OpticStudio provides a number of tools for the design of freeform optics in both Sequential and Non-Sequential modes. In this article, we’ll provide an example of an off-axis parabola using the Chebyshev Polynomial surface in Sequential mode. We’ll also describe filtering tools available in the Lens Data Editor that quickly allow you to find the freeform surface of choice for any desired application from the more than 20 such surfaces that are currently supported in Sequential mode.

Authored By Erin Elliott

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Introduction

Freeform optical components are complex parts with more degrees of freedom than the typical spherical optic. They are more difficult to manufacture but make up for that fact by allowing systems to be reduced in size. Freeform optics may be applied in any system, and are already found in antennas, laser beam shapers, and in the Hubble space telescope - among others.¹

OpticStudio provides several tools for designing freeform optics in Sequential and Non-Sequential mode. This article will provide an example of generating an off-axis parabola using the Chebyshev Polynomial surface in Sequential mode. The article will also discuss how to filter surface types to quickly choose freeform optics for any desired application.

Chebyshev Polynomial Surface

One of the many examples of the freeform surfaces available in OpticStudio is based on Chebyshev polynomials. These 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 the definition of non-rotationally symmetric systems and non-elliptical apertures.

The Chebyshev polynomial of the first kind is given by formula:

The first ten Chebyshev polynomial coefficients 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):  

Using a finite sum of the Chebyshev polynomial terms, the resulting sag equation for this freeform surface is:

Here, a_ij are the coefficients of the Chebyshev polynomial sum; x, y are normalized surface coordinates; N and M are the maximum polynomial orders in x and y dimensions; 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.

Creating an Off-Axis Parabola Using the Chebyshev Polynomial Surface

As a simple example, we’ll use the Chebyshev Polynomial surface to create an off-axis parabola, as shown below in Figure 1. We’ll specify the system as follows:

Parameter	Value	Notes
Focal length	500 mm
Mirror aperture	50 x 50 mm
Decenter distance	60 mm	Large enough that the focal plane lies outside the incoming beam.
Entrance pupil diameter	71 mm	Sized to contain the 50 x 50 mm square mirror
Wavelength	1µm
Field angle(s)	0	Angle for theoretically perfect performance

 

Figure 1: Shaded Model layout of an off-axis parabola created with a Chebyshev Polynomial surface.

A coordinate break on Surface 2 shifts the surface 60 mm in the y direction. Surface 3 contains the Chebyshev Polynomial surface. The thickness of the surface has been set to -500 mm, since we want a focal length of 500 mm. The material type is set to “Mirror” to create a reflective surface. 

Figure 2: The lens data editor, showing the Chebyshev Polynomial surface in line 3.

The Surface 3 Properties window is shown in Figure 3.  The aperture of the Chebyshev Polynomial surface is set to 50x50 mm with a -60 mm decenter. We’ve set the system’s Entrance Pupil Diameter (not shown) to 71 mm so that it fully contains the square mirror.

Figure 3: The Surface 3 Properties window is used to specify the mirror’s square and decentered aperture.

For Chebyshev Polynomial surfaces, the C(2,0) coefficient is associated with the polynomial T₂(x) • T₀(y), or 2x² – 1, while the C(0,2) coefficient is associated with the polynomial T₀(x) • T₂(y), or 2y² – 1. If the values of C(2,0) and C(0,2) are equal, the result is a traditional rotationally-symmetric parabola. Note that the curvature of the base sphere in the Chebyshev Polynomial surface has been set to zero (see Figure 2), allowing the C(2,0) and C(0,2) terms to fully specify the curvature of the surface.   

The maximum term for the Chebyshev Polynomial surface in this case, then, will be 2. The setting is shown in Figure 4. The normalization distances in x and y are also shown in Figure 4. We’ve set both to 85 mm, which is the decenter distance plus the half-width of the mirror, or the radius of the “parent” parabola.

Figure 4: Normalization lengths in x and y have been set to 85 mm, and the maximum orders in x and y have been set to 2.

To create the parabola, the C(2,0) and C(0,2) surfaces are set as variables as shown in Figure 5. We’ve entered a guess for the coefficient values of (-0.0001). A quick look at the Shaded Model plot (Figure 6) shows that the values are not quite correct to achieve focus at the specified image plane.

Figure 5: To create a parabolic surface using the Chebyshev Polynomial surface, the C(2,0) and C(0,2) coefficients must be specified. If the two values are equal, the result is a traditional rotationally-symmetric parabola.

Figure 6: Shaded Model plot showing that the Chebyshev polynomial coefficient values aren’t yet correct to achieve focus at the desired image plane.

We can use the default merit function to optimize the system, provided we’ve chosen the correct settings for the square mirror aperture. We use an additional REAY operand as a shortcut, because if the parabola is specified correctly in the y-coordinate direction, the chief ray must fall at y=0 on the image plane. The merit function setup and the resulting merit function are shown in Figure 7 and Figure 8.

After optimization, the expected off-axis parabola results as shown in Figure 9 and Figure 10. The correct values for the C(2,0) and C(0,2) coefficients are -2.5e-4. The spot diagram shows the expected perfect performance.

Figure 7: Setup of the default merit function for the system.

Figure 8: The first few lines of the merit function for the system.

Figure 9: The optimization routine finds the correct values of C(2,0) and C(0,2) for the Chebyshev Polynomial surface.

Figure 10: The 3D Layout plot and spot diagrams show that the off-axis parabola is behaving as expected.

Finding Your Freeform

Filtering capabilities are provided in the Lens Data Editor to help you find the right surface quickly. When you want to see the freeform surface types available in OpticStudio, simply select the Freeform radio button to see the more than 20 surfaces included in this group. If an Even Asphere or Q-Type will suit your application, select “Conventional” surfaces to see the most common surface types.

Figure 11: Surface type may be filtered by category in the Lens Data Editor.

The full list of freeform surfaces includes surfaces represented by polynomial descriptions, diffractive surfaces based on polynomials, and surfaces that are fit from control points.

Polynomial Freeform Surfaces

• Biconic Zernike
• Chebyshev Polynomial
• Cylinder Fresnel
• Extended Fresnel
• Generalized Fresnel
• Odd Asphere and Extended Odd Asphere
• Polynomial and Extended Polynomial
• Superconic
• Zernike Annular Standard Sag
• Zernike Fringe Sag
• Zernike Standard Sag

Diffractive Freeform Surfaces

• Elliptical Grating 1 and Elliptical Grating 2
• Toroidal Grating and Extended Toroidal Grating

Freeform Fits to Control Points

• Cubic Spline and Extended Cubic Spline
• Grid Gradient
• Grid Sag
• Radial NURBS
• Toroidal NURBS

References

1. M. Tricard and D. Bajuk, "Practical Examples of FreeForm Optics," in Renewable Energy and the Environment, OSA Technical Digest (online) (Optical Society of America, 2013), paper FT3B.2. 

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