Integrating Wave-Optics and 5x5 Ray Matrices for More Accurate Optical System Modeling

Transcription

1 Integrating Wave-Optics and 5x5 Ray Matrices for More Accurate Optical System Modeling Justin D. Mansell, Robert Suizu, Robert W. Praus, Brent Strickler, Anthony Seward, and Stephen Coy

2 Motivation Methodology Outline Simplified Ray Tracing 5x5 Ray Matrix Formalism Integration of 5x5 Ray Matrices with Wave-Optics Conclusions & Future Work 2

3 Motivation : Model More Effects Wave-optics is not sufficient to model some optical effects like reflection-induced image inversion and geometric image rotation. Image Inversion Image Rotation jmansell@mza.com 3

4 Motivation 2: 6 DOF Perturbation Analysis Mechanical disturbances of optics and the resulting induced beam jitter control are typically modeled separately from waveoptics. These effects impact waveoptics performance. We want a mechanism for control model integration. jmansell@mza.com 4

5 Our Solution Augment our wave-optics with a 5x5 ray matrix formalism based on simplified ray tracing. Why? Wave-optics infrastructure exists Complete ray tracing is complex and time consuming 5x5 ray matrices simple to implement, fast, sufficient to add desired effects Consistent with beam jitter control matrix development and FEM modeling jmansell@mza.com 5

6 Presentation Limitations Removed some mathematical methods that are excessively time-consuming to present. More time to focus on highlights These details are available in the paper. 6

7 Simplified Ray Tracing 7

8 Ray Trace Procedure Start with a coordinate in a plane. Find the intersection of that coordinate in the next plane. Find next direction through interaction with plane normal plus curvature. s intersection C P i+ i P v i+ i N i+ intersection i+ {, }, {, } line = P v plane = C N s i i i i+ i+ = ( ) intersection ( ) N P C i+ i i+ N i+ P = P + s v v i i jmansell@mza.com 8

9 Curvature-Induced Tilt Additional tilt added due to optic curvature is handled separately by calculating the distance from the center of the optic (called beam walk) and adding the appropriate curvature-induced tilt. f = R / 2 f θlocal = bw f θ = M θ i+ global transform local v = v + θ v i+ i+ global i+ R local jmansell@mza.com 9

10 Sequential Evaluation Motivation Wave-optics systems are peppered with non-ideal optics like deformable mirrors. Wave-optics systems model propagation through optics as a sequential process. jmansell@mza.com

11 Generalizing Ray Tracing We were looking for a way to: apply ray tracing to a general ray and apply it sequentially to an optical system Needed Effects: image rotation image inversion ability to do perturbation analysis jmansell@mza.com

12 Ray Matrix Formalism 2

13 Introduction - ABCD Matrices The most common ray matrix formalism is the 2x2 or ABCD that describes how a ray height, x, and angle, θ x, changes through a system. x θ x A B C D x x ' A B x θx ' = C D θ x x ' = Ax + Bθ θ ' x x = Cx + Dθ x θ x jmansell@mza.com 3

14 2x2 Ray Matrix Examples Propagation x θ x x x ' = x + θil x ' L x θx ' = θ x x Lens θ x x θ x θ ' = θ x / x x x ' x θ x ' = / f θ x f jmansell@mza.com 4

15 Example ABCD Matrices Matrix Type Propagation Lens Curved Mirror (normal incidence) Curved Dielectric Interface (normal incidence) Form L n f 2 R 2 R ( n n ) Variables L = physical length n = refractive index f = effective focal length R = effective radius of curvature n = starting refractive index n 2 = ending refractive index R = effective radius of curvature jmansell@mza.com 5

16 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 offaxis elements. The 4x4 formalism included two-axis operations like axis inversion and image rotation. 3 x 3 4 x 4 A B E x C D F θ x E = Offset F = Added Tilt Ax Bx x Cx D x θ x Ay By y Cy Dy θ y jmansell@mza.com 6

17 5x5 Formalism We developed a 5x5 ray matrix formalism as a combination of the 2x2, 3x3, and 4x4. Ax Bx Ex x Cx Dx F x θ x Ay By E y y Cy Dy Fy θ y jmansell@mza.com 7

18 8 Example 5x5 System Matrices C S C S S C S C y y x x M M M M Reflection Inversion (reflection from a mirror in the plane of the x-axis) M x = magnification along x axis M y = magnification along y axis Magnification C = cos(θ image ) S = sin(θ image ) θ image = image rotation angle Image Rotation Variables Form Matrix Type

19 9 Example 5x5 System Matrices 2 y x y x θ θ θ x = added tilt along the x axis θ y = added tilt along the y axis Tilt Addition x = translation along the x axis y = translation along the y axis Image Translation Variables Form Matrix Type

20 Limitations No Astigmatism axially symmetric curvature only Why? This can be represented, but there may be insufficient degrees of freedom for arbitrary axis astigmatism. Astigmatism can be put into the wave-optics. 2

21 Procedure for Generation of a 5x5 Ray Matrix jmansell@mza.com 2

22 Motivation for Modified Procedure Multiplying 5x5 ray matrices allow for all the desired effects to be accumulated The magnitude of the image rotation cannot be determined with a sequential ray matrix approach A different procedure had to be devised to incorporate this effect jmansell@mza.com 22

23 Simple Ray Trace Procedure 5 probe rays x d Rays modeled as simple geometric ray tracing c a b e thin optic approximation curvature handled separately jmansell@mza.com 23

24 Reduction of Probe Rays to 5x5 Matrix Determine the difference in location and direction from the unperturbed central ray (a). Project these differences onto the x and y beam axes in global coordinates. RESULT: 5x5 matrix v v v v b c d e = v = v = v = v A = P d b e c A5 = P a last last last last x A2 = P x A3 = P x A4 = P x x ( b) v ( c) v ( d ) v ( e) v A B C D E global beam global beam global beam global beam global beam last last last last ( a) ( a) ( a) ( a) A2 B2 C2 D2 E2 A3 B3 C3 D3 E3 B = v d B2 = v B3 = v B4 = v B5 = v P b P c P d P b e c a e = = = = A4 B4 C4 D4 E4 x x x x P P P P last last last last A5 B5 C5 D5 E5 global beam x global beam global beam global beam global beam ( b) P ( c) P ( d ) P ( e) P last last last last ( a) ( a) ( a) ( a) jmansell@mza.com 24

25 Sequential Evaluation of 5x5 Matrices Ray matrices can be multiplied sequentially to establish a system ray matrix. We established a sequential technique that involves: Propagate from the source to a plane a small delta from the optic, which is a virtual sensor. That plane becomes the next virtual source. Target Source Virtual Source/Sensor 2 Virtual Source/Sensor jmansell@mza.com 25

26 Example Systems Analysis using 5x5 Ray Matrices 26

27 Simple Powered Optic y x z [ ] [ δ] R=2,f= Unperturbed Matrix 2δ - - 2δ - NOTE: δ terms will be removed henceforth. jmansell@mza.com 27

28 Retro-Reflector Case Cross-Section 3D [ ] [ δ] Only 2 of 3 bounces shown. Unperturbed Matrix - - jmansell@mza.com 28

29 Focal Plane Inversion Plane Test Case [ ] [ ] R=, f=/ [ δ ] R=, f=/2 [ δ ] - 3 [ ] [ ] f= 3 [ δ ] f= [ δ ] jmansell@mza.com 29

30 3 Set of 3 Image Rotation Matrices /2 [ ] [ ] [ ] [2 ] 3 3 Unperturbed Matrix [ ] [ ] [ ] [2 ] z y x 4 4 Unperturbed Matrix [ ]

31 45 Image Rotation Test Case [2 ] [ ] Unperturbed Matrix y z x [ ] / 2 / / / 2 -/ / / 2 -/ 2 [ ] jmansell@mza.com 3

32 Image Rotation Test Case [ 3 ] xb g [- 2 ] [ 2 ] ybg [ 4 ] [ ] [ 3 ] Unperturbed Matrix [- ] xb g [ ] yb g [-2 ] [ ] z y x jmansell@mza.com 32

33 Rectifying the Wave-Optics Results with 5x5 Ray Matrix Effects 33

34 Rationale for Combining Ray and Wave Results (Rectification) In some locations, wave-optics and ray matrix effects need to be combined. We call this process rectification. Example locations include Deformable Mirrors Aberrated Optics Wavefront Sensors 34

35 Rectification Outline Ray Matrix Effects needing Rectification Effect Magnitude Determination Order of Operations Method of Applying Effects 35

36 Effects Modeled by Ray Matrix Reflection Inversion Image Rotation Magnification Power Translation Tilt Propagation Cannot be done with wave-optics. Traditionally done with wave-optics, but can adversely impact wave-optics mesh parameters Can be done with wave-optics, but adversely affects the wave-optics mesh parameters Needs to be done with wave-optics 36

37 Magnitudes of the Ray Effects Send 9 probe rays through the ray matrix 4 angles, 4 offsets, and center Analysis of each of the rays coming out of the system allows determination of the magnitudes of the effects -y The 9 Probe Rays +x B C G I A E H +y - F -x D jmansell@mza.com 37

38 Rationale for Needing Order of Operations At a rectification plane, many ray matrix effects have been combined into a single ray-matrix The order of application of the effects has been lost Operations happen in parallel Sometimes we only have the ray matrix Large number of sequential operations Any 5x5 ray matrix can be decomposed into 7 effects Minimizing the number of rectifications reduces noise Effects interact EXAMPLE: power and magnification - applying power then magnifying reduces the optical power by the magnification jmansell@mza.com 38

39 Operations Interaction Matrix Rotate Magnify Power Translate Tilt Invert (xaxis) Invert (x-axis) Tilt Translate Power Magnify Key No Problem Column before Row Row before Column Potential Problem This interaction matrix gives 5 possible orders. Rotate jmansell@mza.com 39

40 The 5 Potential Orders Order Operations invert rotate magnify power translate tilt magnify invert rotate power translate tilt invert magnify rotate power translate tilt magnify power invert rotate translate tilt magnify invert power rotate translate tilt invert magnify power rotate translate tilt invert rotate magnify power tilt translate magnify invert rotate power tilt translate invert magnify rotate power tilt translate magnify power invert rotate tilt translate magnify invert power rotate tilt translate invert magnify power rotate tilt translate invert rotate magnify tilt power translate invert magnify rotate tilt power translate invert magnify rotate tilt power translate jmansell@mza.com 4

41 Rectification Implementation Two Options Interpolate the complex field Interpolate the applied phase/magnitude or sensor grid Interpolation of complex numbers tends to create more noise than interpolating real numbers, so generally the latter is preferred. 4

42 Conclusions & Future Work Developed a 5x5 Ray Matrix Formalism Developed a Technique for Integrating the effects modeled by ray matrices with Wave-Optics Future Work: Show how the 5x5 ray matrix can be used to specify the wave-optics propagation exp( jkl) U x y U x y (, ) = (, ) jλb jk ( ( ) ( ) ( )) exp A x + y 2 x x 2 + y y 2 + D x 2 + y 2 dx dy 2B From Siegman s Lasers jmansell@mza.com 42

43 Acknowledgements Thanks to the ABL SPO for their continued support of this development. Special thanks to Dr. Salvatore Cusumano and Capt. Jason Tellez. 43

44 Questions? 44