Simplified Algorithm for Implementing an ABCD Ray Matrix Wave-Optics Propagator

Transcription

1 Simpliied Algorithm or Implementing an ABC Ray atrix Wave-Optics Propagator Justin. ansell, Robert Praus, Anthony Seward, and Steve Coy ZA Associates Corporation

2 Outline Introduction & otivation Ray atrices Siegman ecomposition Algorithm odiications to the Siegman ABC ecomposition Algorithm Simpliication by Removing One Step Addressing egeneracies and etails Comparison o ABC and Sequential Wave-Optics Propagation Conclusions

3 Introduction & otivation odel propagation o a beam through a complex system o simple optics in as ew steps as possible. We developed a technique or using ray matrices to include image rotation and relection image inversion in wave-optics modeling. Here we introduce a technique to prescribe a wave-optics propagation using a ray matrix. Laser 3

4 Ray atrix Formalism 4

5 Introduction - Ray atrices The most common ray matrix ormalism is x a.k.a. ABC matrix It describes how a ray height, x, and angle, θ x, changes through a system. x θ x A B C x x' A B x θ x ' = C θ x x' = Ax+ Bθ θ ' x x = Cx + θ x θ x 5

6 x Ray atrix Examples Propagation x θ x x x' = x+ θ L x' L x θ x ' = θ x x Lens θ x x θ x θ ' = θ x/ x x x' x θx ' = / θ x 6

7 Example ABC atrices atrix Type Form Variables Propagation Lens Curved irror (normal incidence) Curved ielectric Interace (normal incidence) L n R R ( n n ) L = physical length n = reractive index = eective ocal length R = eective radius o curvature n = starting reractive index n = ending reractive index R = eective radius o curvature 7

8 3x3 and 4x4 Formalisms Siegman s Lasers book describes two other ormalisms: 3x3 and 4x4 The 3x3 ormalism added the capability or tilt addition and oaxis elements. The 4x4 ormalism included two-axis operations like axis inversion and image rotation. 3 x 3 4 x 4 A B E x C F θ x E = Oset F = Added Tilt Ax Bx x C θ x x x Ay By y C θ y y y 8 ual-axis ABC

9 5x5 Formalism We use a 5x5 ray matrix ormalism as a combination o the x, 3x3, and 4x4. Previously introduced by Paxton and Latham Allows modeling o eects not in waveoptics. Image Rotation Relection Image Inversion Image Rotation Ax Bx Ex x Cx x F x θ x Ay By Ey y Cy y Fy θ y ual-axis ABC Tilt and Oset 9

10 Ray atrix Wave-Optics Propagation Introduction Introduced a way o applying eects captured by a 5x5 ray matrix model with wave-optics. Image Inversion Image Rotation This relied on a parallel sequential wave-optics model and integration o these eects at the end. We complete the integration technique here by showing how the residual dual-axis ABC matrices embedded in a 5x5 ray matrix can be used to speciy a wave-optics propagation.

11 ABC Ray atrix Wave-Optics Propagator

12 Implementation Options Siegman combined the ABC terms directly in the Huygens integral. Less intuitive Cannot obviously be built rom simple components He then also introduced a way o decomposing any ABC propagation into 5 individual steps. U ( x, y ) U A C ( x, y ) exp( jkl) = jλb L exp jk B B = / / A x x ( + y ) ( x x + y y ) ( + y ) / + dxdy /

13 Siegman ecomposition Algorithm Choose magniications & (= * ) Calculate the eective propagation length and the ocal lengths. L eq A C = = = B L B = B A B / 3

14 odiications to the Siegman ecomposition Algorithm We ound that one o the magniication terms was unnecessary ( =.). We modiied Siegman s algorithm to better address two important situations: image planes and ocal planes. We worked on how add diraction into choosing magniication. A C B = / / / L / / L = A + η λ 4

15 5 Eliminating a agniication Term We determined that one o the two magniication terms that Siegman put into his decomposition was unnecessary. There were ive steps (,,L,, )and our inputs (ABC). = / / / L C B A = / / / / L C B A L L New ecomposition Original ecomposition

16 Image Plane: B= This case is an image plane. There is no propagation involved here, but there is curvature and magniication. / = / / / B L eq = = B = = A B = = / Siegman Our Algorithm C L eq = -/ = = C 6

17 Automated agniication etermination: Problems with the Focal Plane We were trying to automate the selection o the magniication by setting it equal to the A term o the ABC matrix. This minimizes the mesh requirements In doing so, we ound that the decomposition algorithm was problematic at a ocal plane. / Siegman, =A = A = = / Leq = = = = A = = 7

18 Propagation to a Focus: A= / = / For a collimated beam going to a ocus, this ray envelope diameter is zero. To handle this case, we orce the user to speciy the magniication. We also give the user guidance on how to choose magniication when there is substantial diraction Siegman, =A = A = Leq = = = = A = = Siegman, = = B Leq = = B = = A B = = / 8

19 Choosing agniication while Considering iraction We propose here to add a diraction term to the magniication to avoid the case o small. We added a tuning parameter, η, which is the number o eective diraction limited diameters. = = A = A + η = A + Lλ + η Lλ η N 9

20 Common iraction Patterns Airy Sinc Normalized Intensity Gaussian Normalized Radius

21 Integrated Energy Integrated Energy Threshold = - Airy Sinc Gaussian We concluded that η=5 is suicient to capture more than 99% o the integrated energy. η

22 odiied ecomposition Algorithm I at an image plane (B=) =A (possible need or interpolation) Apply ocus Else Speciy, considering diraction i necessary Calculate and apply the eective propagation length and the ocal lengths. L L or =. L eq L B = = B = A B = / L - L

23 Wave-Optics Implementation etails 3

24 Implementing Negative agniication Ater going through a ocus, the magniication is negated. We implement negative magniication by inverting the ield in one or both axes. We consider the dual axis ray matrix propagation using the 5x5 ray matrix ormalism. 4

25 ual Axis Implementation Cylindrical telescopes along the axes are handled by dividing the convolution kernel into separate parts or the two axes. U H = = P F exp ( H F( U )) [ ( )] jπλ z + z F( x) = Fourier Transorm o x P = Phase Factor x x y y 5

26 WaveTrain Implementation 6

27 Example: ABC Propagator 7

28 Example System / Compared sequential and ABC propagation ields 8

29 Field beore the Lens / Field agnitude 9 Field Phase

30 Field Ater Lens by istance / / Field agnitude 3 Field Phase

31 Field Ater Lens by istance / Field agnitude 3 Field Phase

32 Field Ater Lens by istance 3/ / Field agnitude 3 Field Phase

33 Field Ater Lens by istance / Field agnitude 33 Field Phase

34 ABC Ray atrix Fourier Propagation Conclusions We have modiied Siegman s ABC decomposition algorithm to remove one o the magniications and include several special cases such as Image planes Propagation to a ocus This enables complex systems comprised o simple optical elements to be modeled in 4 steps (one Fourier propagation). 34

35 Questions? (55) x 35