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Technical Guide Technical RATINGS G Breault Research Organization, Inc.
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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 Huygens-Fresnel principle Huygens-Fresnel velocity, suchthatat a latertime, the source of secondary wavelets (or sources Huygens proposedthatever havethesamelength. of source rays from a point path lengths theoptical which a contemporary,geometrical sense,wa isama Awavefront secondary wavefronts. and primary upon diffractionbased of a theory proposed Huygens Christian equation. toderive thegrating principle theory, less rigorous yet mathematically intuitive physically a more is, perhaps, This principle. theHuygens-Fresnel from alsobederived can equation The grating use thistechnique since it ASAP, including software andanalysis design programs, Mostoptical technique. with or calculated begenerally easily cannot to thegrating accordingexiting beams The diffractionlight. orders five isapproximately period the grating Swanson notesthatthescalar periodislarger thediffractingof ifthe grating thewavelength valid light. than interacting witha diffractive element.Th reflected an thevarious of propagation diffractionUsing scalar theory, wecan calculation. type this of doesnot DOE. perform ASAP DOE structuraldata, but alsothe physi the diffractive surface prof diffracted intoeachorderiscalled the di as thefractionalpowercontai as well DOE orders transmitted reflected and the various Using vectordiffraction theory,cal wecan vector calculation and Twoapproachesexist commonforcalculati . ELEMENTS FUNDAMENTALS OF scalar calculation. is adaptable to ray tracing. is adaptabletoray ile. This complicated calculation involves not only the the only not involves calculation complicated ile. This y point on a primary wavefront can be considered a be a primarywavefrontcan y pointon are selected by adjustingth approximationcanbeusedinMaxwell’s equations if DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION Fundamentals of Diffrac primary wavefront is the envelope of the of envelope the is wavefront primary DIFFRACTIVE OPTICAL d transmitted orders of a beam ordersofabeam after d transmitted vefronts are mathematical surfaces over cal and electromagnetic propertiesofthe equation. Thediffraction efficiencies calculate theideali calculate derivation than one fro one than derivation . Therefore,weuseHuygens-Fresnel times the wavelength of the diffracting of the wavelength the times ffraction efficiency, ), each with the same frequency and the samefrequency ), eachwith e scalardiffraction culate the directions of propagation of ofpropagation thedirections culate ned ineachorder. The fractionalpower thematical surfaceofconstantphase. In of a beam after interacting with the the with afterinteracting beam of a ngthe diffraction patterns fromDOEs: this method since this is a scalar is a method sincethis this ASAP TechnicalASAP Guide e direction cosines of the cosines of e direction zed directions of zed directions tive Optical Elements tive Optical theory technique is technique theory and is determined by m scalardiffraction
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 NOTE Optics theintroductory theASAP technical material in guide, grating. If light transmits through the di transmits through Iflight grating. phasealtering material If of type some grating. the grating. Young’s transmission of amplitude slit anexample is a simple double by reflected is not and through passes light if the grating transmission an amplitude apertures areused,the diffraction grating andphaseof a wa amplitude phase, both or periodic structures, produ oftheir nature Diffraction gratings areregular arrays of available in the Knowledge Base at http://www.breault.com/k-base.php. inthe Knowledge available are ASAP in algorithm beam superposition the Gaussian describing papers Several wave diffraction theory. learning about for resource introductory Joseph W. Goodman’s book, theFresnel-Kirchhoffof assumptions scalar diffractionSommerfeld theory, with certain nonphysical did away which theory. Wilhelm Johannes the Rayleigh- Arnold developed later Sommerfeld principle, which eventuallybecame the Fr mathema a developed Kirchhoff Gustav fields. optical arbitrary simulate tos functions asthe basis amplitudes Fresnel. However, ASAP uses quadrat to Huygens- issimilar in ASAP algorithm The Gaussianbeam superposition sum amplitudes. isalsousedtoalgebraically Superposition the wavefront. wavescomprising of individual the summations algebraic arethe onawavefront perturbations theresultant that states superposition inthe functions asthe basis wavefronts using spherical principle isa superposition principle The Huygens-Fresnel reflection grating. reflection phase transmission Formore information regarding amplitudesphases, and see .
grating. If light reflectsoff Iflight grating. itiscalled a phase grating, the An Introduction FourierOptics to DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION Fundamentals of Diffrac uperpose individual Gaussian to beams Gaussian uperpose individual scalarwavediffraction theory. ic phase fronts and Gaussianbeam is used, the diffractiis used, ffraction grating,th superposition. The principle of wavefront ofwavefront The principle superposition. tical theory of theHuygens-Fresnel ce periodic perturbations in the amplitude, amplitude, the in perturbations ce periodic apertures orphase structures that, by the Huygens-Fresnel principal andscalar principal the Huygens-Fresnel is an amplitude grating and,specifically, grating an amplitude is esnel-Kirchhoff scalar wavediffraction vefront incident on the structure. If structure. on the incident vefront ASAP TechnicalASAP Guide tive Optical Elements tive Optical on grating is on grating a phase e grating is called a Wave Wave , isan excellent
11 . . . . . DIFFRACTION GRATINGS AND DOES Fundamentals of Diffractive Optical Elements
Generally, phase gratings are more efficient at diffracting monochromatic light than amplitude gratings, and surface relief phase gratings are more efficient at diffracting broadband light than volume phase gratings. Theoretically, a phase grating can diffract 100% of incident monochromatic light into a single diffraction order. A surface relief phase grating has higher diffraction efficiencies than a volume phase grating operating over the same bandwidth. An amplitude grating cannot diffract 100% of the incident monochromatic light into a single diffraction order, because some of the incident light is reflected, absorbed, and so on, by the grating mask. The simplest type of surface relief—phase transmission diffraction grating—is a linear phase transmission grating embossed on a flat, slab substrate. We can gain an intuitive understanding of more complicated diffraction gratings from examining the linear grating. A linear grating is a series of regularly arrayed lines or scratches embossed in a transmissive material known as the substrate. Phase transmission 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 or etched onto a glass substrate. As a simple physical model, the regularly arrayed lines can be considered scratches, troughs, or humps in the substrate. In the case of scratches, each line acts like a linear scattering center. In the case of troughs or humps, each line acts like a very short radius of curvature, highly divergent cylindrical lens. In both cases, the effect is to produce equivalent linear or line sources. Each effective line source is illuminated by the same, or more correctly, different parts of the same wavefront. Therefore, each line source resembles the secondary sources or wavelets of the Huygens-Fresnel principle. See Figure 1 Their superposition at a later time or, equivalently, a later position, will determine the optical field at that point in time or position. A constructive interference of all the wavelets in a particular direction leads to diffraction orders. Some of the light is not scattered or diverged in transmitting through the diffraction grating. This occurs in between the grating lines. This light, basically unperturbed from the specular direction, is called the zeroth order. In other words, the zeroth order of the diffraction grating is specular direction of the incident light. We have constructed a simple graphical model of the Huygens-Fresnel principle as applied to linear phase transmission and reflection gratings in Figure 2.
12 ASAP Technical Guide Here the incident angle is, sineof that the trigonometric note illustration, transmission phase thelinear For diffracted orderangleis, d is the grating line spacing. Similarly, spacing. line grating is the not Specular - Zeroth Order Zeroth - Specular Phase Reflection Grating Reflection Phase Grating Transmission Phase Diffraction Orders m Figure 2 Transmission and reflection gratings DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION P 3 Equation 2 Equation 1 Fundamentals of Diffrac i i P 2 P P 3 4 m i i d P d P e that the trigonom the e that m P P 1 4 2 1 ASAP TechnicalASAP Guide m pclr-Zeroth Order Specular - Diffraction Orders m tive Optical Elements tive Optical etric sineofthe (EQ 2) (EQ 1) (EQ 13 . . . . . DIFFRACTION GRATINGS AND DOES Fundamentals of Diffractive Optical Elements
If the phase difference between lines P1 P2 and P3 P4 is an integer number of multiple wavelengths , all the waves from the secondary wavelets are in phase in the direction corresponding to m. Operationally, this is,
(EQ 3) Equation 3 The above equation is called the grating equation, and m is called the grating or diffraction order. Note that the grating equation is independent of the refractive index of the slab substrate. This can be easily validated with Snell’s law. The grating equation is the same, even in the case of the phase reflection grating. The grating equation is the same as the equation that describes the maximum orders for Young’s double slit experiment. However, there are some fundamental physical differences in the diffraction patterns for each situation: • The diffraction or interference pattern from Young’s double-slit experiment is due to an amplitude transmission grating. It is also more diffuse or less sharp than that from the phase diffraction grating. This is primarily due to the larger size of the apertures in Young’s double slit experiment as compared to the relatively small line sources in the diffraction grating. In Young’s experiment, different points across the two apertures, other than the center points, can be in phase with each other and constructively interfere at slightly different locations than the center points causing a broadening of the diffraction pattern in that order. • The linear phase transmission grating behaves more like a Fabry-Perot etalon or filter.
Blazed gratings In many applications that use linear diffraction gratings, we are not interested in the light in the zeroth order. This light is often wasted. It is useful to transfer light out of the zeroth order and into another order or arbitrary direction other than the specular direction. Many modern diffraction gratings, especially phase reflection gratings are blazed to accomplish this task. John William Strutt Rayleigh, or Lord Rayleigh suggested a clever way to transfer energy out of the zeroth order into other orders. This involved changing the angle
14 ASAP Technical Guide gives the result for diffgives theresultfor gratings can becalculated usingscalar di The diffraction efficiency of a given orders area thegrating spacing of function The keytoablazedgratingisthatthean Such a grating iscalledablazedgrating. of or tilting the surfaces thatthespecular the higher 3. the higher diffractionFigure orders. grating not but output, ofthe specular thedirection lines, change we thegrating between grating plane andnotthebl , andtheangleofincidence Blazed Grating Blazed raction efficiencies as, Figure3 Blazedtr azed (tilted) surface. If we If azed (tilted)surface. i . However, DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION Index n Index hasRefractive Grating Fundamentals of Diffrac order forregular and blazed linearphase d gular directionofthenon-zerodiffraction ansmission grating ffraction theory.Swanson, forexample, light sees inbetweenthegratinglines. Blaze AngleBlaze d m , the wavelength of the incident light light incident of the wavelength the , and Period Grating ASAP TechnicalASAP Guide i aremeasuredfrom the tilt the specularsurfaces in tive Optical Elements tive Optical
15 . . . . . DIFFRACTION GRATINGS AND DOES Fundamentals of Diffractive Optical Elements
(EQ 4) Equation 4 Here, m is the diffraction order, T is the grating period, and :
(EQ 5) Equation 5 where n is the refractive index of the grating, d is the blaze thickness, and is the wavelength of light. From the grating equation and these equations, we determine that the diffraction angles and efficiencies are highly dependent on wavelength and blaze thickness. Light from large bandwidth sources will be diffracted into many orders. This is a type of scatter, but it is deterministic scatter since its direction and diffraction or scatter efficiency can be exactly computed with the above equations.
NOTE Deterministic scatter is different from the surface scatter due to random surface variations that are covered in the ASAP technical guide, Scattering. The concept of the linear grating in this context can be used to develop a simple but useful physical model of random scatter. You can consider a random scattering surface as being composed of many randomly oriented linear gratings, whose combined grating spacing and orientation yield the surface scatter pattern.
Phase function Blazed 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 the
16 ASAP Technical Guide cycle measured in radians. phase of the dimension linear the convert A blazed A blazed grating has a tran propagating through the prism. prism. the through propagating unaltered was but the amplitude phaseis that changed, Here, weassumed to, equal A prismhasa function transmittance A(x,y) Generally, as, defined is function transmittance the ASAP. in it model a of diffractive to phasefunction know element the later thatwemust optical We the component. through of propagating andphase a wavefront amplitude learn aperture. Mathematically, inside is one function transmittance whose is perh function transmittance The simplest is the amplitude function and function istheamplitude smittance to, smittance function equal o is a constant term of the ofthe phase function. term is a constant k is the wave number whose physicaleffectto the wavenumber isis DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION Equation 9 Equation 8 Equation 7 Equation 6 Fundamentals of Diffrac (x,y) function into a fraction of an oscillatory ofanoscillatory a fraction into function the aperture and zero outside of the theaperture ofthe and zero outside is the phase func phase is the aps thatofaone-dimensionalaperture ASAP TechnicalASAP Guide tive Optical Elements tive Optical tion of the element. (EQ 9) (EQ 8) (EQ 7) (EQ 6) (EQ 17 . . . . . DIFFRACTION GRATINGS AND DOES Fundamentals of Diffractive Optical Elements
Here is 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 the modulated phase. The common, constant-depth Fresnel lens is an example of a type of kinoform whose modulo operation is much larger than a wavelength of light. Its kinoform is a surface relief profile of the actual physical surface or surface sag, and not the phase change (transmittance function) through the element. See Figure 4.
Phase Function of a Prism
(x)
x
(x) Modulo or
x
Phase Function of a Blazed Grating
Figure 4 Phase functions of a prism and blazed grating
18 ASAP Technical Guide continuous and is a surface of is a continuous and a point inspace,whichth remaining region thelens. around Mathematically, region remaining is, this ex 5The wavefront 10.Figure The geometry for a radiallysymmetric conve lens. distance inamaterial.Therefor agiven it travels time same inthe travelsinavacuum light that is thedistance at is lessthan ray attheedgeoflens wavefront. The magn to the is perpendicular bythesame amount also delayed because it more center at isdelayed thelensthan the the at ray is its of edge, the wavefront We light. Since aray of byexamining phenomena this understand can equivalently additi “slowed” downbythe center of thewavefrontwhenitexits the the of than time in the same amount further has edge of wavefront propagated the collimated light,for example, delays thickness, center thickness is lessthanits lens whose outer edge biconvex lens, as a Apositive such thelens. of thickness to the in proportion the wavefront the phase of time, inthe sense thatit in delays, lensis transformer A thin a phase lens. ofthe otherside on the at thesamepoint compared toitsdiamet lensissmall very whosethickness is Athin one lens. a thin that of simple Wea mo examine position to in a arenow diffractive counterpart’s phasefunction. modulo 2 modulo blazed grating isthediffrac Mathematically, thedifference The transmittance function operation on the of onthe phase function operation er. A ray intersecting one side of a thin lens essentially exits exits er. essentially lens a thin of side one rayintersecting A tive counterpart ofthe refrac of the prism looks simila the prismlooks of is particularcase,th onal refractive lensmateri onal constant phase, the wavefr the constantphase, more atthe center of the lens than atthe edge.The periences a total phase delay due lensandthe phasedelaydue to the periences a total is manifest in the modulo operation. Physically,operation. modulo in the is manifest the e, the edge ray is bent toward the focal point of the of towardthefocalpoint rayis bent e, the edge DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION Fundamentals of Diffrac the the center. ofa path length ray The optical center of the lens, re complicated phase behavior;namely, is related to andinfactthe wavefront itude of the optical path length ofthe pathlength of itude the optical x-plano lens is illustrated in Equation Equation in illustrated lens is x-plano a refractive elementyieldsits e focal point of thelens. ASAP TechnicalASAP Guide r to theblazed grating. al. Since the wavefrontis tive prism.In general, a ont must converge toward tive Optical Elements tive Optical because it is not
19 . . . . . DIFFRACTION GRATINGS AND DOES Fundamentals of Diffractive Optical Elements
(EQ 10) Equation 10
20 ASAP Technical Guide the sphericalsurfacewith a quadra wecanreplace theorem, binomial with the paraxial approximation the By invoking Figure 5 Phase-transforming Modulo Phase of the Quadratic Phase Function Quadratic the Phase of Modulo Quadratic (Parabolic) Phase Function from Lens from Function Phase (Parabolic) Quadratic d modulo 2 lens, resulting quadratic phas Plano-Convex Lens with a Surface Spherical with Lens Plano-Convex DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION tic approximation.This leads to, R Equation 11 Fundamentals of Diffrac phasefunction ASAP TechnicalASAP Guide e function,and tive Optical Elements tive Optical ( ) diffractive
21 . . . . . DIFFRACTION GRATINGS AND DOES Fundamentals of Diffractive Optical Elements
Our phase function then becomes,
Equation 12 We recognize the last term of the previous equations as a form of the lens maker’s equation for a convex-plano lens with focal length f.
(EQ 13) Equation 13 Therefore, the phase function becomes,
(EQ 14) Equation 14 The first term is a constant phase term of little interest, and the second term is a quadratic approximation to a spherical wavefront. It is a quadratic phase function of a focused beam. The lens transforms the planar phase (wavefront) of the incident collimated beam into a converging quadratic phase (wavefront) focusing at the focal point of the lens. If modulo 2 operation is carried out on the phase function, we obtain the diffractive counterpart to the refractive lens. Arbitrary phase functions and their diffractive counterparts are obtained in similar manners, usually with the aid of lens design software. However, the wavefronts and, subsequently, the phase functions of these elements can be much more complicated than the plane waves and quadratic waves we examined for the linear grating and a diffractive lens.
22 ASAP Technical Guide Binary optics Binary levels and producediffractionlevels and above efficiencies of 95%andhigher. efficiency. Two diffraction levelsproduce The oflevels. a thenumber of function 6The in Figure isillustrated This concept in profileoverthe2 arecontinuous intheprevious section The diffractivediscussed kinoforms phase efficiencies. vector diffraction theorytoproperl We apply these equations. properly to impossible usually to resorting recommend across canvaryradically grating spacing problem is that withmorecomplica li simple the for developed relationships diffraction efficiencies are stillgov perhaps a or oflight, beam pencil small equa grating the in used spacing grating to determine, in a local sense, thedirectio beused still can equation grating the phasefunctions, complicated more Even with approximate the continuous profile of the modulo 2 modulo of the profile continuous the approximate or carvingstair-step struct tochiseling is optics similar binary processusedtogenerate The IC manufacturing optics. term andfieldcalled binary diffractive opticalelement’ a to toproduce approximations wereadapted (IC)industry circuit integrated the continuoussurface profiles. ofdiffractivethan those opt larger ofmagnitudes orders many are intervals and depths kinoform lenses, whose Fresnel most Even intervals. small oversuch profile a continuous in manufactured phase depth. However, theseprofiles cannot be ures intoa substrate that s kinoform. The manufacturing technique has led to the tothe hasled technique manufacturing The s kinoform. ical elements,typicallyuse Fortunately,in thelast DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION y accountforthecanon erned by the wavelength and depth depth and thewavelength erned by ted phasesurfaces, theoveralldepthand Fundamentals of Diffrac more levels, the higherdiffraction tion is the local grating spacing wherea ray, intersectsthediffractive element.The near grating. However,near grating. fundamental the n of the n ofthe orders of diffractedthe light. The the element, making the element, diffraction orderis efficiency agiven for efficiencies on theorderof 40%. Eight , on a microscopicscale, ASAP TechnicalASAP Guide phase function. phasefunction. decade the techniques of thedecadeof thetechniques linear approximations to linear approximations ical diffraction tive Optical Elements tive Optical it impracticaland
23 . . . . . DIFFRACTION GRATINGS AND DOES Fundamentals of Diffractive Optical Elements
Phase function of a blazed grating
Multi-level phase structure or binary optic equivalent
A more complicated phase function
Multi-level phase structure or binary optic equivalent
Figure 6 Four-level binary optic representation of some phase functions A series of lithographic masks are made whose total number is determined by the number of etch levels and, therefore, the desired diffraction efficiency. The steps are listed below. The masks are like zero order amplitude gratings, which transmit and reflect a certain 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 the photoresist is exposed to light, it is developed and washed away from the substrate.
24 ASAP Technical Guide 7 6 5 4 3 the substrate. See Figure 7. the substrate.See Figure et have untilyou repeated processis This depth. special to a and th isremoved exposed photoresist The turn, isilluminated. in combination, the and coatedsubstrate the above positioned nextmask is The with photoresist. coated again is substrate and the entire is ofthe removed photoresist rest The process. to the etching impervious phot unexposed The step. abinary leaving photoresist without substrate areas The DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION Fundamentals of Diffrac are then etched to a special depth depth special a to etched then are ched the desired number of levels into oflevels number desired ched the oresist remaining onthesubstrateis oresist remaining at portion of the substrate is etched ofthesubstrate is etched portion at ASAP TechnicalASAP Guide tive Optical Elements tive Optical
25 . . . . . DIFFRACTION GRATINGS AND DOES Fundamentals of Diffractive Optical Elements
Amplitude mask for second level
Substrate recoated with photoresist
Remove exposed photoresist
Etch bare substrate to required depth
Remove remaining photoresist
Resulting Binary Optic (only 4 levels)
Figure 7 Binary optic fabrication process
26 ASAP Technical Guide modulo modulo Design of Multi-level Diffractive Multi-level Design of Opti ofTechnologyInstitute “BinaryOptics Technology: (MIT)titled The Theory and depths are ingreatdetail in discussed The entire binary optics process as well multiple modulo modulo multiple TRANSITION POINTS TRANSITION the blazed grating, thewidthof the maskon The width of themasksforeach step is a f modulo modulo begins step,ortransition the firstetch, mask. Fora circul ofthe previous that is intervals half mask insubsequent Thewidth phase interval. etch depths are shown in Equation 15 and Equation 15. 15andEquation Equation in areshown etch depths phase depth. Subsequent radial transi radial Subsequent phasedepth. phase depth. On the second mask, the transitions occur mask,thetransitions at integer second On the phasedepth. arly symmetric case,like the thin /2 phase depths, and so on. and depths, phase /2 DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION Equation 15 Fundamentals of Diffrac Swanson’s report from theMassachusetts cal Elements”.The at the radialvaluecorrespondingtoa as requiredtransition pointsand etch unction of the type of phasesurface.For type unction ofthe the first etch is half of the modulo 2 of the is half thefirst etch tion points occur at multiples of the ofthe atmultiples occur points tion lens, the width of themaskon the width lens, ASAP TechnicalASAP Guide transition points points and transition tive Optical Elements tive Optical (EQ 15) 27 . . . . . DIFFRACTION GRATINGS AND DOES Fundamentals of Diffractive Optical Elements
ETCH DEPTHS
Equation 16 Even though binary optics are manufactured in discrete steps, their diffractive behavior in ASAP is still simulated by defining the grating lines and using the grating equation. The actual stair-step phase structures are not modeled. The resulting diffraction efficiencies are adjusted according to the number of etch levels or steps.
Holographic optical elements Holographic optical elements are actually the result of a diffractive application of holography. Holography is a process where an interference pattern of a three- dimensional object is produced that contains not only amplitude information about the object but also the phase information. The two-dimensional pictures that we view, including this page and these words, are irradiance maps of objects. An irradiance map contains information only about the amplitude of the object and not its phase. In this sense, we get only half the picture, as the phase information about the object is missing. Perhaps this lead Dennis Gabor to name the phenomena he discovered 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 reference wavefront with a split portion of itself reflected from an object. The hologram is typically recorded on a photographic emulsion. The recording is “played back” by re-illuminating the hologram with the reference beam. The hologram can be a transmission 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 an angle 1 with respect to the plate normal, and another plane wave—a split portion of the first plane wave—incident on the same plate but at an angle 2. The electric field of the two waves is,
28 ASAP Technical Guide If the phasedifference See Figure 8. See Figure k k Figure 8 Thekvectorsand fringes 2 1
2 The k Vectors of 2 Interfering Plane Plane Waves 2 kThe Interfering Vectors of 1 1 - 2 is a constant, the two wavesaremutuallycoherent. the constant, is a DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION Y Y Equation 17 Fundamentals of Diffrac of twointerfering plane waves ASAP TechnicalASAP Guide rgtFringes Bright tive Optical Elements tive Optical Z Z (EQ 17) 29 . . . . . DIFFRACTION GRATINGS AND DOES Fundamentals of Diffractive Optical Elements
We can write the phase relationships in Equation 18at the recording plane, based upon the propagation vectors of the two interfering plane waves.
Equation 18 The last equation is of a line. This implies that the phase function of the two interfering beams is linear. Recall that the phase function of a prism was linear and a 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 between the two waves varies by 2 in between adjacent bright fringes. Furthermore, the fringe spacing in the special case of z and 1-2 = 0 is,
(EQ 19) Equation 19 The irradiance pattern from this superposition is,
30 ASAP Technical Guide NOTE behaviorASAP. in portionof the hologram, as thisisneededto model diffractive its fringe pattern at all, and resemble aspeck at resemble all, pattern and fringe a look like conventional not may In holograms fact, some pattern. interference In general, themore complicatedtheretu like a diffractive lens. Iftwo spherical fringes are spacedjust like a sphericalwaveplane waveand interfere, to theorycanbeextended Our fringe/phase spacing. thefringe in manifest is of hologram, which phase information the for Wehigher orders. information andamplitude phase contain againneed the efficiency ofthe order. of or contrast visibility the in manifest diffraction Theam of orders. angles the hologram ismanifestin the localand gl of grating. The phaseinformation the likeasinusoidal just behaves beam, The interferencepattern,orhologram,when two and diffractiongrating orderfrom a thezeroth justlike thehologram, through reference beam passing therefe with illuminated is hologram this that afringepatternexistsateverypoi a arestraight-lined fringes arethat the irradian results The important components. cosine termwhose argument is the vectors results in an additional component ofthe polarization The dotproduct first orders, because the hologrambehaves However,optics. zerothand the only wesimulate thecaseof plane waves, in two ofdiffraction gratings and binary the behavior weusedtomodel that equation thesamegrating with simulated can be holograms that imply above Our results Inthistechnicalare guide,we interested the onlyin phase nd equally spaced. Thefringes those found in a Fresnelwhich behaves in phaseplate, those found additional first-order beams. first-order additional DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION angle between the linear polarization Equation 20 Fundamentals of Diffrac nt where the twoplanewavesoverlap.When where nt waves interfere, hyperbolic fringes result. fringes hyperbolic interfere, waves the fringes, and determines thediffraction anddetermines the fringes, plitude information of the hologram is of hologram information the plitude obal fringe spacing, and determines the and determines spacing, obal fringe rn wavefront, the mo the wavefront, rn rence beam,wesee part of the original ce pattern varies cosinusoidally, but the le (acoherent scatter le pattern pattern). circular fringes result. Thecircular morecomplicatedfringe patterns. If a like a sinusoidal gr like a re-illuminated with the reference with the re-illuminated ASAP TechnicalASAP Guide are also non-localized in are alsonon-localized tive Optical Elements tive Optical re complicatedthe ating anddoes not (EQ 20) 31 . . . . . DIFFRACTION GRATINGS AND DOES Fundamentals of Diffractive Optical Elements
Lens design codes commonly use two point sources to generate holographic optical elements, which in turn are used as diffractive lenses or corrective elements. The first-order beams of a hologram are also referred to as side bands. In complicated holographic pictures, one of the side bands contains the original amplitude distribution, but a negative phase term. This results in the famous inside-out image. The other side band contains the original amplitude distribution and the normal phase term. Its image accurately reproduces the object as it appeared during recording. Holographic optical elements are found in a large number of different types of optical systems, such as scanners and heads-up displays (HUDs). In one sense, interference coatings might be considered as holographic optical elements operating only in the zeroth order.
Multiple exposure holograms Multiple exposure holograms are a single hologram whose recording media contain more than one exposed holographic fringe pattern. The most common type of multiple exposure hologram is the double-exposure hologram. One type of double-exposure hologram is a composite hologram made of an unperturbed object and its resulting fringe pattern, and a perturbed object and its resulting fringe pattern. The composite fringe pattern actually represents the difference in perturbation between the unperturbed object and perturbed object. The perturbation might be, for example, the displacement (vibration) or distortion of an object from its normal state. A multiple exposure hologram exposed over a long period of time is called a time- averaged hologram. Its interference pattern is actually a superposition of many interference patterns. The resultant interference pattern is really a standing wave whose contours represent areas of constant perturbation. In most instances, the perturbation is a vibration. Multiple exposure holograms are not volume holograms.
Volume holograms The interference pattern produced by a hologram is non-localized. This means that it exists everywhere in space. Our previous example of two interfering plane waves is an example of non-localized fringes. In other words, the interference pattern produced via holography is three-dimensional. In this respect, it is a three-
32 ASAP Technical Guide A volume hologram has a thickness is athickness that has hologram A volume interference filter. diffraction It can al grating. dimensional the incident wavelength. Similarly, wavelength. the incident you Because ofthis property, you canstore only a specific wavelength is diffracted Volume thediffractionand wavelength words, de-couple angle. In other holograms order.the first effici Infact, isdiffracted light themore hologram, into the volume The thicker pattern. fringe tens of microns. The larger is thickness orderof the perhapson thin, very is still it then, Even counterpart. holographic thin volume and surface hol volume andsurface assume technique Thissimulation 1969). from, forexample, However, theysometimes include approxim diffraction efficiencies. Mostotherlens adjusted appropriately with holograms as thin holograms volume ASAP simulates model. coating this with option COATING MODELS However, wemustset up appropriate di but holograms, volume simulate ASAP can hologr volume sense,the this li is atspecialwavelengths Only wavelength. not accountedfor, thereplay Volume su arehighly holograms achieve thesameeffect. Kogelnik’s Systems’ (Bell theory option inASAP, and specifically ograms isthebehavior of th encies of100% are achievable. wavelength mustbeshorter than the recording am actslike a spectralfilter. sceptible to shrinkage.Th sceptible DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION Fundamentals of Diffrac at a specificangle in the hologram. many hologramsatone design codes also use this technique. technique. this use also codes design needed to recordth needed ffraction efficiency can changethe angl so be regarded as a three-dimensional a three-dimensional as regarded be so s that the primarydifference between much greater than th much greaterthan ated diffraction effi it does so using the grating equation. equation. grating sousing the it does ght diffracted atapa e diffraction efficiencies. ASAP TechnicalASAP Guide Technical Journal usetheinterface diffract erefore, if shrinkage is tive Optical Elements tive Optical models with the the with models e three-dimensional e three-dimensional e of incidenceto e ciency calculations time. Justchange e thickness of its e thickness rticular angle. In angle. rticular , 48,2909,
33 . . . . . DIFFRACTION GRATINGS AND DOES Simulating Diffractive Optical Elements in ASAP
SIMULATING DIFFRACTIVE OPTICAL .ELEMENTS ...... IN. . . .ASAP ...... ASAP simulates diffractive optical elements (DOEs) by using the phase function of the DOE to generate a series of grating lines. Therefore, as a first step in simulating DOEs in ASAP, you must know the mathematical form of the phase function of the DOE or the grating line function. This information is easily obtained by knowing the line spacing for simple DOEs, such as linear and circular phase diffraction gratings. More complicated phase functions are usually obtained from lens design codes in the form of polynomial equations. ASAP automatically replicates grating lines from information it obtains from the phase function. In effect, ASAP automatically performs the modulo 2 phase operation on the phase function. From the preceding section, we know that a grating line occurs whenever the phase function goes through 2 radians. The resultant replication is then associated with the base entity of an object. After intersecting the base object, ASAP computes the local grating spacing, and applies the 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 or USERFUNC surface commands. ASAP surfaces must be modified to convert the phase function into a modulo 2 phase function and, ultimately, the grating line representation. This is done with the surface modifier, MULTIPLE (see the section, “MULTIPLE command”). MULTIPLE creates multiple sheets of a phase function to simulate the grating lines. The replicated grating line spacing, originally obtained from the phase function, is then assigned to an object’s interface with the INTERFACE command. The phase surfaces of an optical field exiting a DOE may exhibit 2 step discontinuities. Although the wavefront shows the phase steps, SPREAD and FIELD removes the discontinuities in the process of synthesizing the optical field.
34 ASAP Technical Guide MULTIPLE command optical element. MULTIPLE different manner than theASAP surface modifier, objects. Itcreates multiplesurfaces, but surfaces,surface intomultipleparallel appropriate surfacedefinition.The f n where is, The syntax system geometries. mu elements thancreatingfor replaces itwith, mathematical manner. If the orig The surface isconverted into multiple EXPONENT p x yz d The ' is the number ofsheetstobegenerated, isthenumber isthedistance between original and first sheets, is an additive constant to the original function, tothe original constant isanadditive MULTIPLE is an arbitrary pointon isan surfacemodifierto cr is the exponent to which sheet number israised. number sheet towhich the exponent is commandisa MULTIPLE theoriginalsurface, and is more often used for modeling diffractive usedfor optical often modeling ismore ltiple parallelsurfacesthat SURFACE inal surface isgivenby f(x,y,z), DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION Simulating Diffractive Optical Elements in ASAP eate arbitrarygrating li MULTIPLE parallel surfaces which may beusedtocreate repetitive modifier.come afteran Therefore,itmust in a fundamentallyand mathematically command converts a designated command convertsa ARRAY ASAP TechnicalASAP Guide in the following are used purely foroptical are nes ofadiffractive . We. the willalso use MULTIPLE
35 . . . . . DIFFRACTION GRATINGS AND DOES Simulating Diffractive Optical Elements in ASAP
If 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 surface in the case of a diffractive optical element because of its special reference on the INTERFACE command where j specifies the diffractive order numbers. Depending upon the nature of the surface, the value of the exponent p, and the additive 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 evenly spaced 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 to simulate a diffractive optical element? It is an equation describing the grating line spacing. You can derive this equation from the phase function, or you can compute it directly. To derive f(x,y,z) from the phase function, we must understand that it is a 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 incident wave is altered by an optical element. A modulo 2 phase function is the phase function of a diffractive optical element.
NOTE The function f(x,y,z) is the function obtained when the phase function of a given order m is divided by 2m uniquely specifies the grating line equation. The grating line spacing is a physical spacing, which does not change for other orders. Other orders are diffracted according to this spacing. In other words, a physical grating line is obtained whenever a specific phase goes though 2. In general, the function f(x,y,z) on the MULTIPLE command, as well as its other terms, can be obtained from an arbitrary phase function by dividing the phase function by 2m or where m is the order number corresponding to that phase function. We can mathematically demonstrate this relationship by re-examining Figure 2 “Transmission and reflection gratings” on page 13. The local phase change within a specific period in one dimension is,
36 ASAP Technical Guide grating lines per unit length. perunit grating lines and ofposition, a function Note here that But the difference in sine terms isequal to d(y) indicates that the local spatial pe indicateslocal thatthe n(y) is a spatial frequency denoting the number of local thenumber denoting isa frequency spatial DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION Simulating Diffractive Optical Elements in ASAP Equation 22 Equation 21 riod may,riod as change ingeneral, ASAP TechnicalASAP Guide (EQ 22) (EQ 21) 37 . . . . . DIFFRACTION GRATINGS AND DOES Simulating Diffractive Optical Elements in ASAP
We can equate the above equations, rearrange, and integrate to yield,
(EQ 23) Equation 23 Our results can be extended to rotationally symmetric systems or to three dimensions 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 MULITIPLE command is not needed normally. If you choose to, you can derive the grating line equation directly. In this case, you must 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) on the original surface to the first sheet is d. It does this by evaluating the expression,
(EQ 24) Equation 24 This second expression is most useful for diffractive optical elements, such as linear gratings that have equally spaced grating lines. You are not specifying the phase function, but rather the direction in which parallel replications of the original surface are generated. At each distance d along the original surface’s normal,
38 ASAP Technical Guide command, when thegrating lines areequallyspaced. e of theabove similarity Note the are spaceda distance spaced atadistance options of options the Before weexamineseveral DOEexamples surface. Inthecaseof a plane, resulting pl f(x,y,z), of thegradient original from the of computed version areplicated get you to anobject. INTERFACE d apart.Inthecase of d apart. This special spaci This special apart. commandfor assigning a di DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION quation to thespecialcase of the Simulating Diffractive Optical Elements in ASAP Equation 25 cylinders,resulting anes arestacked on , we must learnabout the necessary ng is the grating line line spacing. ng isthegrating ASAP TechnicalASAP Guide ffractiveinterface optical top ofeachother but concentric cylinders MULTIPLE (EQ 25)
39 . . . . . DIFFRACTION GRATINGS AND DOES Simulating Diffractive Optical Elements in ASAP
INTERFACE command for DOEs The INTERFACE command assigns specular reflection and transmission properties to the interface of an object. We also use the INTERFACE command to assign diffractive optical properties to it. The INTERFACE command has several different syntaxes. We are most interested in those that allow us to assign diffractive properties to an interface, which are listed 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, and coat coat' are the names of a given coating property. The first part of the INTERFACE command, preceding any of the DIFFRACT options, is exactly the same as you have seen before. You can reference a coating from the coating database and assign it to the object, as well as refractive indices on either side of the geometry of the object. If the object is an optical boundary through which rays are to be traced, the optical properties of the interface must be specified using the INTERFACE command after the definition command for that object. If an INTERFACE command does not follow an edge or surface object definition, the surface is assumed to be perfectly absorbing, and all rays reaching the surface are trapped there. If your interface has non-zero reflection and transmission coefficients and in addition is a diffractive interface, you will see reflected and transmitted diffraction orders. When you use the first form of the INTERFACE DIFFRACT syntax, grating lines are created by the intersection of the object surface with the different sheets of a MULTIPLE surface i. For example, a ruled linear phase grating is created if i is a plane, 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 is
40 ASAP Technical Guide efficiencies. Ifa di the vect have equation anddoesnot the diffraction efficiencies the diffractioncompute does not efficiencies has no bearing on this application.Specification of the sign of number specification. The asthe last beinthesameunits thespacing is assumed to negative, The simplest coating model is model the coating The simplest complex amplitudes. a it isname “coat”, syntax is, syntax from the ASAP scatter model set. diffractionefficiency discrete envelopeor diffraction efficiencies ataninterface.In for polarization properties the diffractedASAP generates rays/beams for the diffraction given by order numbers phase function. of definition the ASAPand in units the with COATING MODELS efficiencies; thatis, reflection andtransmission coefficients ar atintermedia values the to obtain values efficienciesof asafunction you to specify a table of reflection and MODEL However, of form thesecond with diffraction efficiency of a given order can compute diffractionASAP cannot effici orders. j 's withrelativeefficiencies give can beusedto specify the angula ffraction efficiency, COATING PROPERTY allows you to to specify th allows you S and P reflectionandtransmission coefficients or of sheets entered on the of sheetsenteredonthe as a function of order. of a function as AS wavelength. ASAP linearly ASAP wavelength. DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION Simulating Diffractive Optical Elements in ASAP COATING MODELS COATING PROPERTIES INTERFACE or capability tocompute diffraction n by the corresponding positive positive corresponding the by n e transmission coefficients ordiffraction encies onthefly. Furthermore,the , is entered as a negative number or as a or as a negative number is enteredas , te wavelengths. In our current case, the case, our current te wavelengths. In r variation of the diffraction order our current case, we use it tospecifycurrent case, weuse the our diffraction efficiencies.uses models It e diffraction effici change as afunction of incident angle. f(x,y,z), your from is calculated which that possibly contains polychromatic polychromatic contains thatpossibly as a function of order. asafunction You enter must e angular,e dispersion,and
DIFFRACT MULTIPLE ASAP TechnicalASAP Guide AP uses the scalar grating has two syntax forms.Its has syntax two interpolates betweenthese encies of specific , a named , anamed . This model allows allows . Thismodel i surface command surface must be consistent be must consistent WAVELENGTH e 's. ASAP ASAP 's. COATING 41
. . . . . DIFFRACTION GRATINGS AND DOES Simulating Diffractive Optical Elements in ASAP
Here, k is the starting coating number, r r' r" ... are the real energy (or complex amplitude) reflectance efficiencies, and t t' t" ... are the real energy (or complex amplitude) transmittance efficiencies. Starting with coating number k, coatings are entered with real energy (or complex amplitude) reflectance r and transmittance t diffraction efficiencies for the zeroth order. 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 using previously defined scatter MODELS where i is the model for reflected S polarization, j is the model for reflected P polarization, m is the model for transmitted S polarization, and n is the model for transmitted P polarization. We can specify different reflection and transmission diffraction efficiencies for the zeroth order for different wavelengths, as specified on the last WAVELENGTH command. ASAP uses the normalized scatter model data and the reflection and transmission diffraction efficiencies for the zeroth order to account for the angularly dependent nature of the diffraction grating. If you are not using a functional scatter model such as USERBSDF, but rather data from BSDFDATA, ASAP linearly interpolates in logarithmic amplitude space to determine reflection and transmission diffraction efficiencies at other orders than those specified in the data set. However, with this syntax the same angularly dependent polarization models are used for all sources at different wavelengths. Therefore, this form of COATING MODELS should be used for diffraction gratings that are non- dispersive. Alternatively, in the second syntax, groups of six numbers can be entered to account for grating dispersion and volume holograms. Each group corresponds to a wavelength entered on the last multiple WAVELENGTH(S) command. The first number is the reflection diffraction efficiency of the zeroth order corresponding to the first wavelength on the last multiple WAVELENGTH command. The next two numbers are the angularly dependent S and P reflection diffraction efficiency models, as discussed previously, at that wavelength. The fourth number is the
42 ASAP Technical Guide amplitude, energy coefficients ofthe where BSDF is the bi-directional bi-directional is the where BSDF In theseequations, be, angle actual di asin Equation 26.The example, in defined those between wavelengths diffr transmission and The reflection is unavailable. function data Again, angular diffractiontransmission effi diffraction transmission S andP efficiency dependent by its angularly followed a and a wavelength and a wavelength r, r r, is linearlyinterpolated in lo ’ , t,t b diffraction efficiencies. ’ between the first two ciency models, and so on for different andsoonfor ciency models, wavelengths. are thecomplex amplitudesor DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION scattering distribution function. distribution scattering Simulating Diffractive Optical Elements in ASAP Equation 27 Equation 26 action efficiencies forsources with COATING MODELS ffraction efficiencies at an incidence garithmic amplitudespacewhena WAVELENGTH f(i,a) ASAP TechnicalASAP Guide isthe normalized angular square rootsofthereal areinterpolated,for ( S ) w w' (EQ 27) (EQ 26) would would 43 . . . . . DIFFRACTION GRATINGS AND DOES Simulating Diffractive Optical Elements in ASAP
TIP The key to implementing a COATING MODEL lies in understanding how to set up the data. If your data is not in a functional form, such as that which can be used with USERBSDF or another functional ASAP scatter model, it will be in the form of reflection and transmission diffraction efficiencies as a function of incident angle. The most commonly used format for entering data in this form is the BSDFDATA scatter model format, since it allows you to enter this type of data directly. ASAP linearly interpolates in logarithmic amplitude space to determine coefficients between those entered in BSDFDATA. Regardless of the scatter model you choose, you must first run the MODELS keyword command, just like you would with COATING 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 angles a b [a' b' ...] are spherical ANGLES from and around normal for other orders; f [f' ...] are the diffraction efficiencies; and ao' bo' are the second specular direction. The data entered on the lines with two entries only, indicated by o’s following the letters a, b and so on, defines one incident specular direction for the sets of triplets to follow on the next lines. In the case of a COATING MODEL, the first two numbers 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 they specify the angle from the normal. The bo 's and b 's can be set to 0 as they are the angles around the normal in plane data. The f parameters are the diffraction
44 ASAP Technical Guide specifically, to theotherorders to diffractiona or, andefficiencies envelope angles corresponding order once. For example, For order once. the of form third the using by Finally, modeled be can holograms exposure multiple functions. scattering deterministic discrete, produce gratings diffraction efficiencies. The data. diffraction orderefficiencies directly to specific correspond only this you have if efficiencywavelength function atthis of diffraction enter ofanenvelope the numbers you are discrete numbers the ot simulating triplets sets of multiple specified byasingle set of triplets. Ho like a diffraction grating operat What then aretheadditional andsoon. triplets, itsassociated with direction specular defines the efficiencies.next two numbers thathasonly Thenextline INTERFACE DIFFRACT COATING MODELS USERBSDF / USERBSDF a ' ive syntax. However, syntax. ive the zeroth only you specify b ofadiffraction grating. ing only in the zeroth order ( order zeroth the in only ing ' f DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION Simulating Diffractive Optical Elements in ASAP ' , a is a general scatter model anddiffraction model isa scatter general her diffractionorder efficiencies. The " and angle of incidence. Thenumbers can wever, a diffraction can grating have b technique is anappr " b "
terms, and so on? They are the and so terms, ASAP TechnicalASAP Guide In one sense, a coatingis opriate waymodel to a, b, f ), which is
45 . . . . . DIFFRACTION GRATINGS AND DOES Simulating Diffractive Optical Elements in ASAP
DOE examples The following examples illustrate how to set up various diffraction gratings, binary optical elements, and holograms. The examples move from the simple to the complex and demonstrate various forms of the MULTIPLE command and the INTERFACE…DIFFRACT command.
LINEAR PHASE GRATING In our first example, we will simulate a simple linear phase grating with both forms of the MULTIPLE command. When we use the first form of the multiple command, we must know the mathematical description of the phase surface of our linear grating. We know that the grating spacing is d and that the grating frequency is 1/d. But we also have, from our previous work, the relationship between the grating frequency, the phase function, and the grating line equation,
(EQ 28) Equation 28 Substituting 1/d in the above equation for the grating frequency yields the following relationships,
(EQ 29) Equation 29 Clearly, f(y)=y and f’=d. We also could have specified the coefficient of y as 1/d and set f’ equal to 1. ASAP now has all the necessary information to compute the local grating line spacing for a ray incident on any point of the base surface. It then uses the grating equation to generated diffracted rays according to the orders specified on the INTERFACE...DIFFRACT command. ASAP is splitting rays at the diffractive interface. Some of these rays will become child rays. If there are more diffraction gratings in the optical system, you may have 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 surface is both reflecting and tr reflecting surface isboth PROPERTIES from the independent surface of intersectionisa surfaceperpendi the Note how present. ordersaresimulated, first and plus/minus GENERAL database. See Example 1. SeeExample database. MULTIPLE GENERAL command isusedtodefine command ansmitting, as defined in the DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION Simulating Diffractive Optical Elements in ASAP even though more orders are physically arephysically orders more even though cular to the z axis. Its geometryis surface. Finally, notethatthe base ASAP TechnicalASAP Guide f(x,y,z) , and how the base the how , and COATING
47 . . . . . DIFFRACTION GRATINGS AND DOES Simulating Diffractive Optical Elements in ASAP
!!ASAP EXAMPLE OF A PHASE DIFFRACTION GRATING !!USING THE FIRST FORM OF THE MULTIPLE COMMAND SYSTEM NEW RESET
$DATIM OFF OFF
PI=4*ATAN(1) LAMBDA=0.025 D=0.1
COATING PROPERTIES 0.4 0.6 'DOE'
SURFACE GENERAL 0 0 0 Y 1 MULTIPLE 1 (D) PLANE Z 0 RECTANGLE 1 grating base surface OBJECT 'LINEAR_GRATING' INTERFACE COATING DOE AIR AIR DIFFRACT 0.2 -1st order 0.25, 0th 0.5, 1st 0.25
SURFACE PLANE Z 10 RECT 1 detector plane OBJECT 'DETECTOR' ROTATE X ASIN[.25] 0 0
BEAMS INCOHERENT GEOMETRIC WAVELENGTH (LAMBDA)
GRID ELLIPTIC Z -1 -4@1 1 11 SOURCE DIRECTION 0 0 1
WINDOW Y Z WINDOW 1.4 TITLE 'GRATING W/-1,0,1st ORDERS IN REFLECTION AND TRANSMISSION' MISSED 10
48 ASAP Technical Guide RETURN !!$VIEW 0-2 -2.25,0.25,.25 0,1'-1storder' 01.75 -2.25,0.25, .250,1'+1storder' 01 8.75, 0.25,.25 0,1'0thorder' 02.8 4.5,0.25,.25 0,1'+1storder' 0-2.6 10,0.25,.25 0,1'-1storder' TEXT TRACE PLOT PLOT FACETS 3OVERLAY Output thePlotViewerfrom Output Figure 9. isshown in Figure 9 Example 1: Diffracti Example 1. Linear phasExample 1.Linear the first form of the the firstformof e diffraction gratingwi on orders fromreflection a an DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION Simulating Diffractive Optical Elements in ASAP MULTIPLE th the first form of the firstform th command ASAP TechnicalASAP Guide d transmission grating, using d MULTIPLE
49 . . . . . DIFFRACTION GRATINGS AND DOES Simulating Diffractive Optical Elements in ASAP
We can also simulate the same diffraction grating with the second form of the MULTIPLE command. Recall that in this second form you specify a distance between the original function or surface and the first replica. ASAP automatically computes the f’ value from,
(EQ 30) Equation 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 are generated. We know that the grating lines are perpendicular to the y axis, so we can use a plane whose normal is collinear with this axis to denote the gradient of f(x,y,z). The MULTIPLE terms then include the grating spacing, which is just the distance between the original function or surface and the first replica, and a point on the original function. Our plane, whose normal is collinear with the y axis, is replicated to form a series of parallel planes whose spacing is the grating line spacing.
The second form of the MULTIPLE syntax illustrates the ASAP syntax for this case. See Example 2.
50 ASAP Technical Guide PLOT FACETS 3 OVERLAY MISSED 10 TITLE 'GRATING W/-1,0,1st ORDERS USING2ndFORM OFMULTIPLE' WINDOW 1.4 WINDOW YZ SOURCE DIRECTION01 GRID ELLIPTIC Z-1-4@11 11 WAVELENGTH 0.025 GEOMETRIC BEAMS INCOHERENT ROTATE XASIN[.25]0 OBJECT 'DETECTOR' PLANE Z10RECT1detector plane SURFACE OBJECT 'LINEAR_GRATING' PLANE Z0RECT1grating basesurface MULTIPLE 1,gratingspacing 0.1,point0 0 PLANE Y0gratinglinegenerating surface SURFACE 0.4 0.6 'DOE' COATING PROPERTIES $DATIM OFF RESET SYSTEM NEW !!USING THESECONDFORMOF THEMULTIPLECOMMAND !!ASAP EXAMPLEOFAPHASE DIFFRACTION GRATING 0.25 INTERFACE COATING DOE AIR DIFFRACT 0.2 -1storder 0.25, 0th 0.5,1st DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION Simulating Diffractive Optical Elements in ASAP ASAP TechnicalASAP Guide
51 . . . . . DIFFRACTION GRATINGS AND DOES Simulating Diffractive Optical Elements in ASAP
TRACE PLOT TEXT 0 -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'
RETURN Example 2. Linear phase diffraction grating with the second form of MULTIPLE Again, only the zero and plus/minus first orders are simulated, even though more orders are physically present. See Figure 10.
Figure 10 Example 2: Diffraction orders from a reflection and transmission grating, using the second form of the MULTIPLE command
52 ASAP Technical Guide functional form easily implemented with the the with implemented easily form functional However, a line func grating wenow have MULTIPLE where cont in Fourierspace edge structure. orphase edge a that of smooth the Fourieraspectsof diffraction atash first orde and zero diffracted the into only is light Asa consequence, discrete steps. instead of phase structure asinusoid is that its is the exception phase phase like with grating a linear grating, A sinusoidal SINUSOIDAL PHASEGRATING represent this grating line line fu this grating represent form, the of is function phase the grating, sinusoidal a In the firstorder. sinusoidal phase structureand, diffraction in the aremissing subsequent These frequency orders. components d isagain the gratingperiod. Theorder command is then, ribute to the higher-order nction?with answer is The therefore, soarethediff DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION Simulating Diffractive Optical Elements in ASAP Equation 32 Equation 31 arp edge or phasestructure comparedto Higher frequency components of a sharp ofasharp components Higher frequency rs. This can be understood by examining examining by beunderstood can rs. This tion that is not a polynomial function, a function, is apolynomial that not tion GENERAL m is one. The function f(x,y,z) isone. function The the of diffraction anglesand ASAP TechnicalASAP Guide raction orders higher than than higher orders raction command.How can we USERFUNC . (EQ 32) (EQ 31) 53 . . . . . DIFFRACTION GRATINGS AND DOES Simulating Diffractive Optical Elements in ASAP
The 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 or cosine, or it can be one created with the internally defined macro $FCN. Its syntax is as follows,
Here x y z are the global coordinates of the reference point; fcn is a user-defined function; and c c' c" ... are user-defined coefficients of the function. USERFUNC specifies a user-defined function with reference point (x,y,z) and double-precision coefficients. You can define its value and gradient at any point in the macro, $FCN named fcn or the Fortran function USERFUNC. If the function is continuous in both value and gradient everywhere in space, there are no restrictions on the use of this function in ASAP, except possibly the application of non-orthogonal transformations to it; that is, SCALE or SKEW or non-isotropic SCALE. If the fcn is specified, the local (x,y,z) coordinates are passed in the _1, _2, and _3 registers. You can also define other parameters of the function and set them, with up to 63 coefficients c c' c" ..., in the registers _4, _5,. . ._66. If four or more values are returned, the last four entries of the function that was run must be the functional value and its gradient vector. For example, a sphere of radius 10 centered about the reference point is done as follows,
Otherwise, the default USERFUNC is an aspheric conicoid. You can examine the other parameters of the default user function in the ASAP online help.
54 ASAP Technical Guide gradient. The gradient of our grating line function is, line function our of grating Thegradient gradient. These values areenteredon USERFUNC In our example,weuse have set it equal 1/2 to have setitequal 11 Figure diffractionin areshown orders We chose , we must enter the functional of form functional the enter , wemust USERFUNC . USERFUNC DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION Simulating Diffractive Optical Elements in ASAP Equation 33 to define the sinusoidal phase function. On function. phase thesinusoidal define to as illustrated in Example 3. The Example 3. as illustrated in the grating line function and its its and function line grating the f ASAP TechnicalASAP Guide ’ to be 1, but we could also could we but 1, be to (EQ 33) 55 . . . . . DIFFRACTION GRATINGS AND DOES Simulating Diffractive Optical Elements in ASAP
!!ASAP EXAMPLE OF A PHASE DIFFRACTION GRATING !!USING THE FIRST FORM OF THE MULTIPLE COMMAND SYSTEM NEW RESET
$DATIM OFF OFF
PI=4*ATAN(1) LAMBDA=0.025 D=0.1
COATING PROPERTIES 0.4 0.6 'DOE'
$FCN GRATING Y=_2, SIN(2*PI*Y/D)/(2*PI) 0, (1/D)*COS(2*PI*Y/D) 0
SURFACE USERFUNC 0 0 0 GRATING MULTIPLE 1 1 PLANE Z 0 RECT 1 grating base surface OBJECT 'SINUSOIDAL_GRATING' INTERFACE COATING DOE AIR AIR DIFFRACT 0.2 -1st order 0.25, 0th 0.5, 1st 0.25
SURFACE PLANE Z 10 RECT 4 detector plane OBJECT 'DETECTOR'
BEAMS INCOHERENT GEOMETRIC WAVELENGTH (LAMBDA)
GRID ELLIPTIC Z -1 -4@1 1001 1001 RANDOM 1 SOURCE DIRECTION 0 0 1
WINDOW Y Z WINDOW 1.4 TITLE 'SINUSODIAL GRATING W/-1,0,1st ORDERS IN REFLECTION AND TRANSMISSION'
56 ASAP Technical Guide RETURN PICTURE AVERAGE DISPLAY SPOTS POSITION EVERY137 CONSIDER ONLYDETECTOR PIXELS 101 WINDOW XY 0-2 -2.25,0.25,.25 0,1'-1storder' 01.75 -2.25,0.25, .250,1'+1storder' 0.2 10.1,0.25,.25 0,1'0thorder' 02.8 4.5,0.25,.25 0,1'+1storder' 0-2.8 4.5,0.25,.25 0,1'-1storder' TRACE PLOT 11111TEXT PLOT FACETS 3OVERLAY MISSED 10 Example 3. Sinusoi dal phasediffract MULTIPLE ion gratingwith DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION Simulating Diffractive Optical Elements in ASAP
command USERFUNC ASAP TechnicalASAP Guide and the first and formofthe
57 . . . . . DIFFRACTION GRATINGS AND DOES Simulating Diffractive Optical Elements in ASAP
The following illustrations are all part of Figure 11.
58 ASAP Technical Guide Figure 11 Example3: Diffraction specifically relatedto Recall thatthephasesurface and thegrat obtain the phase surface andf(x,y,z). to frequency theradial can integrate we a similar technique, Following is 1/d. This grating,therefore,has The circulardiffraction grating is alinear CIRCULAR DIFFRACTIONGRATING orders from asinusoida orders first form of the the radial frequency by MULTIPLE l reflection and transm circularly concentric grating lines. Its radial frequency frequency Its radial lines. grating concentric circularly DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION Simulating Diffractive Optical Elements in ASAP command Equation 34 ing line equation are related, and equation arerelated, ing line grating bent around anaxis of around symmetry.bent grating ission grating, using ission grating,
ASAP TechnicalASAP Guide USERFUNC and the (EQ 34) 59 . . . . . DIFFRACTION GRATINGS AND DOES Simulating Diffractive Optical Elements in ASAP
Substituting 1/d in the above equation for the grating frequency yields the following relationships.
(EQ 35) Equation 35
(EQ 36) Equation 36 and f’=d. However, our relationship for the grating line equation contains a square root. Therefore, it is best to use USERFUNC, since a square root cannot be directly entered on the GENERAL command, and then MULTIPLEd. Remember that when you use USERFUNC, you should configure it to return four values, the first being the value of the function and the last three its gradient. In our case, the function and its gradient are,
60 ASAP Technical Guide diffracted first order for the circular grating is illustrated in 12. diffracted in Figure illustrated is grating order first for the circular These values areenteredinthe
USERFUNC DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION Simulating Diffractive Optical Elements in ASAP Equation 37 command as shown in Example4. The ASAP TechnicalASAP Guide (EQ 37) 61 . . . . . DIFFRACTION GRATINGS AND DOES Simulating Diffractive Optical Elements in ASAP
!!ASAP EXAMPLE OF A CIRCULAR DIFFRACTION GRATING !!USING THE FIRST FORM OF THE MULTIPLE COMMAND SYSTEM NEW RESET
$DATIM OFF OFF
PI=4*ATAN(1) D=0.058476 grating line spacing
$FCN CIRC SQRT(_1^2+_2^2), _1/SQRT(_1^2+_2^2), _2/SQRT(_1^2+_2^2), 0
COATING PROPERTIES 0 1 'TRANS'
BEAMS INCOHERENT GEOMETRIC WAVELENGTH 0.02
SYSRAY { 2 RAYS 0 GRID ELLIPTIC Z -1 -4@1 #1 #2 SOURCE DIRECTION 0 0 1 } ENTER NUMBER OF RAYS ALONG X-AXIS ENTER NUMBER OF RAYS ALONG Y-AXIS
SURFACE USERFUNC 0 0 0 CIRC MULTIPLE 1 (D) PLANE Z 0 ELLIP 1 OBJECT 'CIRCULAR_GRATING' INTERFACE COATING TRANS AIR AIR DIFFRACT 0.2 -1
SURFACE PLANE Z 4 RECT 1.5
62 ASAP Technical Guide !!$VIEW RETURN TRACE PLOT PLOT FACETS 3OVERLAY OBLIQUE WINDOW 1.5 WINDOW Y-22Z-13 $SYSRAY 511 $IO VECTOR REWIND RETURN TRACE PLOT PLOT FACETS 90OVERLAY WINDOW Y-22Z-13 $SYSRAY 111 MISSED ARROW3.5 TITLE 'CIRCULAR GRATING| 1st FORMOFMULTIPLE' OBJECT 'DETECTOR' Example 4. phasCircular e diffraction gratingwi DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION Simulating Diffractive Optical Elements in ASAP th the first form of the firstform th ASAP TechnicalASAP Guide MULTIPLE
63 . . . . . DIFFRACTION GRATINGS AND DOES Simulating Diffractive Optical Elements in ASAP
Figure 12 Example 4: First diffraction order from a circular phase grating, using USERFUNC and the first form of MULTIPLE
64 ASAP Technical Guide Therefore, spacing between twoadjacent grating circles as, d They allalsousedthe default line function. the grating to generate function phase the ormodifying function weregene point to this examples All our and then we must identify we identify and then must We must nowputthis equati which is also a case where the case where is also a which We directly, function line grating the forderiving a demonstrate technique now squaring both sides of the previous equation we get, we equation previous of the sides both squaring We can directly writetheequati A circulardiffraction gratin We one, equal to default. the again return is againthegratingcircle spacing.This isa constant. the appropriate terms of the the appropriatetermsof g has concentric,equallyspaced grating circles (lines). on intoa form that ASAP cansimulate as asurface, EXPONENT EXPONENT on of thesegrating lines as, DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION Simulating Diffractive Optical Elements in ASAP Equation 40 Equation 39 Equation 38 rated by integrating the radial frequency frequency theradial rated by integrating can be verified by computing the grating the grating computing by can beverified to the circular grating for this example. grating forthis to thecircular on the onthe on the on MULTIPLE MULTIPLE ASAP TechnicalASAP Guide MULTIPLE command. command is not command. By (EQ 40) (EQ 39) (EQ 38) 65 . . . . . DIFFRACTION GRATINGS AND DOES Simulating Diffractive Optical Elements in ASAP
When we compare this with the form of the MULTIPLE command, we see that,
(EQ 41) Equation 41 The MULTIPLE syntax with the EXPONENT option for this case is illustrated in Example 5. Note that the grating spacing d is 0.058476, and the wavelength is 0.02—both are in arbitrary units. We can compute the diffraction angle for this element by using the grating equation, sin(q)=ml/d, and we see that it is 20, the same as in the previous circular grating example.
66 ASAP Technical Guide $IO VECTOR REWIND TRACE PLOT PLOT FACETS 9 0OVERLAY WINDOW Y-2 2 Z-13 $SYSRAY 1 MISSED ARROW3.5 TITLE 'CIRCULAR GRATING| EXPONENTIAL OPTION OF MULTIPLE' ENTER NUMBER OFRAYSALONG X-AXIS> } SOU DIR 01 GRID ELL Z-1-4@1#111 WAVELENGTH 0.02 GEOMETRIC BEAMS INCOHERENT RAYS 0 SYSRAY {1 INTERFACE COATINGTRANS AIR DIFFRACT0.2 -1 OBJECT 'CIRCULAR_GRATING' PLANE Z0ELLIP1 MULTIPLE 0D*DEXPONENT 2 GENERAL 00;X21;Y2 1 SURFACE 01 'TRANS' COATING PROPERTIES D=0.058476 gratinglinespacing $DATIM OFF RESET SYSTEM NEW !!USING THEEXPONENTIALOPTION OFTHEMULTIPLE COMMAND !!ASAP EXAMPLEOFACIRCULAR DIFFRACTIONGRATING DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION Simulating Diffractive Optical Elements in ASAP ASAP TechnicalASAP Guide
67 . . . . . DIFFRACTION GRATINGS AND DOES Simulating Diffractive Optical Elements in ASAP
$SYSRAY 5 WINDOW 1.5 OBLIQUE
PLOT FACETS 9 9 0 OVERLAY TRACE PLOT
RETURN Example 5. Circular phase diffraction grating with the EXPONENT option of the MULTIPLE command The plots for the above script are shown in Figure 13.
68 ASAP Technical Guide Figurediffraction 13Example5:First or equation, the by lines arecreated lines. These grating lines represent2 These grating lines. multiple surfaces whose inte simulateDOEs. Even with more compli the which equation, line grating the determine must the samet complicated DOEs,westilluse We now examineamore complicated diffr MULTIPLE to simple deliberately They weremade haveb theexamples point, all Up to this DOE LENS options of the der from acircularphase grating, using command and the techniques need techniques andthe command MULTIPLE MULTIPLE rsection withthebasesu DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION Simulating Diffractive Optical Elements in ASAP command command, they must be defined by bedefinedby the they must command, phasesteps. Finally, sincethesegrating demonstrate thedifferent forms of the cated phase structures,wewillcreate een simplelinearandcircular gratings. echniqueswe have developed so far. We active element,a DOE lens. With more
GENERAL ed tosimulateDOEelements. ASAP TechnicalASAP Guide MULTIPLE rface specifies thegrating rface and the command uses to uses to command EXPONENTIAL 69 . . . . . (EQ 42) Equation 42 where j and p are integers and f’ is a constant. In general, the process requires knowing the phase function at a particular order m and converting that phase function into the grating line equation by dividing by 2m. These terms can then be related directly to the grating line equation and the constant and exponential terms that are needed in the MULTIPLE command. We will model a DOE lens that produces a simple quadratic phase transformation on a collimated input beam. We have already determined the phase function of this lens in the previous section. Its phase function for the minus first order was,
(EQ 43) Equation 43 Recall that the first term was a constant phase term of little interest, and the second term was a quadratic approximation to a spherical wavefront. The phase function is related to the grating line function,
(EQ 44) Equation 44 Therefore, the grating line function is,
(EQ 45) Equation 45 constant terms can be obtained by rearranging terms as follows. and equation line grating the describing equations valid equally other that Note MULTIPLE is thelens of length focal The paraxial termsas, DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION Simulating Diffractive Optical Elements in ASAP Equation 46 f . We can writef(x,y,z) and the other ASAP TechnicalASAP Guide (EQ 46) 71 . . . . . DIFFRACTION GRATINGS AND DOES Simulating Diffractive Optical Elements in ASAP
(EQ 47) Equation 47 ASAP syntax that is needed to simulate the DOE lens is shown in Example 6. The script is shown in two parts due to its length. Output from this example illustrate the 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 WINDOW XY CONSIDER ONLY DETECTOR TITLE 'F/5 DOE LENSWITHSIMPLE QUADRATICPHASE' PIXELS 255 WINDOW YZ SOURCE DIRECTION01 GRID ELLIPTIC Z-1-4@2301 301RANDOM1 WAVELENGTH (L) GEOMETRIC BEAMS INCOHERENT OBJECT 'DETECTOR' PLANE Z20ELLIPSE3 SURFACE INTERFACE COATINGTRANS AIR DIFFRACTsurface 1,-1storder OBJECT 'DOE' PLANE Z0ELLIPSE2 MULTIPLE n1,fprime2*L*F Y21 !!f(x,y,z) X21 GENERAL 0 SURFACE 01 'TRANS' COATING PROPERTIES L=0.6328E-4 F=20 !!$GUI CHARTS_OFF!!forPlot Viewer $GUI CHARTS_ON !!forChart Viewer $DATIM OFF RESET SYSTEM NEW DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION Simulating Diffractive Optical Elements in ASAP ASAP TechnicalASAP Guide
73 . . . . . DIFFRACTION GRATINGS AND DOES Simulating Diffractive Optical Elements in ASAP
RAYS 0 BEAMS COHERENT DIFFRACT WAVELENGTH (L) PARABASAL 4 WIDTHS 1.6
GRID ELLIPTIC Z -1 [email protected] 101 101 RANDOM 1 SOURCE DIRECTION 0 0 1, 0 SIN[1] COS[1]
CONSIDER ALL WINDOW Y Z !!PLOT FACETS 3 3 0 OVERLAY TRACE PLOT 71
CONSIDER ONLY DETECTOR SELECT ONLY SOURCE 1
FOCUS MOVE
WINDOW Y -.005 .005 X -.005 .005 WINDOW .7 PIXELS 201
SPREAD NORMAL DISPLAY ISOMETRIC 2 'IRRADIANCE DISTRIBUTION (ON-AXIS)' PICTURE 'ON-AXIS SOURCE' RETURN
SELECT ONLY SOURCE 2 FOCUS MOVE WINDOW Y .344 .354 X -.005 .005 WINDOW .7 PIXELS 201
SPREAD NORMAL DISPLAY ISOMETRIC 2 'IRRADIANCE DISTRIBUTION (OFF-AXIS)'
74 ASAP Technical Guide RETURN PICTURE 'OFF-AXISSOURCE' Figurelens 14 .Example6:DOE FigureIrradi Example 6: 15 Example 6. DOE lens input (above) DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION Simulating Diffractive Optical Elements in ASAP ance distribution (on-axis) with simple quadraticphase with ASAP TechnicalASAP Guide
75 . . . . . DIFFRACTION GRATINGS AND DOES Simulating Diffractive Optical Elements in ASAP
Figure 16 Example 6: On-axis source
Figure 17 Example 6: Irradiance distribution (off-axis)
76 ASAP Technical Guide Figure 18 Example6:Display Viewer output, off-axis source DIFFRACTION GRATINGS AND DOES GRATINGSAND DIFFRACTION Simulating Diffractive Optical Elements in ASAP ASAP TechnicalASAP Guide
77 . . . . . DIFFRACTION GRATINGS AND DOES References
.REFERENCES ...... • Optics, Hecht, Second Addition. • Introduction to Fourier Optics, Goodman. • Binary Optics Technology: The Theory and Design of Multi-level Diffractive Optical 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