photographic emulsion. The recording is “played back” byre-illuminating the hologram with the reference beam. The hologram can be atransmission or reflection hologram.We can generate a simple hologram by interfering two plane waves together.Imagine a linearly polarized plane wave incident on a photographic plate at anangle 1 with respect to the plate normal, and another plane wave—a split portionof the first plane wave—incident on the same plate but at an angle 2 . The electricfield of the two waves is,28 ASAP Technical Guide

DIFFRACTION GRATINGS AND DOESFundamentals of Diffractive Optical Elements. . . . .Equation 17(EQ 17)If the phase difference 1 - 2 is a constant, the two waves are mutually coherent.See Figure 8.Y 2k 2 1Zk 1The k Vectors of 2 Interfering Plane WavesYBright FringesZFigure 8 The k vectors and fringes of two interfering plane wavesASAP Technical Guide 29

DIFFRACTION GRATINGS AND DOESFundamentals of Diffractive Optical ElementsWe can write the phase relationships in Equation 18at the recording plane, basedupon the propagation vectors of the two interfering plane waves.Equation 18The last equation is of a line. This implies that the phase function of the twointerfering beams is linear. Recall that the phase function of a prism was linear anda modulo 2 operation on it resulted in its diffractive counterpart, a blazed grating.A bright fringe occurs when m. In other words, the relative phase betweenthe two waves varies by 2 in between adjacent bright fringes. Furthermore, thefringe spacing in the special case of z and 1 - 2 = 0 is,Equation 19The irradiance pattern from this superposition is,(EQ 19)30 ASAP Technical Guide

DIFFRACTION GRATINGS AND DOESFundamentals of Diffractive Optical Elements. . . . .Equation 20The dot product of the polarization component vectors results in an additionalcosine term whose argument is the angle between the linear polarizationcomponents.(EQ 20)The important results are that the irradiance pattern varies cosinusoidally, but thefringes are straight-lined and equally spaced. The fringes are also non-localized inthat a fringe pattern exists at every point where the two plane waves overlap. Whenthis hologram is illuminated with the reference beam, we see part of the originalreference beam passing through the hologram, just like the zeroth order from adiffraction grating and two additional first-order beams.The interference pattern, or hologram, when re-illuminated with the referencebeam, behaves just like a sinusoidal grating. The phase information of thehologram is manifest in the local and global fringe spacing, and determines thediffraction angles of the orders. The amplitude information of the hologram ismanifest in the visibility or contrast of the fringes, and determines the diffractionefficiency of the order.NOTE In this technical guide, we are interested only in the phaseportion of the hologram, as this is needed to model its diffractivebehavior in ASAP.Our results above imply that holograms can be simulated with the same gratingequation that we used to model the behavior of diffraction gratings and binaryoptics. However, in the case of two plane waves, we simulate only the zeroth andfirst orders, because the hologram behaves like a sinusoidal grating and does notcontain phase and amplitude information for higher orders. We again need thephase information of the hologram, which is manifest in the fringe spacing.Our fringe/phase theory can be extended to more complicated fringe patterns. If aplane wave and a spherical wave interfere, circular fringes result. The circularfringes are spaced just like those found in a Fresnel phase plate, which behaveslike a diffractive lens. If two spherical waves interfere, hyperbolic fringes result.In general, the more complicated the return wavefront, the more complicated theinterference pattern. In fact, some holograms may not look like a conventionalfringe pattern at all, and resemble a speckle pattern (a coherent scatter pattern).ASAP Technical Guide 31

DIFFRACTION GRATINGS AND DOESFundamentals of Diffractive Optical ElementsLens design codes commonly use two point sources to generate holographicoptical elements, which in turn are used as diffractive lenses or correctiveelements.The first-order beams of a hologram are also referred to as side bands. Incomplicated holographic pictures, one of the side bands contains the originalamplitude distribution, but a negative phase term. This results in the famousinside-out image. The other side band contains the original amplitude distributionand the normal phase term. Its image accurately reproduces the object as itappeared during recording.Holographic optical elements are found in a large number of different types ofoptical systems, such as scanners and heads-up displays (HUDs). In one sense,interference coatings might be considered as holographic optical elementsoperating only in the zeroth order.Multiple exposure hologramsMultiple exposure holograms are a single hologram whose recording mediacontain more than one exposed holographic fringe pattern. The most common typeof multiple exposure hologram is the double-exposure hologram.One type of double-exposure hologram is a composite hologram made of anunperturbed object and its resulting fringe pattern, and a perturbed object and itsresulting fringe pattern. The composite fringe pattern actually represents thedifference in perturbation between the unperturbed object and perturbed object.The perturbation might be, for example, the displacement (vibration) or distortionof an object from its normal state.A multiple exposure hologram exposed over a long period of time is called a timeaveragedhologram. Its interference pattern is actually a superposition of manyinterference patterns. The resultant interference pattern is really a standing wavewhose contours represent areas of constant perturbation. In most instances, theperturbation is a vibration.Multiple exposure holograms are not volume holograms.Volume hologramsThe interference pattern produced by a hologram is non-localized. This means thatit exists everywhere in space. Our previous example of two interfering planewaves is an example of non-localized fringes. In other words, the interferencepattern produced via holography is three-dimensional. In this respect, it is a three-32 ASAP Technical Guide

DIFFRACTION GRATINGS AND DOESFundamentals of Diffractive Optical Elements. . . . .dimensional diffraction grating. It can also be regarded as a three-dimensionalinterference filter.A volume hologram has a thickness that is much greater than the thickness of itsthin holographic counterpart. Even then, it is still very thin, perhaps on the order oftens of microns. The larger thickness is needed to record the three-dimensionalfringe pattern. The thicker the volume hologram, the more light is diffracted intothe first order. In fact, efficiencies of 100% are achievable.Volume holograms de-couple wavelength and the diffraction angle. In other words,only a specific wavelength is diffracted at a specific angle in the hologram.Because of this property, you can store many holograms at one time. Just changethe incident wavelength. Similarly, you can change the angle of incidence toachieve the same effect.Volume holograms are highly susceptible to shrinkage. Therefore, if shrinkage isnot accounted for, the replay wavelength must be shorter than the recordingwavelength. Only at special wavelengths is light diffracted at a particular angle. Inthis sense, the volume hologram acts like a spectral filter.ASAP can simulate volume holograms, but it does so using the grating equation.However, we must set up appropriate diffraction efficiency models with theCOATING MODELS option in ASAP, and specifically use the interface diffractoption with this coating model.ASAP simulates volume holograms as thin holograms with appropriately adjusteddiffraction efficiencies. Most other lens design codes also use this technique.However, they sometimes include approximated diffraction efficiency calculationsfrom, for example, Kogelnik’s theory (Bell Systems’ Technical Journal, 48, 2909,1969). This simulation technique assumes that the primary difference betweenvolume and surface holograms is the behavior of the diffraction efficiencies.ASAP Technical Guide 33

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAPSIMULATING DIFFRACTIVE OPTICALELEMENTS IN ASAP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASAP simulates diffractive optical elements (DOEs) by using the phase functionof the DOE to generate a series of grating lines. Therefore, as a first step insimulating DOEs in ASAP, you must know the mathematical form of the phasefunction of the DOE or the grating line function. This information is easilyobtained by knowing the line spacing for simple DOEs, such as linear and circularphase diffraction gratings. More complicated phase functions are usually obtainedfrom lens design codes in the form of polynomial equations.ASAP automatically replicates grating lines from information it obtains from thephase function. In effect, ASAP automatically performs the modulo 2 phaseoperation on the phase function. From the preceding section, we know that agrating line occurs whenever the phase function goes through 2 radians.The resultant replication is then associated with the base entity of an object. Afterintersecting the base object, ASAP computes the local grating spacing, and appliesthe grating equation to compute the direction of propagation for defined orders.Since ASAP can split rays, it can also propagate several diffracted orders,including transmitted and reflected, simultaneously after the intersection.The phase functions in ASAP can be described by any of the ASAP SURFACES.However, the most common way is to describe it with the GENERAL orUSERFUNC surface commands. ASAP surfaces must be modified to convert thephase function into a modulo 2 phase function and, ultimately, the grating linerepresentation. This is done with the surface modifier, MULTIPLE (see thesection, “MULTIPLE command”). MULTIPLE creates multiple sheets of a phasefunction to simulate the grating lines.The replicated grating line spacing, originally obtained from the phase function, isthen assigned to an object’s interface with the INTERFACE command.The phase surfaces of an optical field exiting a DOE may exhibit 2 stepdiscontinuities. Although the wavefront shows the phase steps, SPREAD andFIELD removes the discontinuities in the process of synthesizing the optical field.34 ASAP Technical Guide

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAP. . . . .MULTIPLE commandThe MULTIPLE command is a SURFACE modifier. Therefore, it must come after anappropriate surface definition. The MULTIPLE command converts a designatedsurface into multiple parallel surfaces, which may be used to create repetitiveobjects. It creates multiple surfaces, but in a fundamentally and mathematicallydifferent manner than the ASAP surface modifier, ARRAY. We will also use theMULTIPLE surface modifier to create arbitrary grating lines of a diffractiveoptical element. MULTIPLE is more often used for modeling diffractive opticalelements than creating for multiple parallel surfaces that are used purely for opticalsystem geometries. The syntax is,wheren is the number of sheets to be generated,f' is an additive constant to the original function,d is the distance between original and first sheets,x y z is an arbitrary point on the original surface, andEXPONENT p is the exponent to which sheet number is raised.The surface is converted into multiple parallel surfaces in the followingmathematical manner. If the original surface is given by f(x,y,z), MULTIPLEreplaces it with,ASAP Technical Guide 35

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAPIf MULTIPLE is used to define a diffraction grating, the value of n is irrelevant,and can even be zero. ASAP automatically knows how to use the multiple surfacein the case of a diffractive optical element because of its special reference on theINTERFACE command where j specifies the diffractive order numbers.Depending upon the nature of the surface, the value of the exponent p, and theadditive constant f' to the original surface, the sheets may not be equally spaced.The exponent p is defaulted to 1, but can be used, for example, to get evenlyspaced cylinders or spheres (p=2). The zeroth sheet is just the original surface.What is the function f(x,y,z) when used with the first syntax of MULTIPLE tosimulate a diffractive optical element? It is an equation describing the grating linespacing. You can derive this equation from the phase function, or you can computeit directly. To derive f(x,y,z) from the phase function, we must understand that it isa normalized form of the phase function that we described in the section,“Fundamentals of Diffractive Optical Elements” on page 9.Recall that the phase function, in general, describes how the phase of an incidentwave is altered by an optical element. A modulo 2 phase function is the phasefunction of a diffractive optical element.NOTE The function f(x,y,z) is the function obtained when the phasefunction of a given order m is divided by 2m uniquely specifiesthe grating line equation.The grating line spacing is a physical spacing, which does not change for otherorders. Other orders are diffracted according to this spacing. In other words, aphysical grating line is obtained whenever a specific phase goes though 2. Ingeneral, the function f(x,y,z) on the MULTIPLE command, as well as its otherterms, can be obtained from an arbitrary phase function by dividing the phasefunction by 2m or where m is the order number corresponding to that phasefunction.We can mathematically demonstrate this relationship by re-examining Figure 2“Transmission and reflection gratings” on page 13. The local phase change withina specific period in one dimension is,36 ASAP Technical Guide

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAP. . . . .Equation 21But the difference in sine terms is equal to(EQ 21)Equation 22(EQ 22)Note here that d(y) indicates that the local spatial period may, in general, change asa function of position, and n(y) is a spatial frequency denoting the number of localgrating lines per unit length.ASAP Technical Guide 37

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAPWe can equate the above equations, rearrange, and integrate to yield,Equation 23(EQ 23)Our results can be extended to rotationally symmetric systems or to threedimensions without loss of generality. If f(y) and f’ are determined in this manner,f’ is commonly set equal to one, and the exponent term of the MULITIPLEcommand is not needed normally.If you choose to, you can derive the grating line equation directly. In this case, youmust derive f(x,y,z) and determine f’ and the exponent p of the j term.Alternatively, ASAP can calculate f’ such that the distance from a point (x,y,z) onthe original surface to the first sheet is d. It does this by evaluating the expression,Equation 24(EQ 24)This second expression is most useful for diffractive optical elements, such aslinear gratings that have equally spaced grating lines. You are not specifying thephase function, but rather the direction in which parallel replications of the originalsurface are generated. At each distance d along the original surface’s normal,38 ASAP Technical Guide

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAP. . . . .computed from the gradient of f(x,y,z), you get a replicated version of the originalsurface. In the case of a plane, resulting planes are stacked on top of each other butspaced at a distance d apart. In the case of cylinders, resulting concentric cylindersare spaced a distance d apart. This special spacing is the grating line spacing.Note the similarity of the above equation to the special case of the MULTIPLEcommand, when the grating lines are equally spaced.Equation 25(EQ 25)Before we examine several DOE examples, we must learn about the necessaryoptions of the INTERFACE command for assigning a diffractive optical interfaceto an object.ASAP Technical Guide 39

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAPINTERFACE command for DOEsThe INTERFACE command assigns specular reflection and transmissionproperties to the interface of an object. We also use the INTERFACE command toassign diffractive optical properties to it.The INTERFACE command has several different syntaxes. We are most interestedin those that allow us to assign diffractive properties to an interface, which arelisted below.Here,DIFFRACT is a flag to assign a diffractive interface to an object,i is MULTIPLE surface number representing the grating line spacing,j j' ... are diffraction order number(s),e e' ... are relative efficiencies of the corresponding orders, andcoat coat' are the names of a given coating property.The first part of the INTERFACE command, preceding any of the DIFFRACToptions, is exactly the same as you have seen before. You can reference a coatingfrom the coating database and assign it to the object, as well as refractive indiceson either side of the geometry of the object.If the object is an optical boundary through which rays are to be traced, the opticalproperties of the interface must be specified using the INTERFACE commandafter the definition command for that object. If an INTERFACE command does notfollow an edge or surface object definition, the surface is assumed to be perfectlyabsorbing, and all rays reaching the surface are trapped there. If your interface hasnon-zero reflection and transmission coefficients and in addition is a diffractiveinterface, you will see reflected and transmitted diffraction orders.When you use the first form of the INTERFACE DIFFRACT syntax, grating linesare created by the intersection of the object surface with the different sheets of aMULTIPLE surface i. For example, a ruled linear phase grating is created if i is aplane, a zone phase plate is created if the surface is a cylinder, etc. If i is positive,the multiple sheet spacing is taken to be the grating spacing in system units. If i is40 ASAP Technical Guide

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAP. . . . .negative, the spacing is assumed to be in the same units as the last WAVELENGTHspecification. The number of sheets entered on the MULTIPLE surface commandhas no bearing on this application. Specification of the sign of i must be consistentwith the units in ASAP and the definition of f(x,y,z), which is calculated from yourphase function.ASAP generates diffracted rays/beams for the diffraction order numbers given bythe j's with relative efficiencies given by the corresponding positive e's. ASAPdoes not compute the diffraction efficiencies as a function of order. You must enterthe diffraction efficiencies as a function of order. ASAP uses the scalar gratingequation and does not have the vector capability to compute diffractionefficiencies. If a diffraction efficiency, e, is entered as a negative number or as aname “coat”, it is a COATING PROPERTY that possibly contains polychromaticcomplex amplitudes.The simplest coating model is the COATING PROPERTIES. This model allowsyou to specify a table of reflection and transmission coefficients or diffractionefficiencies as a function of wavelength. ASAP linearly interpolates between thesevalues to obtain the values at intermediate wavelengths. In our current case, thereflection and transmission coefficients are diffraction efficiencies of specificorders.ASAP cannot compute diffraction efficiencies on the fly. Furthermore, thediffraction efficiency of a given order can change as a function of incident angle.However, with the second form of INTERFACE DIFFRACT, a named COATINGMODEL can be used to specify the angular variation of the diffraction orderefficiencies; that is,COATING MODELS allows you to specify the angular, dispersion, andpolarization properties for S and P reflection and transmission coefficients ordiffraction efficiencies at an interface. In our current case, we use it to specify thediffraction efficiency envelope or discrete diffraction efficiencies. It uses modelsfrom the ASAP scatter model set. COATING MODELS has two syntax forms. Itssyntax is,ASAP Technical Guide 41

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAPHere,k is the starting coating number,r r' r" ... are the real energy (or complex amplitude) reflectance efficiencies, andt t' t" ... are the real energy (or complex amplitude) transmittance efficiencies.Starting with coating number k, coatings are entered with real energy (or complexamplitude) reflectance r and transmittance t diffraction efficiencies for the zerothorder. The default value for k is one more than the largest coating number defined,and is set to one at the start of program execution.In the first syntax, separate angular properties of the coating are specified by usingpreviously defined scatter MODELS wherei is the model for reflected S polarization,j is the model for reflected P polarization,m is the model for transmitted S polarization, andn is the model for transmitted P polarization.We can specify different reflection and transmission diffraction efficiencies for thezeroth order for different wavelengths, as specified on the last WAVELENGTHcommand. ASAP uses the normalized scatter model data and the reflection andtransmission diffraction efficiencies for the zeroth order to account for theangularly dependent nature of the diffraction grating. If you are not using afunctional scatter model such as USERBSDF, but rather data from BSDFDATA,ASAP linearly interpolates in logarithmic amplitude space to determine reflectionand transmission diffraction efficiencies at other orders than those specified in thedata set. However, with this syntax the same angularly dependent polarizationmodels are used for all sources at different wavelengths. Therefore, this form ofCOATING MODELS should be used for diffraction gratings that are nondispersive.Alternatively, in the second syntax, groups of six numbers can be entered toaccount for grating dispersion and volume holograms. Each group corresponds to awavelength entered on the last multiple WAVELENGTH(S) command. The firstnumber is the reflection diffraction efficiency of the zeroth order corresponding tothe first wavelength on the last multiple WAVELENGTH command. The next twonumbers are the angularly dependent S and P reflection diffraction efficiencymodels, as discussed previously, at that wavelength. The fourth number is the42 ASAP Technical Guide

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAP. . . . .transmission diffraction efficiency followed by its angularly dependent S and Ptransmission diffraction efficiency models, and so on for different wavelengths.Again, angular data is linearly interpolated in logarithmic amplitude space when afunction is unavailable.The reflection and transmission diffraction efficiencies for sources withwavelengths between those defined in COATING MODELS are interpolated, forexample, as in Equation 26. The actual diffraction efficiencies at an incidenceangle a and a wavelength b between the first two WAVELENGTH(S) w w' wouldbe,Equation 26(EQ 26)In these equations, r, r’, t, t’ are the complex amplitudes or square roots of the realenergy coefficients of the diffraction efficiencies. f(i,a) is the normalized angularamplitude,Equation 27where BSDF is the bi-directional scattering distribution function.(EQ 27)ASAP Technical Guide 43

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAPTIP The key to implementing a COATING MODEL lies inunderstanding how to set up the data. If your data is not in afunctional form, such as that which can be used with USERBSDF oranother functional ASAP scatter model, it will be in the form ofreflection and transmission diffraction efficiencies as a function ofincident angle.The most commonly used format for entering data in this form is the BSDFDATAscatter model format, since it allows you to enter this type of data directly. ASAPlinearly interpolates in logarithmic amplitude space to determine coefficientsbetween those entered in BSDFDATA. Regardless of the scatter model you choose,you must first run the MODELS keyword command, just like you would withCOATING or MEDIA.The BSDFDATA syntax for a coating model is then as follows,Here,k is the model data base number;ANGLES specifies spherical angle coordinates in degrees;ao bo are the first specular direction, polar and spherical anglesa b [a' b' ...] are spherical ANGLES from and around normal for other orders;f [f' ...] are the diffraction efficiencies; andao' bo' are the second specular direction.The data entered on the lines with two entries only, indicated by o’s following theletters a, b and so on, defines one incident specular direction for the sets of tripletsto follow on the next lines. In the case of a COATING MODEL, the first twonumbers of the first set of triplets are the incident specular angles repeated again.For in-plane data, you need enter only the ao, ao', and accompanying a 's as theyspecify the angle from the normal. The bo 's and b 's can be set to 0 as they are theangles around the normal in plane data. The f parameters are the diffraction44 ASAP Technical Guide

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAP. . . . .efficiencies. The next line that has only two numbers defines the next speculardirection with its associated triplets, and so on.What then are the additional a' b' f', a" b" b" terms, and so on? They are thediffraction angles and efficiencies corresponding to a diffraction envelope or,specifically, to the other orders of a diffraction grating. In one sense, a coating islike a diffraction grating operating only in the zeroth order (a, b, f), which isspecified by a single set of triplets. However, a diffraction grating can havemultiple sets of triplets simulating other diffraction order efficiencies. Thenumbers you enter are the discrete numbers of an envelope of the diffractionefficiency function at this wavelength and angle of incidence. The numbers cancorrespond directly to specific diffraction order efficiencies if you have only thisdata.The COATING MODELS/ USERBSDF technique is an appropriate way to modeldiffraction efficiencies. USERBSDF is a general scatter model and diffractiongratings produce discrete, deterministic scattering functions.Finally, multiple exposure holograms can be modeled by using the third form ofthe INTERFACE DIFFRACTive syntax. However, you specify only the zerothorder once. For example,ASAP Technical Guide 45

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAPDOE examplesThe following examples illustrate how to set up various diffraction gratings, binaryoptical elements, and holograms. The examples move from the simple to thecomplex and demonstrate various forms of the MULTIPLE command and theINTERFACE…DIFFRACT command.LINEAR PHASE GRATINGIn our first example, we will simulate a simple linear phase grating with bothforms of the MULTIPLE command. When we use the first form of the multiplecommand, we must know the mathematical description of the phase surface of ourlinear grating. We know that the grating spacing is d and that the grating frequencyis 1/d. But we also have, from our previous work, the relationship between thegrating frequency, the phase function, and the grating line equation,Equation 28Substituting 1/d in the above equation for the grating frequency yields thefollowing relationships,(EQ 28)Equation 29(EQ 29)Clearly, f(y)=y and f’=d. We also could have specified the coefficient of y as 1/dand set f’ equal to 1. ASAP now has all the necessary information to compute thelocal grating line spacing for a ray incident on any point of the base surface. It thenuses the grating equation to generated diffracted rays according to the ordersspecified on the INTERFACE...DIFFRACT command.ASAP is splitting rays at the diffractive interface. Some of these rays will becomechild rays. If there are more diffraction gratings in the optical system, you mayhave to set SPLIT to more than one to generate these other diffracted rays.Example 1 illustrates the ASAP syntax for configuring this grating. Only the zero,46 ASAP Technical Guide

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAP. . . . .and plus/minus first orders are simulated, even though more orders are physicallypresent.Note how the GENERAL command is used to define f(x,y,z), and how the basesurface of intersection is a surface perpendicular to the z axis. Its geometry isindependent from the MULTIPLE GENERAL surface. Finally, note that the basesurface is both reflecting and transmitting, as defined in the COATINGPROPERTIES database. See Example 1.ASAP Technical Guide 47

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAP!!ASAP EXAMPLE OF A PHASE DIFFRACTION GRATING!!USING THE FIRST FORM OF THE MULTIPLE COMMANDSYSTEM NEWRESET$DATIM OFF OFFPI=4*ATAN(1)LAMBDA=0.025D=0.1COATING PROPERTIES0.4 0.6 'DOE'SURFACEGENERAL 0 0 0Y 1MULTIPLE 1 (D)PLANE Z 0 RECTANGLE 1 grating base surfaceOBJECT 'LINEAR_GRATING'INTERFACE COATING DOE AIR AIR DIFFRACT 0.2 -1st order 0.25, 0th 0.5, 1st0.25SURFACEPLANE Z 10 RECT 1 detector planeOBJECT 'DETECTOR'ROTATE X ASIN[.25] 0 0BEAMS INCOHERENT GEOMETRICWAVELENGTH (LAMBDA)GRID ELLIPTIC Z -1 -4@1 1 11SOURCE DIRECTION 0 0 1WINDOW Y ZWINDOW 1.4TITLE 'GRATING W/-1,0,1st ORDERS IN REFLECTION AND TRANSMISSION'MISSED 1048 ASAP Technical Guide

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAP. . . . .PLOT FACETS 3 3 OVERLAYTRACE PLOT TEXT0 -2.6 10, 0 0 .25, 0 .25 0, 1 '-1st order'0 2.8 4.5, 0 0 .25, 0 .25 0, 1 '+1st order'0 1 8.75, 0 0 .25, 0 .25 0, 1 '0th order'0 1.75 -2.25, 0 0 .25, 0 .25 0, 1 '+1st order'0 -2 -2.25, 0 0 .25, 0 .25 0, 1 '-1st order'!!$VIEWRETURNExample 1. Linear phase diffraction grating with the first form of MULTIPLEOutput from the Plot Viewer is shown in Figure 9.Figure 9 Example 1: Diffraction orders from a reflection and transmission grating, usingthe first form of the MULTIPLE commandASAP Technical Guide 49

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAPWe can also simulate the same diffraction grating with the second form of theMULTIPLE command. Recall that in this second form you specify a distancebetween the original function or surface and the first replica. ASAP automaticallycomputes the f’ value from,Equation 30(EQ 30)In our case, the gradient of the function f(x,y,z), obtained from the phase function,specifies the direction in which parallel replications of the original surface aregenerated. We know that the grating lines are perpendicular to the y axis, so we canuse a plane whose normal is collinear with this axis to denote the gradient off(x,y,z). The MULTIPLE terms then include the grating spacing, which is just thedistance between the original function or surface and the first replica, and a pointon the original function. Our plane, whose normal is collinear with the y axis, isreplicated to form a series of parallel planes whose spacing is the grating linespacing.The second form of the MULTIPLE syntax illustrates the ASAP syntax for this case.See Example 2.50 ASAP Technical Guide

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAP. . . . .!!ASAP EXAMPLE OF A PHASE DIFFRACTION GRATING!!USING THE SECOND FORM OF THE MULTIPLE COMMANDSYSTEM NEWRESET$DATIM OFF OFFCOATING PROPERTIES0.4 0.6 'DOE'SURFACEPLANE Y 0 grating line generating surfaceMULTIPLE 1, grating spacing 0.1, point 0 0 0PLANE Z 0 RECT 1 grating base surfaceOBJECT 'LINEAR_GRATING'INTERFACE COATING DOE AIR AIR DIFFRACT 0.2 -1st order 0.25, 0th 0.5, 1st0.25SURFACEPLANE Z 10 RECT 1 detector planeOBJECT 'DETECTOR'ROTATE X ASIN[.25] 0 0BEAMS INCOHERENT GEOMETRICWAVELENGTH 0.025GRID ELLIPTIC Z -1 -4@1 1 11SOURCE DIRECTION 0 0 1WINDOW Y ZWINDOW 1.4TITLE 'GRATING W/-1,0,1st ORDERS USING 2nd FORM OF MULTIPLE'MISSED 10PLOT FACETS 3 3 OVERLAYASAP Technical Guide 51

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAPTRACE PLOT TEXT0 -2.6 10, 0 0 .25, 0 .25 0, 1 '-1st order'0 2.8 4.5, 0 0 .25, 0 .25 0, 1 '+1st order'0 1 8.75, 0 0 .25, 0 .25 0, 1 '0th order'0 1.75 -2.25, 0 0 .25, 0 .25 0, 1 '+1st order'0 -2 -2.25, 0 0 .25, 0 .25 0, 1 '-1st order'RETURNExample 2. Linear phase diffraction grating with the second form of MULTIPLEAgain, only the zero and plus/minus first orders are simulated, even though moreorders are physically present. See Figure 10.Figure 10 Example 2: Diffraction orders from a reflection and transmission grating, usingthe second form of the MULTIPLE command52 ASAP Technical Guide

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAP. . . . .SINUSOIDAL PHASE GRATINGA sinusoidal phase grating is like a linear phase grating, with the exception that itsphase structure is a sinusoid instead of discrete steps. As a consequence, light isdiffracted only into the zero and first orders. This can be understood by examiningthe Fourier aspects of diffraction at a sharp edge or phase structure compared tothat of a smooth edge or phase structure. Higher frequency components of a sharpedge in Fourier space contribute to the higher-order diffraction angles andsubsequent diffraction orders. These frequency components are missing in thesinusoidal phase structure and, therefore, so are the diffraction orders higher thanthe first order.In a sinusoidal grating, the phase function is of the form,Equation 31(EQ 31)where d is again the grating period. The order m is one. The function f(x,y,z) of theMULTIPLE command is then,Equation 32(EQ 32)However, we now have a grating line function that is not a polynomial function, afunctional form easily implemented with the GENERAL command. How can werepresent this grating line function? The answer is with USERFUNC.ASAP Technical Guide 53

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAPThe USERFUNC command creates a surface specified by a user-defined function.The user-defined function can be one of the ASAP intrinsic functions like sine orcosine, or it can be one created with the internally defined macro $FCN. Its syntaxis as follows,Herex y z are the global coordinates of the reference point;fcn is a user-defined function; andc c' c" ... are user-defined coefficients of the function.USERFUNC specifies a user-defined function with reference point (x,y,z) anddouble-precision coefficients. You can define its value and gradient at any point inthe macro, $FCN named fcn or the Fortran function USERFUNC. If the function iscontinuous in both value and gradient everywhere in space, there are norestrictions on the use of this function in ASAP, except possibly the application ofnon-orthogonal transformations to it; that is, SCALE or SKEW or non-isotropicSCALE.If the fcn is specified, the local (x,y,z) coordinates are passed in the _1, _2, and _3registers. You can also define other parameters of the function and set them, withup to 63 coefficients c c' c" ..., in the registers _4, _5,. . ._66. If four or morevalues are returned, the last four entries of the function that was run must be thefunctional value and its gradient vector. For example, a sphere of radius 10centered about the reference point is done as follows,Otherwise, the default USERFUNC is an aspheric conicoid. You can examine theother parameters of the default user function in the ASAP online help.54 ASAP Technical Guide

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAP. . . . .In our example, we use USERFUNC to define the sinusoidal phase function. OnUSERFUNC, we must enter the functional form of the grating line function and itsgradient. The gradient of our grating line function is,Equation 33(EQ 33)These values are entered on USERFUNC as illustrated in Example 3. Thediffraction orders are shown in Figure 11 We chose f’ to be 1, but we could alsohave set it equal to 1/2.ASAP Technical Guide 55

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAP!!ASAP EXAMPLE OF A PHASE DIFFRACTION GRATING!!USING THE FIRST FORM OF THE MULTIPLE COMMANDSYSTEM NEWRESET$DATIM OFF OFFPI=4*ATAN(1)LAMBDA=0.025D=0.1COATING PROPERTIES0.4 0.6 'DOE'$FCN GRATING Y=_2,SIN(2*PI*Y/D)/(2*PI) 0,(1/D)*COS(2*PI*Y/D) 0SURFACEUSERFUNC 0 0 0 GRATINGMULTIPLE 1 1PLANE Z 0 RECT 1 grating base surfaceOBJECT 'SINUSOIDAL_GRATING'INTERFACE COATING DOE AIR AIR DIFFRACT 0.2 -1st order 0.25, 0th 0.5, 1st0.25SURFACEPLANE Z 10 RECT 4 detector planeOBJECT 'DETECTOR'BEAMS INCOHERENT GEOMETRICWAVELENGTH (LAMBDA)GRID ELLIPTIC Z -1 -4@1 1001 1001 RANDOM 1SOURCE DIRECTION 0 0 1WINDOW Y ZWINDOW 1.4TITLE 'SINUSODIAL GRATING W/-1,0,1st ORDERS IN REFLECTION AND TRANSMISSION'56 ASAP Technical Guide

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAP. . . . .MISSED 10PLOT FACETS 3 3 OVERLAYTRACE PLOT 11111 TEXT0 -2.8 4.5, 0 0 .25, 0 .25 0, 1 '-1st order'0 2.8 4.5, 0 0 .25, 0 .25 0, 1 '+1st order'0 .2 10.1, 0 0 .25, 0 .25 0, 1 '0th order'0 1.75 -2.25, 0 0 .25, 0 .25 0, 1 '+1st order'0 -2 -2.25, 0 0 .25, 0 .25 0, 1 '-1st order'WINDOW X YPIXELS 101CONSIDER ONLY DETECTORSPOTS POSITION EVERY 137DISPLAYAVERAGEPICTURERETURNExample 3. Sinusoidal phase diffraction grating with USERFUNC and the first form of theMULTIPLE commandASAP Technical Guide 57

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAPThe following illustrations are all part of Figure 11.58 ASAP Technical Guide

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAP. . . . .Figure 11 Example 3: Diffraction orders from a sinusoidal reflection and transmission grating, using USERFUNC and thefirst form of the MULTIPLE commandCIRCULAR DIFFRACTION GRATINGThe circular diffraction grating is a linear grating bent around an axis of symmetry.This grating, therefore, has circularly concentric grating lines. Its radial frequencyis 1/d. Following a similar technique, we can integrate the radial frequency toobtain the phase surface and f(x,y,z).Recall that the phase surface and the grating line equation are related, andspecifically related to the radial frequency byEquation 34(EQ 34)ASAP Technical Guide 59

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAPSubstituting 1/d in the above equation for the grating frequency yields thefollowing relationships.Equation 35(EQ 35)Equation 36(EQ 36)and f’=d. However, our relationship for the grating line equation contains a squareroot. Therefore, it is best to use USERFUNC, since a square root cannot be directlyentered on the GENERAL command, and then MULTIPLEd. Remember that whenyou use USERFUNC, you should configure it to return four values, the first beingthe value of the function and the last three its gradient. In our case, the functionand its gradient are,60 ASAP Technical Guide

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAP. . . . .Equation 37(EQ 37)These values are entered in the USERFUNC command as shown in Example 4. Thediffracted first order for the circular grating is illustrated in Figure 12.ASAP Technical Guide 61

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAP!!ASAP EXAMPLE OF A CIRCULAR DIFFRACTION GRATING!!USING THE FIRST FORM OF THE MULTIPLE COMMANDSYSTEM NEWRESET$DATIM OFF OFFPI=4*ATAN(1)D=0.058476 grating line spacing$FCN CIRC SQRT(_1^2+_2^2),_1/SQRT(_1^2+_2^2),_2/SQRT(_1^2+_2^2),0COATING PROPERTIES0 1 'TRANS'BEAMS INCOHERENT GEOMETRICWAVELENGTH 0.02SYSRAY { 2RAYS 0GRID ELLIPTIC Z -1 -4@1 #1 #2SOURCE DIRECTION 0 0 1}ENTER NUMBER OF RAYS ALONG X-AXISENTER NUMBER OF RAYS ALONG Y-AXISSURFACEUSERFUNC 0 0 0 CIRCMULTIPLE 1 (D)PLANE Z 0 ELLIP 1OBJECT 'CIRCULAR_GRATING'INTERFACE COATING TRANS AIR AIR DIFFRACT 0.2 -1SURFACEPLANE Z 4 RECT 1.562 ASAP Technical Guide

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAP. . . . .OBJECT 'DETECTOR'TITLE 'CIRCULAR GRATING | 1st FORM OF MULTIPLE'MISSED ARROW 3.5$SYSRAY 1 11WINDOW Y -2 2 Z -1 3PLOT FACETS 9 9 0 OVERLAYTRACE PLOTRETURN$IO VECTOR REWIND$SYSRAY 5 11WINDOW Y -2 2 Z -1 3WINDOW 1.5OBLIQUEPLOT FACETS 3 3 OVERLAYTRACE PLOTRETURN!!$VIEWExample 4. Circular phase diffraction grating with the first form of MULTIPLEASAP Technical Guide 63

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAPFigure 12 Example 4: First diffraction order from a circular phase grating, using USERFUNC and the first form ofMULTIPLE64 ASAP Technical Guide

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAP. . . . .All our examples to this point were generated by integrating the radial frequencyfunction or modifying the phase function to generate the grating line function.They all also used the default EXPONENT on the MULTIPLE command.We now demonstrate a technique for deriving the grating line function directly,which is also a case where the EXPONENT on the MULTIPLE command is notequal to one, the default. We again return to the circular grating for this example.A circular diffraction grating has concentric, equally spaced grating circles (lines).We can directly write the equation of these grating lines as,Equation 38(EQ 38)d is again the grating circle spacing. This can be verified by computing the gratingspacing between two adjacent grating circles as,Therefore, is a constant.Equation 39(EQ 39)We must now put this equation into a form that ASAP can simulate as a surface,and then we must identify the appropriate terms of the MULTIPLE command. Bysquaring both sides of the previous equation we get,Equation 40(EQ 40)ASAP Technical Guide 65

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAPWhen we compare this with the form of the MULTIPLE command, we see that,Equation 41(EQ 41)The MULTIPLE syntax with the EXPONENT option for this case is illustrated inExample 5. Note that the grating spacing d is 0.058476, and the wavelength is0.02—both are in arbitrary units. We can compute the diffraction angle for thiselement by using the grating equation, sin(q)=ml/d, and we see that it is 20, thesame as in the previous circular grating example.66 ASAP Technical Guide

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAP. . . . .!!ASAP EXAMPLE OF A CIRCULAR DIFFRACTION GRATING!!USING THE EXPONENTIAL OPTION OF THE MULTIPLE COMMANDSYSTEM NEWRESET$DATIM OFF OFFD=0.058476 grating line spacingCOATING PROPERTIES0 1 'TRANS'SURFACEGENERAL 0 0 0; X2 1; Y2 1MULTIPLE 0 D*D EXPONENT 2PLANE Z 0 ELLIP 1OBJECT 'CIRCULAR_GRATING'INTERFACE COATING TRANS AIR AIR DIFFRACT 0.2 -1SYSRAY { 1RAYS 0BEAMS INCOHERENT GEOMETRICWAVELENGTH 0.02GRID ELL Z -1 -4@1 #1 11SOU DIR 0 0 1}ENTER NUMBER OF RAYS ALONG X-AXIS>TITLE 'CIRCULAR GRATING | EXPONENTIAL OPTION OF MULTIPLE'MISSED ARROW 3.5$SYSRAY 1WINDOW Y -2 2 Z -1 3PLOT FACETS 9 9 0 OVERLAYTRACE PLOT$IO VECTOR REWINDASAP Technical Guide 67

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAP$SYSRAY 5WINDOW 1.5OBLIQUEPLOT FACETS 9 9 0 OVERLAYTRACE PLOTRETURNExample 5. Circular phase diffraction grating with the EXPONENT option of theMULTIPLE commandThe plots for the above script are shown in Figure 13.68 ASAP Technical Guide

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAP. . . . .Figure 13 Example 5: First diffraction order from a circular phase grating, using GENERAL and the EXPONENTIALoptions of the MULTIPLE commandDOE LENSUp to this point, all the examples have been simple linear and circular gratings.They were made deliberately simple to demonstrate the different forms of theMULTIPLE command and the techniques needed to simulate DOE elements.We now examine a more complicated diffractive element, a DOE lens. With morecomplicated DOEs, we still use the same techniques we have developed so far. Wemust determine the grating line equation, which the MULTIPLE command uses tosimulate DOEs. Even with more complicated phase structures, we will createmultiple surfaces whose intersection with the base surface specifies the gratinglines. These grating lines represent 2 phase steps. Finally, since these gratinglines are created by the MULTIPLE command, they must be defined by theequation,ASAP Technical Guide 69

Equation 42(EQ 42)where j and p are integers and f’ is a constant. In general, the process requiresknowing the phase function at a particular order m and converting that phasefunction into the grating line equation by dividing by 2m. These terms can thenbe related directly to the grating line equation and the constant and exponentialterms that are needed in the MULTIPLE command.We will model a DOE lens that produces a simple quadratic phase transformationon a collimated input beam. We have already determined the phase function of thislens in the previous section. Its phase function for the minus first order was,Equation 43(EQ 43)Recall that the first term was a constant phase term of little interest, and the secondterm was a quadratic approximation to a spherical wavefront. The phase functionis related to the grating line function,Therefore, the grating line function is,Equation 44(EQ 44)Equation 45(EQ 45)

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAP. . . . .The paraxial focal length of the lens is f. We can write f(x,y,z) and the otherMULTIPLE terms as,Equation 46Note that other equally valid equations describing the grating line equation andconstant terms can be obtained by rearranging terms as follows.(EQ 46)ASAP Technical Guide 71

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAPEquation 47(EQ 47)ASAP syntax that is needed to simulate the DOE lens is shown in Example 6. Thescript is shown in two parts due to its length. Output from this example illustratethe geometric and diffractive output from the DOE lens for on- and off-axis beams.See Figure 14 , Figure 15, Figure 16, Figure 17, and Figure 18.72 ASAP Technical Guide

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAP. . . . .SYSTEM NEWRESET$DATIM OFF OFF$GUI CHARTS_ON !!for Chart Viewer!!$GUI CHARTS_OFF !!for Plot ViewerL=0.6328E-4 F=20COATING PROPERTIES0 1 'TRANS'SURFACEGENERAL 0 0 0X2 1Y2 1 !!f(x,y,z)MULTIPLE n 1, f prime 2*L*FPLANE Z 0 ELLIPSE 2OBJECT 'DOE'INTERFACE COATING TRANS AIR AIR DIFFRACT surface 1, -1st orderSURFACEPLANE Z 20 ELLIPSE 3OBJECT 'DETECTOR'BEAMS INCOHERENT GEOMETRICWAVELENGTH (L)GRID ELLIPTIC Z -1 -4@2 301 301 RANDOM 1SOURCE DIRECTION 0 0 1WINDOW Y ZPIXELS 255TITLE 'F/5 DOE LENS WITH SIMPLE QUADRATIC PHASE'CONSIDER ONLY DETECTORWINDOW X YASAP Technical Guide 73

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAPRAYS 0BEAMS COHERENT DIFFRACTWAVELENGTH (L)PARABASAL 4WIDTHS 1.6GRID ELLIPTIC Z -1 -4@1.8 101 101 RANDOM 1SOURCE DIRECTION 0 0 1, 0 SIN[1] COS[1]CONSIDER ALLWINDOW Y Z!!PLOT FACETS 3 3 0 OVERLAYTRACE PLOT 71CONSIDER ONLY DETECTORSELECT ONLY SOURCE 1FOCUS MOVEWINDOW Y -.005 .005 X -.005 .005WINDOW .7PIXELS 201SPREAD NORMALDISPLAYISOMETRIC 2 'IRRADIANCE DISTRIBUTION (ON-AXIS)'PICTURE 'ON-AXIS SOURCE'RETURNSELECT ONLY SOURCE 2FOCUS MOVEWINDOW Y .344 .354 X -.005 .005WINDOW .7PIXELS 201SPREAD NORMALDISPLAYISOMETRIC 2 'IRRADIANCE DISTRIBUTION (OFF-AXIS)'74 ASAP Technical Guide

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAP. . . . .PICTURE 'OFF-AXIS SOURCE'RETURNExample 6. DOE lens input (above)Figure 14 .Example 6: DOE lens with simple quadratic phaseFigure 15 Example 6: Irradiance distribution (on-axis)ASAP Technical Guide 75

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAPFigure 16 Example 6: On-axis sourceFigure 17 Example 6: Irradiance distribution (off-axis)76 ASAP Technical Guide

DIFFRACTION GRATINGS AND DOESSimulating Diffractive Optical Elements in ASAP. . . . .Figure 18 Example 6: Display Viewer output, off-axis sourceASAP Technical Guide 77

DIFFRACTION GRATINGS AND DOESReferencesREFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .• Optics, Hecht, Second Addition.• Introduction to Fourier Optics, Goodman.• Binary Optics Technology: The Theory and Design of Multi-level DiffractiveOptical Elements, Swanson.• Fundamentals of Polarized Light A Statistical Approach, Brosseau.• Polarized Light Production and Use, Shurcliff.• Optical Waves in Crystals, Yariv and Yeh.• Principles of Optics, Born and Wolf.• Optical Thin Films Users Handbook, Rancourt.• Optical Scattering Measurement and Analysis, Stover.78 ASAP Technical Guide