Design & Implementation of a New Freeform Surface Based on Chebyshev Polynomials Presenting Author: Betsy Goodwin, (Zemax LLC) Contributing Authors from: Zemax LLC: Erin Elliott, Sanjay Gangadhara, Vladimir Smagley, Alison Yates Asphericon GmbH: Ulrike Fuchs, Sven Kiontke Abstract

The addition of a new 2-D freeform surface in Zemax OpticStudio based on Chebyshev polynomials enables greater flexibility in the design of freeform optics and facilitates various manufacturing techniques. We present designs including a reflector for generating uniform illumination from Gaussian input as well as a TMA telescope. Comparison is made between the Chebyshev polynomial, XY polynomial, and Zernike surface descriptions. In each case, the Chebyshev surface yields equivalent or superior results.

Introduction

The use of freeform surfaces for the design of optical and illumination systems is becoming increasingly prevalent as design requirements become more complex. While optical design software platforms offer the user many types of freeform surfaces, the uniqueness of the Chebyshev polynomial set available in Zemax OpticStudio lends particular advantages to many design scenarios. These two-dimensional polynomials are by definition orthogonal over a normalized square aperture, meaning that the coefficients which define the surface geometry are linearly independent. As such, optimization of the surface geometry generally does not suffer from the presence of local minima, resulting in a much more straightforward design process than is often the case when using other types of aspheric surfaces.

Additionally, the Chebyshev polynomial set is derived in Cartesian coordinates, which – unlike many other polynomial freeform surfaces used to describe rotationally-symmetric systems – allows for straightforward definition of anamorphic or non-rotationally symmetric systems and non-elliptical apertures. If the user is unsure in the early stages of a design about what surface type to use (as it may not be known in advance how much the final design will deviate from symmetry) the Chebyshev Polynomial surface can be a solid choice. It is capable of modeling various degrees of asymmetry, as needed.

A further advantage to the usage of Chebyshev polynomials is that surface form will not diverge on the edges to the same extent that surfaces described by non-orthogonal polynomials often will. During the manufacturing process, often aperture sizes must be enlarged, so having a surface description that does not diverge wildly on the edges can make the manufacturing process much easier.

Chebyshev Polynomials

The Chebyshev polynomial of the first kind is given by the formula: Tn (x) = cos(n cos-1(x)) , n = 0…∞, x ∈ [-1,1] The first ten Chebyshev polynomials are the following: T0 (x) = 1 T1 (x) = x T2 (x) = 2x2 - 1 T3 (x) = 4x3 - 3x T4 (x) = 8x4 - 8x2 + 1 T5 (x) = 16x5- 20x3 + 5x T6 (x) = 32x6 - 48x4 + 18x2 - 1 T7 (x) = 64x7 - 112x5 + 56x3 - 7x T8 (x) = 128x8 - 256x6 + 160x4 - 32x2 + 1 T9 (x) = 256x9 - 576x7 + 432x5 - 120x3 + 9x T10 (x) = 512x10 - 1280x8 + 1120x6 - 400x4 + 50x2 - 1 It is possible to transition to a two-dimensional Chebyshev polynomial set by using the product basis tij (x,y): tij (x,y) = Ti (x)∙Tj (y) , i,j = 0...∞, x ∈ [-1,1], y ∈ [-1,1] Using a finite sum of the Chebyshev polynomial terms, the resulting sag equation for this freeform surface is:

𝑧𝑧 =𝑐𝑐(𝑥𝑥2 + 𝑦𝑦2)

1 + �1 − 𝑐𝑐2(𝑥𝑥2 + 𝑦𝑦2)+ ��𝑐𝑐𝑖𝑖𝑖𝑖𝑇𝑇𝑖𝑖(�̅�𝑥)𝑇𝑇𝑖𝑖(𝑦𝑦�)

𝑀𝑀

𝑖𝑖=0

𝑁𝑁

𝑖𝑖=0

Here, cij are the coefficients of the Chebyshev polynomial sum; �̅�𝑥, 𝑦𝑦� are normalized surface coordinates; N and M are the maximum polynomial orders in the x and y dimensions; and c is the curvature for the base sphere on top of which the polynomial is added.

While the only limits to the maximum order are governed by performance speed, 10 orders is typically sufficient to describe most surfaces. It can be seen that as the orders of the polynomials increase, so does the spatial frequency of the ripples present on the surface. Here are the first 5x5 terms, shown graphically.

Implementation of Chebyshev Polynomials in Zemax OpticStudio

In Zemax OpticStudio, the Chebyshev Polynomial surface is a freeform surface in sequential mode. It is described by a base radius of curvature and a sequence of Chebyshev polynomials, as described above, with maximum orders in X and Y of 14. The arguments of the 2D Chebyshev polynomials are defined on the unit interval. To make interpolation valid for an arbitrary interval we use normalized �̅�𝑥 and 𝑦𝑦� as the arguments of the polynomials. The user specifies the Normalization Lengths in X and Y to be used for the normalization. Polynomial coefficients in the Lens Data Editor are normalized by a product of X and Y normalization lengths.

Example: Reflector for Converting Gaussian to Uniform Rectangular Illumination using the Chebyshev Polynomial Surface

In this example, we will look at how a reflective Chebyshev Polynomial surface can be used to produce uniform rectangular illumination from a Gaussian input beam. We begin by specifying the entrance pupil diameter and Gaussian apodization. Starting with a flat mirror, the illumination at the image surface looks Gaussian, like this:

The next step is to change the mirror’s surface type from Standard to Chebyshev Polynomial. The user can specify the maximum orders in X and Y of the polynomials to use. In this case, only a subset of the available Chebyshev Polynomials was used, but the maximum orders in X and Y were set at 14 so this subset of polynomial terms could be chosen from all the available orders.

The other parameters specified on the Chebyshev Polynomial surface are the Normalization Lengths in X and Y. For this example, we have specified a normalization length of 16, to normalize the coefficients on a surface slightly larger than the entrance pupil diameter of 30.

We have also placed a rectangular aperture on the Chebyshev surface, with an X Half Width of 16 and a Y Half Width of 16.

A merit function was created to take rays incident on the entrance pupil at various locations and map them to the desired Y heights on the image surface, using REAY operands to target the real ray heights in Y.

Because the final design form was expected to be similar to an Even Asphere surface, only the zeroth and even orders of Chebyshev polynomials were used as variables in the optimization, due to their similarity to the Even Asphere terms.

For optimization, one variable was set on c(2,0), which is the Chebyshev 2x2-1 term. The value of c(0,2), which is the 2y2-1 term, was picked up from the value of c(2,0) because X-Y symmetry was desired. Because these second order terms were set as variables, the base radius of curvature was not used as a variable for optimization and was left as infinite.

Additional variables were added on c(4,0) with a pickup on c(0,4), on c(6,0) with a pickup on c(0,6) and so forth, up to the 14th order, for a total of seven variables and 7 pickups.

After optimizing with Damped Least Squares through the 14th order polynomials, the desired result was achieved. The illumination of the image plane can be seen as uniform and rectangular:

The resulting Chebyshev surface had the following prescription. Note that symmetry has been enforced- the c(2,0) value equals the c(0,2) value, the c(4,0) value equals the c(0,4) value, etc. Polynomial terms with a coefficient of 0 have been omitted for brevity.

The sag of the Chebyshev reflector is depicted below. The lack of rotational symmetry is apparent:

The final Chebyshev reflector system looks like this:

Comparison with Equivalent Extended Polynomial Surface

For comparison purposes, the same optimization was performed, in the same manner but with an Extended Polynomial surface instead of a Chebyshev Polynomial surface. The Extended Polynomial surface consists of a base radius of curvature, conic constant, and polynomial terms which are a power series in x and y. The first term is x, then y, then x*x, x*y, y*y, etc.

The sag of the Extended Polynomial surface can be described by the equation:

𝑧𝑧 =𝑐𝑐𝑟𝑟2

1 + �1 − (1 + 𝑘𝑘)𝑐𝑐2𝑟𝑟2+ �𝐴𝐴𝑖𝑖𝐸𝐸𝑖𝑖(𝑥𝑥,𝑦𝑦)

𝑁𝑁

𝑖𝑖=1

where N is the number of polynomial coefficients in the series, and Ai is the coefficient on the ith extended polynomial term.

Radius of Curvature Infinity

Maximum term # 119 Normalization Radius 16

Coefficient on X2Y0 -4.96666

Coefficient on X0Y2 -4.96666

Coefficient on X4Y0 10.56936

Coefficient on X0Y4 10.56936

Coefficient on X6Y0 -26.40825 Coefficient on X0Y6 -26.40825

Coefficient on X8Y0 45.46581

Coefficient on X0Y8 45.46581

Coefficient on X10Y0 -48.60423

Coefficient on X0Y10 -48.60423

Coefficient on X12Y0 28.77962 Coefficient on X0Y12 28.77962

Coefficient on X14Y0 -7.17780

Coefficient on X0Y14 -7.17780

The resulting Extended Polynomial surface had the following prescription; polynomial terms with a coefficient of 0 have been omitted for brevity:

The results of this optimization were nearly identical to the results of the Chebyshev optimization.

After optimizing through the 14th order polynomials, the illumination of the image plane was uniform and rectangular:

The surface sag of the Extended Polynomial reflector is depicted below. It is nearly identical to that of the Chebyshev Polynomial reflector.

The Extended Polynomial system looks nearly identical to that obtained with the Chebyshev Polynomial surface:

This shows that comparable results can be obtained between the Chebyshev Polynomial surface and the Extended Polynomial surface for this case.

Example: Three Mirror Anastigmat Design using the Chebyshev Polynomial Surface

In this example, we will look at a Three Mirror Anastigmat (TMA) design. First, we will look at the performance of a TMA with all spherical surfaces. Then, we can improve this design using Zernike coefficients, and finally we can improve the design using Chebyshev polynomials and compare the results.

All-Spherical TMA Design

The starting point for this example is a TMA which incorporates only spherical surfaces, shown above. We can see that the performance of this design leaves some room for improvement.

The scale for the Spot Diagrams shown below is 10 mm:

TMA Design with Zernike Fringe Sag Surface for Second Mirror

We have found that we can improve the performance of the system by introducing a freeform surface for Mirror 2. In this case, we used a Zernike Fringe Sag surface. The Zernike Fringe Sag surface in Zemax is defined by the same polynomial as the Even Aspheric surface (which supports planes, spheres, conics, and polynomial aspheres) plus additional aspheric terms defined by the Zernike Fringe coefficients. The surface sag is of the form:

𝑧𝑧 =𝑐𝑐𝑟𝑟2

1 + �1 − (1 + 𝑘𝑘)𝑐𝑐2𝑟𝑟2+ �𝛼𝛼𝑖𝑖𝑟𝑟2𝑖𝑖 + �𝐴𝐴𝑖𝑖𝑍𝑍𝑖𝑖(𝜌𝜌,𝜑𝜑)

𝑁𝑁

𝑖𝑖=1

8

𝑖𝑖=1

where N is the number of Zernike coefficients in the series, Ai is the coefficient on the ith Zernike Fringe polynomial, r is the radial ray coordinate, ρ is the normalized radial ray coordinate, and φ is the angular ray coordinate.

The deviation of the freeform from spherical is shown in the sag plot below. The freeform nature of the surface can be seen.

Radius of Curvature -160 Zernike Term 4 2.56147E-02 Zernike Term 5 2.70871E-02 Zernike Term 8 -2.33753E-02 Zernike Term 9 1.33160E-04 Zernike Term 11 2.68662E-03 Zernike Term 12 4.02365E-05 Zernike Term 15 -9.07766E-05 Zernike Term 16 -8.56130E-07 Zernike Term 17 -1.26045E-05 Zernike Term 20 -1.12562E-05 Zernike Term 21 1.47691E-06 Zernike Term 24 -8.68686E-07 Zernike Term 25 -1.82192E-08 Zernike Term 27 5.43139E-06 Zernike Term 28 4.64378E-08 Zernike Term 31 1.01767E-07 Zernike Term 32 1.02875E-08 Zernike Term 35 3.06499E-08 Zernike Term 36 2.72167E-08

In this case, 36 Zernike terms were used, with 19 terms variable. Optimization using a merit function to minimize the RMS Spot Size yielded a system identical to the All-Spherical TMA design but with the addition of the following non-zero Zernike terms on the second mirror (terms which have a value of zero have been omitted for brevity):

This freeform surface has greatly improved the performance of the TMA. The scale of the Spot Diagram below has been reduced by a factor of 10X and is now 1 mm:

Radius of Curvature -160 Maximum Order in X 5 Maximum Order in Y 5

Normalization Length in X 25 Normalization Length in Y 25

c(0,0) 6.846533e-005 c(2,0) 5.6597464e-005 c(3,0) 6.8495512e-010 c(4,0) 1.8267885e-007 c(1,1) 2.5264952e-009 c(2,1) -5.0488952e-005 c(4,1) -7.8491421e-008 c(0,2) 1.2814272e-005 c(1,2) 1.7504961e-009 c(2,2) 6.7628862e-007 c(4,2) 3.0845063e-008 c(0,3) -2.9697643e-005 c(0,4) 1.3489457e-007 c(2,4) 2.9257892e-008 c(4,4) 1.2646488e-008

TMA Design with Chebyshev Polynomial Surface for Second Mirror

Now we will look at what happens when we use a Chebyshev Polynomial surface instead of a Zernike Fringe Sag surface for the second mirror in the TMA. Using maximum orders of 5 in both X and Y and 14 variable polynomial terms, we obtain the following surface description (with values of 0 omitted for brevity):

The deviation of the Chebyshev surface from spherical is shown at the same scale as the Zernike TMA sag. The Chebyshev surface exhibits greater sag, shown in red, along the edges than the Zernike surface.

This led to a slight improvement in performance, as compared to the Zernike TMA design. Note that the scale is 1 mm, and the RMS Spot radii have decreased very slightly throughout the field of view:

It is likely that during the manufacturing of this mirror, the part will be made with a larger diameter than the final diameter. Let’s look at the surface sag using a diameter of 60 mm instead of 50 mm, for the Chebyshev formulation (top, below), and compare it to the same diameter enlargement for the Zernike design (bottom, below):

The surface sag plots above are shown at the same scale, and it is evident that at the edges of the part, outside the aperture the surface was designed for, there is a much greater deviation in sag when the Zernike surface description is used than when the Chebyshev Polynomial description is used. The Chebyshev Polynomial description holds this significant advantage over the Zernike description from a manufacturability viewpoint, as it provides the user more control over the sag at the edges of the element.

Conclusion

We have provided examples of the usage of the new Chebyshev Polynomial surface in Zemax OpticStudio in the design of a reflector for uniform illumination as well as a TMA. The illuminator design incorporating the Chebyshev Polynomial surface was shown to be comparable to the same design utilizing an XY Polynomial surface. The all-spherical TMA design was improved by using a Zernike surface type, and was then further improved using a Chebyshev Polynomial surface for better manufacturability. In addition to improving manufacturability, the Chebyshev Polynomial surface also provides the user with an option supporting varying degrees of asymmetry, which can be helpful for cases where the final design form is not known at the start of the optimization process.

References

A. Sommariva, M. Vianello, and R. Zanovello, “Adaptive bivariate Chebyshev approximation”, Numerical Algorithms Vol. 38, Issue 1-3, pp 79-94, 2005.

Fei Liu ; Brian M. Robinson ; Patrick J. Reardon and Joseph M. Geary, "Analyzing optics test data on rectangular apertures using 2-D Chebyshev polynomials", Opt. Eng. 50(4), 043609 (April 29, 2011).

http://proceedings.spiedigitallibrary.org/volume.aspx?conferenceid=3419&volumeid=16825

http://en.wikipedia.org/wiki/Chebyshev_polynomials

Acknowledgement

This work has been supported by the German government within the fo+ freeform optics project.

Contributing Authors: Asphericon GmbH: Ulrike Fuchs, Sven Kiontke Zemax LLC: Betsy Goodwin, Erin Elliott, Sanjay Gangadhara, Vladimir Smagley, Alison Yates