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**MZA Associates Corporation**

Simplified Method of Modeling Complex Systems of Simple Optics using Wave-Optics: An Engineering Approach Dr. Justin D. Mansell, Steve Coy, Liyang Xu, Anthony Seward, and Robert Praus MZA Associates Corporation

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**Outline Introduction Aperture Imaging Into Input Space**

Ray Matrices Lens Law Fourier Optics Aperture Imaging Into Input Space Finding the Field Stop & Aperture Stop Mesh Determination for Complex Systems Conclusions

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Ray Matrix Formalism

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**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. θx θx’ x x’

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2x2 Ray Matrix Examples Propagation θx x’ x Lens θx θx’ x

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**Example ABCD Matrices Matrix Type Form Variables Propagation**

L = physical length n = refractive index Lens f = effective focal length Curved Mirror (normal incidence) R = effective radius of curvature Curved Dielectric Interface n1 = starting refractive index n2 = ending refractive index

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3x3 and 4x4 Formalisms Siegman’s Lasers book describes two other formalisms: 3x3 and 4x4 The 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 3 E = Offset F = Added Tilt 4 x 4

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5x5 Formalism We use a 5x5 ray matrix formalism as a combination of the 2x2, 3x3, and 4x4. Previously introduced by Paxton and Latham

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Lens Law This equation governs the location of the formation of an image. object image f d2 d1

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Huygens Principle In 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.

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**Huygens-Fresnel Integral**

1 2

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**Fresnel Approximation**

Fresnel found that in modeling longer propagations, the cosine term could be neglected and the spherical term could be approximated by a r2 term. Huygens-Fresnel Integral Fresnel Approximation

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**Notation Simplification**

Quadratic Phase Factor (QPF): Equivalent to the effect a lens has on the wavefront of a field. Fourier Transform Multiplicative Phase Factor: Takes into account the overall phase shift due to propagation

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**Fresnel Approximation Validity**

spherical parabolic radius

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**Fresnel Approximation Validity Examples**

a (cm) z >> (π/4 a4/λ)1/3 Goodman Nf max 100 92 m 10838 10 4.3 m 2335 1 0.2 m 503 0.1 9.2 mm 108 0.01 430 μm 23 0.001 20 μm 5 λ = 1.0 μm

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**Implementing Fourier Propagation**

One-Step Fourier Transform Single Fourier integral Convolution Propagator Short distances Two Cascaded One-Step Propagations Longer distances

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**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

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**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 function Matlab Script x=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);

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**One-Step Fourier Propagator**

Steps: Multiply by QPF Fourier Transform Multiply by QPF & Phase Factor Comments Least computationally expensive Offers no control over the resulting mesh spacing

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**Convolution Propagator**

Steps: Fourier transform multiplication by the Fourier transformed kernel an inverse Fourier transform Advantage: Maintains the mesh spacing

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**Two-Step Forward Propagator**

Steps: Fourier Propagate Comments: 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 transform δ1 δi δ2 z1 z2

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**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.

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**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. object image f d2 d1

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**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). δ1 δ2 z Rref

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**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

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**Mathematical Comparison of Convolution & Double Propagation**

Convolution Propagator with SWP Double Propagator With the SWP addition, the Double Propagator and the Convolution Propagator are equivalent.

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Conclusions We use the convolution propagator in WaveTrain with spherical-reference wave propagation. This allows us to solve any general wave-optics problem.

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**Computer Modeling of Fourier Propagation**

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**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) Samples Inadequate Sampling Adequate Sampling k = 1; R = 16; x = -63:1:64; p = exp(j*k* (x.^2) ./ (2*R)); plot(angle(p),'b*-')

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**Methods of Determining Sampling**

Diffraction Limit Angular Bandwidth

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**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. λ 2δ

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**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. z D1 D2

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**Mesh Sampling: Angular Bandwidth**

z D1 D2

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**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.

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**Mesh Size: Equal Sized Apertures**

D+θz D z Dup Ddown To 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.

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Unequal Apertures For 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.

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**Fresnel Number The Fresnel number is**

z The Fresnel number is the 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.

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**Mesh Size and Fresnel Number**

Equal Sized Apertures For 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

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**Summary of Mesh Determination**

Mesh Sample Spacing Mesh Size Derived by either: Diffraction limit OR Required angular bandwidth Derived based on eliminating overlap between adjacent virtual apertures and the region of interest.

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Conclusions For 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 at how phase aberrations impact the mesh choice, and how this can be extended to a system of multiple apertures.

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**Impact of Phase Aberrations on Mesh Determination**

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**Sinusoidal Phase Grating**

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**Turbulence-Induced Aperture Blurring**

Gaussian PSF Blurred Aperture Turbulence acts to increase the size of the point spread function This 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.

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**Blur Effect on Aperture Size**

Blurred Edge of a Hard Aperture Turbulence 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.

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**Procedure to Determine the Mesh while Considering Turbulence**

z D1’ D1 D2 D2’ 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.

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**Determining Fourier Propagation Mesh Parameters for Complex Optical Systems of Simple Optics**

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Introduction The 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.

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**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.

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**Example System Optical System D=15 D=15 D=1 D=5 f=100 f=100 A1 L1 L2**

150 200 50 Input Plane 2 Plane 3 Plane 4 15 15 Input Space 5 1 -50 150 A1 A2 L2 L1

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**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.

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**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. 15 15 5 1 -50 150 A1 A2 L2 L1

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**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. 15 15 5 1 -50 150 A1 A2 L2 L1

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**Example: Fourier Propagation**

Input Space 15 15 5 1 -50 150 A1 A2 L2 L1 D1 = 1 mm, D2 = 15 mm, λ = 1 μm, z = 0.15 m Minimal Mesh = 400 x μm = 3.75 mm

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**Example System Optical System D=15 D=15 D=1 D=5 f=100 f=100 A1 L1 L2**

150 200 50 Input Plane 2 Plane 3 Plane 4

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N=1024, δ=6.6 μm Over-Sampled Plane 2 Plane 3 Plane 4 Input

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N=512, δ=9.4 μm Minimal Sampling Plane 2 Plane 3 Plane 4 Input

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N=256, δ=13.3 μm Under Sampled Plane 2 Plane 3 Plane 4 Input

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**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…

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**ABCD Ray Matrix Wave-Optics Propagator**

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**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.

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**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.

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**Siegman Decomposition Algorithm**

Choose magnifications M1 & M2 (M=M1*M2) Calculate the effective propagation length and the focal lengths.

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**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.

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**Image Plane: B=0 This case is an image plane.**

There is no propagation involved here, but there is curvature and magnification. Siegman Our Algorithm

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Focal Plane Case We 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 requriements In doing so, we found that the decomposition algorithm was problematic at a focal plane. Siegman, M=A

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**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=A Siegman, M=1

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**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.

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**Common Diffraction Patterns**

Airy Sinc Gaussian Normalized Intensity Normalized Radius

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Integrated Energy We concluded that η=5 is sufficient to capture more than 99% of the 1D integrated energy. Threshold = 10-10 Integrated Energy Airy Sinc Gaussian η

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**Modified Decomposition Algorithm**

If at an image plane (B=0) M=A (possible need for interpolation) Apply focus Else Specify M, considering diffraction if necessary Calculate and apply the effective propagation length and the focal lengths.

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**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.

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**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.

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**Example: ABCD Propagator**

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Example System f/2 D f 2f 2f

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Magnitude z=2f before lens z=2f after lens z=2.5f z=3f z=3.5f z=4f

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**Unwrapped Phase z=2f before lens z=2f after lens z=2.5f z=3f z=3.5f**

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**ABCD Ray Matrix Fourier Propagation Conclusions**

We have modified Siegman’s ABCD decomposition algorithm to include several special cases, including: Image planes Propagation to a focus This enables complex systems comprised of simple optical elements to be modeled in 4 steps.

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