Using Skew Rays to Model Gaussian Beams

Using Skew Rays to Model Gaussian Beams Host: Paul Colbourne, Lumentum Zemax, LLC 2016 1

Topics we ll cover today: Using skew rays to represent a Gaussian beam. Use of User-Defined Surfaces to generate skew rays, view circular Gaussian beams on 3D layout diagrams, and optimize systems with circular Gaussian beams. What is general astigmatism? Use of a merit function macro to optimize systems with elliptical or generally astigmatic Gaussian beams. Calculating the properties of a generally astigmatic Gaussian beam in an optical system using a ZPL macro. Zemax, LLC 2016 2

Conventional ray tracing Conventional ray tracing is done using rays originating from a point and converging to a point. Input Output (End view of launched rays)

Conventional ray tracing Conventional ray tracing is done using rays originating from a point and converging to a point. However, the resulting rays do not properly represent the propagation of Gaussian beams The output Gaussian beam is out of focus in this example. (End view of launched rays) Gaussian beam profile

Skew ray tracing By generating skew rays, the rays properly represent a propagating Gaussian beam. (End view of launched rays)

Skew ray tracing This is not a new concept. Figures from J. Arnaud, Representation of Gaussian beams by complex rays, Appl. Opt. vol. 24, no. 4, pp. 538-543, 1985.

Skew rays generated using a user-defined surface An OpticStudio user-defined surface us_gskew.dll was created which adds an offset to an incident ray to convert it to a skew ray. The beam waist radius ω 0 is specified as a parameter of the UDS. The (circular) beam can then be viewed on 3D layout diagrams using the Ring ray pattern. m is ray slope toward y axis xx = xx + mmmm 0 θθ 0 yy = yy llll 0 θθ 0 mm = dddd dddd l is ray slope toward x axis ll = dddd dddd Userdefined surface

Optimizing with skew rays (circular beam) By specifying a negative waist radius for a second user-defined surface placed at the output, the reverse offset is made, which can make the rays converge to a point at the output (if the beam is in focus). The system can then be optimized in the normal way. This optimization method only works for circular beams (more on that later). xx = xx mmmm 0 θθ 0 yy = yy + llll 0 θθ 0 User-defined surface

Optimization example with skew rays (circular beam) UDS to generate skew rays 5 μm input beam radius UDS to undo skew rays 5 μm output beam radius (note negative radius entered) 3D layout with Ring ray pattern

Optimization example with skew rays (circular beam) DELETE this slide, for reference only. To OpticStudio demo with file skew_opt_with_uds_paraxial_lens.zmx Begin optimized with rays, w0=0. FICL=1, POP=0.874. Optimize with POP (4 seconds), refresh 3D layout (looks unfocused). FICL=0.745, POP=1. Is this really the correct focus? Yes, but you may have trouble convincing your boss. Is the beam doing what you expect at intermediate points? Hard to say. Cannot do spot diagrams, ray fans, and other ray-based analyses. Return to optimized with rays (undo). Refresh 3D layout. Set w0=0.005. Refresh 3D layout. Can now see the beam profile and can see that the beam is not focused at output. (viewing Ring ray pattern) Optimize with rays (default MF RMS spot size) (0.3 seconds). Refresh 3D layout. FICL=1, POP=1. Now we are confident that we have the correct focus, and we can clearly see the beam size at all points in the system. Note: need a large aperture setting to get accurate FICL, smaller aperture setting to see the correct beam size on 3D layout. Reverse sign of w0 at output for each mirror in the system. With real lenses, can optimize conic constant etc. to minimize aberrations.

Skew ray tracing for elliptical beams Skew rays can also represent elliptical beams in simply astigmatic systems (separable into x and y components). UDS us_gaussxy.dll created to generate skew rays for elliptical beams.

Evolution of an elliptical ray bundle An elliptical skew ray bundle remains elliptical but rotates as the rays propagate. The ray bundle (squares) matches the profile of the beam (triangles) only when viewed along x or y directions. At other view angles, the wrong beam size is seen. At waist 200 μm from waist 500 μm from waist

Size of the elliptical beam at other view angles Rays can be generated with either left skew or right skew. If both left skew and right skew rays are propagated, the projected size of the propagating elliptical beam A(φ) is equal to the RMS average of the projected sizes of the left skew ray bundle A L (φ) and right skew ray bundle A R (φ). 2A L (φ) 2A(φ) y φ AA φφ = AA RR φφ 2 + AA LL φφ 2 2 2ω y 2A R (φ) ω 2 ω 1 θ θ x This holds even for general astigmatism (optics not aligned to x and y axes). 2ω x

What is General Astigmatism? Simple astigmatism describes an elliptical Gaussian beam where all cylindrical optical elements are aligned with the beam axes. The x and y components of the simply astigmatic beam can be treated separately, as if each was a circular Gaussian beam propagating through a symmetric optical system. If an elliptical Gaussian beam is incident on a cylinder lens not aligned with the beam, general astigmatism results. The intensity profile of the generally astigmatic beam is elliptical, and the phase surface is ellipsoidal or hyperboloidal, just like a simply astigmatic beam. However, the phase surface is not aligned with the intensity ellipse. The intensity ellipse rotates in space as the beam propagates. A generally astigmatic beam has no clearly defined waist.

Illustration of simple and general astigmatism Simply astigmatic beam Two waists can be defined. Beam is always aligned with x-y axes. Generally astigmatic beam No defined waist. Beam rotates as it propagates. E. Kochkina, G. Wanner, D. Schmelzer, M. Tröbs, and G. Heinzel, Modeling of the general astigmatic Gaussian beam and its propagation through 3D optical systems, Applied Optics, Vol. 52, No. 24 p. 6030 (2013).

Do I need to worry about General Astigmatism? Most optical systems do not intentionally create general astigmatism. However, there are possible sources of general astigmatism to be aware of: Even if an optical system does not contain any intentionally rotated cylinder lenses, lenses may become rotated during a tolerancing exercise. General astigmatism can result if an elliptical beam hits a spherical lens off-axis, due to astigmatism of the spherical lens. Are these effects significant? Without the ability to account for general astigmatism, you just don t know. If an algorithm is to work for any optical system, without limitations on the orientation of lenses or positions of beams, it needs to accommodate general astigmatism.

Skew ray generation Two rays are sufficient to define the right skew ray bundle, and two rays are sufficient to define the left skew ray bundle. Right skew rays: Left skew rays: xx 1 = ppωω 0xx cos α xx 3 = ppωω 0xx cos β 2 yy 1 = ppωω 0yy sin α yy 3 = ppωω 0yy sin β 3 1 ll 1 = ppθθ 0xx sin α ll 3 = ppθθ 0xx sin β mm 1 = ppθθ 0yy cos α mm 3 = ppθθ 0yy cos β 4 xx 2 = ppωω 0xx cos α + π 2 xx 4 = ppωω 0xx cos β + π 2 (α=0, β=0) yy 2 = ppωω 0yy sin α + π 2 yy 4 = ppωω 0yy sin β + π 2 ll 2 = ppθθ 0xx sin α + π 2 ll 4 = ppθθ 0xx sin β + π 2 mm 2 = ppθθ 0yy cos α + π 2 mm 4 = ppθθ 0yy cos β + π 2. ll = dddd dddd mm = dddd dddd AA φφ = 1 pp xx 1 cos φφ + yy 1 sin φφ 2 + xx 2 cos φφ + yy 2 sin φφ 2 + xx 3 cos φφ + yy 3 sin φφ 2 + xx 4 cos φφ + yy 4 sin φφ 2 2

Skew ray generation Other ray definitions can be used which produce the same results for A(φ). xx 1 = ppωω 0xx cos α cos δ + sin α + γ sin δ yy 1 = ppωω 0yy sin α + γ cos δ cos α sin δ ll 1 = ppθθ 0xx sin α cos δ cos α + γ sin δ mm 1 = ppθθ 0yy cos α + γ cos δ + sin α sin δ xx 2 = ppωω 0xx cos α + π 2 cos δ + sin α + π 2 + γ sin δ yy 2 = ppωω 0yy sin α + π 2 + γ cos δ cos α + π 2 sin δ ll 2 = ppθθ 0xx sin α + π 2 cos δ cos α + π 2 + γ sin δ mm 2 = ppθθ 0yy cos α + π 2 + γ cos δ + sin α + π 2 sin δ xx 3 = ppωω 0xx cos β cos δ + sin β γ sin δ yy 3 = ppωω 0yy sin β γ cos δ cos β sin δ ll 3 = ppθθ 0xx sin β cos δ cos β γ sin δ mm 3 = ppθθ 0yy cos β γ cos δ + sin β sin δ xx 4 = ppωω 0xx cos β + π 2 cos δ + sin β + π 2 γ sin δ yy 4 = ppωω 0yy sin β + π 2 γ cos δ cos β + π 2 sin δ ll 4 = ppθθ 0xx sin β + π 2 cos δ cos β + π 2 γ sin δ mm 4 = ppθθ 0yy cos β + π 2 γ cos δ + sin β + π 2 sin δ. New parameters γ and δ introduced.

Effect of γ and δ on the skew ray bundles γγ = 0 γγ = ππ 4 γγ = ππ 2 δδ = 0 δδ = ππ 8 δδ = ππ 4 All of these ray bundles represent the same elliptical beam. We can launch whatever ray bundle we wish (for example γ=0, δ=0), but at the output we could get any of these ray patterns. An optimization method must be able to work with any of these ray bundles at the output.

Skew rays for optimization with elliptical beams At the output, we may have γ 0 and δ 0 if the beam is not circular. xx 1 = ppωω 0xx cos α cos δ + sin α + γ sin δ yy 1 = ppωω 0yy sin α + γ cos δ cos α sin δ ll 1 = ppθθ 0xx sin α cos δ cos α + γ sin δ mm 1 = ppθθ 0yy cos α + γ cos δ + sin α sin δ xx 2 = ppωω 0xx cos α + π 2 cos δ + sin α + π 2 + γ sin δ yy 2 = ppωω 0yy sin α + π 2 + γ cos δ cos α + π 2 sin δ ll 2 = ppθθ 0xx sin α + π 2 cos δ cos α + π 2 + γ sin δ mm 2 = ppθθ 0yy cos α + π 2 + γ cos δ + sin α + π 2 sin δ xx 3 = ppωω 0xx cos β cos δ + sin β γ sin δ yy 3 = ppωω 0yy sin β γ cos δ cos β sin δ ll 3 = ppθθ 0xx sin β cos δ cos β γ sin δ mm 3 = ppθθ 0yy cos β γ cos δ + sin β sin δ xx 4 = ppωω 0xx cos β + π 2 cos δ + sin β + π 2 γ sin δ yy 4 = ppωω 0yy sin β + π 2 γ cos δ cos β + π 2 sin δ ll 4 = ppθθ 0xx sin β + π 2 cos δ cos β + π 2 γ sin δ mm 4 = ppθθ 0yy cos β + π 2 γ cos δ + sin β + π 2 sin δ. m (for the same ray) was used to offset x at the output for circular beams. xx mmmm 0 θθ 0 = 0 only when γ = 0. These two parameters have the same dependence on α, γ, δ. xx 1 + ll 2 ωω 0xx θθ 0xx = 0 for any α, γ, δ. So we can use this calculation for optimizing elliptical beams.

Merit function macro for optimization OpticStudio macro ZPL31.zpl created. Callable from within the merit function. Propagates left and right skew rays, 3 rings of 8 arms, plus central ray (no UDS needed). Offsets rays at output so they converge to a point if focused. Note: each ray offset is calculated based on the angle of a different ray at α+π/2. UDS cannot do this. Returns RMS spot radius. xx 1bb = xx 1 + ll 2 ωω 0xx yy 1bb = yy 1 + mm 2 ωω 0yy θθ 0xx θθ 0yy Macro calculation

Example optimization using ZPL31.ZPL Input is a Gaussian beam with ω 0x = 10 μm and ω 0y = 20 μm. The rotated cylinder lens at surface 3 creates general astigmatism. The subsequent series of 4 cylinder lenses attempts to convert the beam to a circular beam with ω 0 = 10 μm, located 1.5 mm from the input.

Example optimization using ZPL31.ZPL (cont d) The merit function contains only one ZPLM 31 operand. Here the optimization has reduced the RMS spot size to less than 0.01 μm. ω 0x = 10 μm, ω 0y = 20 μm at input ω 0x = 10 μm, ω 0y = 10 μm at output

Example optimization using ZPL31.ZPL (cont d) The output is a circular beam, POP coupling is 99.97%. Target output achieved! It would be very difficult to do this particular optimization any other way!

Notes on the use of ZPL31.ZPL The macro by default computes the RMS spot size at the image surface. To compute at other surfaces, use IMSF before ZPLM to change the image surface. The macro by default computes the RMS spot size at the primary wavelength. To compute at other wavelengths, use PRIM before ZPLM to change the primary wavelength. If the output ω 0x =0, the macro computes the RMS spot size in the y direction only, or if ω 0y =0, the macro computes the RMS spot size in the x direction only. This enables optimizing such that the x and y waists lie in different planes. Another macro ZPL32.zpl traces more rays (with γγ = ππ 2, δδ = 0 and δδ = ππ 4) for more accurate results. Rays with γγ = ππ 2 have no skew component, so may give more accurate results with refractive surfaces. Ensure maximum filling of the aperture with rays.

Calculation of intensity ellipse 1. Propagate skew rays through the optical system. 2. Calculate projected beam size at 3 view angles. 3. Extract orientation and major and minor axes of intensity ellipse. AA 0 = AA 0 = 1 pp AA ππ/2 = AA ππ 2 AA ππ/4 = AA ππ 4 = 1 pp = 1 pp xx 2 1 +xx 2 2 +xx 2 3 +xx 2 4 2 yy 2 1 +yy 2 2 +yy 2 3 +yy 2 4 2 xx 1 +yy 1 2 + xx 2 +yy 2 2 + xx 3 +yy 3 2 + xx 4 +yy 4 2 4. 2A π/2 2A π/4 y y x x θθ = 1 2AA 2 tan 1 ππ/4 2 AA 2 2 0 AA ππ/2 AA 2 2 0 AA ππ/2 ωω xx = 1 pp ωω yy = 1 pp xx 1 2 +xx 2 2 +xx 3 2 +xx 4 2 2 yy 2 1 +yy 2 2 +yy 2 3 +yy 2 4 2 2A 0

Calculation of wavefront The wavefront curvature (C = 1/ROC) can be calculated using da/dz along the major or minor axis of the intensity ellipse (x and y ): z The wavefront sag is of the form SS = CC xx xx 2 + CCyy yy 2 + bbxx yy 2 With one more calculation of da/dz at a different view angle we can solve for b and thus complete the description of the propagating generally astigmatic Gaussian beam. For more details, see GenAstigGaussianBeam.zpl or Proc. SPIE 9293, IODC 2014, 92931S

ZPL macro to calculate beam parameters A macro GenAstigGaussianBeam.zpl was written which calculates the properties of a generally astigmatic Gaussian beam propagating through an optical system. 16 rays traced instead of 4, for greater accuracy. Executing C:\Documents and Settings\col\My Documents\ZEMAX\MACROS\GenAstigGaussianBeam.ZPL. Generally astigmatic Gaussian beam property calculation at each surface Greynolds generally astigmatic beam.zmx Greynolds generally astigmatic beam Configuration number 1 Field number 1 Start surface 1 End surface 11 8/24/2016 Input beam parameters: X waist size (um): 1128.38 Y waist size (um): 564.19 Wavelength (um): 10.00000 p = 0.8710 orient is the orientation of the intensity ellipse relative to the surface x-y coordinates. orient is measured perpendicular to the beam. wx and wy are the size of the intensity ellipse (major and minor axis) at the surface, perpendicular to the beam zx and zy are the distances to the minimum beam size measured in intensity ellipse coordinates w0x and w0y are the minimum beam sizes in intensity ellipse coordinates ph_orient is the orientation of the phase surface relative to the intensity ellipse. If ph_orient is non-zero, the beam has some general astigmatism (if Cx and Cy are different, and wx and wy are different). Cx and Cy are wavefront curvatures; ROCx and ROCy are radii of curvature of the wavefront (0 = infinity). All parameters are AFTER refraction from the surface, within the surface material. Surface orient(deg) wx(um) wy(um) w0x(um) w0y(um) zx(um) zy(um) ph_orient(deg) Cx(1/mm) Cy(1/mm) ROCx(um) ROCy(um) 1 0.00 1128.38 564.19 1128.38 564.19 0 0 45.00-0.00000 0.00000 0 0 2 us_gaussxy.dll 0.00 1128.38 564.19 1128.38 564.19 0 0 45.00-0.00000 0.00000 0 0 3 start POP here 0.00 1128.38 564.19 1128.38 564.19 0 0 45.00-0.00000 0.00000 0 0 5 Rotated cyl lens f=100 mm 45.00 564.19 1128.38 531.92 651.47-22222 -133333 45.00-0.01000 0.00000-100000 0 7 50 mm 27.50 954.65 427.59 924.20 417.64-53168 -9213 36.89 0.00374-0.00991 267539-100872 8 100 mm -36.65 425.94 1006.10 414.31 804.88 10178 120726-39.31 0.01000-0.00172 100000-580001 9 150 mm -15.23 597.01 1238.37 524.31 619.71 34229 151932-38.92 0.00996 0.00166 100436 603073 10 200 mm 0.00 797.88 1595.77 651.47 531.92 66667 177778-45.00 0.00750 0.00250 133333 400000 11 250 mm 9.39 996.00 2020.16 746.18 486.15 110618 214199 41.02 0.00262 0.00574 381457 174127

Previous calculation methods Previous methods for calculating the properties of generally astigmatic Gaussian beams make use of complex rays (1 complex ray = 1 real ray and 1 imaginary ray) and complex matrix calculations. J. A. Arnaud, Nonorthogonal Optical Waveguides and Resonators, Bell System Technical Journal, vol. 49, no. 9 (Nov. 1970), pp. 2311-2348. Alan W. Greynolds, Propagation of generally astigmatic Gaussian beams along skew ray paths, SPIE vol. 560 (1985), pp. 33-50. Baida Lü, Guoying Feng, Bangwei Cai, Complex ray representation of the astigmatic Gaussian beam propagation, Optical and Quantum Electronics vol. 25 (1993) pp. 275-284. The method presented here requires no complex rays and no matrix calculations. Both methods require 4 propagated rays (plus the chief ray).

Comparison with previously published results* Complex matrix calculations 0 27.51 53.35 74.77 90.00 99.39 Skew ray calculations of intensity ellipse orientation θ ω 0x = (2/ π) mm ω 0y = (1/ π) mm f = 100 mm cyl. lens at 45 *Alan W. Greynolds, Propagation of generally astigmatic Gaussian beams along skew ray paths, SPIE vol. 560 (1985), pp. 33-50.

Future Work While the methods developed so far are useful, more could be done: Calculation of coupling loss using skew rays for elliptical beams. Calculation of aberrations (spherical aberration, coma, etc.). Improve convergence for optimization of elliptical and generally astigmatic beams. Compiled program instead of OpticStudio macro. Multi-threaded computation. Better optimization algorithm? Tolerance to ray vignetting at apertures. Methods for high NA beams (NA>0.15). Calculation of Guoy shift (phase shift as the beam passes through focus) for generally astigmatic beams. Mathematical proof of formula for A(φφ) for generally astigmatic beams. I hope others will contribute to make this even better! Zemax User Forum could be a good forum for exchange of ideas and contributions.

Summary A method was presented of using skew rays to optimize an optical system to obtain the correct Gaussian beam focus and minimize aberrations. Implemented using a User-Defined Surface for circular beams. Implemented using an OpticStudio merit function macro for elliptical beams. A method was presented of using 4 skew rays to calculate the propagation of a generally astigmatic Gaussian beam. Implemented in an OpticStudio macro GenAstigGaussianBeam.zpl

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