Abstract

We report on an analytic model of a laser-controlled thermally deformable mirror. The model shows the spatial low pass behavior of such a mirror system regarding the intensity distribution that controls the temperature distribution and the optical phase difference distribution of the deformable mirror. The model is validated using the data of measurements described by Schmid and Mahnke [J. Opt. Soc. Am. B 35, 2661 (2018) [CrossRef]  ].

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References

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  1. E. Schmid and P. Mahnke, “Thermally deformable mirror to compensate for phase aberrations in high-power laser systems,” J. Opt. Soc. Am. B 35, 2661–2666 (2018).
    [Crossref]
  2. C. Pruss, M. Matic, D. Graupner, and W. Osten, “Optically Addressed Thermally Activated Adaptive Optics,” in The Institut für Technische Optik Universität Stuttgart Annual Report (2007/2008), p. 53.
  3. R. Lange and D. Kolbe, “Laser-controlled adaptive optics for beam quality improvements in a multi-pass thin-disk amplifier,” Opt. Lett. 43, 3453–3456 (2018).
    [Crossref]
  4. K. Schmidt, P. Wittmuess, S. Piehler, M. Abdou Ahmed, T. Graf, and O. Sawodny, “Modeling and simulating the thermoelastic deformation of mirrors using transient multilayer models,” Mechatronics 53, 168–180 (2018).
    [Crossref]
  5. N. Hodgson and A. Caprara, “Semi-analytical solution for the temperature profiles in solid-state laser disks mounted on heat spreaders,” Appl. Opt. 55, 5110–5117 (2016).
    [Crossref]
  6. Schott AG, “TIE-31 Mechanical and thermal properties of optical glass” (2018), https://www.schott.com/d/advanced_optics/d08c2fb9-c2f2-4861-a57b-18495ef5a4fd/1.2/schott_tie-31_mechanical_and_thermal_properties_of_optical_eng.pdf .
  7. Schott AG, “RG-1000 Datasheet” (2008), https://microsites.schott.com/d/us-lidar/1f11815a-6def-4b9d-8bf3-e6df0009e3c6/20190925140417/schott-microsites-datasheet-rg1000-english-25092019.pdf .

2018 (3)

2016 (1)

Abdou Ahmed, M.

K. Schmidt, P. Wittmuess, S. Piehler, M. Abdou Ahmed, T. Graf, and O. Sawodny, “Modeling and simulating the thermoelastic deformation of mirrors using transient multilayer models,” Mechatronics 53, 168–180 (2018).
[Crossref]

Caprara, A.

Graf, T.

K. Schmidt, P. Wittmuess, S. Piehler, M. Abdou Ahmed, T. Graf, and O. Sawodny, “Modeling and simulating the thermoelastic deformation of mirrors using transient multilayer models,” Mechatronics 53, 168–180 (2018).
[Crossref]

Graupner, D.

C. Pruss, M. Matic, D. Graupner, and W. Osten, “Optically Addressed Thermally Activated Adaptive Optics,” in The Institut für Technische Optik Universität Stuttgart Annual Report (2007/2008), p. 53.

Hodgson, N.

Kolbe, D.

Lange, R.

Mahnke, P.

Matic, M.

C. Pruss, M. Matic, D. Graupner, and W. Osten, “Optically Addressed Thermally Activated Adaptive Optics,” in The Institut für Technische Optik Universität Stuttgart Annual Report (2007/2008), p. 53.

Osten, W.

C. Pruss, M. Matic, D. Graupner, and W. Osten, “Optically Addressed Thermally Activated Adaptive Optics,” in The Institut für Technische Optik Universität Stuttgart Annual Report (2007/2008), p. 53.

Piehler, S.

K. Schmidt, P. Wittmuess, S. Piehler, M. Abdou Ahmed, T. Graf, and O. Sawodny, “Modeling and simulating the thermoelastic deformation of mirrors using transient multilayer models,” Mechatronics 53, 168–180 (2018).
[Crossref]

Pruss, C.

C. Pruss, M. Matic, D. Graupner, and W. Osten, “Optically Addressed Thermally Activated Adaptive Optics,” in The Institut für Technische Optik Universität Stuttgart Annual Report (2007/2008), p. 53.

Sawodny, O.

K. Schmidt, P. Wittmuess, S. Piehler, M. Abdou Ahmed, T. Graf, and O. Sawodny, “Modeling and simulating the thermoelastic deformation of mirrors using transient multilayer models,” Mechatronics 53, 168–180 (2018).
[Crossref]

Schmid, E.

Schmidt, K.

K. Schmidt, P. Wittmuess, S. Piehler, M. Abdou Ahmed, T. Graf, and O. Sawodny, “Modeling and simulating the thermoelastic deformation of mirrors using transient multilayer models,” Mechatronics 53, 168–180 (2018).
[Crossref]

Wittmuess, P.

K. Schmidt, P. Wittmuess, S. Piehler, M. Abdou Ahmed, T. Graf, and O. Sawodny, “Modeling and simulating the thermoelastic deformation of mirrors using transient multilayer models,” Mechatronics 53, 168–180 (2018).
[Crossref]

Appl. Opt. (1)

J. Opt. Soc. Am. B (1)

Mechatronics (1)

K. Schmidt, P. Wittmuess, S. Piehler, M. Abdou Ahmed, T. Graf, and O. Sawodny, “Modeling and simulating the thermoelastic deformation of mirrors using transient multilayer models,” Mechatronics 53, 168–180 (2018).
[Crossref]

Opt. Lett. (1)

Other (3)

C. Pruss, M. Matic, D. Graupner, and W. Osten, “Optically Addressed Thermally Activated Adaptive Optics,” in The Institut für Technische Optik Universität Stuttgart Annual Report (2007/2008), p. 53.

Schott AG, “TIE-31 Mechanical and thermal properties of optical glass” (2018), https://www.schott.com/d/advanced_optics/d08c2fb9-c2f2-4861-a57b-18495ef5a4fd/1.2/schott_tie-31_mechanical_and_thermal_properties_of_optical_eng.pdf .

Schott AG, “RG-1000 Datasheet” (2008), https://microsites.schott.com/d/us-lidar/1f11815a-6def-4b9d-8bf3-e6df0009e3c6/20190925140417/schott-microsites-datasheet-rg1000-english-25092019.pdf .

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (5)

Fig. 1.
Fig. 1. Setup of the adaptive mirror system. The fiber end of the heating diode laser is imaged to the micromirror array, which laterally modulates the profile of the reflected beam to generate a “heating pattern” in the substrate of the mirror that results in a systematic deformation of the mirror. The resulting OPD of the mirror allows systematic compensation for phase distortions (taken from [1]).
Fig. 2.
Fig. 2. Schematic of the adaptive mirror. The filter glass substrate is glued on a copper heat sink to enable rear-side water cooling.
Fig. 3.
Fig. 3. (a) Normalized transient spatial response of the OPD on the spatial frequencies of the heating laser image. (b) Normalized spatial low frequency response dependent on the relative absorption $\alpha d$ .
Fig. 4.
Fig. 4. (a) Light distribution used for the simulation of the OPD shown in (b). The spatial cut-off frequency $0.17\;{{\rm mm}^{- 1}}$ was chosen, which results in an estimated “thermal thickness” of the disk for a highly absorbing RG1000 filter of approximately 1.3 mm. Therefore, the device has a 3 dB resolution of 0.8 mm. The difference to the real thickness of 0.5 mm RG 1000 substrate results from the insulation from the glue and the thermal distribution in the copper heat sink. The heating intensity was $5.75 \frac{{\rm W}}{{{{{\rm cm}}^2}}}$ .
Fig. 5.
Fig. 5. Simulated and measured OPD. The simulated annular light distribution is also displayed for reference.

Equations (15)

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ρ c p T t ( λ T ) = q v t ,
( λ T ) = q v t .
λ Δ T ( x , y , z ) = I ( x , y ) α e α z ,
T ( x , y , z ) = T d + n = 1 T x y n ( x , y ) cos ( 2 π ( 2 n 1 ) 4 d z ) .
λ ( ( 2 x 2 + 2 y 2 + 2 z 2 ) ( T d + n = 1 T x y n ( x , y ) cos ( 2 π ( 2 n 1 ) 4 d z ) ) ) = I ( x , y ) α e α z .
λ ( n = 1 ( 2 x 2 + 2 y 2 ( 2 π ( 2 n 1 ) 4 d ) 2 ) × ( T x y n ( x , y ) cos ( 2 π ( 2 n 1 ) 4 d z ) ) ) = I ( x , y ) n = 1 f n ( α ) cos ( 2 π ( 2 n 1 ) 4 d z ) ,
f n ( α ) = 8 α 2 d + 4 ( 1 ) n e d α α π ( 1 2 n ) 4 d 2 α 2 + ( 1 2 n ) 2 π 2 .
λ ( n = 1 ( ω x 2 + ω y 2 + ( 2 π ( 2 n 1 ) 4 d ) 2 ) × ( T x y n ( ω x , ω y ) cos ( 2 π ( 2 n 1 ) 4 d z ) ) ) = I ( ω x , ω y ) n = 1 f n ( α ) cos ( 2 π ( 2 n 1 ) 4 d z ) .
T x y n ( ω x , ω y ) = f n ( α ) λ ( ( 2 π ( 2 n 1 ) 4 d ) 2 + ( ω x 2 + ω y 2 ) ) I ( ω x , ω y ) .
O P D ( x , y ) = k 0 d ( T ( x , y , z ) T d ) d z ,
O P D ( ω x , ω y ) = k 0 d n = 1 T x y n ( ω x , ω y ) cos ( 2 π ( 2 n 1 ) 4 d z ) d z
= I ( ω x , ω y ) n = 1 k f n ( α ) sin ( π n π 2 ) λ ( 2 π ( 2 n 1 ) 4 d ) ( ( 2 π ( 2 n 1 ) 4 d ) 2 + ( ω x 2 + ω y 2 ) )
= I ( ω x , ω y ) n = 1 k f n ( α ) ( 1 ) n λ ( 2 π ( 2 n 1 ) 4 d ) ( ( 2 π ( 2 n 1 ) 4 d ) 2 + ( ω x 2 + ω y 2 ) ) .
O P D ( ω x , ω y ) I ( ω x , ω y ) k 16 d 2 α + 8 α 2 e d α π λ π ( 4 d 2 α 2 + π 2 ) ( ( 2 π 4 d ) 2 + ( ω x 2 + ω y 2 ) ) ,
O P D ( ω x , ω y ) I ( ω x , ω y ) ( 2 π 4 d ) 2 + ω 2 .