Abstract

An analysis is presented of a TGG crystal rod under high power laser operation. A semianalytical thermal analysis is investigated to obtain the temperature profile and thermal lensing effect in a TGG crystal upon incidence of a high power laser light. By solving the heat transfer equation for the TGG crystal and taking the Gaussian beam transverse intensity profile as the heat source, the optical path difference due to induced thermal effects was obtained. Moreover, a detailed model for the dependence of thermal lensing and beam degradation which takes into account up to the fifth-order spherical aberration is presented. Based on this model, it is shown that up to a critical value of the beam power the degradation of the beam is not significant. The experimental results on thermal lensing and degradation on beam quality of a high power laser passing through a TGG crystal rod are in agreement with the main results from our model.

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  1. U. O. Farrukh, A. M. Buoncristiani, and C. E. Byvik, “An analysis of the temperature distribution in finite solid-state laser rods,” IEEE J. Quantum Electron.24, 2253–2263 (1988).
    [CrossRef]
  2. M. Innocenzi, H. Yura, C. Fincher, and R. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett.56, 1831–1833 (1990).
    [CrossRef]
  3. C. Pfistner, R. Weber, H. P. Weber, S. Merazzi, and R. Gruber, “Thermal beam distortions in end-pumped Nd:YAG, Nd:GSGG, and Nd:YLF rods,” IEEE J. Quantum Electron.30, 1605–1615 (1994).
    [CrossRef]
  4. S. Fan, X. Zhang, Q. Wang, S. Li, S. Ding, and F. Su, “More precise determination of thermal lens focal length for end-pumped solid-state lasers,” Opt. Commun.266, 620–626 (2006).
    [CrossRef]
  5. P. Shi, W. Chen, L. Li, and A. Gan, “Semianalytical thermal analysis of thermal focal length on Nd:YAG rods,” Appl. Opt.46, 6655–6661 (2007).
    [CrossRef] [PubMed]
  6. E. Khazanov, N. Andreev, A. Babin, A. Kiselev, O. Palashov, and D. H. Reitze, “Suppression of self-induced depolarization of high-power laser radiation in glass-based Faraday isolators,” J. Opt. Soc. Am. B17, 99–102 (2000).
    [CrossRef]
  7. E. Khazanov, N. Andreev, A. Mal’shakov, O. Palashov, A. Poteomkin, A. Sergeev, A. Shaykin, V. Zelenogorsky, I. Ivanov, R. Amin, G. Mueller, D. Tanner, and D. Reitze, “Compensation of thermally induced modal distortions in Faraday isolators,” IEEE J. Quantum Electron.40, 1500–1510 (2004).
    [CrossRef]
  8. V. Zelenogorsky, O. Palashov, and E. Khazanov, “Adaptive compensation of thermally induced phase aberrations in Faraday isolators by means of a DKDP crystal,” Opt. Commun.278, 8–13 (2007).
    [CrossRef]
  9. The Vigro Collaboration, “In-vacuum optical isolation changes by heating in a faraday isolator,” Appl. Opt.47, 5853–5861 (2008).
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  13. J. Alda, Laser and Gaussian Beam Propagation and Transformation (Marcel Dekker, Inc, 2003), pp. 999–1013, Encyclopedia of Optical Engineering.
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    [CrossRef]
  15. Y. Chen, T. Huang, C. Kao, C. Wang, and S. Wang, “Optimization in scaling fiber-coupled laser-diode end-pumped lasers to higher power: influence of thermal effect,” IEEE J. Quantum Electron.33, 1424–1429 (1997).
    [CrossRef]
  16. V. N. Mahajan, “Strehl ratio of a Gaussian beam,” J. Opt. Soc. Am. A22, 1824–1833 (2005).
    [CrossRef]
  17. M. A. Porras, J. Alda, and E. Bernabeu, “Complex beam parameter and ABCD law for non-Gaussian and non-spherical light beams,” Appl. Opt.31, 6389–6402 (1992).
    [CrossRef] [PubMed]
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    [CrossRef]
  19. NanoModeScan operational manual (Photon Inc, www.ophiropt.com , 2008).

2008 (1)

2007 (2)

P. Shi, W. Chen, L. Li, and A. Gan, “Semianalytical thermal analysis of thermal focal length on Nd:YAG rods,” Appl. Opt.46, 6655–6661 (2007).
[CrossRef] [PubMed]

V. Zelenogorsky, O. Palashov, and E. Khazanov, “Adaptive compensation of thermally induced phase aberrations in Faraday isolators by means of a DKDP crystal,” Opt. Commun.278, 8–13 (2007).
[CrossRef]

2006 (1)

S. Fan, X. Zhang, Q. Wang, S. Li, S. Ding, and F. Su, “More precise determination of thermal lens focal length for end-pumped solid-state lasers,” Opt. Commun.266, 620–626 (2006).
[CrossRef]

2005 (1)

2004 (1)

E. Khazanov, N. Andreev, A. Mal’shakov, O. Palashov, A. Poteomkin, A. Sergeev, A. Shaykin, V. Zelenogorsky, I. Ivanov, R. Amin, G. Mueller, D. Tanner, and D. Reitze, “Compensation of thermally induced modal distortions in Faraday isolators,” IEEE J. Quantum Electron.40, 1500–1510 (2004).
[CrossRef]

2002 (1)

2001 (1)

J. Alda, “Quality improvement of a coherent and aberrated laser beam by using an optimum and smooth pure phase filter,” Opt. Commun.192, 199–204 (2001).
[CrossRef]

2000 (1)

1997 (1)

Y. Chen, T. Huang, C. Kao, C. Wang, and S. Wang, “Optimization in scaling fiber-coupled laser-diode end-pumped lasers to higher power: influence of thermal effect,” IEEE J. Quantum Electron.33, 1424–1429 (1997).
[CrossRef]

1994 (1)

C. Pfistner, R. Weber, H. P. Weber, S. Merazzi, and R. Gruber, “Thermal beam distortions in end-pumped Nd:YAG, Nd:GSGG, and Nd:YLF rods,” IEEE J. Quantum Electron.30, 1605–1615 (1994).
[CrossRef]

1993 (1)

1992 (1)

1990 (1)

M. Innocenzi, H. Yura, C. Fincher, and R. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett.56, 1831–1833 (1990).
[CrossRef]

1988 (1)

U. O. Farrukh, A. M. Buoncristiani, and C. E. Byvik, “An analysis of the temperature distribution in finite solid-state laser rods,” IEEE J. Quantum Electron.24, 2253–2263 (1988).
[CrossRef]

1976 (1)

Alda, J.

J. Alda, “Quality improvement of a coherent and aberrated laser beam by using an optimum and smooth pure phase filter,” Opt. Commun.192, 199–204 (2001).
[CrossRef]

M. A. Porras, J. Alda, and E. Bernabeu, “Complex beam parameter and ABCD law for non-Gaussian and non-spherical light beams,” Appl. Opt.31, 6389–6402 (1992).
[CrossRef] [PubMed]

J. Alda, Laser and Gaussian Beam Propagation and Transformation (Marcel Dekker, Inc, 2003), pp. 999–1013, Encyclopedia of Optical Engineering.

Amin, R.

E. Khazanov, N. Andreev, A. Mal’shakov, O. Palashov, A. Poteomkin, A. Sergeev, A. Shaykin, V. Zelenogorsky, I. Ivanov, R. Amin, G. Mueller, D. Tanner, and D. Reitze, “Compensation of thermally induced modal distortions in Faraday isolators,” IEEE J. Quantum Electron.40, 1500–1510 (2004).
[CrossRef]

Andreev, N.

E. Khazanov, N. Andreev, A. Mal’shakov, O. Palashov, A. Poteomkin, A. Sergeev, A. Shaykin, V. Zelenogorsky, I. Ivanov, R. Amin, G. Mueller, D. Tanner, and D. Reitze, “Compensation of thermally induced modal distortions in Faraday isolators,” IEEE J. Quantum Electron.40, 1500–1510 (2004).
[CrossRef]

E. Khazanov, N. Andreev, A. Babin, A. Kiselev, O. Palashov, and D. H. Reitze, “Suppression of self-induced depolarization of high-power laser radiation in glass-based Faraday isolators,” J. Opt. Soc. Am. B17, 99–102 (2000).
[CrossRef]

Babin, A.

Bernabeu, E.

Buoncristiani, A. M.

U. O. Farrukh, A. M. Buoncristiani, and C. E. Byvik, “An analysis of the temperature distribution in finite solid-state laser rods,” IEEE J. Quantum Electron.24, 2253–2263 (1988).
[CrossRef]

Byvik, C. E.

U. O. Farrukh, A. M. Buoncristiani, and C. E. Byvik, “An analysis of the temperature distribution in finite solid-state laser rods,” IEEE J. Quantum Electron.24, 2253–2263 (1988).
[CrossRef]

Chen, W.

Chen, Y.

Y. Chen, T. Huang, C. Kao, C. Wang, and S. Wang, “Optimization in scaling fiber-coupled laser-diode end-pumped lasers to higher power: influence of thermal effect,” IEEE J. Quantum Electron.33, 1424–1429 (1997).
[CrossRef]

Ding, S.

S. Fan, X. Zhang, Q. Wang, S. Li, S. Ding, and F. Su, “More precise determination of thermal lens focal length for end-pumped solid-state lasers,” Opt. Commun.266, 620–626 (2006).
[CrossRef]

Fan, S.

S. Fan, X. Zhang, Q. Wang, S. Li, S. Ding, and F. Su, “More precise determination of thermal lens focal length for end-pumped solid-state lasers,” Opt. Commun.266, 620–626 (2006).
[CrossRef]

Farrukh, U. O.

U. O. Farrukh, A. M. Buoncristiani, and C. E. Byvik, “An analysis of the temperature distribution in finite solid-state laser rods,” IEEE J. Quantum Electron.24, 2253–2263 (1988).
[CrossRef]

Fields, R.

M. Innocenzi, H. Yura, C. Fincher, and R. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett.56, 1831–1833 (1990).
[CrossRef]

Fincher, C.

M. Innocenzi, H. Yura, C. Fincher, and R. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett.56, 1831–1833 (1990).
[CrossRef]

Gan, A.

Gruber, R.

C. Pfistner, R. Weber, H. P. Weber, S. Merazzi, and R. Gruber, “Thermal beam distortions in end-pumped Nd:YAG, Nd:GSGG, and Nd:YLF rods,” IEEE J. Quantum Electron.30, 1605–1615 (1994).
[CrossRef]

Huang, T.

Y. Chen, T. Huang, C. Kao, C. Wang, and S. Wang, “Optimization in scaling fiber-coupled laser-diode end-pumped lasers to higher power: influence of thermal effect,” IEEE J. Quantum Electron.33, 1424–1429 (1997).
[CrossRef]

Innocenzi, M.

M. Innocenzi, H. Yura, C. Fincher, and R. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett.56, 1831–1833 (1990).
[CrossRef]

Ivanov, I.

E. Khazanov, N. Andreev, A. Mal’shakov, O. Palashov, A. Poteomkin, A. Sergeev, A. Shaykin, V. Zelenogorsky, I. Ivanov, R. Amin, G. Mueller, D. Tanner, and D. Reitze, “Compensation of thermally induced modal distortions in Faraday isolators,” IEEE J. Quantum Electron.40, 1500–1510 (2004).
[CrossRef]

Kao, C.

Y. Chen, T. Huang, C. Kao, C. Wang, and S. Wang, “Optimization in scaling fiber-coupled laser-diode end-pumped lasers to higher power: influence of thermal effect,” IEEE J. Quantum Electron.33, 1424–1429 (1997).
[CrossRef]

Kennedy, C. J.

Khazanov, E.

V. Zelenogorsky, O. Palashov, and E. Khazanov, “Adaptive compensation of thermally induced phase aberrations in Faraday isolators by means of a DKDP crystal,” Opt. Commun.278, 8–13 (2007).
[CrossRef]

E. Khazanov, N. Andreev, A. Mal’shakov, O. Palashov, A. Poteomkin, A. Sergeev, A. Shaykin, V. Zelenogorsky, I. Ivanov, R. Amin, G. Mueller, D. Tanner, and D. Reitze, “Compensation of thermally induced modal distortions in Faraday isolators,” IEEE J. Quantum Electron.40, 1500–1510 (2004).
[CrossRef]

E. Khazanov, N. Andreev, A. Babin, A. Kiselev, O. Palashov, and D. H. Reitze, “Suppression of self-induced depolarization of high-power laser radiation in glass-based Faraday isolators,” J. Opt. Soc. Am. B17, 99–102 (2000).
[CrossRef]

Kiselev, A.

Koechner, W.

W. Koechner, Solid State Laser Engineering (Springer, 1988).
[CrossRef]

Li, L.

Li, S.

S. Fan, X. Zhang, Q. Wang, S. Li, S. Ding, and F. Su, “More precise determination of thermal lens focal length for end-pumped solid-state lasers,” Opt. Commun.266, 620–626 (2006).
[CrossRef]

Mahajan, V. N.

Mal’shakov, A.

E. Khazanov, N. Andreev, A. Mal’shakov, O. Palashov, A. Poteomkin, A. Sergeev, A. Shaykin, V. Zelenogorsky, I. Ivanov, R. Amin, G. Mueller, D. Tanner, and D. Reitze, “Compensation of thermally induced modal distortions in Faraday isolators,” IEEE J. Quantum Electron.40, 1500–1510 (2004).
[CrossRef]

Merazzi, S.

C. Pfistner, R. Weber, H. P. Weber, S. Merazzi, and R. Gruber, “Thermal beam distortions in end-pumped Nd:YAG, Nd:GSGG, and Nd:YLF rods,” IEEE J. Quantum Electron.30, 1605–1615 (1994).
[CrossRef]

Mueller, G.

E. Khazanov, N. Andreev, A. Mal’shakov, O. Palashov, A. Poteomkin, A. Sergeev, A. Shaykin, V. Zelenogorsky, I. Ivanov, R. Amin, G. Mueller, D. Tanner, and D. Reitze, “Compensation of thermally induced modal distortions in Faraday isolators,” IEEE J. Quantum Electron.40, 1500–1510 (2004).
[CrossRef]

Noll, R. J.

Palashov, O.

V. Zelenogorsky, O. Palashov, and E. Khazanov, “Adaptive compensation of thermally induced phase aberrations in Faraday isolators by means of a DKDP crystal,” Opt. Commun.278, 8–13 (2007).
[CrossRef]

E. Khazanov, N. Andreev, A. Mal’shakov, O. Palashov, A. Poteomkin, A. Sergeev, A. Shaykin, V. Zelenogorsky, I. Ivanov, R. Amin, G. Mueller, D. Tanner, and D. Reitze, “Compensation of thermally induced modal distortions in Faraday isolators,” IEEE J. Quantum Electron.40, 1500–1510 (2004).
[CrossRef]

E. Khazanov, N. Andreev, A. Babin, A. Kiselev, O. Palashov, and D. H. Reitze, “Suppression of self-induced depolarization of high-power laser radiation in glass-based Faraday isolators,” J. Opt. Soc. Am. B17, 99–102 (2000).
[CrossRef]

Pfistner, C.

C. Pfistner, R. Weber, H. P. Weber, S. Merazzi, and R. Gruber, “Thermal beam distortions in end-pumped Nd:YAG, Nd:GSGG, and Nd:YLF rods,” IEEE J. Quantum Electron.30, 1605–1615 (1994).
[CrossRef]

Porras, M. A.

Poteomkin, A.

E. Khazanov, N. Andreev, A. Mal’shakov, O. Palashov, A. Poteomkin, A. Sergeev, A. Shaykin, V. Zelenogorsky, I. Ivanov, R. Amin, G. Mueller, D. Tanner, and D. Reitze, “Compensation of thermally induced modal distortions in Faraday isolators,” IEEE J. Quantum Electron.40, 1500–1510 (2004).
[CrossRef]

Reitze, D.

E. Khazanov, N. Andreev, A. Mal’shakov, O. Palashov, A. Poteomkin, A. Sergeev, A. Shaykin, V. Zelenogorsky, I. Ivanov, R. Amin, G. Mueller, D. Tanner, and D. Reitze, “Compensation of thermally induced modal distortions in Faraday isolators,” IEEE J. Quantum Electron.40, 1500–1510 (2004).
[CrossRef]

Reitze, D. H.

Sergeev, A.

E. Khazanov, N. Andreev, A. Mal’shakov, O. Palashov, A. Poteomkin, A. Sergeev, A. Shaykin, V. Zelenogorsky, I. Ivanov, R. Amin, G. Mueller, D. Tanner, and D. Reitze, “Compensation of thermally induced modal distortions in Faraday isolators,” IEEE J. Quantum Electron.40, 1500–1510 (2004).
[CrossRef]

Shaykin, A.

E. Khazanov, N. Andreev, A. Mal’shakov, O. Palashov, A. Poteomkin, A. Sergeev, A. Shaykin, V. Zelenogorsky, I. Ivanov, R. Amin, G. Mueller, D. Tanner, and D. Reitze, “Compensation of thermally induced modal distortions in Faraday isolators,” IEEE J. Quantum Electron.40, 1500–1510 (2004).
[CrossRef]

Shi, P.

Siegman, A. E.

Su, F.

S. Fan, X. Zhang, Q. Wang, S. Li, S. Ding, and F. Su, “More precise determination of thermal lens focal length for end-pumped solid-state lasers,” Opt. Commun.266, 620–626 (2006).
[CrossRef]

Tanner, D.

E. Khazanov, N. Andreev, A. Mal’shakov, O. Palashov, A. Poteomkin, A. Sergeev, A. Shaykin, V. Zelenogorsky, I. Ivanov, R. Amin, G. Mueller, D. Tanner, and D. Reitze, “Compensation of thermally induced modal distortions in Faraday isolators,” IEEE J. Quantum Electron.40, 1500–1510 (2004).
[CrossRef]

Wang, C.

Y. Chen, T. Huang, C. Kao, C. Wang, and S. Wang, “Optimization in scaling fiber-coupled laser-diode end-pumped lasers to higher power: influence of thermal effect,” IEEE J. Quantum Electron.33, 1424–1429 (1997).
[CrossRef]

Wang, Q.

S. Fan, X. Zhang, Q. Wang, S. Li, S. Ding, and F. Su, “More precise determination of thermal lens focal length for end-pumped solid-state lasers,” Opt. Commun.266, 620–626 (2006).
[CrossRef]

Wang, S.

Y. Chen, T. Huang, C. Kao, C. Wang, and S. Wang, “Optimization in scaling fiber-coupled laser-diode end-pumped lasers to higher power: influence of thermal effect,” IEEE J. Quantum Electron.33, 1424–1429 (1997).
[CrossRef]

Weber, H. P.

C. Pfistner, R. Weber, H. P. Weber, S. Merazzi, and R. Gruber, “Thermal beam distortions in end-pumped Nd:YAG, Nd:GSGG, and Nd:YLF rods,” IEEE J. Quantum Electron.30, 1605–1615 (1994).
[CrossRef]

Weber, R.

C. Pfistner, R. Weber, H. P. Weber, S. Merazzi, and R. Gruber, “Thermal beam distortions in end-pumped Nd:YAG, Nd:GSGG, and Nd:YLF rods,” IEEE J. Quantum Electron.30, 1605–1615 (1994).
[CrossRef]

Yura, H.

M. Innocenzi, H. Yura, C. Fincher, and R. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett.56, 1831–1833 (1990).
[CrossRef]

Zelenogorsky, V.

V. Zelenogorsky, O. Palashov, and E. Khazanov, “Adaptive compensation of thermally induced phase aberrations in Faraday isolators by means of a DKDP crystal,” Opt. Commun.278, 8–13 (2007).
[CrossRef]

E. Khazanov, N. Andreev, A. Mal’shakov, O. Palashov, A. Poteomkin, A. Sergeev, A. Shaykin, V. Zelenogorsky, I. Ivanov, R. Amin, G. Mueller, D. Tanner, and D. Reitze, “Compensation of thermally induced modal distortions in Faraday isolators,” IEEE J. Quantum Electron.40, 1500–1510 (2004).
[CrossRef]

Zhang, X.

S. Fan, X. Zhang, Q. Wang, S. Li, S. Ding, and F. Su, “More precise determination of thermal lens focal length for end-pumped solid-state lasers,” Opt. Commun.266, 620–626 (2006).
[CrossRef]

Appl. Opt. (5)

Appl. Phys. Lett. (1)

M. Innocenzi, H. Yura, C. Fincher, and R. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett.56, 1831–1833 (1990).
[CrossRef]

IEEE J. Quantum Electron. (4)

C. Pfistner, R. Weber, H. P. Weber, S. Merazzi, and R. Gruber, “Thermal beam distortions in end-pumped Nd:YAG, Nd:GSGG, and Nd:YLF rods,” IEEE J. Quantum Electron.30, 1605–1615 (1994).
[CrossRef]

Y. Chen, T. Huang, C. Kao, C. Wang, and S. Wang, “Optimization in scaling fiber-coupled laser-diode end-pumped lasers to higher power: influence of thermal effect,” IEEE J. Quantum Electron.33, 1424–1429 (1997).
[CrossRef]

E. Khazanov, N. Andreev, A. Mal’shakov, O. Palashov, A. Poteomkin, A. Sergeev, A. Shaykin, V. Zelenogorsky, I. Ivanov, R. Amin, G. Mueller, D. Tanner, and D. Reitze, “Compensation of thermally induced modal distortions in Faraday isolators,” IEEE J. Quantum Electron.40, 1500–1510 (2004).
[CrossRef]

U. O. Farrukh, A. M. Buoncristiani, and C. E. Byvik, “An analysis of the temperature distribution in finite solid-state laser rods,” IEEE J. Quantum Electron.24, 2253–2263 (1988).
[CrossRef]

J. Opt. Soc. Am. (1)

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

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

Opt. Commun. (3)

V. Zelenogorsky, O. Palashov, and E. Khazanov, “Adaptive compensation of thermally induced phase aberrations in Faraday isolators by means of a DKDP crystal,” Opt. Commun.278, 8–13 (2007).
[CrossRef]

J. Alda, “Quality improvement of a coherent and aberrated laser beam by using an optimum and smooth pure phase filter,” Opt. Commun.192, 199–204 (2001).
[CrossRef]

S. Fan, X. Zhang, Q. Wang, S. Li, S. Ding, and F. Su, “More precise determination of thermal lens focal length for end-pumped solid-state lasers,” Opt. Commun.266, 620–626 (2006).
[CrossRef]

Other (3)

NanoModeScan operational manual (Photon Inc, www.ophiropt.com , 2008).

J. Alda, Laser and Gaussian Beam Propagation and Transformation (Marcel Dekker, Inc, 2003), pp. 999–1013, Encyclopedia of Optical Engineering.

W. Koechner, Solid State Laser Engineering (Springer, 1988).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Schematic diagram of the experimental setup for M2 measurement

Fig. 2
Fig. 2

An ideal lens (dashed line) and a sextic phase aberration (thin line) are fitted to the actual thermal induced phase difference (thick line). The vertical line at 0.65 mm represents a guide line for laser beam radius

Fig. 3
Fig. 3

Degradation in beam quality factor, M2, as a function of input laser power for a TGG rod crystal. The theoretical curve was calculated from Eq. (10)

Tables (1)

Tables Icon

Table 1 Noll representation of Zernike polynomials [18]

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

ω 2 ( z ) = ω 0 2 ( 1 + z 2 z R 2 ) ,
T ( r , z ) = n , l A n l J 0 ( ν n r / a ) sin ( λ l z + β l ) ,
Δ ϕ ( r ) = k 0 L d n d T T ( r , z ) d z ,
𝒮 = | 0 a e i ( Δ ϕ ( r ) Δ ϕ ( 0 ) + k r 2 ) / 2 f T e r 2 / ω 0 2 r d r | 2 | 0 e r 2 / ω 0 2 r d r | 2 ,
Δ ϕ ( r ) = k L d n d T n A n 0 J 0 ( ν n r / a ) .
M 2 = π λ ( W 2 Θ 2 S 2 ) ,
Δ ϕ ( r ) = 2 π λ n = 1 N C n Z n ( r , θ ) ,
Δ ϕ ( r ) = 2 π λ ( b 2 r 2 + b 4 r 4 + b 6 r 6 ) ,
b 2 = 2 3 C 4 6 5 C 11 + 12 7 C 22 , b 4 = 6 5 C 11 30 7 C 22 , b 6 = 20 7 C 22 .
M 2 = ( M 0 2 ) 2 + ( Δ M a b 2 ) 2 ,
( Δ M a b 2 ) 2 = 1440 π 2 λ 2 [ ( C 11 2 2 35 C 11 C 22 ) W 8 + 6 35 C 11 C 22 W 10 ] .
ω 2 ( z ) = C z 2 + B z + A ,

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