November 27, 2020

Korsch telescope

Industry: Optics | Software: pSeven| Company: THALES

Objective

Korsch-type telescopes are optical systems often used in space telescopes and Earth observation satellites. A Korsch telescope consists of three mirrors and an image plane, as shown in Fig. 1. Korsch telescopes are typically designed with a specified focal length and the objective is to minimize the average spot area. Ideally, a spot area of zero would indicate that a point is truly represented as a point on the image plane.

In the use case presented here, the longitudinal positions of the three mirrors and the image plane are fixed. The design variables of the problem are the radius of curvature and the conic constant value of each mirror, representing a total of 6 design variables.

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Figure 1 – Ray tracing diagram of the Korsch telescope.

To analyze the quality of an optical design, several rays of light coming from different positions are simulated (ray tracing). In this case, CODE V is used as the optical simulation tool. The pSeven platform is used to automate the creation of optical designs and their evaluation in CODE V and perform the optimization of the system. First, we use an analytical relationship between the radii of curvature to suppress the equality constraint on the focal length and eliminate one radius of curvature as a design variable. Then, a new parameterization for the remaining radii is introduced, which allows to tackle the problem and find solutions of the required quality in terms of the average spot area. This problem has a best known design point, whose coordinates and objective value will be used to assess the quality of the proposed methodology. It will be used for comparison with the results of the optimization method that we propose, where ray tracing is considered as a blackbox.

Challenges

  • When considering the original independent parameters (3 radii and 3 conicities) within the initial prescribed bounds, the range of the function goes from values of the order of 1.e-3 to the order of 1.e8. Therefore, this problem presents multiple scales.
  • Within the initial bounds there is a fraction of about 14% of points without a defined physical output (NaN) for the objective function.
  • The constraint function of the focal length exhibits infinite discontinuities.
  • The optimization problem is multimodal: several local minima exist, even within the scale that presents the most promising objective function values.

Solution

Process Automatiom

CODE V simulations are driven from pSeven’s PythonScript block using the Python API available in CODE V. The PythonScript enables to open a single CODE V instance and feed it with the automatically generated input files, avoiding the overhead of launching a new CODE V instance for each function call. By using the “Keep globals” option in pSeven’s PythonScript block, allows to keep the same instance of the Python object used to communicate with CODE V. This allows to perform up to four CODE V calls per second, compared to starting up a new CODE V instance for each call, which takes more than one second per call.

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Figure 2 – CODE V integration in pSeven.

Problem Description

The Korsch telescope consists of 3 mirrors. Each mirror i has 2 design parameters: the radius of curvature \(R_{i}\) (or \(1/k_{i})\), where \(k_{i}\) is the curvature, and the conic constant \(c_{i}\) . This is illustrated in Fig. 3. The conic constant is related to the eccentricity of the conic section representing the optical surface. For example, values greater than -1 represent an elliptical surface (with the special case of the spherical surface for a value of zero), a value of -1 represents a parabolic surface, and values lower than -1 represent hyperbolic surfaces [2].

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Figure 3 – Lens diagram of the Korsch telescope.

In this particular case, the distances between the mirrors are fixed. The goal is to minimize the average of the RMS spot size values (denoted as f) computed at different values of the field angle (0° and 0.5°), which is the angle between the chief ray and the main axis. At the same time, it must be ensured that the focal length is equal to 12000 mm.

Focal length constraint

Since the focal length (F) considered in this problem is obtained using the paraxial approximation, it can be determined analytically as a function [1] of the radii of curvature:

\(F=\frac{1}{2}\frac{R_{1}R_{2}R_{3}}{\tilde{R}_{1}R_{2} - \tilde{R}_{1}\tilde{R}_{3} + R_{2}\tilde{R}_{3}}\),

with \(\tilde{R}_{1} = R_{1} + 2d_{1}\), and \(\tilde{R}_{3} = R_{3} + 2d_{3}\), where \(d_{1} = 500\) and \(d_{3} = 1000\) are constants that depend on the distance between the mirrors.

This allows to choose one radius (\(R_{3}\), for example) and express it as a function of the target focal length and the remaining radius. In that manner, \(R_{1}\) is no longer an independent variable and the constraint on the focal length (in the paraxial order) is automatically satisfied.

Known solution

The coordinates of the known analytical solution are:

\(R_{1}\) -1306.17 mm
\(R_{2}\) -400.10 mm
\(R_{3}\) -564.65 mm
\(c_{1}\) -0.987812
\(c_{2}\) -2.4957816
\(c_{3}\) -0.6815093

This configuration gives a performance of f = 4.27e-3 mm.

Initial Parametrization and Preliminary Design Space Exploration

For this use case, the radii of curvature of the mirrors can vary between -3000 and -100 mm (the sign of the radius indicates the orientation of the curvature). In this case, the conic constant of each mirror is allowed to vary between -3 and 0. special.

Since, in the considered design space, the response function f can take values that range from the order of 1.e-3 (the order of magnitude of the best known performance) up to observed values of the order of 1.e8, it is convenient to apply a logarithmic transformation (in this case, the natural logarithm).

Before actually optimizing the problem, we perform a preliminary exploration of the design space to show the extreme range of the objective function. For this preliminary exploration, and to be able to visualize the function, we use two variables only: the curvatures (\(k_{1} = 1/R_{i})\) of mirrors 2 and 3. The curvature of the first mirror is determined using the focal length equation. The remaining variables (the conicities) are all set to zero. On Fig. 4, we can see how the transformed output (natural logarithm applied) ranges from values around 1 to values around 18. On Fig. 5, we zoom into the low values of the \(k_{3}\) curvature and plot slices of the objective function for several constant values of \(k_{2}\). Note that, on Fig. 5, the bottom of the valley is not smooth, which shows multimodality at the low scale level.

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Figure 4 – Natural logarithm of the objective function as a function of  \(k_{2}\) and  \(k_{3}\).

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Figure 5 – Slices across the valley observed in Fig. 4. zoomed in for low values of  \(k_{3}\).

Effective Parametrization and Optimization After this preliminary design space exploration to understand the behavior of the objective function, we move forward to the actual optimization of the problem. For the optimization, the radius of curvature of the third mirror is set as the dependent one (through the focal length equation). The remaining curvatures are not used directly as design variables, but a new parameterization for these two variables, which uses information from the focal length equation, is used. The conicities are used directly as design variables.

Results

Using this approach, the best point obtained is:

\(R_{1}\) -1300.14 mm
\(R_{2}\) -390.46 mm
\(R_{3}\) -569.24 mm
\(c_{1}\) -0.9875281
\(c_{2}\) -2.4556663
\(c_{3}\) -0.6934065

The best found configuration has a performance of f = 1.39e-3 mm, which is lower than the best known configuration (f = 4.27e-3 mm), while ensuring the focal length constraint of 12000 mm.

Conclusion

  • pSeven is used to automate the creation of optical designs and their evaluation in CODE V and perform the optimization of the system.
  • CODE V can be integrated in pSeven using CODE V’s API, enabling fast simulation updates with negligible overhead thanks to the ability of pSeven’s PythonScript block to keep Python object instances between runs.
  • Initial global design of experiments study revealed a very special behavior of the responses and an effective parametrization was proposed to address multiscale and multimodal nature of the spot size function.
  • Two steps of optimization search, global and localized, were performed in an automated way to address the specific function shape, using the different optimization methods from the pSeven collection.
  • The obtained solution performs better than the best known point while ensuring the equality of the focal length constraint.
  • The developed methodology could be applied to similar problems if the analytical paraxial approximation of the focal length is available.

By Joan Mas Colomer, Application Engineers at pSeven SAS and Sébastien Héron, Optical Engineer at Thales Group

Acknowledgement:

Special acknowledgement to Thales Research & Technology and Sebastien Héron for the selection of this case study, their active contribution and partial financial support to the development of the methodology and its implementation. Contact: sebastien.heron(at)thalesgroup.com

References

- [1] GERRARD, Anthony et BURCH, James M. Introduction to matrix methods in optics. Courier Corporation, 1994.

- [2] SMITH, Warren J. Modern optical engineering. Tata McGraw-Hill Education, 2008.

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