Abstract

A folded resonator requires an oblique angle of incidence on the folded curved mirror, which introduces astigmatic distortions that limit the performance of the lasers. We present a simple method to compensate the astigmatism of folded resonator without Brewster windows for the first time to the best of our knowledge. Based on the theory of the propagation and transformation of Gaussian beams, the method is both effective and reliable. Theoretical results show that the folded resonator can be compensated astigmatism completely when the following two conditions are fulfilled. Firstly, when the Gaussian beam with a determined size beam waist is obliquely incident on an off-axis concave mirror, two new Gaussian beam respectively in the tangential and sagittal planes are formed. Another off-axis concave mirror is located at another intersection point of the two new Gaussian beams. Secondly, adjusting the incident angle of the second concave mirror or its focal length can make the above two Gaussian beam coincide in the image plane of the second concave mirror, which compensates the astigmatic aberration completely. A side-pumped continues-wave (CW) passively mode locked Nd:YAG laser was taken as an example of the astigmatically compensated folded resonators. The experimental results show good agreement with the theoretical predictions. This method can be used effectively to design astigmatically compensated cavities resonator of high-performance lasers.

© 2014 Optical Society of America

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References

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  1. H. Kogelnik, E. P. Ippen, A. Dienes, C. V. Shank, “Astigmatically compensated cavities for CW dye lasers,” IEEE J. Quantum Electron. 8(3), 373–379 (1972).
    [CrossRef]
  2. D. Kane, “Astigmatism compensation in off-axis laser resonators with two or more coupled foci,” Opt. Commun. 71(3–4), 113–118 (1989).
    [CrossRef]
  3. N. Jamasbi, J.-C. Diels, L. Sarger, “Study of a linear femtosecond laser in passive and hybrid operation,” J. Mod. Opt. 35(12), 1891–1906 (1988).
    [CrossRef]
  4. T. Skettrup, T. Meelby, K. Faerch, S. L. Frederiksen, C. Pedersen, “Triangular laser resonators with astigmatic compensation,” Appl. Opt. 39(24), 4306–4312 (2000).
    [CrossRef] [PubMed]
  5. T. Skettrup, “Rectangular laser resonators with astigmatic compensation,” J. Opt. A Pure Appl. Opt. 7(11), 645–654 (2005).
    [CrossRef]
  6. S. Yefet, V. Jouravsky, A. Pe’er, “Kerr lens mode locking without nonlinear astigmatism,” J. Opt. Soc. Am. B 30(3), 549–551 (2013).
    [CrossRef]
  7. U. Keller, D. A. Miller, G. D. Boyd, T. H. Chiu, J. F. Ferguson, M. T. Asom, “Solid-state low-loss intracavity saturable absorber for Nd:YLF lasers: an antiresonant semiconductor Fabry-Perot saturable absorber,” Opt. Lett. 17(7), 505–507 (1992).
    [CrossRef] [PubMed]
  8. D. Burns, M. Hetterich, A. Ferguson, E. Bente, M. Dawson, J. Davies, S. Bland, “High-average-power (> 20-W) Nd:YVO lasers mode locked by strain-compensated saturable Bragg reflectors,” J. Opt. Soc. Am. B 17(6), 919–926 (2000).
    [CrossRef]
  9. R. Paschotta, J. Aus der Au, G. J. Spühler, F. Morier-Genoud, R. Hövel, M. Moser, S. Erhard, M. Karszewski, A. Giesen, U. Keller, “Diode-pumped passively mode-locked lasers with high average power,” Appl. Phys. B Lasers Opt. 70(S1), S25–S31 (2000).
    [CrossRef]
  10. S. W. Tsai, Y. P. Lan, S. C. Wang, K. F. Huang, Y. F. Chen, “High-power diode-end-pumped passively mode-locked Nd: YVO4 laser with a relaxed saturable Bragg reflector,” Proc. SPIE 4630, 17–23 (2002).
    [CrossRef]
  11. K. K. Li, A. Dienes, J. R. Whinnery, “Stability and astigmatic compensation analysis of five-mirror cavity for mode-locked dye lasers,” Appl. Opt. 20(3), 407–411 (1981).
    [CrossRef] [PubMed]
  12. K. K. Li, “Stability and astigmatic analysis of a six-mirror ring cavity for mode-locked dye lasers,” Appl. Opt. 21(5), 967–970 (1982).
    [CrossRef] [PubMed]
  13. E. Cojocaru, T. Julea, N. Herisanu, “Stability and astigmatic compensation analysis of five- and six- or seven-mirror cavities for mode-locked dye lasers,” Appl. Opt. 28(13), 2577–2580 (1989).
    [CrossRef] [PubMed]
  14. H. Kegelnik, T. Li, “Imaging of optical modes-resonators with internal lenses,” Bell Syst. Tech. J. 44, 40 (1965).
  15. D. G. Lancaster, J. M. Dawes, “Thermal-lens measurement of a quasi steady-state repetitively flashlamppumped Cr, Tm, Ho:YAG laser,” Opt. Laser Technol. 30(2), 103–108 (1998).
    [CrossRef]
  16. F. Song, C. Zhang, X. Ding, J. Xu, G. Zhang, M. Leigh, N. Peyghambarian, “Determination of thermal focal length and pumping radius in gain medium in laser-diode-pumped Nd:YVO4 lasers,” Appl. Phys. Lett. 81(12), 2145–2147 (2002).
    [CrossRef]
  17. A. Geng, C. Zhao, B. Yong, Y. Lu, Z. Xu, “A method for measuring thermal focal length of LD-side -pumped laser crystal (in Chinese),” Acta Phys. Sin. 57, 6987–6991 (2008).

2013

2008

A. Geng, C. Zhao, B. Yong, Y. Lu, Z. Xu, “A method for measuring thermal focal length of LD-side -pumped laser crystal (in Chinese),” Acta Phys. Sin. 57, 6987–6991 (2008).

2005

T. Skettrup, “Rectangular laser resonators with astigmatic compensation,” J. Opt. A Pure Appl. Opt. 7(11), 645–654 (2005).
[CrossRef]

2002

S. W. Tsai, Y. P. Lan, S. C. Wang, K. F. Huang, Y. F. Chen, “High-power diode-end-pumped passively mode-locked Nd: YVO4 laser with a relaxed saturable Bragg reflector,” Proc. SPIE 4630, 17–23 (2002).
[CrossRef]

F. Song, C. Zhang, X. Ding, J. Xu, G. Zhang, M. Leigh, N. Peyghambarian, “Determination of thermal focal length and pumping radius in gain medium in laser-diode-pumped Nd:YVO4 lasers,” Appl. Phys. Lett. 81(12), 2145–2147 (2002).
[CrossRef]

2000

1998

D. G. Lancaster, J. M. Dawes, “Thermal-lens measurement of a quasi steady-state repetitively flashlamppumped Cr, Tm, Ho:YAG laser,” Opt. Laser Technol. 30(2), 103–108 (1998).
[CrossRef]

1992

1989

1988

N. Jamasbi, J.-C. Diels, L. Sarger, “Study of a linear femtosecond laser in passive and hybrid operation,” J. Mod. Opt. 35(12), 1891–1906 (1988).
[CrossRef]

1982

1981

1972

H. Kogelnik, E. P. Ippen, A. Dienes, C. V. Shank, “Astigmatically compensated cavities for CW dye lasers,” IEEE J. Quantum Electron. 8(3), 373–379 (1972).
[CrossRef]

1965

H. Kegelnik, T. Li, “Imaging of optical modes-resonators with internal lenses,” Bell Syst. Tech. J. 44, 40 (1965).

Asom, M. T.

Aus der Au, J.

R. Paschotta, J. Aus der Au, G. J. Spühler, F. Morier-Genoud, R. Hövel, M. Moser, S. Erhard, M. Karszewski, A. Giesen, U. Keller, “Diode-pumped passively mode-locked lasers with high average power,” Appl. Phys. B Lasers Opt. 70(S1), S25–S31 (2000).
[CrossRef]

Bente, E.

Bland, S.

Boyd, G. D.

Burns, D.

Chen, Y. F.

S. W. Tsai, Y. P. Lan, S. C. Wang, K. F. Huang, Y. F. Chen, “High-power diode-end-pumped passively mode-locked Nd: YVO4 laser with a relaxed saturable Bragg reflector,” Proc. SPIE 4630, 17–23 (2002).
[CrossRef]

Chiu, T. H.

Cojocaru, E.

Davies, J.

Dawes, J. M.

D. G. Lancaster, J. M. Dawes, “Thermal-lens measurement of a quasi steady-state repetitively flashlamppumped Cr, Tm, Ho:YAG laser,” Opt. Laser Technol. 30(2), 103–108 (1998).
[CrossRef]

Dawson, M.

Diels, J.-C.

N. Jamasbi, J.-C. Diels, L. Sarger, “Study of a linear femtosecond laser in passive and hybrid operation,” J. Mod. Opt. 35(12), 1891–1906 (1988).
[CrossRef]

Dienes, A.

K. K. Li, A. Dienes, J. R. Whinnery, “Stability and astigmatic compensation analysis of five-mirror cavity for mode-locked dye lasers,” Appl. Opt. 20(3), 407–411 (1981).
[CrossRef] [PubMed]

H. Kogelnik, E. P. Ippen, A. Dienes, C. V. Shank, “Astigmatically compensated cavities for CW dye lasers,” IEEE J. Quantum Electron. 8(3), 373–379 (1972).
[CrossRef]

Ding, X.

F. Song, C. Zhang, X. Ding, J. Xu, G. Zhang, M. Leigh, N. Peyghambarian, “Determination of thermal focal length and pumping radius in gain medium in laser-diode-pumped Nd:YVO4 lasers,” Appl. Phys. Lett. 81(12), 2145–2147 (2002).
[CrossRef]

Erhard, S.

R. Paschotta, J. Aus der Au, G. J. Spühler, F. Morier-Genoud, R. Hövel, M. Moser, S. Erhard, M. Karszewski, A. Giesen, U. Keller, “Diode-pumped passively mode-locked lasers with high average power,” Appl. Phys. B Lasers Opt. 70(S1), S25–S31 (2000).
[CrossRef]

Faerch, K.

Ferguson, A.

Ferguson, J. F.

Frederiksen, S. L.

Geng, A.

A. Geng, C. Zhao, B. Yong, Y. Lu, Z. Xu, “A method for measuring thermal focal length of LD-side -pumped laser crystal (in Chinese),” Acta Phys. Sin. 57, 6987–6991 (2008).

Giesen, A.

R. Paschotta, J. Aus der Au, G. J. Spühler, F. Morier-Genoud, R. Hövel, M. Moser, S. Erhard, M. Karszewski, A. Giesen, U. Keller, “Diode-pumped passively mode-locked lasers with high average power,” Appl. Phys. B Lasers Opt. 70(S1), S25–S31 (2000).
[CrossRef]

Herisanu, N.

Hetterich, M.

Hövel, R.

R. Paschotta, J. Aus der Au, G. J. Spühler, F. Morier-Genoud, R. Hövel, M. Moser, S. Erhard, M. Karszewski, A. Giesen, U. Keller, “Diode-pumped passively mode-locked lasers with high average power,” Appl. Phys. B Lasers Opt. 70(S1), S25–S31 (2000).
[CrossRef]

Huang, K. F.

S. W. Tsai, Y. P. Lan, S. C. Wang, K. F. Huang, Y. F. Chen, “High-power diode-end-pumped passively mode-locked Nd: YVO4 laser with a relaxed saturable Bragg reflector,” Proc. SPIE 4630, 17–23 (2002).
[CrossRef]

Ippen, E. P.

H. Kogelnik, E. P. Ippen, A. Dienes, C. V. Shank, “Astigmatically compensated cavities for CW dye lasers,” IEEE J. Quantum Electron. 8(3), 373–379 (1972).
[CrossRef]

Jamasbi, N.

N. Jamasbi, J.-C. Diels, L. Sarger, “Study of a linear femtosecond laser in passive and hybrid operation,” J. Mod. Opt. 35(12), 1891–1906 (1988).
[CrossRef]

Jouravsky, V.

Julea, T.

Kane, D.

D. Kane, “Astigmatism compensation in off-axis laser resonators with two or more coupled foci,” Opt. Commun. 71(3–4), 113–118 (1989).
[CrossRef]

Karszewski, M.

R. Paschotta, J. Aus der Au, G. J. Spühler, F. Morier-Genoud, R. Hövel, M. Moser, S. Erhard, M. Karszewski, A. Giesen, U. Keller, “Diode-pumped passively mode-locked lasers with high average power,” Appl. Phys. B Lasers Opt. 70(S1), S25–S31 (2000).
[CrossRef]

Kegelnik, H.

H. Kegelnik, T. Li, “Imaging of optical modes-resonators with internal lenses,” Bell Syst. Tech. J. 44, 40 (1965).

Keller, U.

R. Paschotta, J. Aus der Au, G. J. Spühler, F. Morier-Genoud, R. Hövel, M. Moser, S. Erhard, M. Karszewski, A. Giesen, U. Keller, “Diode-pumped passively mode-locked lasers with high average power,” Appl. Phys. B Lasers Opt. 70(S1), S25–S31 (2000).
[CrossRef]

U. Keller, D. A. Miller, G. D. Boyd, T. H. Chiu, J. F. Ferguson, M. T. Asom, “Solid-state low-loss intracavity saturable absorber for Nd:YLF lasers: an antiresonant semiconductor Fabry-Perot saturable absorber,” Opt. Lett. 17(7), 505–507 (1992).
[CrossRef] [PubMed]

Kogelnik, H.

H. Kogelnik, E. P. Ippen, A. Dienes, C. V. Shank, “Astigmatically compensated cavities for CW dye lasers,” IEEE J. Quantum Electron. 8(3), 373–379 (1972).
[CrossRef]

Lan, Y. P.

S. W. Tsai, Y. P. Lan, S. C. Wang, K. F. Huang, Y. F. Chen, “High-power diode-end-pumped passively mode-locked Nd: YVO4 laser with a relaxed saturable Bragg reflector,” Proc. SPIE 4630, 17–23 (2002).
[CrossRef]

Lancaster, D. G.

D. G. Lancaster, J. M. Dawes, “Thermal-lens measurement of a quasi steady-state repetitively flashlamppumped Cr, Tm, Ho:YAG laser,” Opt. Laser Technol. 30(2), 103–108 (1998).
[CrossRef]

Leigh, M.

F. Song, C. Zhang, X. Ding, J. Xu, G. Zhang, M. Leigh, N. Peyghambarian, “Determination of thermal focal length and pumping radius in gain medium in laser-diode-pumped Nd:YVO4 lasers,” Appl. Phys. Lett. 81(12), 2145–2147 (2002).
[CrossRef]

Li, K. K.

Li, T.

H. Kegelnik, T. Li, “Imaging of optical modes-resonators with internal lenses,” Bell Syst. Tech. J. 44, 40 (1965).

Lu, Y.

A. Geng, C. Zhao, B. Yong, Y. Lu, Z. Xu, “A method for measuring thermal focal length of LD-side -pumped laser crystal (in Chinese),” Acta Phys. Sin. 57, 6987–6991 (2008).

Meelby, T.

Miller, D. A.

Morier-Genoud, F.

R. Paschotta, J. Aus der Au, G. J. Spühler, F. Morier-Genoud, R. Hövel, M. Moser, S. Erhard, M. Karszewski, A. Giesen, U. Keller, “Diode-pumped passively mode-locked lasers with high average power,” Appl. Phys. B Lasers Opt. 70(S1), S25–S31 (2000).
[CrossRef]

Moser, M.

R. Paschotta, J. Aus der Au, G. J. Spühler, F. Morier-Genoud, R. Hövel, M. Moser, S. Erhard, M. Karszewski, A. Giesen, U. Keller, “Diode-pumped passively mode-locked lasers with high average power,” Appl. Phys. B Lasers Opt. 70(S1), S25–S31 (2000).
[CrossRef]

Paschotta, R.

R. Paschotta, J. Aus der Au, G. J. Spühler, F. Morier-Genoud, R. Hövel, M. Moser, S. Erhard, M. Karszewski, A. Giesen, U. Keller, “Diode-pumped passively mode-locked lasers with high average power,” Appl. Phys. B Lasers Opt. 70(S1), S25–S31 (2000).
[CrossRef]

Pe’er, A.

Pedersen, C.

Peyghambarian, N.

F. Song, C. Zhang, X. Ding, J. Xu, G. Zhang, M. Leigh, N. Peyghambarian, “Determination of thermal focal length and pumping radius in gain medium in laser-diode-pumped Nd:YVO4 lasers,” Appl. Phys. Lett. 81(12), 2145–2147 (2002).
[CrossRef]

Sarger, L.

N. Jamasbi, J.-C. Diels, L. Sarger, “Study of a linear femtosecond laser in passive and hybrid operation,” J. Mod. Opt. 35(12), 1891–1906 (1988).
[CrossRef]

Shank, C. V.

H. Kogelnik, E. P. Ippen, A. Dienes, C. V. Shank, “Astigmatically compensated cavities for CW dye lasers,” IEEE J. Quantum Electron. 8(3), 373–379 (1972).
[CrossRef]

Skettrup, T.

Song, F.

F. Song, C. Zhang, X. Ding, J. Xu, G. Zhang, M. Leigh, N. Peyghambarian, “Determination of thermal focal length and pumping radius in gain medium in laser-diode-pumped Nd:YVO4 lasers,” Appl. Phys. Lett. 81(12), 2145–2147 (2002).
[CrossRef]

Spühler, G. J.

R. Paschotta, J. Aus der Au, G. J. Spühler, F. Morier-Genoud, R. Hövel, M. Moser, S. Erhard, M. Karszewski, A. Giesen, U. Keller, “Diode-pumped passively mode-locked lasers with high average power,” Appl. Phys. B Lasers Opt. 70(S1), S25–S31 (2000).
[CrossRef]

Tsai, S. W.

S. W. Tsai, Y. P. Lan, S. C. Wang, K. F. Huang, Y. F. Chen, “High-power diode-end-pumped passively mode-locked Nd: YVO4 laser with a relaxed saturable Bragg reflector,” Proc. SPIE 4630, 17–23 (2002).
[CrossRef]

Wang, S. C.

S. W. Tsai, Y. P. Lan, S. C. Wang, K. F. Huang, Y. F. Chen, “High-power diode-end-pumped passively mode-locked Nd: YVO4 laser with a relaxed saturable Bragg reflector,” Proc. SPIE 4630, 17–23 (2002).
[CrossRef]

Whinnery, J. R.

Xu, J.

F. Song, C. Zhang, X. Ding, J. Xu, G. Zhang, M. Leigh, N. Peyghambarian, “Determination of thermal focal length and pumping radius in gain medium in laser-diode-pumped Nd:YVO4 lasers,” Appl. Phys. Lett. 81(12), 2145–2147 (2002).
[CrossRef]

Xu, Z.

A. Geng, C. Zhao, B. Yong, Y. Lu, Z. Xu, “A method for measuring thermal focal length of LD-side -pumped laser crystal (in Chinese),” Acta Phys. Sin. 57, 6987–6991 (2008).

Yefet, S.

Yong, B.

A. Geng, C. Zhao, B. Yong, Y. Lu, Z. Xu, “A method for measuring thermal focal length of LD-side -pumped laser crystal (in Chinese),” Acta Phys. Sin. 57, 6987–6991 (2008).

Zhang, C.

F. Song, C. Zhang, X. Ding, J. Xu, G. Zhang, M. Leigh, N. Peyghambarian, “Determination of thermal focal length and pumping radius in gain medium in laser-diode-pumped Nd:YVO4 lasers,” Appl. Phys. Lett. 81(12), 2145–2147 (2002).
[CrossRef]

Zhang, G.

F. Song, C. Zhang, X. Ding, J. Xu, G. Zhang, M. Leigh, N. Peyghambarian, “Determination of thermal focal length and pumping radius in gain medium in laser-diode-pumped Nd:YVO4 lasers,” Appl. Phys. Lett. 81(12), 2145–2147 (2002).
[CrossRef]

Zhao, C.

A. Geng, C. Zhao, B. Yong, Y. Lu, Z. Xu, “A method for measuring thermal focal length of LD-side -pumped laser crystal (in Chinese),” Acta Phys. Sin. 57, 6987–6991 (2008).

Acta Phys. Sin.

A. Geng, C. Zhao, B. Yong, Y. Lu, Z. Xu, “A method for measuring thermal focal length of LD-side -pumped laser crystal (in Chinese),” Acta Phys. Sin. 57, 6987–6991 (2008).

Appl. Opt.

Appl. Phys. B Lasers Opt.

R. Paschotta, J. Aus der Au, G. J. Spühler, F. Morier-Genoud, R. Hövel, M. Moser, S. Erhard, M. Karszewski, A. Giesen, U. Keller, “Diode-pumped passively mode-locked lasers with high average power,” Appl. Phys. B Lasers Opt. 70(S1), S25–S31 (2000).
[CrossRef]

Appl. Phys. Lett.

F. Song, C. Zhang, X. Ding, J. Xu, G. Zhang, M. Leigh, N. Peyghambarian, “Determination of thermal focal length and pumping radius in gain medium in laser-diode-pumped Nd:YVO4 lasers,” Appl. Phys. Lett. 81(12), 2145–2147 (2002).
[CrossRef]

Bell Syst. Tech. J.

H. Kegelnik, T. Li, “Imaging of optical modes-resonators with internal lenses,” Bell Syst. Tech. J. 44, 40 (1965).

IEEE J. Quantum Electron.

H. Kogelnik, E. P. Ippen, A. Dienes, C. V. Shank, “Astigmatically compensated cavities for CW dye lasers,” IEEE J. Quantum Electron. 8(3), 373–379 (1972).
[CrossRef]

J. Mod. Opt.

N. Jamasbi, J.-C. Diels, L. Sarger, “Study of a linear femtosecond laser in passive and hybrid operation,” J. Mod. Opt. 35(12), 1891–1906 (1988).
[CrossRef]

J. Opt. A Pure Appl. Opt.

T. Skettrup, “Rectangular laser resonators with astigmatic compensation,” J. Opt. A Pure Appl. Opt. 7(11), 645–654 (2005).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

D. Kane, “Astigmatism compensation in off-axis laser resonators with two or more coupled foci,” Opt. Commun. 71(3–4), 113–118 (1989).
[CrossRef]

Opt. Laser Technol.

D. G. Lancaster, J. M. Dawes, “Thermal-lens measurement of a quasi steady-state repetitively flashlamppumped Cr, Tm, Ho:YAG laser,” Opt. Laser Technol. 30(2), 103–108 (1998).
[CrossRef]

Opt. Lett.

Proc. SPIE

S. W. Tsai, Y. P. Lan, S. C. Wang, K. F. Huang, Y. F. Chen, “High-power diode-end-pumped passively mode-locked Nd: YVO4 laser with a relaxed saturable Bragg reflector,” Proc. SPIE 4630, 17–23 (2002).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Configuration of cavity for SESAM mode-locked laser.

Fig. 2
Fig. 2

A special transformation of Gaussian beam pass through two off-axis lenses sequence.

Fig. 3
Fig. 3

The astigmatic values (defined as the absolute value of w 2s w 2t and l 2s l 2t ) vary with the focal length of M2 and the angle θ 2 of incidence at M2. The differences of beam waist sizes of the two planes are marked by the red solid line. The differences of beam waist location are marked by the blue solid line. (a) f2 = 250mm, (b) f2 = 200mm, (c) θ 2 =11.9° , (d) θ 2 =10.7° .

Fig. 4
Fig. 4

output spot Intensity profiles from the terminal arms on different observation locations: (a) output spot intensity profiles from the arm between M2 and M4 at location close to M4, (b) output spot intensity profiles from the arm between M2 and M4 at location far from M4, (c) output spot intensity profiles from the arm between M1 and M3 at location close to M3, (d) output spot intensity profiles from the arm between M1 and M3 at location far from M3.

Fig. 5
Fig. 5

Oscillograms of the continuously mode-locked laser trains.

Equations (5)

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

f s = f cosθ
f t =fcosθ
1 w ' 1 2 = 1 w 1 2 ( 1 l 1 f ) 2 + 1 f 2 ( π w 1 λ ) 2
l 1 f=( l 1 f ) f 2 ( l 1 f ) 2 + ( π w 1 2 λ ) 2
w ( z ) 2 = w 2 [ 1+ ( λz π w 2 ) 2 ]

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