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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 18 — Sep. 9, 2013
  • pp: 20497–20505
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Orientation dependent wavefront correction system under grazing incidence

Xingkun Ma, Lei Huang, Mali Gong, Qiao Xue, Zexin Feng, Ping Yan, and Qiang Liu  »View Author Affiliations


Optics Express, Vol. 21, Issue 18, pp. 20497-20505 (2013)
http://dx.doi.org/10.1364/OE.21.020497


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Abstract

Making use of the stretching effect of grazing incident laser beam, a novel method of wavefront correction was promoted. Without adding any extra beam expanding components, aberrations of wavefront could achieve satisfying correction by two grazing reflections along orthogonal directions on the deformable mirrors. The stretching effect expanded the beam size along grazing direction and the orientation dependent varying aberrations were well compensated as more actuators took effect in the correction process. Analysis showed that the fitting coefficient of all the first 30 order Zernike polynomials could be controlled within 5% by this method.

© 2013 Optical Society of America

1. Introduction

Proceeding from the purpose to enlarge the input beam, the simple idea of grazing incidence occurred to us. This opinion was first employed to continuously control the fringe separation in the interferometry [19

19. N. Abramson, “The interferoscope: a new type of interferometer with variable fringe separation,” Optik (Stuttg.) 30, 56–71 (1969).

]. Benefitting from the grazing incidence, great improvements were achieved in last decades [20

20. T. E. Carlsson, N. H. Abramson, and K. H. Fischer, “Automatic measurement of surface height with the interferoscope,” Opt. Eng. 35(10), 2938–2942 (1996). [CrossRef]

23

23. H. Nüger and J. Schwider, “Measurement of curvature and thickness variations of plane surfaces by grazing incidence interferometry,” Optik (Stuttg.) 111, 319–327 (2000).

]. It also drew attention in the design of solid state laser where compactness and good beam quality were accomplished [24

24. H. Zimer, K. Albers, and U. Wittrock, “Grazing-incidence YVO4-Nd:YVO4 composite thin slab laser with low thermo-optic aberrations,” Opt. Lett. 29(23), 2761–2763 (2004). [CrossRef] [PubMed]

,25

25. F. He, M. Gong, L. Huang, Q. Liu, Q. Wang, and X. Yan, “Compact TEM00 grazing-incidence Nd:GdVO4 laser using a folded cavity,” Appl. Phys. B 86(3), 447–450 (2007). [CrossRef]

]. In this paper, the application of grazing incidence in wavefront correction was introduced. One distorted wavefront could get satisfying correction after two times grazing reflected on DMs along orthogonal directions. Without adding any extra components, this method brought about concise configuration as well as better system stability.

2. Grazing incidence wavefront correction system

When the input beam reflected on the DM at a large incidence angle, the beam size along the incident direction would be stretched, which was showed in Fig. 1
Fig. 1 Diagram of grazing incidence
. By changing the grazing incidence angle θ, the amplifying factor γ can be expressed as

γ=1/cosθ
(1)

Theoretically, as the incidence angle θ changing, the incident beam could be continuously stretched to infinite size as the incidence angle approaching 90°. In practical application, the amplifying factor within 10 could be reached as the incidence angle changed from 0° to 84.3°

Since the incidence angle was distinctly greater than 0°, the optical path difference (OPD) after reflection was dependent on the incidence angle. As showed in Fig. 2
Fig. 2 Optical path difference of grazing incidence in step-shaped DM model.
, the surface of DM could be modeled as steps. Two light beams were reflected on the step-shaped DM surface of which the difference in height was Δh.

The dash-lines AB and DG indicated the incident and reflected wavefront respectively. Obviously, the OPD Δφ between the two beams was |EG|-|CD|

{|EG|=Δh/cosθ|CD|=|EF|=|EG|sinα=|EG|sin(2θπ2),
(2)
Δφ=|EG||CD|=2Δhcosθ.
(3)

Compared with vertical incidence condition, the OPD was reduced by the factor cosθ. Realization of the enlargement of incident beam sacrificed the modulation depth of DM. However, the modified range were among several micrometers which was small compared with the deformation range of DM. Thus, scarification within certain range in the modulation depth was acceptable in practical application.

Fig. 3
Fig. 3 Configuration of the twice grazing reflected AO system.
. showed the configuration of twice grazing reflected correction system. The input beam was directed to be grazing reflected on DM1 and DM2 consecutively. A 4f optical system was employed to arrange the two reflection planes in conjugate positions. The grazing angles were θ1 and θ2 and the grazing directions were along X-axis and Y-axis respectively. The Hartmann Shack senser was employed to detected the output wavefront. After necessary data analysis, the control signals of DMs were generated and the whole close-loop AO system was established.

For explaining the stretching effect, diagrams of beam shapes under different incidence conditions were presented in Fig. 4
Fig. 4 Stretching effect under different incidence conditions.
, where the amplification factor in both directions was 3 times. Taking the central part of the input beam for analyzing, in Fig. 4(a), 4 actuators (painted black) mainly decided the correction result of that area. While under grazing incidence along X-direction, 8 actuators took effect according to the stretching along X-axis in Fig. 4(b). The X-directional varying spatial frequencies which would have not been corrected under vertical incidence would possibly been corrected now. Similar improvement of correction ability occurred along Y-axis as well when the grazing incidence was shifted to the orthogonal direction which was showed in Fig. 4(c).

The direct-gradient wavefront control algorithm [26

26. W. Jiang and H. Li, “Hartmann-Shack wavefront sensing and wavefront control algorithm,” SPIE 1271, 82–93 (1990). [CrossRef]

,27

27. X. Li, C. Wang, H. Xian, X. Wu, and W. Jiang, “Zernike modal compensation analysis for an adaptive optics system using direct-gradient wavefront reconstruction algorithm,” SPIE 3762, 116–124 (1999). [CrossRef]

] was applied in the system. At first the incident wavefront was meshed into discrete pixels. The influence of actuator located at (xa,ya) on each pixel (xi,yi) could be expressed as

ha(xi,yi,θ)=cosθexp[ln(α)((xixar)2+(yiyar)2)β]
(4)

Where θ was the grazing incidence angle, r was the influence radius of actuators, α and β were parameters measured from experiment. The influence on pixels induced by each actuator was set in one column, by conjugating all columns together, the influence matrix H was obtained. The control signal A was a column vector with elements equal to the number of actuators. The relationship between control signal and gradient vector W measured by wavefront senser can be expressed as

Hm×nAn×1=Wm×1
(5)

Where m was the pixels’ number of incident beam and n was the actuators’ number. As the pixels’ number m was greatly larger than the actuators’ number n, in most cases, the linear equations were inconsistent and had no exact solution. However, solution at the least-square sense still could be found as

A*=H+W
(6)

Where H+ was the left inverse of H, A* was the least square solution to Eq. (5). If one laser beam with wavefront of W passed through the grazing reflected AO system with incidence angles of θx and θy, the control signal Ax* and residual wavefront error ex after grazing reflection along X-axis can be written as

Ax*(θx)=Hx+(θx)W,
(7)
ex(θx)=Hx(θx)Ax*(θx)W=[Hx(θx)Hx+(θx)I]W.
(8)

After the second grazing reflection along Y-axis, the control signal Ay* was the function of both grazing angles
Ay*(θx,θy)=Hy+(θy)ex(θx)
(9)
The residual error after two-sections correction can be written as the function of θx and θy

exy(θx,θy)=Hy(θy)Ay*(θx,θy)ex(θx)=[Hy(θy)Hy+(θy)I][Hx(θx)Hx+(θx)I]W.
(10)

3. Characteristics of grazing incidence wavefront correction system

Since the Zernike polynomials are orthogonal within the unit circle, the distortion of wavefront could be conveniently decomposed into the sums of Zernike polynomials. We supposed the aberration could be expressed by the first 30 order Zernike polynomials and investigated the correction effects on each Zernike polynomial. Parameters used in the following calculations were listed in Table 1

Table 1. Parameters Used in Analysis

table-icon
View This Table
.

3.1 Orientation dependent characteristics of grazing incidence

At first, investigations were carried out under one reflection condition to reveal the characteristics of grazing incidence. Since this method was strongly orientation dependent, we picked out the 3th, 6th and 7th Zernike polynomials, which respectively represented radially, X-axis and Y-axis varying wavefront, to see the correction results after grazing reflection along X-direction.

The comparison of the three Zernike polynomials was listed in Fig. 5
Fig. 5 Comparison of correction results of 3th, 6th and 7th Zernike polynomials.
. The first column was the original wavefronts, the second column was the corrected wavefronts under vertical incidence condition and the last column was residual wavefronts after grazing reflection along X-axis at optimal incidence angle. The orientational characteristic was obvious, the 6th Zernike polynomial was well compensated under grazing incidence since its varying orientation was parallel to the grazing reflection direction, the radially varying 3th Zernike polynomial just obtained little improvement while improvement was barely observed in the correction of Y-axis varying 6th Zernike polynomial. In order to compare the correction results at different grazing angle, we defined the improvement factor along X-axis as

δx(θx)=RMS[ex(θx)]RMS[ex(0)]
(11)

Where exx) and ex(0°) were the residual wavefront errors defined in Eq. (8). RMS was the root mean square value of the residual errors. The relationship between improvement factor and grazing angle was demonstrated in Fig. 6
Fig. 6 Improvement factor to grazing angle of 3th, 6th and 7th Zernike polynomials.
.

As the grazing angle increased, the improvement factor of radially varying 3th Zernike polynomial deteriorated at first and then performed certain descent. The X-axis varying 6th Zernike polynomial experienced a clear improvement process and achieved the best correction at the largest grazing angle, while the Y-axis varying 7th Zernike polynomial was totally insensitive to the change of grazing angle.

From the comparison, we can see the strong orientation dependent characteristic of the grazing incidence AO system. Since the actuators were arranged in rectangular shape, the correction process was effective along certain direction and comparatively independent between orthogonal directions, thus there was potential to achieve satisfying correction by two times grazing correction in orthogonal directions.

3.2 Effectiveness after correction in orthogonal directions

The correction results of 3th, 6th and 7th Zernike polynomials were shown in Fig. 7
Fig. 7 Correction results in different processes for 3th, 6th and 7th Zernike polynomials.
. The three columns respectively presented the residual wavefront errors under vertical incidence, X-direction grazing incidence alone and both direction grazing incidence. The improvement in the control of residual error was obvious after both directions’ correction. Since the 6th and 7th Zernike polynomials were orientation dependent aberrations themselves, the wavefronts after correction were close to plane waves. The radially varying 3th Zernike polynomial also benefited from the two times’ stretching effect since more actuators took effect and more flexibilities were available in the direct-gradient algorithm.

As the grazing angle ranging from 0° to 81° in this calculation, an iterative program was applied to find out the optimal permutation of grazing angles to achieve the least RMS of residual wavefronts. In summary of the comparison among the three incidence conditions, the merit chosen to evaluate the correction was defined as the fitting coefficient ξ

ξjZernike=i=1me2(xi,yi)/i=1mZj2(xi,yi)
(12)

Where Zj(x,y) was the jth order Zernike polynomial and e(x,y) was the residual wavefront error after correction. The fitting coefficient ξ measured the ratio of uncompensated aberration to the total aberration of original Zernike polynomial.

The fitting coefficients of the first 30 order Zernike polynomials in three different conditions were summarized in Fig. 8
Fig. 8 Fitting coefficient of the first 30 order of Zernike polynomials.
. It was obvious that one time grazing correction was not effective to all Zernike polynomials since it was strongly orientation dependent. While after grazing reflections along both directions, all the first 30 order Zernike polynomials were well corrected and the fitting coefficients were effectively controlled below 5%. The improvement was especially significant in the correction of some high order Zernike polynomials, such as the 23th,30th Zernike polynomials, where the residual distortions were not pleasing enough after conventional DM correction. Compared with the “Combinational-deformable-mirror” reported in [28

28. H. Yang, G. Liu, C. Rao, Y. Zhang, and W. Jiang, “Combinational-deformable-mirror adaptive optics system for compensation of high-order modes of wavefront,” Chin. Opt. Lett. 5, 435–437 (2007).

], where two DMs were used in wavefront correction as well, the residual error after twice grazing incidence was better and the enhancement was even greater in corrections of high order Zernike polynomials.

Results of the analysis proved the possibility and effectiveness of the grazing incidence wavefront correction method. By simply stretching the size of the input beam along orthogonal directions, the correction ability of the AO system can be improved remarkably under certain actuators’ distribution density. Besides, none extra beam expanding components were needed which benefitted the stability of system as well.

4. Conclusion

Acknowledgments

Supported by the National Natural Science Foundation of China (No. 61275146), the Research Fund for the Doctoral Program of Higher Education of China (No. 20120002110066) and the Special Program of the Co-construction with Beijing Municipal Government of China (No. 20121000302).

References and links

1.

M. L. Gong, Y. Qiu, L. Huang, Q. Liu, P. Yan, and H. T. Zhang, “Beam quality improvement by joint compensation of amplitude and phase,” Opt. Lett. 38(7), 1101–1103 (2013). [CrossRef] [PubMed]

2.

W. Lubeigt, G. Valentine, and D. Burns, “Enhancement of laser performance using an intracavity deformable membrane mirror,” Opt. Express 16(15), 10943–10955 (2008). [CrossRef] [PubMed]

3.

H. Baumhacker, G. Pretzler, K. J. Witte, M. Hegelich, M. Kaluza, S. Karsch, A. Kudryashov, V. Samarkin, and A. Roukossouev, “Correction of strong phase and amplitude modulations by two deformable mirrors in a multistaged Ti:sapphire laser,” Opt. Lett. 27(17), 1570–1572 (2002). [CrossRef] [PubMed]

4.

S. Piehler, B. Weichelt, A. Voss, M. A. Ahmed, and T. Graf, “Power scaling of fundamental-mode thin-disk lasers using intracavity deformable mirrors,” Opt. Lett. 37(24), 5033–5035 (2012). [CrossRef] [PubMed]

5.

F. Druon, G. Chériaux, J. Faure, J. Nees, M. Nantel, A. Maksimchuk, G. Mourou, J. C. Chanteloup, and G. Vdovin, “Wave-front correction of femtosecond terawatt lasers by deformable mirrors,” Opt. Lett. 23(13), 1043–1045 (1998). [CrossRef] [PubMed]

6.

C. Valentin, J. Gautier, J.-P. Goddet, C. Hauri, T. Marchenko, E. Papalazarou, G. Rey, S. Sebban, O. Scrick, P. Zeitoun, G. Dovillaire, X. Levecq, S. Bucourt, and M. Fajardo, “High-order harmonic wave fronts generated with controlled astigmatic infrared laser,” J. Opt. Soc. Am. B 25(7), 161–166 (2008). [CrossRef]

7.

X. Lei, B. Xu, P. Yang, L. Dong, W. Liu, and H. Yan, “Beam cleanup of a 532-nm pulsed solid-state laser using a bimorph mirror,” Chin. Opt. Lett. 10(2), 021401 (2012). [CrossRef]

8.

X. Lei, S. Wang, H. Yan, W. Liu, L. Dong, P. Yang, and B. Xu, “Double-deformable-mirror adaptive optics system for laser beam cleanup using blind optimization,” Opt. Express 20(20), 22143–22157 (2012). [CrossRef] [PubMed]

9.

S. Hu, B. Xu, X. Zhang, J. Hou, J. Wu, and W. Jiang, “Double-deformable-mirror adaptive optics system for phase compensation,” Appl. Opt. 45(12), 2638–2642 (2006). [CrossRef] [PubMed]

10.

R. Zacharias, E. Bliss, S. Winters, R. Sacks, M. Feldman, A. Grey, J. Koch, C. Stolz, J. Toeppen, L. Van Atta, and B. Woods, “Wavefront control of high-power laser beams in the National Ignition Facility (NIF),” Proc. SPIE 3889, 332–343 (2000). [CrossRef]

11.

R. Zacharias, E. Bliss, M. Feldman, A. Grey, M. Henesian, J. Koch, J. Lawson, R. Sacks, T. Salmon, J. Toeppen, L. Van Atta, S. Winters, B. Woods, C. Lafiandra, and D. G. Bruns, “The National Ignition Facility(NIF) wavefront control system,” Proc. SPIE 3492, 678–692 (1999). [CrossRef]

12.

O. Solgaard, F. S. A. Sandejas, and D. M. Bloom, “Deformable grating optical modulator,” Opt. Lett. 17(9), 688–690 (1992). [CrossRef] [PubMed]

13.

T. Sato, H. Ishida, and O. Ikeda, “Adaptive PVDF piezoelectric deformable mirror system,” Appl. Opt. 19(9), 1430–1434 (1980). [CrossRef] [PubMed]

14.

P. Yang, Y. Liu, W. Yang, M.-W. Ao, S.-J. Hu, B. Xu, and W.-H. Jiang, “Adaptive mode optimization of a continuous-wave solid-state laser using an intracavity piezoelectric deformable mirror,” Opt. Commun. 278(2), 377–381 (2007). [CrossRef]

15.

Q. Xue, L. Huang, P. Yan, M. Gong, Z. Feng, Y. Qiu, T. Li, and G. Jin, “Research on the particular temperature-induced surface shape of a National Ignition Facility deformable mirror,” Appl. Opt. 52(2), 280–287 (2013). [CrossRef] [PubMed]

16.

M. Kasprzack, B. Canuel, F. Cavalier, R. Day, E. Genin, J. Marque, D. Sentenac, and G. Vajente, “Performance of a thermally deformable mirror for correction of low-order aberrations in laser beams,” Appl. Opt. 52(12), 2909–2916 (2013). [CrossRef] [PubMed]

17.

M. A. Arain, W. Z. Korth, L. F. Williams, R. M. Martin, G. Mueller, D. B. Tanner, and D. H. Reitze, “Adaptive control of modal properties of optical beams using photothermal effects,” Opt. Express 18(3), 2767–2781 (2010). [CrossRef] [PubMed]

18.

D. Brousseau, E. F. Borra, and S. Thibault, “Wavefront correction with a 37-actuator ferrofluid deformable mirror,” Opt. Express 15(26), 18190–18199 (2007). [CrossRef] [PubMed]

19.

N. Abramson, “The interferoscope: a new type of interferometer with variable fringe separation,” Optik (Stuttg.) 30, 56–71 (1969).

20.

T. E. Carlsson, N. H. Abramson, and K. H. Fischer, “Automatic measurement of surface height with the interferoscope,” Opt. Eng. 35(10), 2938–2942 (1996). [CrossRef]

21.

X. Colonna de Lega, J. F. Biegen, D. Stephenson, and P. J. de Groot, “Characterization of a geometrically desensitized interferometer for flatness testing,” Proc. SPIE 3520, 284–292 (1998). [CrossRef]

22.

P. de Groot, “Diffractive grazing-incidence interferometer,” Appl. Opt. 39(10), 1527–1530 (2000). [CrossRef] [PubMed]

23.

H. Nüger and J. Schwider, “Measurement of curvature and thickness variations of plane surfaces by grazing incidence interferometry,” Optik (Stuttg.) 111, 319–327 (2000).

24.

H. Zimer, K. Albers, and U. Wittrock, “Grazing-incidence YVO4-Nd:YVO4 composite thin slab laser with low thermo-optic aberrations,” Opt. Lett. 29(23), 2761–2763 (2004). [CrossRef] [PubMed]

25.

F. He, M. Gong, L. Huang, Q. Liu, Q. Wang, and X. Yan, “Compact TEM00 grazing-incidence Nd:GdVO4 laser using a folded cavity,” Appl. Phys. B 86(3), 447–450 (2007). [CrossRef]

26.

W. Jiang and H. Li, “Hartmann-Shack wavefront sensing and wavefront control algorithm,” SPIE 1271, 82–93 (1990). [CrossRef]

27.

X. Li, C. Wang, H. Xian, X. Wu, and W. Jiang, “Zernike modal compensation analysis for an adaptive optics system using direct-gradient wavefront reconstruction algorithm,” SPIE 3762, 116–124 (1999). [CrossRef]

28.

H. Yang, G. Liu, C. Rao, Y. Zhang, and W. Jiang, “Combinational-deformable-mirror adaptive optics system for compensation of high-order modes of wavefront,” Chin. Opt. Lett. 5, 435–437 (2007).

OCIS Codes
(010.3310) Atmospheric and oceanic optics : Laser beam transmission
(140.3300) Lasers and laser optics : Laser beam shaping

ToC Category:
Optical Devices

History
Original Manuscript: May 28, 2013
Revised Manuscript: August 16, 2013
Manuscript Accepted: August 16, 2013
Published: August 26, 2013

Citation
Xingkun Ma, Lei Huang, Mali Gong, Qiao Xue, Zexin Feng, Ping Yan, and Qiang Liu, "Orientation dependent wavefront correction system under grazing incidence," Opt. Express 21, 20497-20505 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-18-20497


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References

  1. M. L. Gong, Y. Qiu, L. Huang, Q. Liu, P. Yan, and H. T. Zhang, “Beam quality improvement by joint compensation of amplitude and phase,” Opt. Lett.38(7), 1101–1103 (2013). [CrossRef] [PubMed]
  2. W. Lubeigt, G. Valentine, and D. Burns, “Enhancement of laser performance using an intracavity deformable membrane mirror,” Opt. Express16(15), 10943–10955 (2008). [CrossRef] [PubMed]
  3. H. Baumhacker, G. Pretzler, K. J. Witte, M. Hegelich, M. Kaluza, S. Karsch, A. Kudryashov, V. Samarkin, and A. Roukossouev, “Correction of strong phase and amplitude modulations by two deformable mirrors in a multistaged Ti:sapphire laser,” Opt. Lett.27(17), 1570–1572 (2002). [CrossRef] [PubMed]
  4. S. Piehler, B. Weichelt, A. Voss, M. A. Ahmed, and T. Graf, “Power scaling of fundamental-mode thin-disk lasers using intracavity deformable mirrors,” Opt. Lett.37(24), 5033–5035 (2012). [CrossRef] [PubMed]
  5. F. Druon, G. Chériaux, J. Faure, J. Nees, M. Nantel, A. Maksimchuk, G. Mourou, J. C. Chanteloup, and G. Vdovin, “Wave-front correction of femtosecond terawatt lasers by deformable mirrors,” Opt. Lett.23(13), 1043–1045 (1998). [CrossRef] [PubMed]
  6. C. Valentin, J. Gautier, J.-P. Goddet, C. Hauri, T. Marchenko, E. Papalazarou, G. Rey, S. Sebban, O. Scrick, P. Zeitoun, G. Dovillaire, X. Levecq, S. Bucourt, and M. Fajardo, “High-order harmonic wave fronts generated with controlled astigmatic infrared laser,” J. Opt. Soc. Am. B25(7), 161–166 (2008). [CrossRef]
  7. X. Lei, B. Xu, P. Yang, L. Dong, W. Liu, and H. Yan, “Beam cleanup of a 532-nm pulsed solid-state laser using a bimorph mirror,” Chin. Opt. Lett.10(2), 021401 (2012). [CrossRef]
  8. X. Lei, S. Wang, H. Yan, W. Liu, L. Dong, P. Yang, and B. Xu, “Double-deformable-mirror adaptive optics system for laser beam cleanup using blind optimization,” Opt. Express20(20), 22143–22157 (2012). [CrossRef] [PubMed]
  9. S. Hu, B. Xu, X. Zhang, J. Hou, J. Wu, and W. Jiang, “Double-deformable-mirror adaptive optics system for phase compensation,” Appl. Opt.45(12), 2638–2642 (2006). [CrossRef] [PubMed]
  10. R. Zacharias, E. Bliss, S. Winters, R. Sacks, M. Feldman, A. Grey, J. Koch, C. Stolz, J. Toeppen, L. Van Atta, and B. Woods, “Wavefront control of high-power laser beams in the National Ignition Facility (NIF),” Proc. SPIE3889, 332–343 (2000). [CrossRef]
  11. R. Zacharias, E. Bliss, M. Feldman, A. Grey, M. Henesian, J. Koch, J. Lawson, R. Sacks, T. Salmon, J. Toeppen, L. Van Atta, S. Winters, B. Woods, C. Lafiandra, and D. G. Bruns, “The National Ignition Facility(NIF) wavefront control system,” Proc. SPIE3492, 678–692 (1999). [CrossRef]
  12. O. Solgaard, F. S. A. Sandejas, and D. M. Bloom, “Deformable grating optical modulator,” Opt. Lett.17(9), 688–690 (1992). [CrossRef] [PubMed]
  13. T. Sato, H. Ishida, and O. Ikeda, “Adaptive PVDF piezoelectric deformable mirror system,” Appl. Opt.19(9), 1430–1434 (1980). [CrossRef] [PubMed]
  14. P. Yang, Y. Liu, W. Yang, M.-W. Ao, S.-J. Hu, B. Xu, and W.-H. Jiang, “Adaptive mode optimization of a continuous-wave solid-state laser using an intracavity piezoelectric deformable mirror,” Opt. Commun.278(2), 377–381 (2007). [CrossRef]
  15. Q. Xue, L. Huang, P. Yan, M. Gong, Z. Feng, Y. Qiu, T. Li, and G. Jin, “Research on the particular temperature-induced surface shape of a National Ignition Facility deformable mirror,” Appl. Opt.52(2), 280–287 (2013). [CrossRef] [PubMed]
  16. M. Kasprzack, B. Canuel, F. Cavalier, R. Day, E. Genin, J. Marque, D. Sentenac, and G. Vajente, “Performance of a thermally deformable mirror for correction of low-order aberrations in laser beams,” Appl. Opt.52(12), 2909–2916 (2013). [CrossRef] [PubMed]
  17. M. A. Arain, W. Z. Korth, L. F. Williams, R. M. Martin, G. Mueller, D. B. Tanner, and D. H. Reitze, “Adaptive control of modal properties of optical beams using photothermal effects,” Opt. Express18(3), 2767–2781 (2010). [CrossRef] [PubMed]
  18. D. Brousseau, E. F. Borra, and S. Thibault, “Wavefront correction with a 37-actuator ferrofluid deformable mirror,” Opt. Express15(26), 18190–18199 (2007). [CrossRef] [PubMed]
  19. N. Abramson, “The interferoscope: a new type of interferometer with variable fringe separation,” Optik (Stuttg.)30, 56–71 (1969).
  20. T. E. Carlsson, N. H. Abramson, and K. H. Fischer, “Automatic measurement of surface height with the interferoscope,” Opt. Eng.35(10), 2938–2942 (1996). [CrossRef]
  21. X. Colonna de Lega, J. F. Biegen, D. Stephenson, and P. J. de Groot, “Characterization of a geometrically desensitized interferometer for flatness testing,” Proc. SPIE3520, 284–292 (1998). [CrossRef]
  22. P. de Groot, “Diffractive grazing-incidence interferometer,” Appl. Opt.39(10), 1527–1530 (2000). [CrossRef] [PubMed]
  23. H. Nüger and J. Schwider, “Measurement of curvature and thickness variations of plane surfaces by grazing incidence interferometry,” Optik (Stuttg.)111, 319–327 (2000).
  24. H. Zimer, K. Albers, and U. Wittrock, “Grazing-incidence YVO4-Nd:YVO4 composite thin slab laser with low thermo-optic aberrations,” Opt. Lett.29(23), 2761–2763 (2004). [CrossRef] [PubMed]
  25. F. He, M. Gong, L. Huang, Q. Liu, Q. Wang, and X. Yan, “Compact TEM00 grazing-incidence Nd:GdVO4 laser using a folded cavity,” Appl. Phys. B86(3), 447–450 (2007). [CrossRef]
  26. W. Jiang and H. Li, “Hartmann-Shack wavefront sensing and wavefront control algorithm,” SPIE1271, 82–93 (1990). [CrossRef]
  27. X. Li, C. Wang, H. Xian, X. Wu, and W. Jiang, “Zernike modal compensation analysis for an adaptive optics system using direct-gradient wavefront reconstruction algorithm,” SPIE3762, 116–124 (1999). [CrossRef]
  28. H. Yang, G. Liu, C. Rao, Y. Zhang, and W. Jiang, “Combinational-deformable-mirror adaptive optics system for compensation of high-order modes of wavefront,” Chin. Opt. Lett.5, 435–437 (2007).

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