Transcript Slide
Modeling and Simulation of Beam Control Systems Part 2. Modeling Optical Effects 1 Agenda Introduction & Overview Part 1. Foundations of Wave Optics Simulation Part 2. Modeling Optical Effects Lunch Part 3. Modeling Beam Control System Components Part 4. Modeling and Simulating Beam Control Systems Discussion 2 Part 2. Modeling Optical Effects In Part 2 we will apply the basic theory presented in Part 1 to the practical problems of modeling optical phenomena using wave optics simulation techniques. 3 Modeling Optical Effects Overview Modeling the Light Leaving a Source Modeling Localized Optical Effects Modeling Optical Propagation Through Vacuum Modeling Optical Propagation Through Aberrating Media A Spreadsheet for Choosing Valid Mesh Parameters Special Topics 4 Modeling Optical Effects - Overview In wave optics simulation light is modeled as being made up of what we shall call “waves”, each representing a portion of monochromatic or quasimonochromatic light of limited transverse extent, with a phasefront approximating a specified plane wave or a spherical wave, called its reference wave. Each wave has an associated scalar field u=Aeif, represented by a rectangular complex mesh spanning the transverse extent of the wave. The complex phase at each mesh point represents a phase difference, relative to the specified reference wave: fmesh=f-fref Each wave is initially created to model all or part of the light being transmitted from a particular light source at some instant in time. Waves are propagated from plane to plane by numerically evaluating the Fresnel diffraction integral using the discrete Fourier transform. Optical effects are modeled by operating on waves – either the complex mesh, the reference wave, or both – at various planes along the optical path. Propagate, operate, propagate, operate, and so on. 5 Modeling Optical Effects - Overview In order To correctly to correctly model the model lightthe reaching light reaching a given a receiver given receiver from a givenasource, from given source, it is generally it is sometimes necessary necessary to take into to take account into the physical the account properties physicalofproperties the source, of the the receiver, source, the andreceiver, the entire intervening and the entire optical intervening path, including optical path, any optical including effects any that optical may enter inthat effects at various may enter planes in atalong various theplanes path. along the path. Using that information, we can determine what part of the light leaving the source will (or might) reach the receiver, and then restrict our attention to only that light. This is crucial, because it is often not feasible to model all of the light leaving a source, especially for sources that radiate over wide angles. 6 Modeling Light From Collimated Sources When modeling collimated light and coherent sources, light suchsources, as a lasers, suchit as is a lasers, it is sometimes sometimes feasible to model feasible alltoofmodel the light all leaving of the light the leaving source. the In source. such cases In such it is possible a case the to model sourcethe cansource be modeled correctly correctly withoutwithout knowing anything at about all about the receiver the receiver or theorintervening the opticaloptical path. path. For example, a laser transmitting only one spatial mode can be modeled using just one wave, and we can safely choose the reference wave, mesh spacings and mesh dimensions based on the properties of the source alone. Similarly, a laser transmitting multiple spatial modes can be modeled using multiple waves, one per spatial mode. Some lasers transmit multiple longitudinal modes with slightly different wavelengths within a single spatial mode, resulting in quasimonochromatic light exhibiting temporal partial coherence, while still remaining spatially coherent. Such light can be modeled using a only single wave, provided we also keep track of its coherence properties. 7 Modeling Light From Collimated Sources Beam Waist A collimated light source, by definition, transmits almost all of its energy in a narrow beam. To a good approximation, the transmitted light can be thought of as being made up of only those rays that pass through both the actual aperture at the source and an imaginary aperture at or near the beam waist. 8 Modeling Light From Uncollimated Sources When modeling uncollimated light and/or sources, incoherent suchlight as asources, scene such as a scene illuminated by natural illuminated light, orbylaser natural lightlight, reflected or laser from light anreflected optically from an rough surface, optically it is rough generally surface, not itfeasible is generally to model not feasible all of thetolight model all leaving the ofsource, the lightso leaving it is necessary the source, to determine so it is necessary what part to of determine that light might whatreach part ofthe that given lightreceiver. might reach the given receiver. To do this, we basically need to determine what the image of the receiver would look like as seen from the source, taking into account any intervening optics, and also any physical effects entering in along the path. If the optical path is in vacuum or still air, the image of the receiver as seen from the source will be generally be very sharp. If the path goes through an aberrating medium, the image will be blurred, and may be much larger geometric image. Of all the light leaving the source, only those rays that passes through receiver image will ultimately reach the receiver. 9 Modeling Light From Uncollimated Sources An uncollimated light source, by definition, radiates over a wide angle. In this case, it not feasible to model all of the light radiated by the source, but fortunately it is generally not necessary, because we are only concerned with that portion of the light that will (or might) ultimately reach the given receiver. 10 What Part of the Light Leaving a Source Must be Modeled? Field of View Sensor When modeling an optical sensor with a limited field of view, such as a camera, we can generally restrict our attention to only that part of the light that impinges upon the entrance pupil of the sensor at angles within the sensor field of view, or just slightly outside of it, to take into account diffraction at the edge of pupil. 11 Modeling Localized Optical Effects InInwave waveoptics opticssimulation simulationall alloptical opticaleffects, effects,with withthe thesole soleexception exceptionofof optical opticalpropagation propagationthrough throughvacuum vacuumororan anideal idealdielectric dielectricmedium, medium,are modeled are modeled as if they as if occurred they occurred at discrete at discrete planes. planes. This is This an is an approximation approximationofofcourse, course,since sincemany manyimportant importanteffects, effects,such suchas asthe the optical opticaleffects effectsofofatmospheric atmosphericturbulence, turbulence,do donot notactually actuallyoccur occuratat discrete discreteplanes. planes. However Howeverititisisan anapproximation approximationwhich whichcan cangenerally generally be bemade madeas asaccurate accurateas asrequired, required,albeit albeitatatadditional additionalcomputational computational cost, cost,simply simplyby byusing usingmore moreand andmore moreplanes. planes. Most Mostlocalized localizedoptical opticaleffects effectsare aremodeled modeledby byoperating operatingon onindividual individual waves, waves,modifying modifyingeither eitherthe thecomplex complexmesh, mesh,the thereference referencewave, wave,oror both. both. Most Mostoperations operationson onthe thecomplex complexmesh meshare arejust justmultiplications; multiplications; this thisincludes includesphase phaseperturbations, perturbations,absorption, absorption,and andgain gainmedia. media. Operations on the reference wave include translation and/or scaling transverse to the optical axis, and modification of its tilt (propagation direction) and/or focus (phase curvature). These operations can be used to model many optical effects occurring within an optical system. 12 Modeling Optical Effects Within Optical Systems Within an optical system, the natural coordinate system to use in modeling optical effects is just the nominal optical coordinate system, defined by the system designer. This coordinate system changes (in relationship to any fixed geometric frame) each time the light hits a mirror – the nominal optical axis (z) changes direction, and the transverse axes (x&y) flip about it. And each simple lens or curved mirror imparts a quadratic phase factor (approximately) just like those that appear in the propagation integral. All of these “designed-in” effects can be taken into account simply by adjusting the propagation geometry appropriately. Once this has been done, these effects need not be considered further when choosing mesh spacings and dimensions. 13 Modeling Optical Propagation Through Vacuum It therefore Modeling If But it were it is generally feasible optical behooves propagation to notstick feasible us to to try a through policy to tobe determine so ofvacuum always conservative, what being is the straightforward very because mesh parameters wave optics in principle: yield making simulation correct the onemesh results simply can spacings using evaluates very the time very least the …andconservative, that is where things start to getbe complicated. Fresnel small consuming amount and diffraction of the computation, and mesh resource-expensive integral extents taking numerically, very into large, account even using thewhen problem any DFTs. and really using all a priori the would smallest information be justand that available coarsest straightforward. to meshes us. possible. 14 Modeling Optical Propagation Through Vacuum One-Step DFT Propagation Two-Step DFT Propagation Propagation Artifacts and Filtering Techniques 15 One-Step DFT Propagation DFT propagation is a mathematical algorithm for efficiently computing a discrete approximation to the Fresnel diffraction integral. The Fresnel diffraction integral can be expressed as the composition of three successive operations: multiplication by a quadratic phase factor, followed by a Fourier Transform, followed by multiplication by a second quadratic phase factor. In DFT propagation, the Fourier transform is replaced by a discrete Fourier transform. In one-step DFT propagation, we propagate from the initial plane directly to the final plane, without evaluating the optical field at any intermediate planes. u2 D Pz u1D Qz Fz DQz u1D z z 2 , 1 N1 N 2 16 One-Step DFT Propagation, Special Case: Two Limiting Apertures at DFT Planes u1 θmax θmax u2 2 1 D1 D2 1 2 z1 z 2 2 max2 2 D1 z 1 2 max1 2 D2 17 z2 One-Step DFT Propagation, Special Case: Two limiting Apertures at DFT Planes u2 Nyquist : z 1 2 max1 D2 z 2 2 max 2 D1 Mesh ext ent: N1 D1 u1 θmax θmax 2 1 D1 D2 1 2 N 2 D2 DFT const raint: z D1 D2 N 1 2 z z1 z2 1 z 18 D2 , 2 z D1 , D1D2 N z One-Step DFT Propagation, Special Case Example 1μm, D1 1.0m, D2 1.5m u2 z1 0km, z2 60km, z 60km u1 θmax θmax 1 z D2 1μm 60km 4cm 1.5 m 2 1 D1 D2 1 2 z 1μm 60km 2 6cm D1 1 .0 m N z D1 D2 1.0m 1.5m 25 1 2 z 1μm 60km z1 z2 1 4cm, 2 6cm, N 25 19 Two-Step DFT Propagation One-step DFT propagation is simple, but somewhat inflexible, in that the mesh spacings and mesh dimension cannot be chosen independently, being related by the DFT constraint: 2=z / N1 Suppose instead we carry out the propagation in two steps, first propagating from the initial plane to some intermediate plane, and then from there to the final plane. That gives us an addition degree of freedom: we can choose 1, 2, and N independently, and still satisfy the DFT constraints by adjusting the position of the intermediate plane. The intermediate plane can be placed anywhere: between the initial and final planes, in front of both of them, behind them, or even at “infinity”. (The last can be thought of as a limiting case.) z2 zitm z2 zitm z2 zitm 2 1 z z N itm zitm z1 itm 1 N N1 20 Two-Step DFT Propagation, Special Case: Two limiting apertures located at initial and final DFT planes One-Step Approach 1: two propagation steps in the same direction u2 u1 Letting z z2 z1 , m 2 1 D1 z1 zitm z1 , z2 z2 z1 z1 We obtain z2 u2 z1 m z2 z1 1 , z 1 m D2 u1 Two-Step z2 m z 1 m D1 D2 It can be shown thattheconstraints on 1 , 2 , and N are as follows: 1D2 2 D1 z, N D1 1 D2 2 z1 21 z2 Two-Step DFT Propagation, Special Case: Two limiting apertures located at initial and final DFT planes One-Step Approach 2: two propagation steps in opposite directions u2 u1 Letting z z2 z1 , m 2 1 D1 z1 zitm z1 , z2 z2 z1 z1 We obtain z1 m z2 z1 1 , z 1 m D2 z2 Two-Step z2 m z 1 m u1 D1 u2 D2 Once again, it can be shown thatthe constraints on 1 , 2 , and N are as follows: 1D2 2 D1 z, N D1 1 D2 z1 2 22 z2 Two-Step DFT Propagation, Special Case Example 1μm, D1 1.0m, D2 1.5m u2 z1 0km, z2 60km, z 60km u1 D1 D2 One-Step T o minimizeN , we must set m 1 z 2 D2 2cm, 2 z 2 D1 2 D equal t o 2 , yielding: 1 D1 3cm, N D1 1 D2 2 z1 z2 u2 100 u1 D1 D2 Comparet his t o our previousresult s for one- st ep propagat ion : 1 z D2 4cm, 2 z D1 6cm, N D1 D2 25 z z1 Two-Step Compared to one-step propagation, the mesh spacings need to be twice as small, N needs to be four times as large, and you have to do two one-step propagations, not one. The number of floating point operations required increases by more than a factor of 32. 23 z2 -oru1 D1 z1 u2 D2 z2 Why is Two-Step DFT Propagation Useful? 1. It makes it feasible to propagate waves relatively small distances, e.g. from phase screen to phase screen, where one-step propagation fails, because the mesh dimension, N, varies inversely with the propagation distance. This is a direct consequence of the different curvatures of the quadratic phase factors used in the two techniques. 2. More generally, it makes it possible to “custom-fit” the propagation geometry to the particular problem to be modeled. 3. Finally, it makes it possible to independently choose the mesh spacings at the source and receiver. (And also at other planes, if desired) 24 Mesh Constraints for Propagation Through Vacuum Two - Step Propagation One - Step Propagation z 1 2 max1 D2 z 2 2 max 2 D1 N1 D1 N 2 D2 z D1 D2 N 1 2 z No Filtering : 1 D2 2 D1 z D1 D2 z N 21 2 2 21 2 Optimal Filtering : 1 D2 2 D1 z N 25 D1 1 D2 2 Propagation Through Vacuum Switching Intermediate Planes Outside intermediate plane, used for inside propagations Inside intermediate plane, used for outside propagations 26 Propagation Artifacts and Filtering Techniques In DFT modeling of optical propagation, unless appropriate precautions are taken, the propagated field may exhibit certain kinds of simulation artifacts, i.e. features appearing in the propagated field that do not correctly reflect the physics of the propagation problem being modeled, but instead reflect approximation errors related to the properties of the discrete Fourier transform. There are two main kinds of artifacts that can affect DFT modeling of propagation through vacuum: wraparound and Fresnel ringing. There is an additional kind of artifact that can affect DFT modeling of propagation through aberrating media, related to wraparound occurring at the intermediate plane. All of these artifacts can be eliminated by the use of appropriate filtering techniques. These can include spatial filtering, absorbing boundaries, conjugate beacon techniques, and hybrid techniques 27 Propagation Artifacts and Filtering Techniques Wraparound Initial field at source plane consists of three point sources 28 Propagation Artifacts and Filtering Techniques Eliminating Wraparound Initial field at source plane consists of three point sources 29 Propagation Artifacts and Filtering Techniques Wraparound and Fresnel Ringing Propagated Amplitude from a Discrete Point Source at Various Planes Wraparound Fresnel Ringing 30 Wraparound Propagation Artifacts and Filtering Techniques Eliminating Fresnel Ringing Before Filtering After Filtering 31 Filtering Techniques Used in Wave Optics Simulation Spatial Filtering Absorbing Boundaries Conjugate Beacon Techniques Hybrid Techniques 32 Modeling Optical Propagation Through Aberrating Media In many When wave many optics cases phase simulation of interest, screens optical such are as used, propagation propagation the raysthrough paths through approximate aberrating atmospheric media the turbulence, kinds is modeled of the curved ensemble using pathsphase one of possible would screens, expect raytwo-dimensional paths in the willcontinuous be unbounded, real-valued limit. meshes, strictly speaking, each of which because represents therethrough is no thehard integrated cutoff on optical the range path To correctly model propagation aberrating media it is of difference possible bending (OPD) for angles. a the specific section propagation path. necessary to consider ensemble ofof allthe possible ray paths based Nonetheless, Each on thetime statistics a ray it isof passes generally the aberrating through possible aeffects. phase - andscreen necessary it bends - to abruptly, find a finite and then resumes envelope boundingpropagating all rays carrying alongsignificant a straight amounts line. of energy. 33 Encircled Energy vs. Cutoff Angle for Propagation through Atmospheric Turbulence 34 Ray Envelopes for A Single Initial Ray Earth to Space Space to Earth In thedistance The case ofapropagation ray is deflected through as athe result earth’s of a atmosphere, given bend angle most is of proportional the turbulence to is theconcentrated distance theclose ray travels to the after ground. it is bent. As a result, a ray upward from effects Earth toalong space is path deflected much more Thepropagating distribution of aberrating the can strongly than a ray propagating affectisthe shape of the raydownward envelope.from space to Earth. 35 Ray Envelopes Connecting Two Points Earth to Space Space to Earth …and The envelope the envelope of raysofconnecting rays connecting two specific the original pointssource at the point two ends to of apoint the propagation where the path two through tilted single-ray aberratingenvelopes medium can meet beisobtained just the from the region in single-ray between the envelopes two single-ray for the envelopes. same path: The take same two copies answer of theobtained is single-ray regardless envelope, ofthen which tiltend them of apart the path untilone they starts just touch from. … 36 Ray Envelopes Connecting Two Limiting Apertures For the Now The net diameter suppose case resultofof we ispropagation the this: wish effective attoeither model through aperture end optical ofturbulence, the atpropagation the path, each the the end set through diameter of is rays the an sum that of the of aberrating we the blur must diameters conebe atmedium prepared each of (1) end between the to ofactual model the path two aperture islimiting equivalent is inversely and apertures, (2) toproportional that a blur in one an cone atanalogous to each defined theend of the vacuum by spherical the path. tangents propagation, wave TheFried toset a point-to-point of coherence where ray paths actual length we ray limiting need envelope evaluated to aperture consider atat the at the is opposite the opposite bounded opposite end. by the end union iscan of replaced theof path. the by envelopes a larger “effective” all pairs of points aperture, within the as shown. apertures. zlimiting z (This be seen from the for diagram.) D1 ' D1 cturb r0 spherical 2 37 , D2 ' D2 cturb r0 spherical 1 Propagation Through Aberrating Media Switching Intermediate Planes z1 z2 zswitch Left intermediate plane, used for z1 < z < zswitch Right intermediate plane, used for zswitch < z <z2 When modeling propagation through aberrating media, the mesh dimension N can sometimes be significantly reduced if we adjust the intermediate plane used for two-step propagation to take into account the changing ray region to be modeled. 38 Mesh Constraints for Propagation Through Aberrating Media D1 ' D1 cturb Two - S te pPropagation Fixe dIn te rm e dia te Plan e 1 D'2 2 D'1 z z r0 spherical 2 D2 ' D2 cturb , z r0 spherical 1 Two - S te pPropagati n o Two - S te pPropagati n o S witch in gItm .Pl an e s S witch in gItm .Pl an e s z1 z zswitch zswitch z z2 No Filtering : N D'1 D'2 z 21 2 2 21 2 1 D'2 2 D1 z No Filtering : Optimal Filtering : N D'1 1 D '2 2 N No Filtering : D1 D'2 z 21 2 2 21 2 Optimal Filtering : N D1 1 39 1 D2 2 D'1 z D '2 2 N D'1 D2 z 21 2 2 21 2 Optimal Filtering : N D'1 1 D2 2 A Spreadsheet for Choosing Valid Mesh Parameters (Part 1) 40 A Spreadsheet for Choosing Valid Mesh Parameters (Part 2) 41 Special Topics Hybrid Filtering Techniques for DFT Propagation The “Coy Filter” A General Methodology for Taking Geometric Constraints into Account Ray Region Diagrams Challenging Modeling Problems Reflection from optically rough surfaces Quasi-monochromatic light / temporal partial coherence Polarization and birefringence, partial polarization Thermal blooming, ultrashort pulses, wide field incoherent imaging …et cetera Once again, we won’t have time to cover all these topics in this workshop, but we’d be happy to discuss them off-line. 42 A General Methodology for Taking Geometric Constraints into Account : Ray Region Diagrams RegionIntercept to be Modeled u2 u1 θmax θmax 2 1 D1 D2 Slope 1 2 z1 D2/ z z2 D1 43 Ray Region Diagrams, Continued “Region of Validity” vs. “Region to be Modeled” A one-step DFT propagation models only a discrete set of rays, namely those connecting the mesh points at the initial plane and final planes. These rays span a ray region constrained by two “limiting apertures”, namely the mesh extents at the initial plane and final planes. We will refer to this ray region as the region of validity for the DFT propagation. Region of Validity Region to be Modeled N 2/ z D2/ z = / 1 Suppose that we wish to model the light propagating from a given source to a given receiver, both with finite apertures. This light corresponds to another ray region, also constrained by two “limiting apertures”, namely the source and receiver apertures. We will refer to this ray region as the region to be modeled for the given propagation problem. N 1 D1 The DFT propagation parameters must be chosen such that the region of validity includes the entire region to be modeled. In each case, the ray regions are centered about the ray connecting the centers of the two limiting apertures. 44 Ray Region Diagrams, Continued “Region of Validity” vs. “Region to be Modeled” Region of Validity The requirement that the region of validity include the entire region to be modeled is exactly equivalent to the constraints derived previously. 1 z D2 , 2 z D1 , N N 2/ z Region to be Modeled D2/ z = / 1 z D1D2 1 2 z N 1 D1 Note that the mesh dimension, N1 AreaROV Widt hROV HeightROV N N, is directly proportional the area of the region of validity. 1 45 One-Step DFT Propagation, General Case: Multiple limiting apertures located at arbitrary planes In some propagation problems the light to be modeled may be constrained by multiple limiting apertures at multiple planes. Typically the source and receiver both have finite apertures. If the source is collimated, the beam waist can be thought of as another limiting aperture. Similarly, if the receiver has a limited field of view, that can be thought of as yet another limiting aperture. In determining the size of the effective apertures related to source collimation and/or the receiver field of view, one must take into account diffraction effects. z1 Valid N 2/ z z2 Valid, Minimizing N N 2/ z For a propagation problem with n limiting apertures, the ray region to be modeled is a convex polygon with between three and 2n sides. N 1 46 N 1 Ray Region Diagrams for Two-Step DFT Propagation, Special Case z1< zitm<z2 zitm<z1< z2 Approach 2: Opposite Directions Approach 1: Same Direction Slope=-1/ z D /z N 2 D /z itm D /z Slope=-1/ z N 2 1 1 1 D /z itm 1 1 The region mesh dimension, of validity for N, afor two-step a two-step propagation propagation is the is the intersection intersection of the of the two regions two regions of validity of validity for the for one-step the one-step propagations propagations fromfrom the initial the initial plane plane to the to intermediate the intermediate plane plane andand fromfrom the intermediate the intermediate plane plane to the to final the final plane. plane. 47 1 Two-Step DFT Propagation, General Case: Multiple limiting apertures located at arbitrary planes z1< zitm<z2 zitm<z1< z2 Slope=-1/ z N /z N 2 /z itm Slope=-1/ z N N /z 1 N 2 /z itm 1 1 N 1 1 For two step propagation the region validity is in general a To minimize N we should placeofthe intermediate hexagon three the pairsbeam of parallel sides, two with plane atwith or near waist for the lightfixed to be slopes, corresponding to the initial and final planes, and one modeled. with variable slope, corresponding to the intermediate plane. 48 1