Performance analysis of a mirror by numerical iterative method

Kwijong Park; Myung Cho; Dae-Hee Lee; Bongkon Moon

doi:10.1364/OE.22.031864

1. Introduction

Optical mirrors are the most important parts in high precision optical telescopes such as a payload on a satellite or an astronomical telescope in an observatory. The performance of an optical system will depend on the deformation of a mirror surface by an external force such as gravity, temperature change, mis-alignment, and so on. Assuming that it is possible to predict the deformation of a mirror, an optimized mirror having the best performance will be able to be designed before fabricating a mirror. An optical performance prediction of a mirror can be made by creating the mirror model from a Finite Element Method (FEM) [1], which is a numerical method for approximate solutions of partial differential equations. After the raw deformation of a mirror surface due to external load is calculated through the FEM, the optical surface can be analyzed numerically or mathematically for high precision optical performance analysis. Zernike polynomials are generally used as a data fitting function to predict the optical performance of a mirror, because they are orthogonal on a unit circle and each optical aberration expressed to the polynomials can be corrected independently. However, when the aperture shape of a mirror is not a circle, Zernike polynomials can be no longer used without modification. Although it can be modified by method like Gram-Schmidt orthogonalization [2,3], this process commonly requires extensive computations. It is because whenever a mirror aperture changes to new shape, the polynomials should be modified.

We developed relatively a simple numerical method for performance prediction of all mirror aperture shapes. We conceived a hint from optical engineers, who measure a wavefront error regardless of the mirror aperture shape with some random adjustments in optical laboratories. The random adjustment like rotating and translating a mirror is to search for the best position at which root mean square (RMS) of the mirror surface is minimized. It can serve as a method to predict the performance of general mirror aperture shapes. Piston, tip, tilt, and defocus (P.T.T.F.) aberrations of a mirror can be corrected by moving the mirror rigidly. In case of mirrors in many telescopes such as TMT [4], GMT [5], and others [6,7], the wavefront error of the mirror meets sufficiently the requirement on only removal of P.T.T.F. aberrations, whereas higher aberrations of a mirror can be corrected by deformation of the mirror surface with an active optics and/or an adaptive optics system [8]. This paper introduces a rotation transformation and paraboloid graph subtraction to make the optical surface RMS minimum, for removal of P.T.T.F. aberrations from the raw surface deformation. This numerical iterative method can be applicable to all mirror aperture shapes, and is very easy to implement, and it does not require a long calculation time. In order to demonstrated the optical performance with corrected results by the numerical iterative method and Zernike polynomial fitting, we performed opto-mechanical design [9,10] for a 30 cm concave circular mirror. In addition, we developed codes for both the numerical iterative operation and Zernike polynomial fitting including the 7th power terms [11]. Furthermore, a concave square mirror with 4 flexures was designed and analyzed to visualize the corrected surface map of a general mirror aperture shape by the iterative method.

2. Iterative numerical method

2.1 Rotation transformation

Compensation of both tip and tilt aberrations from the raw deformation of a mirror is available through rotation transformation matrix. As a rotation matrix multiplies by a vector which consists of each vector component, in order to rotate a point in coordinate, it should be done to all nodes of the optical surface of the mirror obtained from a FEM for removal of tip and tilt aberrations. By rotating all deformed nodes over the mirror optical surface, a rotation angle at which mirror surface RMS is minimized can be iteratively solved. When we set the z axis in Cartesian coordinates to the optical axis, then a tilt motion becomes to a rotation of nodes about the x axis, and a tip motion is analogous to a rotation about the y axis. Therefore, rotation transformation of all the nodes about both the x and y axes is necessary in order to correct the tip and tilt aberrations. A rotation transform matrix that rotates a node about the x axis is defined as

(\begin{matrix} x_{i}^{'} \\ y_{i}^{'} \\ Δ z_{n, i}^{'} \end{matrix}) = [\begin{matrix} 1 & 0 & 0 \\ 0 & \cos θ & - \sin θ \\ 0 & \sin θ & \cos θ \end{matrix}] (\begin{matrix} x_{i} \\ y_{i} \\ Δ z_{n, i} \end{matrix}) .

The $x_{i}^{}$ and $y_{i}^{}$ are the ith original node position of a mirror surface, and the $△ x_{i}^{}$ , $△ y_{i}^{}$ , and $△ z_{i}^{}$ are the ith raw displacement obtained from the FEM. The $△ z_{n, i}^{}$ is the ith $△ z_{i}^{}$ normal to the mirror surface. It is calculated from the inner product of the raw displacement vector ( $△ x_{i}^{}$ , $△ y_{i}^{}$ , $△ z_{i}^{}$ ) and unit vector normal to the mirror surface [12]. The $x_{i}^{'}$ , $y_{i}^{'}$ , and $△ z_{n, i}^{'}$ are vector components after the $x_{i}^{}$ , $y_{i}^{}$ , and $△ z_{n, i}^{}$ are rotated about the x axis.

The rotation transform matrix that rotates a node about the y axis, is defined as

(\begin{matrix} x_{i}^{*} \\ y_{i}^{*} \\ △ z_{n, i}^{*} \end{matrix}) = [\begin{matrix} \cos θ & 0 & \sin θ \\ 0 & 1 & 0 \\ - \sin θ & 0 & \cos θ \end{matrix}] (\begin{matrix} x_{i}^{'} \\ y_{i}^{'} \\ △ z_{n, i}^{'} \end{matrix}) .

The $x_{i}^{*}$ , $y_{i}^{*}$ , and $△ z_{n, i}^{*}$ are vector components after the $x_{i}^{'}$ , $y_{i}^{'}$ , and $△ z_{n, i}^{'}$ are rotated about the y axis. The process steps of moving all nodes on a mirror surface by rotation transformation are shown in Fig. 1. The deformed mirror surface was expressed by plotting all nodes. All nodes are rotated to the optimum angle to obtain the minimum surface RMS, so that tip and tilt aberrations are corrected from the raw deformation of the step 0, as shown in the step I of Fig. 1. The step II shows the mean value of the $△ z_{n, i}^{*}$ is subtracted from the raw deformation, and it removes a piston aberration from the mirror. The step III shows the correction process in which piston, tip, and tilt (P.T.T.) aberrations are all removed.

Fig. 1 P.T.T. correction process steps by rotation transformation.

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2.2 Paraboloid graph subtraction

A defocus aberration can be removed by subtracting a paraboloid graph from mirror surface nodes in which P.T.T. aberrations were removed. The ith paraboloid graph, $z_{f, i}$ is defined as

z_{f, i} = c 1 \cdot r_{i}^{2} + c 0,

where the

c 1

decides the shape of a paraboloid, the

c 0

does translation along the optical axis, and the

r_{i}

is the radius from the center of the mirror to the ith node. The final ith displacement,

z_{p t t f, i}

without P.T.T.F. aberrations is defined as

z_{p t t f, i} = △ z_{n, i}^{*} - z_{f, i}

. The

△ z_{n, i}^{*}

denotes the ith displacement in which P.T.T. aberrations were removed. The coefficient (

c 1

) of the paraboloid graph in Zernike polynomial fitting is solved by a matrix calculation through the least squares method [3]. In this paper, the coefficient is determined by iteratively increasing or decreasing the increment of the paraboloid coefficients,

△ c 1

, as shown in Fig. 2. In other words, it repeats the iteration for all the nodes until finding the optimal paraboloid graph to obtain a minimum RMS, and then the

c 1

is the sum of the paraboloid increment (

△ c 1

). The

c 0

means a translation of a paraboloid graph and corresponds to a mean value of the remainder displacement (

△ z_{n, i}^{*} - c 1 \cdot r_{i}^{2}

) after a paraboloid shape was removed.

Fig. 2 Paraboloid graph according to coefficient increment (Δc1).

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2.3 Flow diagram of numerical iterative method

A flow diagram of rotation transformation and paraboloid graph subtraction is shown in Fig. 3. By rotating nodes iteratively as the tilt incremental angle, $△ a$ , to the plus or minus direction about the x axis, the tilt angle at which the surface RMS is minimized was solved. The tip angle was solved by rotating nodes iteratively as the tip incremental angle, $△ b$ , about the y axis. The total tip or tilt angle was solved from the sum of the incremental angle, $△ a$ or $△ b$ . If a change of the surface RMS by rotation transformation does not occurs, the tip or tilt is calculated to zero. Positive tilt, negative tilt, positive tip, and negative tip are calculated in parallel, and then a computer saves the minimum RMS values to the memory for each rotation direction. After comparing each RMS value for each direction, the angle value and direction to obtain the smallest RMS value are finally selected to the solution of the tip and tilt aberration, respectively. In the next step, a piston aberration was removed by subtracting the mean value of the $△ z_{n, i}^{*}$ from all the $△ z_{n, i}^{*}$ , which is the ith displacement without P.T.T. aberrations. The paraboloid coefficient ( $c 1$ ) is also computed by increasing the shape increment of a parabolid gragh ( $△ c 1$ ), to plus and minus direction in parallel. The $c 1$ is solved from the sum of the increment, and either plus $c 1$ or minus $c 1$ can be calculated as the coefficient of a paraboloid graph to obtain the smallest RMS between each the iteration. After solving the $c 0$ , all the final displacement ( $z_{p t t f, i}$ ), in which P.T.T.F. aberrations were removed from raw displacement of the mirror surface are finally obtained.

Fig. 3 Flow chart of iterative method.

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3. Verification of numerical iterative method with a 30 cm circular mirror

A 30 cm concave circular mirror including a center hole whose radius of curvature is 700 mm and conic constant is −1 was designed in order to compare correction results by the numerical iterative method and Zernike polynomial fitting. Its effective focal length is 350 mm, and optical ray path is shown in Fig. 4.

Fig. 4 Optical ray path of mirror.

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The solid model and finite element model of the mirror with three flexures are shown in Fig. 5. The mass of the light-weight mirror is 2.4 kg, and the material is Zerodur (E = 9.1 × 10¹⁰ N/m², ν = 0.24, ρ = 2530 kg/m², coefficient of thermal expansion (CTE) = 5 × 10⁻⁷/K). When the bottom planes of flexures are fixed as a boundary condition, a FEM was performed for gravity along x, y, z axes, and a Δ10 K temperature change, respectively. The number of nodes of the mirror surface is 11020 as shown in Fig. 5(b). In the iterative method, the incremental angle of rotation transformation ( $△ a$ and $△ b$ ) was set to ± 3.8 × 10⁻⁸ deg., and the increment of paraboloid coefficient ( $△ c 1$ ) was 1.0 × 10⁻¹⁰ m. With these increments, as the arc and translation distance of the mirror moved incrementally are below atomic scale, it is impossible for an optical engineer to align the optical system practically within this tolerance. In other words, a very fine increment was used for an accurate analysis by the iterative method.

Fig. 5 A 30 cm concave circular mirror and three flexures: (a) Solid model, (b) Finite element model.

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In the case that gravity along the y axis, Y-Gravity, acts on the mirror, results of the FEM and corrected deformation maps of the mirror surface are shown in Fig. 6. The deformed configuration of the mirror assembly obtained through the FEM is shown in Fig. 6(a). The total maximum displacement was calculated as 434.1 nm. By increasing or reducing the increment of the rotation transformation angle ( $△ a$ and $△ b$ ) and of its corresponding paraboloid coefficient( $△ c 1$ ), P.T.T.F. aberrations were removed as shown in Fig. 6(c). It shows that the tilt aberration was dominant and removed from the raw deformation map given in Fig. 6(b). The deformation map of a mirror was based on a uniform 500 x 500 array. Figure 6(d) shows a correction map by Zernike polynomial fitting, and it is difficult to distinguish the difference from that by the iterative method. With subtracting the correction map by the iterative method from that by Zernike polynomial fitting, the difference map was made as shown in Fig. 6(e). It shows that the iterative method has a tip-tilt error compared to Zernike polynomial fitting. The Peak to Valley (P-V) of the difference map is 0.396 nm, and RMS of 0.103 nm. Its ratios are 3.0% and 3.9% each other, when compared values of the correction map by the iterative method. The smaller a P-V or RMS value in the correction map is, the smaller that in the difference map is, as well. Generally, RMS error over the optical mirror surface is tens of nanometers, so that the actual different value is very small.

Fig. 6 The analysis results for Y-Gravity: (a) FEM result, (b) Raw deformation map, (c) Correction maps by the iterative method, (d) Correction maps by Zernike polynomial fitting, (e) Difference map.

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The optical deformation map of the mirror for each of X-Gravity, Z-Gravity, and a 10 K temperature change is shown in Fig. 7. Through investigating raw deformation map in Fig. 7(a) and correction map in Fig. 7(b), it is confirmed that a tip-tilt aberration is corrected dominantly in the X-Gravity while a defocus was removed dominantly in both Z-Gravity and 10 K temperature change. Difference maps are shown in Fig. 7(c) with subtracting results of the iterative method from that of Zernike polynomial fitting. The maps show that the iterative method has a tip-tilt error for X-Gravity, and strong defocus error for Z-Gravity and 10 K temperature change. Analysis results by the iterative method for each load condition were summarized in Table 1. Maximum ratio was calculated to 8.4% for RMS under 10 K temperature change. In fact, the increment was too large compared to the corrected RSM and P-V value, which are smaller than 1 nm. If the increment is smaller, the ratio will be reduced.

Fig. 7 Optical deformation map of the mirror for each loading; (a) Raw deformation map: from left, X-Gravity, Z-Gravity, and a 10 K temperature change, (b) Correction map by iterative method: from left, X-Gravity, Z-Gravity, and a 10 K temperature change, (c) Difference map: from left, X-Gravity, Z-Gravity, and a 10 K temperature change.

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Table 1. Surface P-V and RMS by the Iterative Method and Zernike Polynomial Fitting for Each Load Condition

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The calculation time required to correct P.T.T.F. aberrations by the iterative method and Zernike polynomial fitting was summarized in Table 2 according to each load condition. The calculation by Zernike fitting took about 4 sec to solve a matrix equation for each load condition with an i7 CPU. That by the iterative method was very faster than that of Zernike fitting for X-Gravity, Y-Gravity, and 10 K temperature change. On the other hand, the maximum time and the number of iteration were 11.56 sec. and 130001 for Z-Gravity, respectively. It took longer by about 3 times than that of Zernike fitting, but was short enough in the performance analysis of a mirror. If the size of increment is smaller, the time may be much longer. However, a user can adequately control the increment for exact analysis and efficient calculation time, according to the size of raw deformation.

Table 2. Time Taken To Calculate P.T.T.F. Aberration by the Iterative Method and Zernike Polynomial Fitting for Each Load Condition

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4. Performance analysis for a square mirror shape

The iterative method can analyze the optical performance of a mirror regardless of aperture shapes. As illustrated in Fig. 8, a concave square mirror held by 4 flexures was modeled to evaluate the correction map in an arbitrary mirror aperture shape by the iterative method.

Fig. 8 A concave square mirror and four flexures: (a) Solid model, (b) FE model.

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With the boundary condition where bottom planes of flexures were fixed, a FEM was performed for X, Y, Z-Gravity, and a 10 K temperature change as a load condition. The raw deformation map and correction map by Zernike polynomials for a square pupil [13] and the iterative method are shown in Fig. 9, respectively. For X-Gravity and Y-Gravity, the dominant aberration is tip-tilt, while the defocus aberration is dominant for Z-gravity and a 10 K temperature change. Correction maps wherein tip-tilt aberrations were removed are symmetric and maps where a defocus aberration was removed show local deformation at the flexure position. The P-V values of correction maps are different as up to 0 ~5%, and 0 ~6% in RMS, but there was not a distinct difference between correction maps between both methods.

Fig. 9 Optical deformation maps of a square mirror before and after the iterative method and Zernike polynomial fitting, for each load condition: (a) X-Gravity, (b) Y-Gravity, (c) Z-Gravity, (d) 10 K temperature change.

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5. Conclusion

P.T.T.F. aberrations were corrected by rotation transformation and paraboloid graph subtraction. A numerical iterative method, from raw deformed surface data of a 30 cm concave circular mirror was demonstrated for the mirror gravity and a 10 K temperature change, respectively. The optical deformation map corrected by the iterative method is almost the same as that by Zernike polynomial fitting. When results corrected by Zernike polynomials assume the reference value, the error from the numerical iterative method is 8.4% in RMS surface and 5.0% in P-V surface. In general, RMS of a mirror surface is tens of nanometers, so that the actual different value between the two methods is very small. Computational time of the iterative method can be faster or slower than that of Zernike fitting according to the size of increment, but was relatively short in the optical performance analysis. This iterative method is not dependent on the mirror aperture shape as shown in results of the concave square mirror with 4 flexures. In addition, it is possible to calculate the correction map of an asymmetric mirror, off-axis mirror, segmented mirror, etc. Therefore, it is concluded that the iterative method can appropriately evaluate the optical performance of mirrors in arbitrary shapes.

Acknowledgment

This research was supported by development project (No. 2011-381001) of a 30 cm cryogenic space infrared telescope from Korean Research Council of Fundamental Science Technology (KRCF).

References and links

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2. W. Swantner and W. W. Chow, “Gram-Schmidt orthonormalization of Zernike polynomials for general aperture shapes,” Appl. Opt. 33(10), 1832–1837 (1994). [CrossRef] [PubMed]

3. D. S. Watkins, Fundamentals of Matrix Computations (WILEY, 2010).

4. M. Cho, “Performance prediction of the TMT secondary mirror support system,” Proc. SPIE 7018, 70181S (2008). [CrossRef]

5. M. Cho, A. Corredor, C. Dribusch, K. Park, Y. S. Kim, and I. K. Moon, “Design and Development of a Fast Steering Secondary Mirror for the Giant Magellan Telescope,” Proc. SPIE 8125, 812505 (2011). [CrossRef]

6. M. Cho, M. Liang, and D. R. Neill, “Performance prediction of the LSST secondary mirror,” Proc. SPIE 7424, 742407 (2009). [CrossRef]

7. K. Park, B. Moon, D. H. Lee, W. S. Jeong, U. W. Nam, Y. Park, J. Pyo, and W. Han, “Performance Analysis for mirrors of 30 cm cryogenic space infrared telescope,” JASS 29(3), 321–328 (2012).

8. P. M. Hinz, A. Bouchez, M. Johns, S. Shectman, M. Hart, B. McLeod, and P. McGregor, “The GMT adaptive optics system,” Proc. SPIE 7736, 77360C(2010). [CrossRef]

9. P. R. Yoder, Opto-mechanical System Design (Marcel Dekker, 1993).

10. P. Y. Bely, The Design and Construction of Large Optical Telescope (Springer, 2003).

11. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66(3), 207–211 (1976). [CrossRef]

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13. A. H. Al-Hamdani and S. Y. Hasan, “Zernike polynomiales for optical systems with rectangular and square apertures of area equal to π,” IJPAP 51(12), 837–843 (2013).

Loading	Raw map		Correction map				Ratio
Loading	P-V (nm)	RMS (nm)	P-V (nm)	RMS (nm)	P-V (nm)	RMS (nm)	P-V (%)	RMS (%)
X-Gravity	162	39.8	14.7	2.62	0.372	0.0968	2.5	3.7
Y-Gravity	161	39.8	13.2	2.63	0.396	0.103	3.0	3.9
Z-Gravity	487	107	380	70.9	10.7	2.93	2.8	4.1
Δ10 K Change	30.9	8.55	4.15	0.596	0.207	0.0502	5.0	8.4

Load Condition	X-Gravity		Y-Gravity		Z-Gravity		10 K Change
Load Condition	Iterative Method	Zernike Fitting	Iterative Method	Zernike Fitting	Iterative Method	Zernike Fitting	Iterative Method	Zernike Fitting
+ Tilt(sec) (Number of Iteration)	0.01 (1)	-	0.02 (1)	-	0.02 (1)	-	0.02 (2)	-
- Tilt(sec) (Number of Iteration)	0.01 (1)	-	0.12 (764)	-	0.01 (7)	-	0.01 (1)	-
+ Tip(sec) (Number of Iteration)	0.11 (763)	-	0.01 (1)	-	0.01 (1)	-	0.01 (1)	-
- Tip(sec) (Number of Iteration)	0 (1)	-	0.01 (1)	-	0.02 (1)	-	0.01 (2)	-
+ Paraboloid (sec) (Number of Iteration)	0 (1)	-	0 (4)	-	11.5 (130001)	-	0 (1)	-
- Paraboloid (sec) (Number of Iteration)	0 (3)	-	0 (1)	-	0 (1)	-	1.23 (13904)	-
Total Time (sec)	0.13	4.23	0.16	4.14	11.56	4.15	1.23	4.30

Loading	Raw map		Correction map				Ratio
Loading	P-V (nm)	RMS (nm)	P-V (nm)	RMS (nm)	P-V (nm)	RMS (nm)	P-V (%)	RMS (%)
X-Gravity	162	39.8	14.7	2.62	0.372	0.0968	2.5	3.7
Y-Gravity	161	39.8	13.2	2.63	0.396	0.103	3.0	3.9
Z-Gravity	487	107	380	70.9	10.7	2.93	2.8	4.1
Δ10 K Change	30.9	8.55	4.15	0.596	0.207	0.0502	5.0	8.4

Load Condition	X-Gravity		Y-Gravity		Z-Gravity		10 K Change
Load Condition	Iterative Method	Zernike Fitting	Iterative Method	Zernike Fitting	Iterative Method	Zernike Fitting	Iterative Method	Zernike Fitting
+ Tilt(sec) (Number of Iteration)	0.01 (1)	-	0.02 (1)	-	0.02 (1)	-	0.02 (2)	-
- Tilt(sec) (Number of Iteration)	0.01 (1)	-	0.12 (764)	-	0.01 (7)	-	0.01 (1)	-
+ Tip(sec) (Number of Iteration)	0.11 (763)	-	0.01 (1)	-	0.01 (1)	-	0.01 (1)	-
- Tip(sec) (Number of Iteration)	0 (1)	-	0.01 (1)	-	0.02 (1)	-	0.01 (2)	-
+ Paraboloid (sec) (Number of Iteration)	0 (1)	-	0 (4)	-	11.5 (130001)	-	0 (1)	-
- Paraboloid (sec) (Number of Iteration)	0 (3)	-	0 (1)	-	0 (1)	-	1.23 (13904)	-
Total Time (sec)	0.13	4.23	0.16	4.14	11.56	4.15	1.23	4.30

Performance analysis of a mirror by numerical iterative method

Abstract

1. Introduction

2. Iterative numerical method

2.1 Rotation transformation

2.2 Paraboloid graph subtraction

2.3 Flow diagram of numerical iterative method

3. Verification of numerical iterative method with a 30 cm circular mirror

4. Performance analysis for a square mirror shape

5. Conclusion

Acknowledgment

References and links

Cited By

Figures (9)

Tables (2)

Equations (3)

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