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1 MZA Associates Corporation Simplified Method of Modeling Complex Systems of Simple Optics using Wave-Optics: An Engineering ApproachDr. Justin D. Mansell, Steve Coy, Liyang Xu, Anthony Seward, and Robert PrausMZA Associates Corporation
2 Outline Introduction Aperture Imaging Into Input Space Ray MatricesLens LawFourier OpticsAperture Imaging Into Input SpaceFinding the Field Stop & Aperture StopMesh Determination for Complex SystemsConclusions
4 Introduction - ABCD Matrices The most common ray matrix formalism is the 2x2 or ABCD that describes how a ray height, x, and angle, θx, changes through a system.θxθx’xx’
5 2x2 Ray Matrix ExamplesPropagationθxx’xLensθxθx’x
6 Example ABCD Matrices Matrix Type Form Variables Propagation L = physical lengthn = refractive indexLensf = effective focal lengthCurved Mirror(normal incidence)R = effective radius of curvatureCurved Dielectric Interfacen1 = starting refractive indexn2 = ending refractive index
7 3x3 and 4x4 FormalismsSiegman’s Lasers book describes two other formalisms: 3x3 and 4x4The 3x3 formalism added the capability for tilt addition and off-axis elements.The 4x4 formalism included two-axis operations like axis inversion and image rotation.3 x 3E = OffsetF = Added Tilt4 x 4
8 5x5 FormalismWe use a 5x5 ray matrix formalism as a combination of the 2x2, 3x3, and 4x4.Previously introduced by Paxton and Latham
9 Lens LawThis equation governs the location of the formation of an image.objectimagefd2d1
10 Huygens PrincipleIn 1678 Christian Huygens “expressed an intuitive conviction that if each point on the wavefront of a light disturbance were considered to be a new source of a secondary spherical disturbance, then the wavefront at any later instant could be found by constructing the envelope of the secondary wavelets.”-J. Goodman, Introduction to Fourier Optics (McGraw Hill, 1968), p. 31.
12 Fresnel Approximation Fresnel found that in modeling longer propagations,the cosine term could be neglected andthe spherical term could be approximated by a r2 term.Huygens-Fresnel IntegralFresnel Approximation
13 Notation Simplification Quadratic Phase Factor (QPF): Equivalent to the effect a lens has on the wavefront of a field.Fourier TransformMultiplicative Phase Factor: Takes into account the overall phase shift due to propagation
17 Discrete Sample Implementation When implementing Fourier propagation on a computer, the field is sampled at discrete points.The mesh spacing between samples (δ) and the number of mesh points (N) required for accurate modeling are discussed later.The mesh spacing can be different at the beginning of a propagation (δ1) than at then end (δ2)δ
18 Samples, Not Local Area Averages We represent the continuous wave with a series of samples, which is NOT a local average.This can be most easily shown by Fourier transforming a 2D grid of ones.The result is a single non-zero mesh point at the center of a mesh, but adding a guard-band shows that the single non-zero mesh point is really an under-sampled sine functionMatlab Scriptx=1:1:128; x3=1:0.5:128.5;E = ones(1,128); E2 = fftshift(fft(fftshift(E)));E3 = fftshift(fft(fftshift(addGuardBand(E',1))));figure; plot(x,E./max(E),'-o');hold on;plot(x3,abs(E3)./max(abs(E3)),'g*-');plot(x,E2./max(E2),'rx-');zoomOut(0.1);
19 One-Step Fourier Propagator Steps:Multiply by QPFFourier TransformMultiply by QPF & Phase FactorCommentsLeast computationally expensiveOffers no control over the resulting mesh spacing
20 Convolution Propagator Steps:Fourier transformmultiplication by the Fourier transformed kernelan inverse Fourier transformAdvantage:Maintains the mesh spacing
21 Two-Step Forward Propagator Steps:Fourier PropagateComments:Easy to scale the mesh by picking intermediate plane location.Works well in long propagation cases because the quadratic phase factor is applied before the Fourier transformδ1δiδ2z1z2
22 Scaling with the Convolution Propagator One drawback of the convolution propagator is its apparent inability to scale the mesh spacing in a propagation.Scaling the mesh is important when propagating with significant wavefront curvature.The convolution propagator can be modified to model propagation relative to a spherical reference wavefront curvature such that the mesh spacing can follow the curvature of the wavefront.We call this Spherical-Reference Wave Propagation or SWP.
23 Spherical Reference Wave Propagation 1/3 Consider convolution propagation with a built in wavefront curvature relative to the lens law.Based on imaging, light propagated through an ideal lens to a distance d1 will be the same shape as if it were propagated to a distance d2 where f-1=d1-1+d2-1 except for a magnification term.For example: a beam propagating 48 cm toward a focus through a 50cm lens. Based on the lens law, this is equivalent to propagating a 1200 cm without going through the focus.It is significantly easier to propagate 1200 cm than it is to propagate 48cm.objectimagefd2d1
24 Spherical Reference Wave Propagation 2/3 The real advantage of SWP is the ability to scale the mesh to track the evolution of the beam size.The spherical reference wave radius of curvature (Rref) can be determined based on the desired magnification (M).δ1δ2zRref
25 Spherical Reference Wave Propagation 3/3 Procedure:Specify desired magnification (M) and propagation distance (z)Calculate effective reference curvature (Rref)Determine new effective propagation distance (zeff)Propagate the effective distance and then change the sample spacing by the magnification factor
26 Mathematical Comparison of Convolution & Double Propagation Convolution Propagator with SWPDouble PropagatorWith the SWP addition, the Double Propagator and the Convolution Propagator are equivalent.
27 ConclusionsWe use the convolution propagator in WaveTrain with spherical-reference wave propagation.This allows us to solve any general wave-optics problem.
29 Computer Fourier Propagation Modeling In most situations, the most rapidly varying part of the field is the QPF.In a complex field, the phase is reset every wavelength or 2π radians.To achieve proper sampling, sampling theory dictates that we need two samples per wave.Phase (radians)SamplesInadequateSamplingAdequate Samplingk = 1; R = 16; x = -63:1:64; p = exp(j*k* (x.^2) ./ (2*R));plot(angle(p),'b*-')
30 Methods of Determining Sampling Diffraction LimitAngular Bandwidth
31 Mesh Sampling: Whittaker-Shannon Theory We need 2 samples for each wave of amplitude.For a parabolic phase surface, this is most limiting at the edge of the phase surface.λ2δ
32 Mesh Sampling: Diffraction Diffraction is the fundamental limit of our ability to model propagation.If we assume we need 2 samples per diffraction spot radius on BOTH sides of a propagation, we get a maximum value for the mesh spacing.zD1D2
34 Virtual Adjacent Apertures Now that we know the mesh sampling intervals (δ1 and δ2), we need to know how big a mesh we need to use to accurately model the diffraction.The Fourier transform assumes a repeating function at the input.This means that there are effective virtual apertures on all sides of the input aperture.We need a mesh large enough that these virtual adjacent apertures do not illuminate our area of interest.
35 Mesh Size: Equal Sized Apertures D+θzDzDupDdownTo avoid adjacent apertures (Dup and Ddown) from interfering with the output area of interest, the modeled region should be D+θz in diameter.This means that the number of mesh points should be this diameter divided by the mesh spacing (δ).For equal apertures, this means a factor of two guard-band.
36 Unequal AperturesFor unequal apertures (such as spherical-reference wave propagation), the same geometric argument can be made.The resulting form can be thought of as the average of the number of mesh points required to cover each of the input apertures plus a diffractive term.If we set the two mesh spacings to their maximum value, the number of mesh points reduces to a familiar form:the Fresnel number.
37 Fresnel Number The Fresnel number is zThe Fresnel number isthe number of half waves of phase of a parabolic wavefront over the aperture.half the number of diffraction limited spots diameters over the aperture.roughly the number of diffraction ripples across an aperture.
38 Mesh Size and Fresnel Number Equal Sized AperturesFor equal sized apertures, the number of mesh points equals 16 times the Fresnel number.For unequal apertures, the same is true if we define an effective Fresnel number as r2r1/λz.Unequal Sized Apertures
39 Summary of Mesh Determination Mesh Sample SpacingMesh SizeDerived by either:Diffraction limit ORRequired angular bandwidthDerived based on eliminating overlap between adjacent virtual apertures and the region of interest.
40 ConclusionsFor a simple system of two limiting apertures, we have determined a set of inequalities that govern the choice of the mesh.Next we will look athow phase aberrations impact the mesh choice, andhow this can be extended to a system of multiple apertures.
41 Impact of Phase Aberrations on Mesh Determination
43 Turbulence-Induced Aperture Blurring GaussianPSFBlurredApertureTurbulence acts to increase the size of the point spread functionThis effectively blurs the apertures at each end.The blurred apertures can be thought of as being larger if we want to capture most of the energy.
44 Blur Effect on Aperture Size Blurred Edge of a Hard ApertureTurbulence can be thought of as diffracting light as if it were sent through a grating with a period equal to r0, which is Fried’s coherence length.r0 can be though of as the characteristic turbule size.A scaling parameter, cturb, is used to control the amount of energy captured in the calculation.cturb ≈ 4 is for 99% of the energy.
45 Procedure to Determine the Mesh while Considering Turbulence zD1’D1D2D2’Simplify the turbulence distribution along the path into effective r0 values for each effective aperture.Modify the aperture size using the effective r0.Use the new effective aperture sizes to determine the mesh.
46 Determining Fourier Propagation Mesh Parameters for Complex Optical Systems of Simple Optics
47 IntroductionThe determination of mesh parameters for wave-optics modeling can be uniquely determined by a pair of limiting apertures separated by a finite distance.An optical system comprised of a set of ideal optics can be analyzed to determine the two limiting apertures that most restrict rays propagating through the system using field and aperture stop techniques.
48 Definitions of Field & Aperture Stop Aperture Stop = the aperture in a system that limits the cone of energy from a point on the optical axis.Field Stop = the aperture that limits the angular extent of the light going through the system.NOTE: All this analysis takes place in ray-optics space.
49 Example System Optical System D=15 D=15 D=1 D=5 f=100 f=100 A1 L1 L2 15020050InputPlane 2Plane 3Plane 41515Input Space51-50150A1A2L2L1
50 Procedure for Finding Stops 1/3 Find the location and size of each aperture in input space.Find the ABCD matrix from the input of the system to each optic in the system.Solve for the distance (zimage) required to drive the B term to zero by inverting the input-space to aperture ray matrix.This matrix is the mapping from the aperture back to input space.The A term is the magnification (Mimage)of the image of that aperture.
51 Procedure for Finding Stops 2/3 Find the angle formed by the edges of each of the apertures and a point in the middle of the object/input plane. The aperture which creates the smallest angle is the image of the aperture stop or the entrance pupil.151551-50150A1A2L2L1
52 Procedure for Finding Stops 3/3 Find the aperture which most limits the angle from a point in the center of the image of the aperture stop in input space. This aperture is the field stop.151551-50150A1A2L2L1
53 Example: Fourier Propagation Input Space151551-50150A1A2L2L1D1 = 1 mm, D2 = 15 mm, λ = 1 μm, z = 0.15 mMinimal Mesh = 400 x μm = 3.75 mm
54 Example System Optical System D=15 D=15 D=1 D=5 f=100 f=100 A1 L1 L2 15020050InputPlane 2Plane 3Plane 4
58 Conclusions of Complex System Mesh Parameter Determination We have devised a procedure to reduce a complex system comprised of simple optics into a pair of the most restricting apertures.It would be nice to have a way of simplifying a complex system of simple optics so that modeling it is computationally easier…
60 Implementation Options Siegman combined the ABCD terms directly in the Huygens integral.He then also introduced a way of decomposing any ABCD propagation into 5 individual steps.
61 Polishing the Siegman Decomposition Algorithm We found that one of the magnification terms was unnecessary (M1=1.0).Siegman’s algorithm did not address two important situations: image planes and focal planes.We worked a bit more on how to pick an appropriate magnification when considering diffraction.
62 Siegman Decomposition Algorithm Choose magnifications M1 & M2 (M=M1*M2)Calculate the effective propagation length and the focal lengths.
63 Eliminating a Magnification Term We determined that one of the two magnification terms that Siegman put into his decomposition was unnecessary.There were five unknowns and four inputs.
64 Image Plane: B=0 This case is an image plane. There is no propagation involved here, but there iscurvature andmagnification.SiegmanOur Algorithm
65 Focal Plane CaseWe were trying to automate the selection of the magnification by setting it equal to the A term of the ABCD matrix.This minimizes the mesh requriementsIn doing so, we found that the decomposition algorithm was problematic at a focal plane.Siegman, M=A
66 Propagation to a Focus: A=0 For a collimated beam going to a focus, this ray envelope diameter is zero.To handle this case, we augmented the magnification determination with diffraction.Siegman, M=ASiegman, M=1
67 Choosing Magnification while Considering Diffraction We propose here to add a diffraction term to the magnification to avoid the case of M=0.We added a tuning parameter, η, which is the number of effective diffraction limited diameters.We leave the selection of magnification to the user.
68 Common Diffraction Patterns AirySincGaussianNormalized IntensityNormalized Radius
69 Integrated EnergyWe concluded that η=5 is sufficient to capture more than 99% of the 1D integrated energy.Threshold = 10-10Integrated EnergyAirySincGaussianη
70 Modified Decomposition Algorithm If at an image plane (B=0)M=A (possible need for interpolation)Apply focusElseSpecify M, considering diffraction if necessaryCalculate and apply the effective propagation length and the focal lengths.
71 Implementing Negative Magnification After going through a focus, the magnification is negated.We implement negative magnification by inverting the field in one or both axes.We consider the dual axis ray matrix propagation using the 5x5 ray matrix formalism.
72 Dual Axis Implementation In our implementation, we handle the case of cylindrical telescopes along the axes by dividing the convolution kernel into separate parts for the two axes.
75 Magnitudez=2f before lensz=2f after lensz=2.5fz=3fz=3.5fz=4f
76 Unwrapped Phase z=2f before lens z=2f after lens z=2.5f z=3f z=3.5f
77 ABCD Ray Matrix Fourier Propagation Conclusions We have modified Siegman’s ABCD decomposition algorithm to include several special cases, including:Image planesPropagation to a focusThis enables complex systems comprised of simple optical elements to be modeled in 4 steps.