Influence functions for a hysteretic deformable mirror with a high-density 2D array of actuators

A. E. M. Schmerbauch; M. A. Vasquez-Beltran; A. I. Vakis; R. Huisman; B. Jayawardhana

doi:10.1364/AO.397472

Applied Optics
Vol. 59,
Issue 27,
pp. 8077-8088
(2020)
•https://doi.org/10.1364/AO.397472

Influence functions for a hysteretic deformable mirror with a high-density 2D array of actuators

A. E. M. Schmerbauch, M. A. Vasquez-Beltran, A. I. Vakis, R. Huisman, and B. Jayawardhana

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A. E. M. Schmerbauch, M. A. Vasquez-Beltran, A. I. Vakis, R. Huisman, and B. Jayawardhana, "Influence functions for a hysteretic deformable mirror with a high-density 2D array of actuators," Appl. Opt. 59, 8077-8088 (2020)

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Abstract

We present modeling and analysis of a hysteretic deformable mirror where the facesheet interacts with a continuous layer of piezoelectric material that can be actuated distributively by a matrix of electrodes through multiplexing. Moreover, a method to calculate the actuator influence functions is described considering the particular arrangement of electrodes. The results are presented in a semi-analytical model to describe the facesheet’s deformation caused by a high-density array of actuators, and validated in a simulation. The proposed modeling of an interconnection layout of electrodes is used to determine the optimal pressures the actuators must exert to achieve a desired surface deformation.

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Case 1	Case 2	Case 3	Case 4	Case 5
$\bar{r} = 0$	$\bar{r} \leq r_{1}$	$\bar{r} \leq r_{1 e}$	$\bar{r} \leq r_{1}$	$\bar{r} \leq r_{1}$
$\bar{r} \geq r_{1}$	$\bar{r} \geq r_{4}$	$\bar{r} \geq r_{4}$	$\bar{r} \geq r_{4}$	$\bar{r} \geq r_{4}$
$0 < \bar{r} \leq r_{1 e}$	$r_{1} < \bar{r} \leq r_{2}$	$r_{1 e} < \bar{r} \leq r_{1}$	$r_{1} < \bar{r} \leq r_{3}$	$r_{1} < \bar{r} \leq r_{2}$
$r_{1 e} < \bar{r} < r_{1}$	$r_{2} < \bar{r} < r_{4}$	$r_{1} < \bar{r} \leq r_{4 e}$	$r_{3} < \bar{r} \leq r_{2}$	$r_{2} < \bar{r} \leq r_{3}$
		$r_{4 e} < \bar{r} < r_{4}$	$r_{2} < \bar{r} < r_{4}$	$r_{3} < \bar{r} < r_{4}$

ZPs	LD: RMSDs in [%]	HD: RMSDs in [%]
$Z_{1}^{- 1}$	4.426	0.001433
$Z_{1}^{1}$	4.352	0.001373
$Z_{2}^{- 2}$	3.806	0.000298
$Z_{2}^{0}$	12.840	0.065971
$Z_{2}^{2}$	9.498	0.004775
$Z_{3}^{- 3}$	10.138	0.007609
$Z_{3}^{- 1}$	13.937	0.131942
$Z_{3}^{1}$	13.756	0.134622
$Z_{3}^{3}$	9.974	0.007343
$Z_{4}^{- 4}$	11.376	0.005252
$Z_{4}^{- 2}$	12.662	0.082268
$Z_{4}^{0}$	25.917	0.524411
$Z_{4}^{2}$	20.755	0.409452
$Z_{4}^{4}$	13.069	0.011954
$Z_{5}^{- 5}$	13.847	0.017128
$Z_{5}^{- 3}$	18.909	0.406282
$Z_{5}^{- 1}$	18.838	0.675500
$Z_{5}^{1}$	18.674	0.692096
$Z_{5}^{3}$	18.717	0.400501
$Z_{5}^{5}$	13.649	0.016273
$Z_{6}^{- 6}$	14.903	0.009467
$Z_{6}^{- 4}$	20.851	0.364835
$Z_{6}^{- 2}$	18.077	0.346565
$Z_{6}^{0}$	25.742	1.735849
$Z_{6}^{2}$	17.145	1.670824
$Z_{6}^{4}$	16.980	0.729995
$Z_{6}^{6}$	14.790	0.049802
$Z_{10}^{- 10}$	16.216	0.087481
$Z_{10}^{- 8}$	18.114	1.420633
$Z_{10}^{- 6}$	18.901	2.341578
$Z_{10}^{- 4}$	20.592	2.417903
$Z_{10}^{- 2}$	22.680	1.451910
$Z_{10}^{0}$	27.243	6.092243
$Z_{10}^{2}$	21.649	6.225307
$Z_{10}^{4}$	18.076	4.505920
$Z_{10}^{6}$	18.868	3.345968
$Z_{10}^{8}$	17.221	1.468534
$Z_{10}^{10}$	17.303	0.271556

Case 1	Boundaries	Coordinate transformation
$I_{1}$	$0 \leq \bar{r} \leq r_{1 e}$	$0 < \bar{ϕ} < 2 π$
$I_{21}$	$r_{1 e} < \bar{r} < r_{1}$	$\arccos (x_{2} / \bar{r}) < \bar{ϕ} < \arcsin (y_{2} / \bar{r})$
$I_{22}$		$\arcsin (y_{2} / \bar{r}) < \bar{ϕ} < \arccos (x_{1} / \bar{r})$
$I_{23}$		$\arccos (x_{1} / \bar{r}) < \bar{ϕ} < \arcsin (y_{1} / \bar{r})$
$I_{24}$		$\arcsin (y_{1} / \bar{r}) < \bar{ϕ} < \arccos (x_{2} / \bar{r})$
Case 2	Boundaries	Coordinate transformation
$I_{1}$	$r_{1} < \bar{r} \leq r_{2}$	$\arcsin (y_{1} / \bar{r}) < \bar{ϕ} \leq \arccos (x_{1} / \bar{r})$
$I_{2}$	$r_{2} < \bar{r} < r_{4}$	$\arccos (x_{2} / \bar{r}) < \bar{ϕ} < \arcsin (y_{2} / \bar{r})$
Case 3	Boundaries	Coordinate transformation
$I_{11}$	$r_{1 e} < \bar{r} \leq r_{1}$	$0 < \bar{ϕ} \leq \arccos (x_{1} / \bar{r})$
$I_{12}$		$2 π - \arccos (x_{1} / \bar{r}) < \bar{ϕ} < 2 π$
$I_{2}$	$r_{1} < \bar{r} \leq r_{4 e}$	$\arcsin (y_{1} / \bar{r}) < \bar{ϕ} \leq \arcsin (y_{2} / \bar{r})$
$I_{31}$	$r_{4 e} < \bar{r} < r_{4}$	$\arccos (x_{2} / \bar{r}) < \bar{ϕ} < \arcsin (y_{2} / \bar{r})$
$I_{32}$		$\arcsin (y_{1} / \bar{r}) < \bar{ϕ} < \arccos (x_{2} / \bar{r})$
Case 4	Boundaries	Coordinate transformation
$I_{1}$	$r_{1} < \bar{r} \leq r_{3}$	$\arcsin (y_{1} / \bar{r}) < \bar{ϕ} \leq \arccos (x_{1} / \bar{r})$
$I_{2}$	$r_{3} < \bar{r} \leq r_{2}$	$\arccos (x_{2} / \bar{r}) < \bar{ϕ} \leq \arccos (x_{1} / \bar{r})$
$I_{3}$	$r_{2} < \bar{r} < r_{4}$	$\arccos (x_{2} / \bar{r}) < \bar{ϕ} < \arcsin (y_{2} / \bar{r})$
Case 5	Boundaries	Coordinate transformation
$I_{1}$	$r_{1} < \bar{r} \leq r_{2}$	$\arcsin (y_{1} / \bar{r}) < \bar{ϕ} \leq \arccos (x_{1} / \bar{r})$
$I_{2}$	$r_{2} < \bar{r} \leq r_{3}$	$\arcsin (y_{1} / \bar{r}) < \bar{ϕ} \leq \arcsin (y_{2} / \bar{r})$
$I_{3}$	$r_{3} < \bar{r} < r_{4}$	$\arccos (x_{2} / \bar{r}) < \bar{ϕ} < \arcsin (y_{2} / \bar{r})$

Symbol	Formula	Symbol	Formula
$κ_{x_{1}}$	$x_{1} \sqrt{1 - x_{1}^{2} / r^{2}}$	$κ_{y_{1}}$	$y_{1} \sqrt{1 - y_{1}^{2} / r^{2}}$
$κ_{x_{2}}$	$x_{2} \sqrt{1 - x_{2}^{2} / r^{2}}$	$κ_{y_{2}}$	$y_{2} \sqrt{1 - y_{2}^{2} / r^{2}}$
$ϵ_{x_{1}}$	$x_{1} \sqrt{(r^{2} - x_{1}^{2}) / r^{2}}$	$ϵ_{y_{1}}$	$y_{1} \sqrt{(r^{2} - y_{1}^{2}) / r^{2}}$
$ϵ_{x_{2}}$	$x_{2} \sqrt{(r^{2} - x_{2}^{2}) / r^{2}}$	$ϵ_{y_{2}}$	$y_{2} \sqrt{(r^{2} - y_{2}^{2}) / r^{2}}$
$α_{1}$	$\arccos (x_{1} / r)$	$α_{2}$	$\arccos (x_{2} / r)$
$β_{1}$	$\arcsin (y_{1} / r)$	$β_{2}$	$\arcsin (y_{2} / r)$
$γ_{1}$	$\arcsin (x_{1} / r)$	$γ_{2}$	$\arcsin (x_{2} / r)$

Case 1	Case 2	Case 3	Case 4	Case 5
$\bar{r} = 0$	$\bar{r} \leq r_{1}$	$\bar{r} \leq r_{1 e}$	$\bar{r} \leq r_{1}$	$\bar{r} \leq r_{1}$
$\bar{r} \geq r_{1}$	$\bar{r} \geq r_{4}$	$\bar{r} \geq r_{4}$	$\bar{r} \geq r_{4}$	$\bar{r} \geq r_{4}$
$0 < \bar{r} \leq r_{1 e}$	$r_{1} < \bar{r} \leq r_{2}$	$r_{1 e} < \bar{r} \leq r_{1}$	$r_{1} < \bar{r} \leq r_{3}$	$r_{1} < \bar{r} \leq r_{2}$
$r_{1 e} < \bar{r} < r_{1}$	$r_{2} < \bar{r} < r_{4}$	$r_{1} < \bar{r} \leq r_{4 e}$	$r_{3} < \bar{r} \leq r_{2}$	$r_{2} < \bar{r} \leq r_{3}$
		$r_{4 e} < \bar{r} < r_{4}$	$r_{2} < \bar{r} < r_{4}$	$r_{3} < \bar{r} < r_{4}$

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Tables (4)

Equations (62)

Applied Optics