Alignment sensitivity of reﬂective optical elements and analysis of automated alignment methods

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Alignment sensitivity of reflective optical elements and analysis of automated alignment methods

Chris van Ewijk November 2017

1 Introduction

The mission of SRON is to bring about breakthroughs in international space research. Therefore the institute develops pioneering technology and advanced space instruments, which typically operate in vacuum and cryogenic environments. Imaging space instruments and their testing setups have to be aligned in order to create high quality images. Here, alignment is the tweaking of optical elements to reduce the optical aberrations in the signal. The alignment of these space instruments and their testing systems is usually done at room temperature. However, when cooling down to cryogenic temperatures, the elements of a system typically shrink and thus do not maintain their original alignment positions.

Triggered by ESA’s tender ”Novel in-vacuum alignment and assembly tech- nologies for optical assemblies” and experience at SRON with optical alignment in cryogenic conditions, a new method has been proposed by Huisman and Eggens [6] to perform optical alignment with sub-micrometer accuracy in vacuum and cryogenic environments. This method can increase the accuracy of the alignment as it can account for typical changes in alignment observed when cooling down from room temperature to cryogenic temperatures. The method also makes alignment at the operating wavelengths realistic, if this is not possible under atmospheric pressure.

The method utilizes pi¨ezo elements for simultaneous positioning, sensing and control over the friction forces between the mirror and its support structure.

After alignment of the optical system, the same pi¨ezo elements can be used for fixation of the mirror in the cryogenic environment.

The combination of actuation, position sensing and the control over the friction forces creates the possibility for automated alignment in a closed loop configuration when using a wavefront sensor to measure system aberrations. Simultane- ous to our research, three students from the Windesheim University of applied sciences have been working on the hardware of the proposed method.

Computer aided alignment methods using the relation between optical aberrations and alignment state have been developed to make alignment more efficient. One of those methods is the sensitivity table (ST) method [15], where the alignment state of an optical system is determined by a linear sensitivity table and measurements of the wavefront. In order to overcome the non-linearity of aberrations the merit function regression method (MFR) was proposed [8], which determines the alignment state of an optical system by simulation of the optical system in an attempt to reproduce the measured wavefront. When aligning multi-element systems Lee et Al. [11] showed that there exists a coupling effect between optical elements. They introduced the differential wavefront sampling (DWS) method [12] for alignment of centered multi-element systems, which uses second derivative information of the wavefront.

In this thesis a basic understanding of the most important optical aberrations

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is provided, whereafter the sensitivity of standard reflective optical elements to displacements and tilts is investigated to create a general idea of the important degrees of freedom for each element separately. We only focus on the aberrations induced by misalignment of the optical elements. Surface shape errors such as radius of curvature and surface roughness are ignored. Several automated alignment methods are discussed for a multi-element system and where possible tested by simulation in the lens design software Zemax. The final goal of the thesis is to determine under which assumptions and limitations each automated alignment method can be used, in order to determine the functionality in our application.

A recent example of an optical system for which the automated alignment in vacuum and cryogenic environments is applicable is the Safari OGSE Re-Imager.

The optical layout of the system is given in figure 1.1 and the detailed lens data is given in the appendix A.1. This optical system is used to focus the incoming light rays from a point source via two off-axis parabolic mirrors and two fold mirrors onto the entrance focal plane of the Safari instrument for the SPICA space telescope. Therefore it is important that the Safari OGSE Re-Imager does not induce aberrations in the signal. Since a detailed description of the system in the lens design software Zemax was already present, and the system is relevant to our research, the OGSE Re-Imager is used in the simulations done throughout this thesis.

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2 Introduction to optical aberrations

The general concept of an optical system of two lenses and an aperture stop is shown in figure 2.1. The rays leave the point object and propagate through the system to focus on the image point. The limiting diameter which determines the amount of light that reaches the image plane is called the aperture stop.

The entrance pupil is the optical image of the aperture stop, as seen from the object. The corresponding image of the aperture as seen through the image plane is called the exit pupil. The quality of an optical image can be defined by the wavefront aberrations of the system at the exit pupil. For an aberration free image the wavefront at the exit pupil must be spherical, with its center of curvature at the image plane.

In real applications the diffraction limit plays an important role in the image quality, yet it is system and wavelength dependent. Therefore we ignore the diffraction limit in our analysis, where it should be taken as a criteria when aligning an real optical system.

Figure 2.1: Entrance and exit pupils of an optical system consisting of two lenses and an aperture stop. The reference spherical wavefront is shown in black, whereas the aberrated wavefront is shown in red [4].

In figure 2.2 the marginal and chief rays are displayed. The ray leaving the

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origin of the object and passing through the maximum aperture of the system is known as the marginal ray. The chief ray originates from an off-axis point in the object and passes through the center of the system. We also consider paraxial rays, which are rays that make a small angle to the optical axis of the system and lay close to that axis throughout the system.

Figure 2.2: A simple lens system and aperture stop with illustrations of the marginal and chief rays [2].

The wavefront aberration function W (x, y) describes the optical path difference between the aberrated wavefront and the ideal spherical reference wavefront. Aberrations of a rotationally symmetric system can be expressed by expanding the wave aberration function in a power series of pupil and field coordinates, ρ, θ and H:

WIJ K ⇒ H^Iρ^Jcos^K(θ)

W (H, ρ, θ) = W₀₀₀+ W₀₂₀ρ²+ W₁₁₁Hρ cos(θ) + W₀₄₀ρ⁴+ W₁₃₁Hρ³cos(θ) + W₂₂₂H²ρ²cos²(θ) + W₂₂₀H²ρ²+ W₃₁₁H³ρ cos(θ) + O(6)

(2.1) where H is the normalized field coordinate and ρ and θ denote the normalized pupil coordinates as shown in figure 2.3. The other terms in the power series, for example W123, are forbidden by the rotational symmetry. In our study the higher order terms (O(6)) are neglected as their influence on the optical image is negligible. The coefficients of the lower order terms are defined as:

W000: Piston W131: Coma W020: Defocus W222: Astigmatism W : Wavefront tilt W : Field curvature

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element such as surface roughness and radius of curvature. Defocus and wavefront tilt are called first order aberrations, whereas spherical, coma, astigmatism, field curvature and distortion are third order aberrations. In section 2.2 these common aberrations and their dependence on the pupil coordinates are discussed.

Figure 2.3: Normalized field coordinate H and pupil coordinates ρx= ρ sin(θ) and ρy = ρ cos(θ) normalized over the exit pupil. [7].

2.1 Zernike polynomials

Zernike polynomials are a set of circular symmetric orthogonal basis functions defined over a unit circle, named after their inventor Frits Zernike [13]. The polynomials are 2D functions of radial and azimuthal coordinates and are important for optical design as they can describe a systems (aberrated) wavefront over the exit pupil, therefore optical design programs like Zemax make use of the Zernike polynomials. In this research the notation of Zernike polynomials introduced by Noll [14], as given in table 2.1, is used when simulating the wavefront in Zemax.

Polynomial Z_j(ρ, θ) Name

Z1 1 Piston

Z₂ 2ρ cos θ X Tilt

Z3 2ρ sin θ Y Tilt

Z₄ √

3(2ρ²− 1) Defocus Z5

√6ρ²sin 2θ

Primary Astigmatism Z6

√6ρ²cos 2θ

Z₇ √

8(3ρ²− 2ρ) sin θ

Primary Coma

Z₈ √

8(3ρ²− 2ρ) cos θ Z9

√8ρ³sin 3θ

Z₁₀ √

8ρ³cos 3θ Z11

√5(6ρ⁴− 6ρ²+ 1) Primary Spherical

Table 2.1: The first eleven standard Zernike polynomials [14] as a function of the pupil coordinates (ρ, θ).

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2.2 Common aberrations

Here the common aberrations as specified in equation 2.1 are discussed. We consider the dependence of the wavefront aberration function on the field coordinate H and the pupil coordinates (ρ, θ), as defined in figure 2.3. Where relevant the dependence of the wavefront aberration function on ρ is plotted and a wavefront plot is provided. The wavefront plot shows the deviation of the wavefront with respect to the reference sphere in the exit pupil.

2.2.1 Piston

Piston is the aberration where the wavefront is displaced with respect to the reference wavefront. The wavefront itself can be perfectly spherical, however the translation of the wavefront along the optical axis causes the image to be blurred. Piston occurs when the misalignment of optical elements of a system cause a translation of the wavefront along the optical axis. The result of piston is illustrated in figure 2.4. The aberration term W000does not depend on any of the field or pupil coordinates, thus the wavefront of piston is equally displaced with respect to the reference wavefront over the whole pupil, as shown in figure 2.5.

Figure 2.4: Piston aberration, the focal point is translated along the optical axis.

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Figure 2.5: The wavefront plot of piston, the green color indicates a difference between the aberrated and reference wavefront which is constant over the exit pupil.

2.2.2 Defocus

An optical system is in focus when the focal point of the system coincides with the image plane. Defocus is the aberration when a system is out of focus, thus when there is a translation of the focal point along the optical axis away from the image plane, see figure 2.6. This result is similar to the result of piston, however it has a different cause.

Figure 2.6: Defocus of a paraxial lens, the focal point does not coincide with the image plane due to a deviation of the radius of curvature of the wavefront.

The wavefront of a defocused system shows deviation in the radius of curvature with respect to the reference wavefront. When using optical elements such as lenses and spherical or parabolic mirrors, uncertainties in their radius of curvature can cause defocus. The aberration term W₀₂₀has a quadratic dependence on the pupil coordinate ρ, is rotationally symmetric and equal for all field positions. The dependence of the aberration function on the pupil coordinates and a plot of the wavefront aberration function over the exit pupil is shown in figure 2.7.

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Figure 2.7: (left) The dependence of the aberration function on the normalized pupil coordinate and (right) plot of the wavefront aberration function with defocus in the exit pupil [1]. Here the colors indicate the amount of displacement of the aberrated wavefront with respect to the reference wavefront, where blue indicates no difference and red indicates a large difference

2.2.3 Wavefront tilt

Wavefront tilt aberration occurs when one or more of the optical elements of a system is tilted or decentered. This results in a transverse displacement of the image with respect to the original position. The tilt or decenter of an optical element often generates higher order aberrations next to wavefront tilt [13]. The aberration term W111depends linearly on H, ρ and cos(θ). This means that on axis (H = 0), there is no wavefront tilt. The dependence of the aberration function on ρ for H = 1 and a plot of the wavefront aberration function over the exit pupil are shown in figure 2.9.

Figure 2.8: Illustration of wavefront Tilt [16].

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Figure 2.9: (left) The dependence of the aberration function on the normalized pupil coordinate for H = 1 and (right) wavefront plot of vertical tilt in the exit pupil [1].

2.2.4 Spherical aberration

Spherical aberration is the variation of focus positions between paraxial rays and marginal rays. Thus there is a variety of focal points dependent on where the rays pass through the pupil, as shown in figure 2.10. The aberration term W040 has a quartic dependence on the pupil coordinate ρ, appears on axis and its magnitude is independent of the field coordinate H. The dependence of the aberration function on ρ and the wavefront plot of spherical aberration is shown in figure 2.11.

Figure 2.10: Spherical aberration of a spherical mirror for a point source at infinity.

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Figure 2.11: (left) The dependence of the aberration function on the normalized pupil coordinate and (right) wavefront plot of spherical aberration in the exit pupil [1].

2.2.5 Coma

Coma is an off-axis aberration occurring when the magnification depends on the annular zone of the aperture. Coma frequently occurs when parabolic mirrors are used. When the incoming rays are not parallel to the axis of the mirror, the individual rays are not focused on the same point anymore. Point sources imaged with a system that introduces coma appear to have a tail like a comet as illustrated in figure 2.12. Coma does not appear on axis, as the aberration term W131depends linearly on H and cos(θ), whereas it shows cubic dependence on ρ. The dependence of the aberration function on ρ for H = 1 and a plot of the wavefront aberration function of spherical aberration is shown in figure 2.13.

Figure 2.12: Spot size diagram of coma induced by a parabolic mirror for rays incoming at 2^◦off-axis.

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Figure 2.13: (left) The dependence of the aberration function on the normalized pupil coordinate for H = 1 and (right) wavefront plot of vertical coma in the exit pupil [1].

2.2.6 Astigmatism

Astigmatism is an aberration that occurs when tangential rays focus at a different location than sagittal rays. There is no astigmatism on-axis and the magnitude increases with respect to the field coordinate H. For a system to be astigmatism free, the sagittal and tangential image surfaces should coincide with each other. Astigmatism causes a spot diagram of a point source to appear like an ellipse. The aberration term W₂₂₂ has a quadratic dependence on H, ρ and cos(θ), thus there is no astigmatism on axis. The dependence of the aberration function on ρ for H = 1 and a plot of the wavefront aberration function over the exit pupil is shown in figure 2.15.

Figure 2.14: Astigmatism of a lens [10]. Here the tangential and sagittal rays are focused at different points on the optical axis.

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Figure 2.15: (left) The dependence of the aberration function on the normalized pupil coordinate for H = 1 and (right) wavefront plot of astigmatism in the exit pupil [1].

2.2.7 Field curvature

An optical system with spherical surfaces produces an image on a curved surface, as illustrated in figure 2.16. This field curvature is an off-axis aberration as the aberration term W220has a quadratic dependence on ρ and H. The wavefront plot of field curvature looks similar to defocus, yet the magnitude depends on the field coordinate and is zero on axis.

Figure 2.16: The result of field curvature [18].

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Figure 2.17: The dependence of the aberration function on the normalized pupil coordinate for H = 1.

2.2.8 Distortion

Distortion is the aberration where the magnification of the image varies with the field coordinate H. Therefore systems with a large field of view are more sensitive to distortion than systems with a small field of view. There are two basic forms of distortion, called barrel and pincushion distortion. Barrel distortion occurs when the center of the image is magnified more than the edges, whereas pincushion is caused by an increasing magnification away from the center.

Figure 2.18: Barrel and Pincushion Distortion [3]

A combination of barrel and pincushion distortion is called mustache distortion, which starts as barrel distortion in the center of the field and gradually changes to pincushion distortion while moving to the edges of the field. This makes the upper horizontal lines of a uniform grid look like a handlebar mustache, hence the name.

The aberration term W311 has a cubic dependence on field coordinate H and linear dependence on ρ and cos(θ). Generally, design programs do not include

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distortion as part of their wavefront plots [7]. Instead distortion is plotted as a percentage versus field coordinate H, as in figure 2.19. The percentage is calculated from the difference in height between a real and a paraxial chief ray for a given field point.

Figure 2.19: Percentage of distortion versus field coordinate H [7]

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3 Alignment sensitivity analysis of standard mir- ror shapes

For the hardware study on the automation of the alignment process of an optical system it is important to know which degrees of freedom should be used to correct for aberrations. The insensitive degrees of freedom might be ignored to simplify the design of the mechanical interface.

To determine these important degrees of freedom four standard mirror shapes are studied: a parabolic, elliptical, spherical and fold mirror. The Zernike standard coefficients as discussed in section 2.1 are used to quantify optical aberrations. Zemax is used to calculate the sensitivity of the Zernike coefficients to perturbations in six degrees of freedom for each mirror separately. These results are meant to give a general insight on which degrees of freedom to manipulate during the alignment process and should not be used blindly. To determine the correct choice of degrees of freedom for optical elements in a system a detailed analysis of the system is necessary.

The coordinate system in Zemax is defined along the optical axis, thus the z-axis always coincides with the optical axis. The y- and x-axis are defined with respect to a top view of the system, where the y-axis is perpendicular to the z-axis and the x-axis points into the plane. The coordinate system is drawn in the left bottom of each illustration of the elements in Zemax for clarity.

We distinguish between on-axis and off-axis parabolic, spherical and elliptical mirrors. The off-axis mirrors are displaced 40mm along the y-axis, considered sufficient for observation of the off-axis influences. The point of rotation of the mirrors is the incident coordinate of the chief ray. This choice of coordinate systems is based on realistic mechanical interfaces and not the optical element on its own. In our analysis we did not fix the image plane, since in some optical systems the image plane can be corrected for.

The tolerances used are the mechanical fabrication tolerances determined for precision optical elements given in table 3.1.

Quality ∆X ∆Y ∆Z ∆θX ∆θY

Standard ±200µm ±200µm ±200µm ±57⁰18” ±57⁰18”

Precision ±40µm ±40µm ±40µm ±2⁰46” ±2⁰46”

High precision ±10µm ±10µm ±10µm ±41” ±41”

Table 3.1: Optical fabrication tolerances [5]

3.1 Zemax sensitivity analysis

The optical design program Zemax offers the possibility to do a sensitivity analysis on a user defined optical system. In the tolerance data editor the desired

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tolerances can be specified for each surface, where the tolerances for precision optics from table 3.1 are used. To generate the desired outputs the user defined merit function given in figure 3.1 was used. This merit function automatically reads out the requested values during the sensitivity analysis. For each element a sensitivity analysis was run for each Zernike polynomial separately.

Figure 3.1: Merit function used for sensitivity analysis to Z1

3.2 Results

3.2.1 Parabolic mirror

The sensitivity study is conducted on a parabolic mirror with a radius of 100mm and for a wavelength of 550nm. The Zemax model used is given in figure 3.2.

Figure 3.2: Zemax model of the on-axis parabolic mirror, where the point of rotation is the center of the mirror and the image plane is given in orange.

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Figure 3.3: Aberrations induced by fabrication tolerances on an on-axis parabolic mirror

The results of the sensitivity analysis for the on-axis parabolic mirror are given in figure 3.3. Displacement of the on-axis parabolic mirror along the Z- axis induces a significant amount of piston and defocus. Tilts about the X and Y axis induce a number of third order aberrations.

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Figure 3.4: Aberrations induced by fabrication tolerances on an off-axis parabolic mirror

In figure 3.4 the results of the sensitivity analysis for an off-axis parabolic mirror are displayed. A significant amount of first order aberrations are induced by displacement along the optical axis and tilt around the x-axis. Third order aberrations are induced by displacement along the optical axis and tilts about the X and Y axis. A clear difference between on- and off-axis parabolic mirrors is the sensitivity to tilts, especially ∆θ_X.

3.2.2 Spherical mirror

A spherical mirror with radius 100mm was used in the sensitivity analysis for a wavelength of 550nm. The Zemax model used is given in figure 3.5.

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Figure 3.5: Zemax model of the on-axis spherical mirror, where the point of rotation is the center of the mirror and the image plane is given in orange.

Figure 3.6: Aberrations induced by fabrication tolerances on an on-axis spherical mirror

The results shown in figure 3.6 show that again displacement along the optical axis induces piston and defocus. Since the sensitivities for ∆X and ∆θY are similar, freedom in one of those degrees should suffice. The same goes for

∆Y and ∆θX.

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Figure 3.7: Aberrations induced by fabrication tolerances on an off-axis spherical mirror

For an off-axis spherical mirror the important degrees of freedom are determined from figure 3.7. It can easily be seen that the most sensitive degrees are

∆Y , ∆Z and ∆θ_X. Since these perturbations induce the same aberrations one could chose to pick the most sensitive degree of freedom, ∆θ_X. However, if one would like to be able to correct for X-tilt, 45^◦ astigmatism or primary x-coma either ∆X, ∆θY or ∆θZ should be a parameter too. It is easily seen that the off-axis spherical mirror is more sensitive to misalignment’s than the on-axis spherical mirror.

3.2.3 Elliptical mirror

Here an elliptical mirror with a radius 100m and a conic constant of 0.4 was used in the sensitivity analysis for a wavelength of 550nm. The Zemax model used is given in figure 3.8.

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Figure 3.8: Zemax model of the on-axis elliptical mirror, where the point of rotation is the center of the mirror and the image plane is given in orange.

Figure 3.9: Aberrations induced by fabrication tolerances on an on-axis elliptical mirror

As seen from figure 3.9 displacement along the optical axis again induces piston and defocus. The sensitivities for ∆X and ∆θY are similar, freedom in one of those degrees should suffice. The same goes for ∆Y and ∆θ_X.

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Figure 3.10: Aberrations induced by fabrication tolerances on an off-axis elliptical mirror

The off-axis elliptical mirror sensitivities are similar to those of an off-axis spherical mirror yet on a different extend, which is shown in figure 3.10. There- fore the same conclusion can be drawn about the important degrees of freedom.

Again the off-axis variant of the mirror is shown to be more sensitive to misalignment’s.

3.2.4 Fold mirror

The Zemax model of the fold mirror is shown in figure 3.11. The fold mirror has been tilted 45^◦ along ∆θ_X in its original state for clarity.

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Figure 3.11: Zemax model of the fold mirror, where the point of rotation is the center of the mirror and the image plane is given in orange.

Figure 3.12: Aberrations induced by fabrication tolerances on a fold mirror.

As seen in figure 3.12 the most sensitive degrees of freedom are ∆Y and ∆Z.

Since the aberrations induced are the same, only ∆Z should suffice. However, for optical systems where the image plane is fixed, tilts should be considered in order to aim the rays on the image plane.

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3.3 Discussion

From studying the influence of perturbations in six degrees of freedom on aberrations the results shown in table 3.2 were found. Here we determined the degrees of freedom needed to be able to correct for all types of aberrations.

∆X ∆Y ∆Z ∆θ_X ∆θ_Y ∆θ_Z

Parabolic on-axis ! ! !

Parabolic off-axis ! ! !

Spherical on-axis ! ! !

Spherical off-axis ! !

Elliptical on-axis ! ! !

Elliptical off-axis ! !

Fold mirror ! !

Table 3.2: Degrees of freedom of standard mirror shapes needed for compensa- tion of all aberrations

Again it should be stressed that these results are determined for single optical elements and do not apply to all systems. These results can be used to create a general idea about the sensitivity of an optical system to misalignments. When aligning an optical system a sensitivity analysis of the system is required to determine the important degrees of freedom. In this study only the aberrations that effect the shape of the wavefront are considered. When the image plane is fixed the aiming of the rays should be taken into account when determining the important degrees of freedom.

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4 Sensitivity table method

During the alignment of an optical system the aberrations of the wavefront can be related to the alignment state of the elements in the system. Assuming that the Zernike polynomial coefficients are linearly dependent on perturbations of an optical component, the relation between the Zernike coefficients (Z_i) and the alignment parameters (P_i) can be expressed like:

∆Z = A∆P (4.1)

where ∆Z, A & ∆P are given by:

∆Z =







Z1− Zo1

... Zm− Zom





 , A =







∂Z₁

∂P1

. . . ∂Z₁

∂Pn

... . .. ...

∂Zm

∂P1

. . . ∂Zm

∂Pn







, ∆P =







P1− Po1

... Pn− Pon







. (4.2)

Here ∆Z is the difference between the measured and designed Zernike coefficients, m is the order of the Zernike coefficient, A is the sensitivity table of the optical system and ∆P is the amount of offset in the alignment parameters (Pi) of the current system defined as the difference between the measured and ideal alignment parameters. In our case the alignment parameters consist of the six degrees of freedom of an optical element, thus P₁= ∆X, P₂= ∆Y, . . . , P₆ =

∆θ_z. The values of these parameters can be obtained by determining the inverse of the sensitivity matrix and solving equation 4.1.

To use this method we need to determine the sensitivity table A, designed Zernike coefficients and the ideal alignment parameters from the optical system.

This can be done in optical design software such as Zemax. However, a ideal configuration model of the system is required. Oh et al. [15] determined the sensitivity table not with a computational model, but in experiment, thereby already including measurement field pointing uncertainties, apparatus control uncertainties and surface irregularities of the optical elements.

Expected downsides of this method are primarily the dependency on the linear relationship, as it is only accurate as long as the Zernike coefficients sensitivity to he alignment perturbations is linear. Therefore if the misalignment would be large, the non-linearity would cause a large residual error.

Applying the method is done in the following steps:

1. Perturb the optical system by known amounts to calculate the sensitivity of the system. With this data the sensitivity matrix A of an optical element can be constructed.

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2. Determine ∆Z by measuring the wavefront and subtracting the desired wavefront.

3. Solve equation 4.1 to obtain the misalignment parameters.

4.1 Testing method of the sensitivity table

When aligning a multi-element optical system the sensitivity of the optical elements depends on the alignment state of all the others, introduces as the multi-element coupling effect by Lee et al. [11]. Since this method does not take multi-element coupling into account we can only construct the sensitivity matrix for each element separately. Therefore in the alignment procedure we assume that only one element is needed for correction. In this case we use the OGSE model in Zemax from [5]. The sensitivity table is constructed for the parabolic mirror M1 by performing a tolerance analysis in Zemax with the following tolerances: +0.05mm for ∆X & ∆Y and +0.05^◦for ∆θX, ∆θY & ∆θZ. This matrix was then implemented in Matlab. The Zernike standard coefficients are given by Zemax and using the in build function mlrdevide from Matlab the misalignment’s are calculated. As a standard the first eleven Zernike coefficients are used. To evaluate the aberrations in the wavefront at the exit pupil, the merit function shown in figure B.1 was constructed.

4.2 Results of the sensitivity table method

First the parabolic mirror M1 was perturbed by +0.1mm, +0.12mm for respectively ∆X & ∆Y and +0.06^◦ and +0.07^◦ for ∆θX & ∆θZ. The calculated misalignments are given in table 4.1.

Original Calculated by the ST method Difference

M1

∆X 0.1 mm -0.3418 mm -0.4418 mm

∆Y 0.12 mm 0.1189 mm -0.0011mm

∆θX 0.06^◦ 0.06^◦ 0 ^◦

∆θY 0 ^◦ 0.0004^◦ 0.0004^◦

∆θZ 0.07^◦ 0.0289^◦ -0.0411 ^◦

Table 4.1: Sensitivity table test for mirror 1

It is clear that the difference in orientation after alignment for ∆X is ex- ceptionally large. This has two possible explanations: either it is caused by the insensitivity of ∆X as discussed in section 3.2.1, or the displacement of 0.1mm is too large and thus violating the linear relationship. The rest of the parameters are calculated accurately. The merit function given in figure B.1 is used to

−6

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Secondly we perturb both M1 and M2 to investigate whether the method could be used to align a system where multiple elements are misaligned. The corrections will be implemented on M1 only. With the original perturbations given in table 4.2 implementation of the sensitivity table method generated the results shown in table 4.3. The value of the merit function is given to be 2.45 × 10⁻⁶ in the unperturbed state, 4.77 × 10⁻³ in the perturbed state and 3.30 × 10⁻² after alignment. Therefore we conclude that the sensitivity table method is unable to correct the aberrations of the system when two mirrors are perturbed.

∆X ∆Y ∆Z ∆θX ∆θY ∆θZ

M1 0.02mm 0.02mm 0.04mm 0.05^◦ 0.01^◦ 0.02^◦ M2 0.02mm 0.02mm 0.04mm 0.016^◦ 0.08^◦ 0.02^◦

Table 4.2: Start perturbations for the sensitivity table method test on multiple elements

∆X ∆Y ∆Z ∆θX ∆θY ∆θZ

M1 2.8293mm -0.2827mm 0.03557mm 0.08913^◦ 0.3976^◦ -0.05821^◦ Table 4.3: Calculated displacements for M1 by the sensitivity table method

4.3 Discussion

Aligning a system where only one element was perturbed, the sensitivity table method was able to accurately reduce the aberrations in the system. Thus when a simple optical system with only one perturbed element has to be aligned this method would suffice. However, the method was shown to be unable to determine the correct misalignments in the alignment of the system where two elements are perturbed, which is caused by the inter-element coupling effect.

This means that the sensitivities depend on the alignment state of the other elements. When applying the corrections resulting from the sensitivity table method from table 4.3, the aberrations in the system had become even worse.

Therefore we conclude that the sensitivity table method is not applicable for our alignment method.

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5 Merit function regression method

The Merit function is defined as:

M F =

sP Wi(Vi− Ti)² P Wi

(5.1)

where Vi and Ti are the current and target values of the chosen parameter, which in our case are the Zernike coefficients and W_i are the weights. The merit function optimization algorithm is used in lens design software such as Zemax.

This method does not rely on the predetermined sensitivity of the Zernike coefficients. Since the MFR method deals with current and target values only, the estimation of misalignment is not affected by the amount of initial misalignment’s.

For the use of the MFR method a detailed Zemax model of the optical system is required. Effectively this method aims to recreate the misalignment state of a optical system by recreating the measured wavefront, which is called reverse optimization. Zemax has built in optimization algorithms such as the damped least squares and the hammer optimization. These inbuilt functions are used to minimize the merit function.

Expected downsides of this method are first of all the optimization time required. As the system of non-linear equations has to be solved to a global and not a local minimum, this might require some computational time. Secondly there is the possibility that the global minimum is never found.

In order to effectively use MFR, Kim et al. [8] wrote a macro carrying out the following tasks:

1. Read the Zernike coefficients from the interferometric measurements of the real wavefront. They are assigned to Ti which represents the misaligned system wavefront error.

2. Assign the ideal model Zernike coefficients to V_i, which represents the alignment state of the designed optical system.

3. Run the optimization algorithm embedded in the software to minimize the MF value. This operation varies the alignment parameters such that V_i→ Ti as close as possible.

4. When the MF is minimized, the alignment parameters are read out and those indicate the misalignment of the optical system.

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In order to simulate the process of the MFR method the macro given in figure C.2 was written, which uses the merit function given in figure C.1. This merit function tries to recreate the aberrated wavefront by perturbing the vari- ables defined in the macro. While the merit function might be reduced to a small value, the misalignment’s used to recreate this wavefront can still differ from the original misalignment’s. Therefore the criteria to determine the applicability of this method is the difference between the original and calculated misalignment’s. Here we perturbed the elements of M1 and M2 within a range of ±0, 1µm. In order to get a general idea of the applicability of the method a standard damped leased squares optimization was used, considering that a hammer optimization requires more computation time. However, when applied in a real situation the hammer optimization should always be run in order to obtain a global minimum. The merit function includes constraints equal to the practical perturbation range, in order to prevent overcompensation of one element.

5.2 Results of the merit function regression method

Original Calculated by MFR Difference

M1

∆Z 0.0643 mm 0.1 mm 0.0357 mm

∆X -0.0139 mm -0.0819 mm -0.06797 mm

∆Y -0.0369 mm -0.0993 mm -0.06241 mm

∆θX 0.0751^◦ 0.07096^◦ -0.00414^◦

∆θY 0.0103^◦ 0.01256^◦ 0.002258^◦

∆θZ -0.0338^◦ -0.0374^◦ -0.0036^◦

M2

∆Z 0.0676 mm 0.005566 mm -0.06203 mm

∆X 0.0839 mm 0.08781 mm 0.003913 mm

∆Y -0.0866 mm 0.002524 mm 0.08912 mm

∆θX -0.03^◦ -0.03424^◦ -0.00424^◦

∆θY 0.0353^◦ 0.03299^◦ -0.00231^◦

∆θZ -0.0078^◦ -0.02323^◦ -0.01543^◦

Table 5.1: Results of the merit function regression method for two elements with constraints. The calculated values are the misalignments used by the MFR method in order to recreate the alignment state.

In table 5.1 the results of a single run of the macro are displayed. In general the rotational degrees of freedom are calculated more accurately then the lateral displacements. This can be caused by the insensitivity of the displacements as discussed in section 3. Since the aberrations are in general less sensitive to lateral displacements these do not need to be calculated as accurately as the tilts.

The results of another 10 runs of the program were averaged and are displayed in table 5.2. The average difference between the calculated and expected values is in the order of tens of microns, however the standard deviation is typically bigger. Here we see no clear difference between the standard deviation of lat-

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eral displacements and tilts, which means that these differences are not related to the sensitivity of the optical elements. Considering that these results were obtained by the damped least squares algorithm, the large standard deviations are assumed to be a result of the local minima found by the algorithm.

Mirror Parameter Average difference Standard Deviation

M1

∆Z 0.01021 mm 0.03295 mm

∆X 0.007081 mm 0.04283 mm

∆Y -0.00807 mm 0.03146 mm

∆θ_X 0.001056^◦ 0.05464^◦

∆θ_Y 0.01951^◦ 0.04746^◦

∆θ_Z 0.01789^◦ 0.06986^◦

M2

∆Z -0.01699 mm 0.05449 mm

∆X -0.02425 mm 0.1163 mm

∆Y 0.01224 mm 0.06409 mm

∆θX 0.003739^◦ 0.05362^◦

∆θY -0.02709^◦ 0.06936^◦

∆θ_Z 0.04743^◦ 0.1012^◦

Table 5.2: Average and standard deviation of 10 runs of the MFR macro While the merit function was brought to a small value by the optimization algorithm, the global minimum has not been found. In order to find this global minimum the optimization algorithm was changed to the hammer optimization.

The results are shown in table 5.3. As clearly can be seen the hammer optimization did not reach the global minimum either, despite the extra reduction of the merit function.

Mirror Parameter Difference

M1

∆Z 0,03341 mm

∆X -0,02128 mm

∆Y -0,02301 mm

∆θX 0,02495^◦

∆θY -0,09748^◦

∆θ_Z -0,00529^◦

M2

∆Z -0,04163 mm

∆X -0,0863 mm

∆Y -0,01369 mm

∆θX 0,09151^◦

∆θY 0,03789^◦

∆θZ 0,02865^◦

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5.3 Discussion

The merit function regression method had problems calculating the exact misalignment state. Depending on the criteria of the alignment this method can be sufficient. In our case we are subject to a fabrication tolerance of 40µm which is of the same order as the differences resulting from the MFR method. Therefore we can conclude that the merit function regression method is not able to determine the misalignment state sufficiently accurate, which might be overcome by running an optimization algorithm that tries to find the global minimum, where the hammer optimization option of Zemax did not suffice. However, during the optimization the merit function value was reduced to a small value, which indicates that there might be another complication. The optimization in Zemax perturbs the elements one by one, and therefore does not account for the inter-element coupling effect. Thus we can conclude that the merit function regression method is not applicable to our alignment method.

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6 Differential Wavefront Sampling Method

Lee et al. [11] showed that when aligning a multi-element system there exists inter-element coupling effects. These effects cause the aberration sensitivity of one element to depend on the alignment state of other elements. In order to account for the inter-element effects they introduced the differential wavefront sampling method [12]. This method uses second order derivative information of the wavefront to calculate the misalignment state of the optical system. The wavefront (Φ(ui)) as a result of misalignment of optical element i is expressed as:

Φ(ui) = Φ0+

M

X

i=0

∇iΦ · ui+

M

X

i=0 M

X

j=0

ui· (∇^T_i∇jΦ) · u^T_j + O(u³_i) (6.1)

where ui = (dxi, dyi, dzi, dθi, dψi, dβi) is the vector containing the misalignment’s of optical element i and Φ0 is the original unperturbed wavefront. Here θ, ψ and β are rotation about the x, y and z axis respectively. The derivatives define the alignment influence functions λ^m_i,j of order m for each combination of elements. These influence functions contain the wavefront variation at each order over the plane of the pupil. By differentiating equation 6.1 with respect to u_i we get:

∂Φ

∂u_i = ∇iΦ +

M

X

j=0

·(∇^T_i∇jΦ) · uj+ O(u²_i) (6.2)

where the first and second order influence functions generate coma and astigmatism respectively [11]. From equation 6.2 a clear relation between the differ- entiated wavefront and the misalignment’s ui is obtained. This relation seems similar to the sensitivity table method, however, here the relation contains more information in the form of the influence functions.

Neglecting misalignment’s associated with the optical axis, which are insensitive in centered optical systems relative to the other axis, we can express the relation in a matrix for four degrees of freedom like:





∂Φ

∂x1

∂Φ

∂y1

∂Φ

∂θ₁





=





∂²Φ

∂x²₁

∂²Φ

∂x1∂y1

∂²Φ

∂x1∂θ1

∂²Φ

∂x1∂ψ1

∂²Φ

∂y1∂x1

∂²Φ

∂y₁²

∂²Φ

∂y1∂θ1

∂²Φ

∂y1∂ψ1

∂²Φ

∂θ1∂x1

∂²Φ

∂θ1∂y1

∂²Φ

∂θ²₁

∂²Φ

∂θ1∂ψ1









 dx1

dy1

dθ1

dψ₁





 +





∂Φ

∂x₁

0

∂Φ

∂y1

0

∂Φ

∂θ₁

0





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Since the second order term of equation 6.2 contains the alignment state and thus is of more importance, we can replace the wavefront by the Zernike coefficient corresponding to 0^◦ astigmatism. For a centered optical system the hessian matrix is symmetric and the off-axis terms (∂φ

∂ui

)0 are zero [12]. This creates the opportunity to separate equation 6.3 into two equations concerning the same optical axis:

∂Z₆

∂qj

=

M

X

k=1

(∇^T_j∇kΦ)q_k, ∂Z₆

∂pj

=

M

X

k=1

(∇^T_j∇kΦ)p_k (6.4)

where Z6 is the standard Zernike coefficient corresponding to 0^◦ astigmatism and qj= (dxj, dψj), pj = (dyj, dθj) are the vectors corresponding to the optical x and y axis respectively. Now equation 6.4 can be used to determine the alignment state of a centered optical system by the following steps:

1. Determine the first order derivative ∂Z₆

∂ui

by deliberately perturbing the system.

2. Sample the wavefront over a grid of pupil coordinates for different perturbations.

3. Approximate the sampled wavefront set using Chebyshev polynomials to determine the influence functions.

4. Solve equation 6.4 to obtain the current misalignment state.

6.1 Computation method

In order to compute the misalignment’s the influence functions and ∂Z6

∂u_i have to be determined such that we can solve equation 6.4.

First we examine how the partial derivative can be approximated. After replacing the wavefront with Z₆ and introducing a perturbation of δu_i we can expand Z₆(u_i± δu_i) in a Taylor series like

Z6(ui± δui) = Z6(ui) ± (∇iZ6) · δui+ δui· (∇^T_i ∇jZ6) · δu^T_j + O(δu³_i), (6.5) thus

Z6(ui+ δui) − Z6(ui− δui) = 2(∇iZ6) · δui+ O(δu³_i) (6.6) shows that the derivative of the wavefront replaced by Z6 with respect to an alignment parameter can be estimated by:

∂Z₆

∂ui

=Z₆(u_i+ δu_i) − Z₆(u_i− δu_i) 2δui

+ O(δu²_i) (6.7)

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for a known perturbation δui. Thus while neglecting the higher order terms O(δu²_i), we can perturb an optical element back and forth and measure the wavefront for each perturbation to estimate the partial derivative at the current alignment state.

Secondly, in order to determine the influence functions a set of N values of the alignment perturbation for each element is used to create a finer sampling of each perturbation. The i-th point is given by x_i= cos(π(i − 0.5)/N ) in [−1, 1]

and then normalized to the perturbation range. The range of perturbations should be chosen large enough to assure a sufficient sampling around the nominal value. At each perturbation grid point the wavefront of a ray passing through a particular pupil coordinate (ξ, η) has to be evaluated. After sampling over the entire pupil plane, the wavefront Φ can be expressed as a function of the pupil coordinates (ξ, η). A set of sampled wavefronts is then obtained by sampling for each perturbation grid point. The dependency of this set on the alignment perturbations can be determined by approximation in Chebyshev polynomials described by Press et al. [17]. When the subsets of the wavefront corresponding to a particular pupil coordinate are decomposed into Chebyshev polynomials, we obtain a set of coefficients which can then be converted into the influence functions using the method given in [17]. By completing the calculations for the entire set of sampled wavefronts over the entire pupil, the coefficients for each term in equation 6.2 are obtained.

In solving equation 6.4 Lee et al. [12] use a two step procedure. First the directional misalignment is estimated which is then used as a constraint for the non-linear estimation of the axial misalignment’s. This can be done due to larger influence on the alignment driven astigmatism of the second order couplings between directional misalignment’s, which makes it possible to estimate the directional and axial misalignment’s separately.

6.2 Measurements and control uncertainties of DWS

Given certain measurement (σ_m) and control (σ_c) uncertainties, the variance of the differential wavefront perturbed by δu_i is given by Lee et al. [11] to be:

σ_u²_i ≈ ∇iΦ σ_c²

2δu²_i + σ_m² 2N δu²_i +1

2



(∇^T_i∇iΦ)²

1 + 2ui

δui

+ ui

δui 2

+X

j6=i

∇^T_i∇jΦuj

δui

2

σ_c² (6.8)

where the first order term can be dropped when measuring astigmatism, as

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of δui. Thus when the alignment parameters approach their nominal value the limit of the variance of the differential wavefront is given by:

lim

|u|→0σ²_u

i ≈1

2

(∇^T_i ∇_iΦ)²σ_c²+ σ²_m N δu²_i

(6.9)

6.3 DWS for off-axis systems

The differential wavefront method was developed for a centered optical system and the accompanying assumptions. However, considering that not all systems are centered, the off-axis consequences should be investigated.

Starting from equation 6.3 we can no longer assume that the hessian matrix is symmetric and that the off-axis terms are zero. Therefore we can no longer split the equation into two separate equations corresponding to the different axis. The first order influence functions corresponding to the off-axis terms can be obtained in a similar way as the second order influence functions, as described in section 6.1.

Lee et al. [11] studies the influence functions for a centered system which led to the conclusion that second order influence functions relate to astigmatism.

They have found an influence of the symmetry associated with the alignment parameters due to the axial symmetry of surfaces and the field of view. This symmetry would not hold for off-axis systems. Rotational parameters about the optical axis of elements, which are not symmetric for off-axis elements, might play an important role although they have been ignored in their research.

Kim et al. [9] have tried to apply the DWS method to the off-axis Korsch telescope. In their simulations they found that the DWS method was able to find the correct misalignment state of the telescope when adding the off-axis terms. However, during the real alignment they found that the measurement and mounting uncertainties greatly influenced the results and gave an unrealistic result for the alignment state.

6.4 Discussion

The differential wavefront sampling method was introduced to overcome the inter-element coupling effects while aligning a multi-element system. Consider- ing that these effects greatly influence the results of the ST and MFR methods, the DWS method looks promising. In our alignment method an optical system is aligned in vacuum using pi¨ezo elements, which means that we can greatly reduce the control uncertainties and thus error given in equation 6.9 by a combination of actuating and sensing pi¨ezo’s. However, the method was developed for centered optical systems, which means that when applying DWS to off-axis systems we have to be cautious. The first order influence functions have to be

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added to correct for the off-axis elements. In order to correctly use the differential wavefront sampling method further research into the applicability of the method to off-axis systems has to be conducted.

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7 Conclusions

In this thesis first a general insight into the wavefront error sensitivity of the misalignment of reflective optical elements has been acquired. From this, we could determine the important degrees of freedom per element shown in table 3.2. However, when determining the important degrees of freedom per element of an optical system a sensitivity analysis of the complete system is still required. Secondly several computer guided automated alignment processes have been discussed: the sensitivity table method, merit Function Regression method and differential wavefront sampling method.

The sensitivity table method was found to be able to significantly reduce the aberrations in a system where only one element was perturbed. When two elements were perturbed the inability of the method to correct for the inter- element coupling effect resulted in an increase rather than decrease of the aberrations. The merit function regression method worked as expected, however the method was not able to find the global minimum. This might be caused by the optimization algorithm used, or by the inter-element coupling effect. The differential wavefront Sampling method is able to correct for the inter-element coupling effect, which makes it the most promising method for our application.

This method is constructed for centered optical systems, whereas off-axis systems are of more importance to us. Several aspects of the applicability of the DWS method to off-axis alignment have been discussed.

Further research can be conducted on the merit function regression method in order to determine whether the optimization algorithm or the inter-element coupling effect is causing the inaccuracy of the method. The differential wavefront sampling method can be used in the alignment for centered optical systems as shown by Lee et al. [12]. To determine the applicability of the differential wavefront sampling method to our project further research should be conducted regarding the functionality of the method for the alignment of off-axis optical systems.

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Appendices

A Detailed lens data of the OGSE Re-Imager

Figure A.1: Detailed lens data of the OGSE Re-Imager

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B Merit function used in the sensitivity table method

Figure B.1: Merit function used to in the sensitivity table method.

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C Merit function regression method

Figure C.1: Merit function used to in the merit function regression method.

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Figure C.2: Macro used in the merit function regression method.