Aberration influence and active compensation on laser mode properties for asymmetric folded resonators

Optics and Laser Technology 94 (2017) 199–207

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Aberration influence and active compensation on laser mode properties for asymmetric folded resonators Xiang Zhang a, Zhiqiu Hu b,⇑, Wentao Yang a, Likun Su a a b

Key Laboratory of Solid-State Laser and Applied Techniques, School of Photoelectric Technology, Chengdu University of Information Technology, Chengdu 610225, China School of Control Engineering, Chengdu University of Information Technology, Chengdu 610225, China

a r t i c l e

i n f o

Article history: Received 16 December 2016 Accepted 29 March 2017

Keywords: Laser resonator Beam characterization Wavefront reconstruction Aberration compensation

a b s t r a c t We demonstrate the influence on mode features with introducing typical intracavity perturbation and results of aberrated wavefront compensation in a folded-type unstable resonator used in high energy lasers. The mode properties and aberration coefficient with intracavity misalignment are achieved by iterative calculation and Zernike polynomial fitting. Experimental results for the relation of intracavity maladjustment and mode characteristics are further obtained in terms of S-H detection and model wavefront reconstruction. It indicates that intracavity phase perturbation has significant influence on out coupling beam properties, and the uniform and symmetry of the mode is rapidly disrupted even by a slight misalignment of the resonator mirrors. Meanwhile, the far-field beam patterns will obviously degrade with increasing the distance between the convex mirror and the phase perturbation position even if the equivalent disturbation is inputted into such the resonator. The closed-loop device for compensating intracavity low order aberration is successfully fabricated. Moreover, Zernike defocus aberration is also effectively controlled by precisely adjusting resonator length, and the beam quality is noticeably improved. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Excellent beam quality is a key prerequisite in present laser technique and application. The characteristics of unstable oscillators with large and uniformly filled mode volume, uniform optical phase, good fundamental transverse mode selection and convenience of output coupling have made it a prime candidate for high-energy laser systems [1,2]. Such structure can also suppress the high order mode to obtain high beam quality. Relative investigations indicate elsewhere that in many situations the brightness of lasers can be significantly increased by adopting an unstable rather than a stable resonator [2–4]. However, High output power and high beam quality are two contrary characteristics for conventional lasers. The problems always affect the beam properties; mainly include the cavity geometry misalignment and thermal distortion, inhomogeneity and thermal effect of the gain medium, etc. [5–7]. So especially for a high-energy oscillator, the eigenmode features shall be degraded distinctly by such perturbations. Under the premise of limited output energy, some methods are developed to improve mode characteristics such as phase conjuga⇑ Corresponding author. E-mail address: [email protected] (Z. Hu). http://dx.doi.org/10.1016/j.optlastec.2017.03.032 0030-3992/Ó 2017 Elsevier Ltd. All rights reserved.

tion [8], adaptive optics (AO) [9], temporal and spatial filtering [10], and so on [11] in recent years. However, these methods have the limitations by relatively more complex structures and higher cost, e.g. the maximum correction stroke of AO device is usually less than several microns, and mode features will degrade rapidly once AO device reaches its stroke limit. Moreover, the compensation capability for lower order aberration e.g. the Zernike phase tilt is very limited. Because unstable resonators are usually used in high-energy laser, it is very important to illustrate the effect on the beam quality and mode distribution with typical intracavity perturbation such as phase-tilt, defocus and astigmatism, which is the basis to further solve the problem of intracavity compensation and fabricate the aberration control device [12]. However, experiment of unstable resonators with typical intracavity perturbation on the output mode has been few and fairly limited in scope. In fact, it is not very explicit currently because beam wavefront has not been decomposed to Zernike polynomial aberration, which is not beneficial to accurately wavefront compensation [13,14]. We analyze the mode characteristics disturbed with the typical intracavity aberration in a folded unstable resonator with asymmetric circular mirrors firstly, then the corresponding subtle wavefront aberration is achieved by polynomial fitting with first

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Fig. 1. Schematic diagram for asymmetric folded unstable resonators with intracavity phase perturbation.

Fig. 2. Wavefront profiles with k/8 intracavity tilt perturbation. (a) 3-D profiles; PV = 1.037k and RMS = 0.304k; (b) wavefront residual error with PV = 0.654k and RMS = 0.112k. In this case, two directional Zernike tilt aberrations are removed.

X. Zhang et al. / Optics and Laser Technology 94 (2017) 199–207

64-order Zernike coefficients to improve reconstruction precision. The related device for compensating typical intracavity aberration by real time control of resonator mirrors is fabricated. In this way, beam aberration is effectively eliminated, and the mode wavefront is close to the ideal plane wave.

mirrors respectively. The eigenmode properties can be analyzed by adopting related numerical methods [15]. The mode properties with asymmetric circular resonator mirrors is further expressed by [16]

c1 u1 ðxÞ ¼ ilþ1 ðk=LÞ 2. Numerical analysis

c2 u2 ðyÞ ¼ ilþ1 ðk=LÞ

2.1. Mode properties with intracavity perturbation

ikL

ZZ S

Z

a2

0

Z

0

a1

yJl ðkxy=LÞexp½iðk=2LÞ  ðg 1 x2 þ g 2 y2 Þu2 ðyÞdy zJl ðkyz=LÞ exp½iðk=2LÞ  ðg 2 y2 þ g 1 z2 Þu1 ðzÞdz ð2Þ

The application of Kirchhoff-Fresnel diffraction theory to an aligned optical resonator leads to the homogeneous integral for the resonator modes, e.g., for the cavity of symmetry rectangular plane mirror, the equation is given by

ie uðx; yÞ ¼ c kL

201

( " #) ðx  x0 Þ2 ðy  y0 Þ2 exp ik uðx0 ; y0 Þdx0 dy0 þ 2L 2L

ð1Þ where c and u is the eigen-value and eigen-function respectively, and L is the resonator length. k is the wavelength and k ¼ 2p=k. q is the distance of the corresponding points on the two cavity

where a1 and a2 are half apertures of the convex and concave mirror respectively, x, y and z are the coordinates on the two mirrors, k ¼ 2p=k, and Ri represent the curve radius of cavity mirrors. g i ¼ 1  L=Ri (i = 1, 2), positive is defined if the center of curvature lies toward the interior of the cavity. However, integral equation will become more complex with intracavity phase perturbation. We analysis this problem by iterative calculation and further wavefront fitting with Zernike aberration. Fig. 1 illustrates the schematic of the folded asymmetric unstable resonator in confocal condition with cavity perturbation, and the perturbation can be inputted by the cavity mirror itself,

Fig. 3. Distribution of circle energy as a function of the diffraction limit in Fraunhofer far-field.

Fig. 4. Distribution of first 12 order Zernike aberration included in the eigenmode by wavefront fitting.

Fig. 5. Mode PSF distribution with equivalent phase disturbance while at different positions in the resonator cavity. (a) Strehl ratio (SR) = 0.53, b = 2.32; (b) SR = 0.24, b = 3.46.

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wavefront compensation components, etc. The integral equation for phase-tilt disturbed cavity can be further achieved via paraxial ray transfer matrix method with the convex mirror feedback [17]. The equation is given by

h i 9 8 < p 2dM x 1  LMT LR 1 = i x x cuðx2 ; y2 Þ ¼  ei2kLT exp i x2 : k ; 2kLT Mx þ 1 h i 9 8 ZZ < p 2dM x 1  LMT LR 1 = x x uðx1 ; y1 Þ exp i x1  : k ; Mx þ 1 S ( " #) p ðx2  Mx x1 Þ2 ðy2  My y1 Þ2 dx1 dy1 þ  exp i kLT Mx þ 1 My þ 1

ð3Þ where Rx and Ry represent the undisturbed curvature radius of the convex reflector, and Mx and My are the geometric magnifications of unperturbed resonator in the two meridional planes respectively. d denotes the x directional perturbation, and L1 is the length between the convex mirror and the perturbation plane. 2.2. Numerical calculation and discussion We carry out the calculations for confocal unstable resonators with intracavity perturbation. The parameters are given by g 1 ¼ 1:5; g 2 ¼ 0:75, meets confocal conditions g 1 þ g 2 ¼ 2g 1 g 2 . The mirror separation LT is 900 mm, the aperture and curvature radius of the convex mirror is 6.0 mm and 1.8 m respectively. The wavelength and resonator magnification M are 10.6 lm and 2.0 respectively. The equivalent Fresnel number meets N e ¼ ðM  1Þa2 =ð2kLT Þ ¼ 0:75. In Fig. 1, the x directional tilt aberration with a k/8 is inputted into the resonator by the perturbation plane, and the distance from the perturbation plane to the convex mirror is L1 = LT/3. Fig. 2(a) and (b) show the out-coupling mode wavefront properties with a k/8 intracavity tilted perturbation acquired by diffraction iterative algorithm. Further, the wavefront distribution can be fitting by first 64-order aberration with Zernike annular polynomials. In our calculation, Zernike coefficient Z1 (x directional tilt aberration) is 0.291, Z3 (defocus) is 0.040, Z5 (astigmatism) is 0.026, etc. Actually, intracavity perturbation has obvious influence on the uniformity of mode intensity and wavefront profiles. Referring to Fig. 2(a), the wavefront with even a small intracavity tilt perturbation deviates obviously the original planewave distribution for the cavity eigenmode. Fig. 2(b) shows that

small amount of high-order aberration is still consisted in the wavefront even if the main aberration Z1 is removed, which will degrade the beam quality remarkably. By further calculation, the circle energy of the aberrated beam in Fraunhofer far-field is shown in Fig. 3. The curve 1 denotes the distribution of circle energy for the idea plane wavefront, and the curve 2 is the distribution of circle energy for the wavefront with 0.12k phase-tilted perturbation. Beam quality is 2.67 times diffraction limit with the circle energy is 0.8, which clearly reflects the concentration of the far field pattern. The first 12 order Zernike aberration in the eigenmode is illustrated in Fig. 4, which is obtained by using Zernike polynomial to fit the distorted wavefront. It shows that low-order Zernike aberration coefficient (Z1) is the principal component in the aberrated wavefront. Some typical even order Zernike aberration such as Z1, Z2, Z4 and Z8 is approximately equal to zero, however, relative higher-order Zernike aberration, e.g. astigmatism (Z3), coma (Z5), triangular astigmatism (Z7), spherical aberration (Z10), etc. are still in the residual because the effect of low-order phase-tilted perturbation. The mode structure with intracavity phase-tilted disturbed resonator is shown in Fig. 5. It represents the perturbation plane is placed at different intracavity location while the equal perturbation with 0.12k is inputted into the resonator. Assumed condition is that (a) L1 = 200 mm, where is an intracavity position used as the phase perturbation plane, and (b) L1 = 450 mm, where is the intermediate position of the resonant cavity (L1 = LT/2). Fig. 5 shows that Strehl ratio (SR) decreases by 2.2 times from 0.53 to 0.24, and b (diffraction limit factor in the far field) changes from 2.32 to 3.46 with the phase perturbation plane moving along the axis direction of the concave mirror. For the lasers adopted the confocal unstable resonator, Fig. 5 also illustrates that intracavity phase disturbation is an important influential factor for degrading beam quality. If larger intracavity tilt aberration exists in the resonator, the energy profile will become very asymmetric and high-order aberration will increase rapidly.

3. Experimental details 3.1. Wavefront model reconstruction Diffraction integral equation will be very complex with the multiform intracavity phase perturbation for actual laser devices. We use Shack-Hartmann (S-H) detection and wavefront reconstruction to study the problem. The laser beam is divided into

Fig. 6. Experimental configuration of analyzing typical aberration influence and compensation on mode properties.

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203

Fig. 7. Zernike tilt aberration Z1 included in the beam wavefront vs. the perturbation by the resonator mirror.

Fig. 9. Subaperture spot diagram of the beam detected by S-H method with 22.700 perturbation by the cavity convex mirror.

Gjy ¼ ¼

1 sj

ZZ  Sj

l X

  l  ZZ  X @ uðx; yÞ ak @Z k ðx; yÞ dxdy ¼ dxdy @y @y sj Sj j j k¼1

ak Z jky

ð6Þ

k¼1

where sj is the area of the No. j sub-aperture and l indicate Zernike mode number, Z jkx and Z jky is expressed as

Z jkx ¼

Fig. 8. Zernike aberration coefficient Z4 as a function of misalignment angle of the cavity convex mirror.

many sub-apertures by a lenslet array, then the beam centroid error and wavefront slope in each sub-aperture can be calculated. The wavefront properties can be further achieved by Zernike polynomial reconstruction [18,19]. It can be expanded as the combination of linear orthogonal Zernike polynomial in the circular domain from the equation

uðx; yÞ ¼

1 X ak Z k ðx; yÞ

ð4Þ

k¼1

where k is the order of Zernike mode wavefront, ak represents the coefficient of mode and Zk is the k-th order Zernike polynomial mode. The average slope of incident wavefront in No. j subaperture of the microlens array Gjx and Gjy is given by

Gjx ¼ ¼

1 sj

ZZ 

l X k¼1

Sj

  l  ZZ  X @ uðx; yÞ ak @Z k ðx; yÞ dxdy ¼ dxdy @x @x sj Sj j j k¼1

ak Z jkx

1 sj

ZZ  Sj

  ZZ  @Z k ðx; yÞ 1 @Z k ðx; yÞ dxdy Z jky ¼ dxdy @x sj Sj @y j j

Further, the relation between the matrix Z, the mode coefficient A, and the wavefront slope matrix G can be described as

G ¼ ZA

ð8Þ

where G is a row vector of 2N, Z is 2N  L matrix and N is the subaperture amounts of the microlens. So if the matrix G is obtained, the matrix A can be written as

A ¼ Zþ G

ð9Þ

+

where Z is the pseudo-inverse matrix of Z. So the wavefront can be constructed according to Eq. (4). So the beam quality can be understood fully. Since the out coupling beam is usually an annular in unstable resonators, we use the Zernike polynomials of the annular domain to reconstruct the wavefront in order to improve the accuracy. Zernike polynomial of the annular domain is the product of the radial and the angular polynomial defined by m Z ei ðx; yÞ ¼ Q m n ðrÞH ðhÞ

ð10Þ

where e is the central obscuration ratio of the annular domain. The angular polynomial Hm ðhÞ is consistent with the Zernike polynomial of the circular region; the radial polynomial is determined by m m m m m m Rm n ðrÞ ¼ cn;m  Q m ðrÞ þ cn;mþ2  Q mþ2 ðrÞ þ . . . þ cn;n  Q n ðrÞ

ð5Þ

ð7Þ

Rm n ðrÞ

ð11Þ

where is the radial polynomial of Zernike circular polynomial, and the identity of c is given by

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Fig. 10. Wavefront profiles of out coupling mode with different intracavity disturbation inputted by the convex mirror. (a) Tilt = 10.700 ; Strehl ratio = 0.79; PV = 0.45k; (b) Tilt = 36.800 ; Strehl ratio = 0.48; PV = 0.73k.

m 2 n;n Þ

ðc

2 ¼ 1  e2

cmn;mþi ¼

2 1  e2

Z

1

e

Z e

1

" Rm n ðrÞ



nm2 X

#2 m m n;mþi Q mþi ðrÞ

c

rdr

ð12Þ

i¼0

m Rm n ðrÞQ mþi ðrÞrdr

ð13Þ

where cm m;m2 ¼ 0, and i ¼ 0; 2; 4; . . . ; n  2. 3.2. Experimental facility architecture The positive-branch confocal unstable resonator is used in highenergy laser oscillators, e.g. chemical oxygen-iodine or CO2 fluid lasers, and such a structure is a typical folded cavity. Because many perturbation factors shall degrade beam quality obviously in practical applications. In order to simplify problems, the studying on the intracavity perturbations and further beam mode control are only limited to the passive resonator. The experiment schematic

is shown in Fig. 6, and the unstable resonator are fabricated by the convex and concave mirror, the folded plane reflectors with 90° and a beam scraper (BS), which is used as the out coupling mirror. Curvature radius of the convex and concave reflector is 7:5 m and 21:0 m respectively, and the cavity magnification is 2.8. The resonator is adopted by virtual confocal ways, so the resonator length is ðR2  R1 Þ=2. A He-Ne laser is used to propagate the probe beam into the cavity via a small hole with 1.5 mm diameter in the folded reflectors. The beam is continued to expand in the resonant cavity, and finally is extracted by a BS onto a S-H detector. The annular beam diameter is 50 mm, and the beam obscuration ratio is 1:2. The S-H detector here included a microlens with 25  25 matrix is used to measure the lattice distribution of the annular beam near the BS mirror. The wavefront characteristic will be further achieved by Zernike modal reconstruction illustrated as Section 3.1. The tilt offset angle is detected by an optical auto-collimator.

X. Zhang et al. / Optics and Laser Technology 94 (2017) 199–207

3.3. Results and discussion Experiment shows that the low-order aberration Z1 is obviously larger than high-order aberration when equal perturbation is inputted into the resonator by comparing Figs. 7 and 8. The horizontal axis represents the tilt disturbation of the convex mirror in Fig. 7. Zernike coefficient Z1 (x directional tilt) is obtained by model wavefront reconstruction, and the fitted curve presents approximate linear relationship. However, the experimental characteristics of beam higher-order Zernike aberration e.g. Z4 (astigmatism) is relatively complex and nonlinear shown in Fig. 8. Fig. 9 shows that the detected 25  25 light spot deviates obviously from the original annular distribution, and the intensity also presents nonuniform. The intracavity phase-tilted perturbation with 22.700 is inputted into the resonator cavity by the convex mirror. It illustrates that even if the disturbance is small e.g. several arc-seconds, the uniform and symmetry of the out coupling mode will be rapidly degraded. Mode wavefront aberration is not obvious with a 10.700 mirror tilt-angle and the Corresponding PV (peak-to-valley) is 0.45k by comparing with the plane wavefront, which is shown in Fig. 10 (a). Wavefront error is increasing with the augment of the mirror’s maladjustment. However, the wavefront aberration arrives to 0.73k and SR reduces to 0.48 along with 36.800 perturbation of the convex mirror shown in Fig. 10(b). Therefore, the eigenmode degradation is closely associated with the resonator mirrors misalignment. It is unwanted especially in the high energy lasers because tilt perturbation will severely disrupt the eigenmode structure. Experiment shows that such characteristics are more obvious if larger perturbation quantity (40–60 arcsec) is inputted. Based on the analysis above, we studied the compensation of the typical intracavity aberration. Since the intracavity tilt perturbation has approximate linear relation with Zernike tilt coefficient of the wavefront, firstly the control of the phase tilt aberration can be realized. The experimental setup is shown in Fig.6, which is fabricated by the unstable oscillator, S-H wavefront detector, the stepping motor and control unit, and the data processing computer. The result of real-time closed-loop controlling to the convex mirror has been achieved. The image sampling rate of S-H detector is 25 frames per second in the experiment, and intracavity phase disturbation is introduced by a blowing fan. Zernike coefficient Z1 and Z2 denote phase tilt aberration in two orthogonal directions respectively. Z3–Z6 is

Fig. 11. Wavefront properties (Zernike aberration Z3–Z6) with intracavity phase perturbation at the condition of system open loop.

205

defined as defocus, 0/90° and 45/135° astigmatism, low-order coma respectively. Zernike coefficient (y directional tilt Z2) of the wavefront is 1.837 in the system open loop, while the Z2 aberration reduces to 0.124 at the closed loop state by model-reconstruction calculation. Zernike aberration Z3–Z6 of the out coupling mode in the state of open loop and closed-loop to the resonator mirror has been shown in Figs. 11 and 12 respectively. The higher-order Zernike aberration especially Z3 and Z4 in the wavefront is significantly reduced with the elimination of phase tilt aberration in the continuous closed loop. It proves that Zernike low order tilt aberration (Z1 and Z2) is the main component in the wavefront distribution, besides, some higher-order aberration is still included in the mode. The wavefront is reconstructed by adopting first 64-order Zernike polynomials to improve calculation accuracy. Experiment shows that the beam mode is highly uniform and symmetry for the well-aligned case and that this symmetry characteristic and wavefront profiles is rapidly degraded associated with a slight misalignment of one of the resonator mirrors.

Fig. 12. Result of Zernike aberration (Z3–Z6) by intracavity aberrated compensation at the condition of the system continuously closed-loop.

Fig. 13. Beam Zernike defocus aberration (Z3) as a function of the axial movement for the resonator convex mirror.

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Fig. 14. Experimental result of output coupling wavefront with intracavity defocus compensation under the condition of continuously closed-loop to the resonator mirror.

According to the obtained Zernike coefficients, calculation shows that the beam PV and RMS for sampling average of 100 frame images is 1.397k and 0.255k respectively, and Strehl ratio (SR) is less than 0.45 when the perturbation is inputted. However, SR is more than 0.8 when the system is continuously closed-loop, and the typical Zernike aberration in this case is shown in Fig. 12. Therefore, it proves that wavefront aberration is reduced obviously and the beam quality is effectively improved. In the experiment, the two directional intracavity phase-tilted aberrations is significantly decreased, Moreover, Zernike defocus aberration can be effectively controlled by precisely adjusting resonator length via controlling a stepper motor along the optical axis. Fig. 13 shows the variation of Zernike defocus aberration (Z3 coef.) in the beam wavefront is as a function of the optical length change of the resonator cavity, and the relationship between the two parameters is nonlinear. Fig. 14 shows the distribution of out coupling wavefront is similar to that of plane wave, and the wavefront value of PV and RMS (Root-Mean-Square) is 0.246k and 0.091k respectively. The wavefront distribution is given by the average value of the calculation with 100 frame image sampling at 4 s.

even if the equal perturbation is inputted. On the basis, a closelooped real time device for compensating intracavity low order aberration has been fabricated. Results show that typical Zernike aberration such as phase tilt and astigmatism included the mode can be effectively decreased, and misalignment sensitivities of the cavity has been improved. Moreover, Zernike defocus aberration Z3 can be also effectively controlled by precisely adjusting resonator length via controlling a stepper motor on the convex mirror. Therefore, it will be a further reference to establish that fairly simple and inexpensive devices based on intracavity compensation are capable of eliminating many problems arising from phase perturbation in such lasers.

Acknowledgments The study is done within the framework of National Postdoctoral Science Foundation of China (No. 2014M562294), and supported by the Research Project of Education Department in Sichuan Province (No. 2014ZA0080). We also acknowledge partial support by the National Natural Science Fund of China (No. 61275039).

4. Conclusions References In conclusion, we demonstrated an efficient method to analysis the effects of typical intracavity phase perturbation and wavefront aberration compensation for a folded unstable resonator. Both theoretical and experimental investigations were performed to evaluate that intracavity phase perturbation has obvious influence on the uniformity of mode intensity and wavefront profiles. Even a smaller phase perturbation is inputted into the resonator, some corresponding Zernike aberrations will increase rapidly in wavefront, which will remarkably affect the mode properties and farfield PSF distribution. We achieved the wavefront properties accurately in terms of S-H detection and Zernike model reconstruction. Zernike tilt and the intracavity phase disturbation present approximate linear relationship; however, the relation of higher-order Zernike aberrations to the perturbation quantity is more complex nonlinear. Moreover, mode PSF distribution will obviously degrade with increasing the distance between the convex mirror and the perturbation plane

[1] H. Injeyan, G.D. Goodno, High-Power Laser Handbook, fourth ed., McGraw-Hill Companies, New York, 2011. [2] B.D. Lv, Laser Optics and Beam Propagation Technology, third ed., Higher Education Press, Beijing, 2003. [3] Q. Zhuang, F.T. Sang, D.Z. Zhou, Shortwave Chemical Laser, National Defence Industry Press, Beijing, 1997. [4] C. Pargmann, T. Hall, F. Duschek, K.M. Grünewald, J. Handke, COIL emission of a modified negative branch confocal unstable resonator, Appl. Opt. 46 (2007) 7751–7756. [5] D. Wen, D. Li, J. Zhao, Generalized sensitivity factors for optical-axis perturbation in nonplanar ring resonators, Opt. Exp. 19 (2011) 19752–19757. [6] C. Liu, T. Riesbeck, X. Wang, Z. Xiang, J. Chen, H.J. Eichler, Asymmetric TEM00 mode cavity for birefringence-compensated two-rod solid-state lasers, IEEE J. Quant. Electron. 44 (2008) 1107–1115. [7] B.L. Pan, J. Yang, Y.J. Wang, M.L. Li, Thermal effects in high-power double diode-end-pumped Cs vapor lasers, IEEE J. Quant. Electron. 48 (2012) 485–489. [8] D.A. Rockwell, A review of phase-conjugate solid-state lasers, IEEE J. Quant. Electron. 24 (1988) 1124–1140. [9] R. Bauer, A. Paterson, C. Clark, D. Uttamchandani, W. Lubeigt, Output characteristics of Q-switched solid-state lasers using intracavity MEMS micromirrors, IEEE J. Sel. Top. Quant. Electron. 21 (2015) 908–916.

X. Zhang et al. / Optics and Laser Technology 94 (2017) 199–207 [10] S. Szatmari, R. Dajka, A. Barna, B. Gilicze, I.B. Foldes, Improvement of the temporal and spatial contrast of high-brightness laser beams, Laser Phys. Lett. 13 (2016) 075301. [11] O. Antipov, O. Eremeykin, A. Ievlev, A. Savikin, Diode-pumped Nd:YAG laser with reciprocal dynamic holographic cavity, Opt. Exp. 12 (2004) 4313–4319. [12] Y.J. Cheng, C.G. Fanning, A.E. Siegman, Transverse-mode astigmatism in a diode-pumped unstable resonator Nd:YVO4 laser, Appl. Opt. 47 (2008) 1130– 1134. [13] D.L. Yu, F.T. Sang, Y.Q. Jin, Y.Z. Sun, Output beam drift and deformation in highpower chemical oxygen iodine laser, Opt. Laser Technol. 35 (2003) 245–249. [14] J.S. Shin, Y.H. Cha, B.H. Cha, H.C. Lee, H.T. Kim, Simulation of the wavefront distortion and beam quality for a high-power zigzag slab laser, Opt. Commun. 380 (2016) 446–451.

207

[15] W.D. Murphy, M.L. Bernabe, Numerical procedures for solving nonsymmetric eigenvalue problems associated with optical resonators, Appl. Opt. 17 (1978) 2358–2365. [16] A.E. Siegman, H.Y. Miller, Unstable optical resonator loss calculations using the prony method, Appl. Opt. 9 (1970) 2729–2735. [17] A. Gerrand, J.M. Burch, Introduction to Matrix Methods in Optics, Wiley Press, New York, 1975. [18] W.H. Southwell, Wave-front estimation from wave-front slope measurements, J. Opt. Soc. Am. 70 (1980) 998–1006. [19] A.V. Goncharov, M. Nowakowski, M.T. Sheehan, C. Dainty, Reconstruction of the optical system of the human eye with reverse ray-tracing, Opt. Exp. 16 (2008) 1692–1703.