Modeling and Simulation of Beam Control Systems

Modeling and Simulation of Beam Control Systems Part 2. Modeling Optical Effects

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

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.

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

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 quasi-monochromatic light of limited transverse extent, with a phasefront approximating a specified plane wave or a spherical wave, called its reference wave. In wave optics simulation light is modeled as being made up of what we shall call “waves”, each representing a portion of monochromatic or quasi-monochromatic 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 In wave optics simulation light is modeled as being made up of what we shall call “waves”, each representing a portion of monochromatic or quasi-monochromatic 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. In wave optics simulation light is modeled as being made up of what we shall call “waves”, each representing a portion of monochromatic or quasi-monochromatic 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.

Modeling Optical Effects - Overview In order to correctly model the light reaching a given receiver from a given source, it is sometimes necessary to take into account the physical properties of the source, the receiver, and the entire intervening optical path, including any optical effects that may enter in at various planes 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. To correctly model the light reaching a given receiver from a given source, it is generally necessary to take into account the physical properties of the source, the receiver, and the entire intervening optical path, including any optical effects that may enter in at various planes along the path.

Modeling Light From Collimated Sources When modeling collimated and coherent light sources, such as a lasers, it is sometimes feasible to model all of the light leaving the source. In such a case the source can be modeled correctly without knowing anything about the receiver or the intervening optical path. When modeling collimated and coherent light sources, such as a lasers, it is sometimes feasible to model all of the light leaving the source. In such a case the source can be modeled correctly without knowing anything about the receiver or the intervening optical 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. When modeling collimated light sources, such as a lasers, it is sometimes feasible to model all of the light leaving the source. In such cases it is possible to model the source correctly without knowing anything at all about the receiver or the optical 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 quasi-monochromatic light exhibiting temporal partial coherence, while still remaining spatiallycoherent. Such light can be modeled using a only single wave, provided we also keep track of its coherence properties.

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.

Modeling Light From Uncollimated Sources When modeling uncollimated and/or incoherent light sources, such as a scene illuminated by natural light, or laser light reflected from an optically rough surface, it is generally not feasible to model all of the light leaving the source, so it is necessary to determine what part of that light might reach the given receiver. When modeling uncollimated and/or incoherent light sources, such as a scene illuminated by natural light, or laser light reflected from an optically rough surface, it is generally not feasible to model all of the light leaving the source, so it is necessary to determine what part of that light 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. When modeling uncollimated light sources, such as a scene illuminated by natural light, or laser light reflected from an optically rough surface, it is generally not feasible to model all of the light leaving the source, so it is necessary to determine what part of that light 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.

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.

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.

Modeling Localized Optical Effects In wave optics simulation all optical effects, with the sole exception of optical propagation through vacuum or an ideal dielectric medium, are modeled as if they occurred at discrete planes. This is an approximation of course, since many important effects, such as the optical effects of atmospheric turbulence, do not actually occur at discrete planes. However it is an approximation which can generally be made as accurate as required, albeit at additional computational cost, simply by using more and more planes. Most localized optical effects are modeled by operating on individual waves, modifying either the complex mesh, the reference wave, or both. Most operations on the complex mesh are just multiplications; this includes phase perturbations, absorption, and gain 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. In wave optics simulation all optical effects, with the sole exception of optical propagation through vacuum or an ideal dielectric medium, are modeled as if they occurred at discrete planes. This is an approximation of course, since many important effects, such as the optical effects of atmospheric turbulence, do not actually occur at discrete planes. However it is an approximation which can generally be made as accurate as required, albeit at additional computational cost, simply by using more and more planes. In wave optics simulation all optical effects, with the sole exception of optical propagation through vacuum or an ideal dielectric medium, are modeled as if they occurred at discrete planes. This is an approximation of course, since many important effects, such as the optical effects of atmospheric turbulence, do not actually occur at discrete planes. However it is an approximation which can generally be made as accurate as required, albeit at additional computational cost, simply by using more and more planes. Most localized optical effects are modeled by operating on individual waves, modifying either the complex mesh, the reference wave, or both. Most operations on the complex mesh are just multiplications; this includes phase perturbations, absorption, and gain media.

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.

Modeling Optical Propagation Through Vacuum Modeling optical propagation through vacuum is straightforward in principle: one simply evaluates the Fresnel diffraction integral numerically, using DFTs. If it were feasible to stick to a policy of always being very conservative, making the mesh spacings very small and the mesh extents very large, the problem really would be just that straightforward. But it is generally not feasible to be so conservative, because wave optics simulation can be very time consuming and resource-expensive even when using the smallest and coarsest meshes possible. It therefore behooves us to try to determine what the mesh parameters yield correct results using the least amount of computation, taking into account any and all a priori information available to us. …and that is where things start to get complicated.

Modeling Optical Propagation Through Vacuum One-Step DFT Propagation Two-Step DFT Propagation Propagation Artifacts and Filtering Techniques

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.

One-Step DFT Propagation, Special Case:Two Limiting Apertures at DFT Planes

One-Step DFT Propagation, Special Case:Two limiting Apertures at DFT Planes

One-Step DFT Propagation, Special Case Example

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: d2=lDz / Nd1 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 d1, d2, 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.)

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 Two-Step

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 Two-Step

Two-Step DFT Propagation, Special Case Example One-Step -or- 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.

Why is Two-Step DFT Propagation Useful? • 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. • More generally, it makes it possible to “custom-fit” the propagation geometry to the particular problem to be modeled. • Finally, it makes it possible to independently choose the mesh spacings at the source and receiver. (And also at other planes, if desired)

Mesh Constraints for Propagation Through Vacuum

Outside intermediate plane, used for inside propagations Inside intermediate plane, used for outside propagations Propagation Through Vacuum Switching Intermediate Planes

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

Propagation Artifacts and Filtering TechniquesWraparound Initial field at source plane consists of three point sources

Propagation Artifacts and Filtering TechniquesEliminating Wraparound Initial field at source plane consists of three point sources

Propagation Artifacts and Filtering TechniquesWraparound and Fresnel Ringing Propagated Amplitude from a Discrete Point Source at Various Planes Wraparound Fresnel Ringing Wraparound

Propagation Artifacts and Filtering TechniquesEliminating Fresnel Ringing Before Filtering After Filtering

Filtering Techniques Used in Wave Optics Simulation Spatial Filtering Absorbing Boundaries Conjugate Beacon Techniques Hybrid Techniques

Modeling Optical Propagation Through Aberrating Media In wave optics simulation optical propagation through aberrating media is modeled using phase screens, two-dimensional real-valued meshes, each of which represents the integrated optical path difference (OPD) for a specific section of the propagation path. Each time a ray passes through a phase screen it bends abruptly, and then resumes propagating along a straight line. When many phase screens are used, the rays paths approximate the kinds of curved paths one would expect in the continuous limit. To correctly model propagation through aberrating media it is necessary to consider the ensemble of all possible ray paths based on the statistics of the aberrating effects. In many cases of interest, such as propagation through atmospheric turbulence, the ensemble of possible ray paths will be unbounded, strictly speaking, because there is no hard cutoff on the range of possible bending angles. Nonetheless, it is generally possible - and necessary - to find a finite envelope bounding all rays carrying significant amounts of energy.

Encircled Energy vs. Cutoff Angle for Propagation through Atmospheric Turbulence

Ray Envelopes for A Single Initial Ray Earth to Space Space to Earth The distance a ray is deflected as a result of a given bend angle is proportional to the distance the ray travels after it is bent. The distribution of aberrating effects along the path can strongly affect the shape of the ray envelope. In the case of propagation through the earth’s atmosphere, most of the turbulence is concentrated close to the ground. As a result, a ray propagating upward from Earth to space is deflected much more than is a ray propagating downward from space to Earth.

Ray Envelopes Connecting Two Points Earth to Space Space to Earth The envelope of rays connecting two specific points at the two ends of a propagation path through aberrating medium can be obtained from the single-ray envelopes for the same path: take two copies of the single-ray envelope, then tilt them apart until they just touch … …and the envelope of rays connecting the original source point to the point where the two tilted single-ray envelopes meet is just the region in between the two single-ray envelopes. The same answer is obtained regardless of which end of the path one starts from.

Ray Envelopes Connecting Two Limiting Apertures Now suppose we wish to model optical propagation through an aberrating medium between two limiting apertures, one at each end of the path. The set of ray paths we need to consider is bounded by the union of the envelopes for all pairs of points within the apertures. The net result is this: at either end of the path, the set of rays that we must be prepared to model is equivalent to that in an analogous vacuum propagation, where actual limiting aperture at the opposite end is replaced by a larger “effective” limiting aperture, as shown. The diameter of the effective aperture at the each end is the sum of the diameters of (1) the actual aperture and (2) a blur cone defined by the tangents to a point-to-point ray envelope at the opposite end. (This can be seen from the diagram.) For the case of propagation through turbulence, the diameter of the blur cone at each end of the path is inversely proportional to the spherical wave Fried coherence length evaluated at the opposite end of the path.

Left intermediate plane, used for z1 < z < zswitch Right intermediate plane, used for zswitch < z <z2 Propagation Through Aberrating MediaSwitching Intermediate Planes z1 z2 zswitch 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.

Mesh Constraints for Propagation Through Aberrating Media

A Spreadsheet for Choosing Valid Mesh Parameters (Part 1)

A Spreadsheet for Choosing Valid Mesh Parameters (Part 2)

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.

Intercept A General Methodology for Taking Geometric Constraints into Account : Ray Region Diagrams Slope

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. 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. In each case, the ray regions are centered about the ray connecting the centers of the two limiting apertures. The DFT propagation parameters must be chosen such that the region of validity includes the entire region to be modeled.

Note that the mesh dimension, N, is directly proportional the area of the region of validity. Ray Region Diagrams, Continued“Region of Validity” vs. “Region to be Modeled” The requirement that the region of validity include the entire region to be modeled is exactly equivalent to the constraints derived previously.

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. For a propagation problem with n limiting apertures, the ray region to be modeled is a convex polygon with between three and 2n sides.

Ray Region Diagrams for Two-Step DFT Propagation, Special Case z1< zitm<z2 zitm<z1< z2 The region of validity for a two-step propagation is the intersection of the two regions of validity for the one-step propagations from the initial plane to the intermediate plane and from the intermediate plane to the final plane. The mesh dimension, N, for a two-step propagation is the intersection of the two regions of validity for the one-step propagations from the initial plane to the intermediate plane and from the intermediate plane to the final plane.

Two-Step DFT Propagation, General Case:Multiple limiting apertures located at arbitrary planes z1< zitm<z2 zitm<z1< z2 For two step propagation the region of validity is in general a hexagon with three pairs of parallel sides, two with fixed slopes, corresponding to the initial and final planes, and one with variable slope, corresponding to the intermediate plane. To minimize N we should place the intermediate plane at or near the beam waist for the light to be modeled.

Modeling and Simulation of Beam Control Systems