## Full-field unsymmetrical beam shaping for decreasing and homogenizing the thermal deformation of optical element in a beam control system |

Optics Express, Vol. 19, Issue S5, pp. A1037-A1050 (2011)

http://dx.doi.org/10.1364/OE.19.0A1037

Acrobat PDF (1684 KB)

### Abstract

We propose and demonstrate the full-field unsymmetrical beam shaping for decreasing and homogenizing the thermal deformation of optical element in a beam control system. The transformation of square dark hollow beam with unsymmetrical and inhomogeneous intensity distribution into square dark hollow beam with homogeneous intensity distribution is chosen to prove the validity of the technique. Dual deformable mirrors (DMs) based on the stochastic parallel gradient descent (SPGD) controller are used to redistribute the intensity of input beam and generate homogeneous square dark hollow beam with near-diffraction-limited performance. The SPGD algorithm adaptively optimizes the coefficients of Lukosz-Zernike polynomials to form the phase distributions for dual DMs. Based on the finite element method, the thermal deformations of CaF_{2} half transparent and half reflecting mirror irradiated by high power laser beam before and after beam shaping are numerically simulated and compared. The thermal deformations of the mirror irradiated by the laser beam with different powers and the influences of thermal deformation on beam quality are also numerically studied. Results show that full-field beam shaping can greatly decrease and homogenize the thermal deformation of the mirror in the beam control system. The strehl ratios of the high power laser beams passing through the beam control system can be greatly improved by the full-field beam shaping. The technique presented in this paper can provide effective guidance for optimum design of high power laser cavity and beam shaping system.

© 2011 OSA

## 1. Introduction

1. R. Hauck, H. P. Kortz, and H. Weber, “Misalignment sensitivity of optical resonators,” Appl. Opt. **19**(4), 598–601 (1980). [CrossRef] [PubMed]

2. J. L. Remo, “Diffraction losses for symmetrically tilted plane reflectors in open resonators,” Appl. Opt. **19**(5), 774–777 (1980). [CrossRef] [PubMed]

3. C. A. Klein, “Optical distortion coefficients of high-power laser windows,” Opt. Eng. **29**(4), 343–350 (1990). [CrossRef]

11. W. P. Wang, F. L. Tan, B. D. Lü, and C. L. Liu, “Three-dimensional calculation of high-power, annularly distributed, laser-beam-induced thermal effects on reflectors and windows,” Appl. Opt. **44**(34), 7442–7450 (2005). [CrossRef] [PubMed]

17. G. Zhou, X. Yuan, P. Dowd, Y. L. Lam, and Y. C. Chan, “Design of diffractive phase elements for beam shaping: hybrid approach,” J. Opt. Soc. Am. A **18**(4), 791–800 (2001). [CrossRef]

14. J. A. Hoffnagle and C. M. Jefferson, “Design and performance of a refractive optical system that converts a Gaussian to a flattop beam,” Appl. Opt. **39**(30), 5488–5499 (2000). [CrossRef] [PubMed]

17. G. Zhou, X. Yuan, P. Dowd, Y. L. Lam, and Y. C. Chan, “Design of diffractive phase elements for beam shaping: hybrid approach,” J. Opt. Soc. Am. A **18**(4), 791–800 (2001). [CrossRef]

18. H. T. Ma, Z. J. Liu, X. J. Xu, S. H. Wang, and C. H. Liu, “Near-diffraction-limited flattop laser beam adaptively generated by stochastic parallel gradient descent algorithm,” Opt. Lett. **35**(17), 2973–2975 (2010). [CrossRef] [PubMed]

_{2}half transparent and half reflecting mirror irradiated by the high power laser beam before and after beam shaping are numerically simulated and compared. The thermal deformations of the mirror irradiated by the laser beam with different powers and the influences of thermal deformation on beam quality are also numerically studied. Results show that full-field beam shaping can greatly decrease and homogenize the thermal deformation of the mirror in the beam control system. The strehl ratio of the high power laser beam passing through the beam control system can be greatly improved by the full-field beam shaping. To the best of our knowledge, the technique proposed in this paper has never been reported.

## 2. Beam Shaping Principle and Numerical Analysis

19. M. A. Vorontsov and V. P. Sivokon, “Stochastic parallel-gradient-descent technique for high-resolution wave-front phase-distortion correction,” J. Opt. Soc. Am. A **15**(10), 2745–2758 (1998). [CrossRef]

*N*Lukosz-Zernike modes.where

*L*(

_{i}*r,θ*) is the

*i*order Lukosz-Zernike mode,

^{th}*a*is the Lukosz-Zernike expansion coefficient, which is also the control signal. The functions were first derived by Lukosz and, later, independently by Braat. Like Zernike polynomials, the Lukosz-Zernike functions are each expressed as the product of a radial and azimuthal function using the same dual index and numbering scheme [20

_{i}20. D. D’ebarre, M. J. Booth, and T. Wilson, “Image based adaptive optics through optimisation of low spatial frequencies,” Opt. Express **15**(13), 8176–8190 (2007). [CrossRef]

22. M. J. Booth, “Wavefront sensorless adaptive optics for large aberrations,” Opt. Lett. **32**(1), 5–7 (2007). [CrossRef] [PubMed]

*n*and

*m*are the radial and azimuthal indices, respectively, and where

*J*=

*J*(

**) is a function of the coefficient**

*a*

*a**=*{

*a*

_{1},

*a*

_{2},

*…*,

*a*}. The SPGD algorithm is used to optimize the quality metric. The steps for SPGD algorithm can be briefly described as follows [19

_{n}19. M. A. Vorontsov and V. P. Sivokon, “Stochastic parallel-gradient-descent technique for high-resolution wave-front phase-distortion correction,” J. Opt. Soc. Am. A **15**(10), 2745–2758 (1998). [CrossRef]

- 1. Generate statistically independent random perturbations
*δa*_{1},*δa*_{2},*…*,*δa*, where all_{n}*δa*are small values that are typically chosen as statistically independent variables having zero mean and equal variances, <_{i}*δa*>=0, <_{i}*δa*>=_{i}δa_{i}*σ*^{2}*δ*where_{ij}*δ*is the Kronecker symbol._{ij} - 2. Apply the control signal with perturbations and get the metric function from the CCD camera,
*J*(_{+}= J*a*_{1}*+δa*_{1},*a*_{2}*+δa*_{2},*…*,*a*), then apply the control signals with the opposite perturbations and get the metric function,_{n}+δa_{n}*J*(_{-}= J*a*_{1}*-δa*_{1},*a*_{2}*-δa*_{2},*…*,*a*). Calculate the difference between two evaluations of the metric function_{n}-δa_{n}*δJ = J*._{+}-J_{-} - 3. Update the control signals,
*a*,_{i}=a_{i}+γδa_{i}δJ*i*=1, 2,*…*,*n*, where*γ*is the update gain.*γ>*0 and*γ<*0 according to the procedure of maximization and minimization respectively.

23. R. A. Muller and A. Buffington, “Real-time correction of atmospherically degraded telescope images through image sharpening,” J. Opt. Soc. Am. **64**(9), 1200–1210 (1974). [CrossRef]

*I*(

_{farfield}*x,y*) is the intensity distribution of the focal spot, which is recorded by CCD2. According to the analysis of Muller, the global maximum of

*J*corresponds to an undistorted wave front. In closed control loop of the intensity redistribution, the SPGD algorithm is used to minimize the

_{compensation}*J*between the target and the actual beam shape recorded by CCD1. In closed control loop of the wave front compensation, the SPGD algorithm is used to maximize the phase error metric

_{fiterror}*J*. In this paper, taking the transformation of square dark hollow beam with unsymmetrical and nonuniform intensity distribution into near-diffraction-limited square dark hollow beam with uniform intensity distribution as an example, we study the full-field shaping of the beam with unsymmetrical intensity distribution based on the SPGD controller. The intensity distribution of input beam is shown in Fig. 2(a) . The target beam is defined aswhere

_{compensation}*a*

_{1},

*a*

_{2}and

*b*

_{1},

*b*

_{2}determine the beam width in

*x*and

*y*directions.

*p*

_{1},

*p*

_{2}and

*q*

_{1},

*q*

_{2}are integers that specify the steepness of the beam sides.

*x*and

_{o}*y*are the centered positions of the target beam spot.

_{o}*m*for monitoring the far field intensity distribution. Wave front information of the output beam can be obtained by analyzing intensity distribution of the focal spot. The far field intensity distribution of the output beam before being compensated by DM2 is shown in Fig. 5(a) . Because of the phase aberration introduced by intensity redistribution, the energy density is low in the far field. In closed control loop of wave front compensation, precompensation by using the complex conjugate of intensity redistribution phase distribution is used to improve the effect. After precompensation, the SPGD algorithm is also performed for a modal wavefront corrector with Lukosz-Zernike polynomials {

*L*(

_{i}*r*,

*θ*)} (

*i*= 2,…, 200) as a set of influence functions. After phase compensation, the phase distribution and far field intensity distribution of the output beam are shown in Figs. 5(b) and 6(a) . It can be found that the output beam has been compensated successfully. The root-mean-square (

*rms*) value of the wave front within a square dark hollow region with inner side length 0.01

*m*and outer side length 0.046

*m*is 0.045λ, where λ is the wavelength of the input beam. According to the far field intensity distributions shown in Figs. 5(a) and 5(b), we calculate the power-in-the-bucket (PIB) curve of the far field intensity of the beam without compensation, with compensation, and ideal compensation, which are shown in Fig. 6(b). It can be found that the far field energy density of the output beam has been greatly improved after compensation. The power in the main lobe of the far field intensity distribution of the beam after compensation is about 102 times larger than that of the beam without compensation. There exists error between the PIB curves of the adaptive compensated output beam and ideal compensated output beam, which is mainly caused by the system itself. The phase distribution generated by the SPGD algorithm and Lukosz-Zernike polynomials for wave front compensation is not ideal.

## 3. Numerical Analysis of the Thermal Deformation

*r*

_{0}and thickness

*d*shown in Fig. 7 , the temperature deformation

*T*(

*r*,

*φ*,

*z*;

*t*) is given by the following thermal conduction equation [24]

*α*=

*κ*/

*ρC*is the thermal diffusivity.

*κ*is the thermal conductivity.

*ρ*is the density of the medium.

*C*is the special heat.

*h*is the convection coefficient in

*K*.

*T*

_{∞}is the ambient temperature.

*q*(

*r*,

*φ*;

*t*) is the heat flux load on the mirror’s surface. Considering the high transmittance, the CaF

_{2}materials are usually chosen as splitter, and optical windows of high energy lasers. In this paper, the half transparent and half reflecting mirror using CaF

_{2}material as substrate is chosen to study the influence of full-field beam shaping on thermal deformation. Some parameters of the CaF

_{2}mirror are listed in Table 1 . As shown in Fig. 7, the beam is incident on half transparent and half reflecting mirror with tilted angle

*θ*=

*π*/4. The thermal deformation is induced by both surface heat and volume heat source. The surface heat source is the absorption of incident laser by coating and volume heat source is the absorption of transmitted laser by CaF

_{2}mirror itself. Because the thickness of the coating is much smaller than that of the substrate, thermal deformation of the coating can be neglected. The thermal deformation of half transparent and half reflecting mirror can be calculated by thermo-elastic equations of the substrate material [25].

*u*and

_{r}*u*are radial and axial thermal deformations, respectively.

_{z}*ν*is the Poisson’s ratio.

*α*is the linear thermal expansion coefficient.

_{l}*ε*is the thermal strain. Because only the thermal deformation of the mirror along

*z*direction has influences on beam quality, only the thermal deformation on the surface of mirror

*u*(

_{z}*r*,

*φ*,

*d*) is considered.

*cm*×10

*cm*. Considering the oblique incident of the beam (tilted angle

*θ*=

*π*/4), the power densities of the 150kW incident laser beam after and before beam shaping on the front surface

*S*

_{1}of the half transparent and half reflecting mirror are shown in Fig. 8 .

*a*(0.03

*m*, 0

*m*) and position

*b*(0.0068

*m*, 0.024

*m*) for analyzing the thermal deformation along with the irradiation time. The corresponding relative thermal deformations are shown in Fig. 10 . In Figs. 10(a), 10(b) and 10(c), pink and blue curves represent the thermal deformations induced by the incident beam before and after beam shaping respectively. When the irradiation time is 3 seconds and 6 seconds, the difference between the thermal deformation at positions

*a*and position

*b*irradiated by the beam with 50kW power after beam shaping are lowered by 92.3% and 88.3%. The difference between the thermal deformation at positions

*a*and position

*b*irradiated by the beam with 100kW power after beam shaping are lowered by 91.7% and 88.5%. The difference between the thermal deformation at positions

*a*and position

*b*irradiated by the beam with 150kW power after beam shaping are lowered by 91.7% and 88.0%.

*m*≤

*x*≤-0.021

*m*& 0.025

*m*≤

*x*≤0.067

*m*& −0.047

*m*≤

*y*≤-0.018

*m*& 0.016

*m*≤ y ≤0.046

*m*), so beam quality is mainly affected by thermal deformation of mirror in this region. The

*PV*and

*rms*values of the thermal deformation in this region are defined as where

*D*(

*n*),

*D*

_{max},

*D*

_{min}are the displacement of node

*n*, the node with maximum displacement, the node with minimum displacement in the selection region respectively.

*M*is the number of nodes in this region.

*PV*and

*rms*values of thermal deformation in this region are calculated and shown in Fig. 11 . It can be found that

*PV*and

*rms*values of thermal deformation induced by the beam after beam shaping are much smaller than that induced by the beam before beam shaping. When the irradiation time is 6 seconds, the corresponding

*PV*values of thermal deformation induced by the beam with 50kW, 100kW and 150kW power after beam shaping are decreased by 0.6150

*μm*, 1.229

*μm*, and 1.884

*μm*, respectively. The corresponding

*rms*values of thermal deformation induced by the beam with 50kW, 100kW and 150kW power after beam shaping are decreased by 0.1302

*μm*, 0.2598

*μm*, and 0.39

*μm*, respectively.

*θ*is tilted angle of the input beam. In the numerical simulation, the tilted angle

*θ*is chosen as π/4. The beam with intensity distribution shown in Fig. 2 is chosen as the beam without beam shaping. The beam with intensity distribution shown in Fig. 3(a) and phase distribution shown in Fig. 6(a) is chosen as the beam after beam shaping. The intensity distribution area of the beam after and before beam shaping is chosen as 10

*cm*×10

*cm*. The half transparent and half reflecting mirror irradiated by the high power laser beam without and with full-field beam shaping (the irradiation time is 6s) is chosen to study the influences. We use the lens with focal length 150

*m*to focus the beam reflected by the mirror. After being reflected by the half transparent and half reflecting mirror, the far field intensity distributions of 50kW power laser beam without and with beam shaping are shown in Figs. 12(a) and 12(b). The corresponding strehl ratios of the beam without and with beam shaping after being reflected by the mirror are 0.417 and 0.61. The strehl ratio of the beam with beam shaping is 1.46 times as large as that of the beam without beam shaping. The beam qualities of the 100kW power laser beam and 150kW power laser beam reflected by the half transparent and half reflecting mirror have also been studied. The strehl ratios of the 100kW and 150kW power laser beams with beam shaping are 1.32 and 1.45 times as large as that of the beams without beam shaping, respectively. It can be concluded that the beam quality of reflected beam can be greatly improved by full-field beam shaping. If the beam control system consists of more mirrors, the beam quality will get greater increase.

## 4. Conclusion

*PV*and

*rms*values of the thermal deformation of the mirrors have been greatly decreased. The strehl ratio of the high power laser beam after passing through the beam control system can be greatly improved by the full-field beam shaping.

## References and links

1. | R. Hauck, H. P. Kortz, and H. Weber, “Misalignment sensitivity of optical resonators,” Appl. Opt. |

2. | J. L. Remo, “Diffraction losses for symmetrically tilted plane reflectors in open resonators,” Appl. Opt. |

3. | C. A. Klein, “Optical distortion coefficients of high-power laser windows,” Opt. Eng. |

4. | C. A. Klein, “Materials for high-power laser optics figures of merit for thermally induced beam distortions,” Opt. Eng. |

5. | J. D. Mansell, J. Hennawi, E. K. Gustafson, M. M. Fejer, R. L. Byer, D. Clubley, S. Yoshida, and D. H. Reitze, “Evaluating the effect of transmissive optic thermal lensing on laser beam quality with a Shack-Hartmann wave-front sensor,” Appl. Opt. |

6. | Y. F. Peng, Z. H. Cheng, Y. N. Zhang, and J. L. Qiu, “Temperature distributions and thermal deformations of mirror substrates in laser resonators,” Appl. Opt. |

7. | J. B. Chen, Z. J. Liu, Z. P. Jiang, Q. S. Lu, Z. W. Zhang, and Y. J. Zhao, “Heating effect of DF laser unstable cavity window and its effect on far-field optical spot,” High Power Laser Part. Beams |

8. | Y. Y. Ma, Z. H. Cheng, and Y. N. Zhang, “Finite-element method in thermal deformation analysis of high power laser windows,” High Power Laser Part. Beams |

9. | C. A. Klein, “High-energy laser windows: case of fused silica,” Opt. Eng. |

10. | M. S. Sparks, “Optical distortion by heated windows in high-power laser systems,” J. Appl. Phys. |

11. | W. P. Wang, F. L. Tan, B. D. Lü, and C. L. Liu, “Three-dimensional calculation of high-power, annularly distributed, laser-beam-induced thermal effects on reflectors and windows,” Appl. Opt. |

12. | F. M. Dickey, S. C. Holswade, and D. L. Shealy, eds., |

13. | J. M. Auerbach and V. P. Karpenko, “Serrated-aperture apodizers for high-energy laser systems,” Appl. Opt. |

14. | J. A. Hoffnagle and C. M. Jefferson, “Design and performance of a refractive optical system that converts a Gaussian to a flattop beam,” Appl. Opt. |

15. | J. L. Kreuzer, “Coherent light optical system yielding an output beam of desired intensity distribution at a desired equiphase surface,” U.S. patent 3,476,463 (4 November 1969). |

16. | J. H. Li, K. J. Webb, G. J. Burke, D. A. White, and C. A. Thompson, “Design of near-field irregular diffractive optical elements by use of a multiresolution direct binary search method,” Opt. Lett. |

17. | G. Zhou, X. Yuan, P. Dowd, Y. L. Lam, and Y. C. Chan, “Design of diffractive phase elements for beam shaping: hybrid approach,” J. Opt. Soc. Am. A |

18. | H. T. Ma, Z. J. Liu, X. J. Xu, S. H. Wang, and C. H. Liu, “Near-diffraction-limited flattop laser beam adaptively generated by stochastic parallel gradient descent algorithm,” Opt. Lett. |

19. | M. A. Vorontsov and V. P. Sivokon, “Stochastic parallel-gradient-descent technique for high-resolution wave-front phase-distortion correction,” J. Opt. Soc. Am. A |

20. | D. D’ebarre, M. J. Booth, and T. Wilson, “Image based adaptive optics through optimisation of low spatial frequencies,” Opt. Express |

21. | O. Braat, “Polynomial expansion of severely aberrated wave fronts,” J. Opt. Soc. Am. A |

22. | M. J. Booth, “Wavefront sensorless adaptive optics for large aberrations,” Opt. Lett. |

23. | R. A. Muller and A. Buffington, “Real-time correction of atmospherically degraded telescope images through image sharpening,” J. Opt. Soc. Am. |

24. | J. H. Lienhard IV and J. H. Lienhard V, |

25. | H. G. Wang, |

26. | M. Born and E. Wolf, |

**OCIS Codes**

(140.0140) Lasers and laser optics : Lasers and laser optics

(140.3300) Lasers and laser optics : Laser beam shaping

(140.3330) Lasers and laser optics : Laser damage

(140.6810) Lasers and laser optics : Thermal effects

**ToC Category:**

Nuclear Fusion

**History**

Original Manuscript: April 1, 2011

Revised Manuscript: May 19, 2011

Manuscript Accepted: June 9, 2011

Published: July 15, 2011

**Citation**

Haotong Ma, Qiong Zhou, Xiaojun Xu, Shaojun Du, and Zejin Liu, "Full-field unsymmetrical beam shaping for decreasing and homogenizing the thermal deformation of optical element in a beam control system," Opt. Express **19**, A1037-A1050 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-S5-A1037

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### References

- R. Hauck, H. P. Kortz, and H. Weber, “Misalignment sensitivity of optical resonators,” Appl. Opt. 19(4), 598–601 (1980). [CrossRef] [PubMed]
- J. L. Remo, “Diffraction losses for symmetrically tilted plane reflectors in open resonators,” Appl. Opt. 19(5), 774–777 (1980). [CrossRef] [PubMed]
- C. A. Klein, “Optical distortion coefficients of high-power laser windows,” Opt. Eng. 29(4), 343–350 (1990). [CrossRef]
- C. A. Klein, “Materials for high-power laser optics figures of merit for thermally induced beam distortions,” Opt. Eng. 36(6), 1586–1595 (1997). [CrossRef]
- J. D. Mansell, J. Hennawi, E. K. Gustafson, M. M. Fejer, R. L. Byer, D. Clubley, S. Yoshida, and D. H. Reitze, “Evaluating the effect of transmissive optic thermal lensing on laser beam quality with a Shack-Hartmann wave-front sensor,” Appl. Opt. 40(3), 366–374 (2001). [CrossRef] [PubMed]
- Y. F. Peng, Z. H. Cheng, Y. N. Zhang, and J. L. Qiu, “Temperature distributions and thermal deformations of mirror substrates in laser resonators,” Appl. Opt. 40(27), 4824–4830 (2001). [CrossRef] [PubMed]
- J. B. Chen, Z. J. Liu, Z. P. Jiang, Q. S. Lu, Z. W. Zhang, and Y. J. Zhao, “Heating effect of DF laser unstable cavity window and its effect on far-field optical spot,” High Power Laser Part. Beams 6, 243–249 (1994).
- Y. Y. Ma, Z. H. Cheng, and Y. N. Zhang, “Finite-element method in thermal deformation analysis of high power laser windows,” High Power Laser Part. Beams 11, 6–10 (1999).
- C. A. Klein, “High-energy laser windows: case of fused silica,” Opt. Eng. 49(9), 091006 (2010). [CrossRef]
- M. S. Sparks, “Optical distortion by heated windows in high-power laser systems,” J. Appl. Phys. 42(12), 5029–5046 (1971). [CrossRef]
- W. P. Wang, F. L. Tan, B. D. Lü, and C. L. Liu, “Three-dimensional calculation of high-power, annularly distributed, laser-beam-induced thermal effects on reflectors and windows,” Appl. Opt. 44(34), 7442–7450 (2005). [CrossRef] [PubMed]
- F. M. Dickey, S. C. Holswade, and D. L. Shealy, eds., Laser Beam Shaping Applications (CRC Press, 2005).
- J. M. Auerbach and V. P. Karpenko, “Serrated-aperture apodizers for high-energy laser systems,” Appl. Opt. 33(15), 3179–3183 (1994). [CrossRef] [PubMed]
- J. A. Hoffnagle and C. M. Jefferson, “Design and performance of a refractive optical system that converts a Gaussian to a flattop beam,” Appl. Opt. 39(30), 5488–5499 (2000). [CrossRef] [PubMed]
- J. L. Kreuzer, “Coherent light optical system yielding an output beam of desired intensity distribution at a desired equiphase surface,” U.S. patent 3,476,463 (4 November 1969).
- J. H. Li, K. J. Webb, G. J. Burke, D. A. White, and C. A. Thompson, “Design of near-field irregular diffractive optical elements by use of a multiresolution direct binary search method,” Opt. Lett. 31(9), 1181–1183 (2006). [CrossRef] [PubMed]
- G. Zhou, X. Yuan, P. Dowd, Y. L. Lam, and Y. C. Chan, “Design of diffractive phase elements for beam shaping: hybrid approach,” J. Opt. Soc. Am. A 18(4), 791–800 (2001). [CrossRef]
- H. T. Ma, Z. J. Liu, X. J. Xu, S. H. Wang, and C. H. Liu, “Near-diffraction-limited flattop laser beam adaptively generated by stochastic parallel gradient descent algorithm,” Opt. Lett. 35(17), 2973–2975 (2010). [CrossRef] [PubMed]
- M. A. Vorontsov and V. P. Sivokon, “Stochastic parallel-gradient-descent technique for high-resolution wave-front phase-distortion correction,” J. Opt. Soc. Am. A 15(10), 2745–2758 (1998). [CrossRef]
- D. D’ebarre, M. J. Booth, and T. Wilson, “Image based adaptive optics through optimisation of low spatial frequencies,” Opt. Express 15(13), 8176–8190 (2007). [CrossRef]
- O. Braat, “Polynomial expansion of severely aberrated wave fronts,” J. Opt. Soc. Am. A 4(4), 643–650 (1987). [CrossRef]
- M. J. Booth, “Wavefront sensorless adaptive optics for large aberrations,” Opt. Lett. 32(1), 5–7 (2007). [CrossRef] [PubMed]
- R. A. Muller and A. Buffington, “Real-time correction of atmospherically degraded telescope images through image sharpening,” J. Opt. Soc. Am. 64(9), 1200–1210 (1974). [CrossRef]
- J. H. Lienhard IV and J. H. Lienhard V, A Heat Transfer Textbook (Phlogiston Press, 2005).
- H. G. Wang, Conspectus of Thermo-Elasticity (Qinghua University Press, 1989).
- M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).

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