Abstract

A theoretical analysis involving ABCD ray transfer matrices is used to find the self-consistent fundamental spatial mode solutions of self-adaptive laser resonators. The resonators investigated consist of a nonlinear medium in a self-intersecting loop geometry together with a feedback output coupler mirror and additional intracavity elements. A simplified system without intracavity elements is analyzed initially, and an analytic expression for the mode solution is deduced. Addition of an intracavity lens is shown to permit enhancement of the quality of the phase-conjugation process as well as control of the mode size. The theoretical analysis is extended to model an experimental self-adaptive laser oscillator utilizing gain-grating formation in a solid-state Nd:YAG laser amplifier. Good agreement is found between the theory and the experimental results.

© 1998 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. W. Koechner, Solid-State Laser Engineering (Springer-Verlag, Berlin, 1992).
  2. B. Ya. Zel’dovich, V. I. Popvichev, V. V. Ragul’skii, and F. S. Faizullov, “Connection between the wavefronts of the reflected and exciting light in stimulated Mandel’shtam Brillouin scattering,” JETP Lett. 15, 109–113 (1972).
  3. J. O. White, M. Cronin-Golomb, B. Fisher, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450–452 (1982).
    [CrossRef]
  4. J. Auyeung, D. Fekete, D. M. Pepper, and A. Yariv, “A theoretical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
    [CrossRef]
  5. P. Sillard, A. Brignon, and J.-P. Huignard, “Nd:YAG loop resonator with a Cr4+:YAG self-pumped phase-conjugate mirror,” IEEE J. Quantum Electron. QE-33, 483 (1997).
    [CrossRef]
  6. I. M. Bel’dyugin, M. V. Zolotarev, S. E. Kireev, and A. I. Odintsov, “Copper vapor laser with a self-pumped wavefront reversing mirror,” Sov. J. Quantum Electron. 16, 535–537 (1986).
    [CrossRef]
  7. M. J. Damzen, R. P. M. Green, and K. S. Syed, “Self-adaptive solid-state laser oscillator formed by dynamic gain-grating holograms,” Opt. Lett. 20, 1704–1706 (1995).
    [CrossRef] [PubMed]
  8. R. W. Hellwarth, “Generation of time-reversed wave fronts by nonlinear refraction,” J. Opt. Soc. Am. 67, 1–3 (1977).
    [CrossRef]
  9. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
    [CrossRef] [PubMed]
  10. A. E. Siegman, “Unstable optical resonators,” Appl. Opt. 13, 353–367 (1974).
    [CrossRef] [PubMed]
  11. M. Bel’dyugin and E. M. Zemskov, “Calculation of the field in a laser resonator with a wavefront-reversing mirror,” Sov. J. Quantum Electron. 10, 764–765 (1980).
    [CrossRef]
  12. I. M. Bel’dyugin, M. G. Galushin, and E. M. Zemskov, “Properties of resonators with wavefront-reversing mirrors,” Sov. J. Quantum Electron. 9, 20–23 (1979).
    [CrossRef]
  13. M. G. Reznikov and A. I. Khizhnyak, “Properties of resonators with a wavefront-reversing mirror,” Sov. J. Quantum Electron. 10, 633–634 (1980).
    [CrossRef]
  14. P. A. Bélanger, A. Hardy, and A. E. Siegman, “Resonant modes of optical cavities with phase conjugate mirrors,” Appl. Opt. 19, 602–609 (1980); “Resonant modes of optical cavities with phase conjugate mirrors: higher order modes,” Appl. Opt. 19, 479–480 (1980).
    [CrossRef]
  15. J. F. Lam and W. P. Brown, “Optical resonators with phase-conjugate mirrors,” Opt. Lett. 5, 61–63 (1980).
    [CrossRef] [PubMed]
  16. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  17. A. Yariv and P. Yeh, “Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers,” Opt. Commun. 13, 370–374 (1975).
    [CrossRef]
  18. M. J. Damzen, G. J. Crofts, and R. P. M. Green, “Spatial characteristics of a laser oscillator formed by optically-written holographic gain gratings,” Opt. Commun. 110, 152–156 (1994).
    [CrossRef]

1997 (1)

P. Sillard, A. Brignon, and J.-P. Huignard, “Nd:YAG loop resonator with a Cr4+:YAG self-pumped phase-conjugate mirror,” IEEE J. Quantum Electron. QE-33, 483 (1997).
[CrossRef]

1995 (1)

1994 (1)

M. J. Damzen, G. J. Crofts, and R. P. M. Green, “Spatial characteristics of a laser oscillator formed by optically-written holographic gain gratings,” Opt. Commun. 110, 152–156 (1994).
[CrossRef]

1986 (1)

I. M. Bel’dyugin, M. V. Zolotarev, S. E. Kireev, and A. I. Odintsov, “Copper vapor laser with a self-pumped wavefront reversing mirror,” Sov. J. Quantum Electron. 16, 535–537 (1986).
[CrossRef]

1982 (1)

J. O. White, M. Cronin-Golomb, B. Fisher, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450–452 (1982).
[CrossRef]

1980 (3)

M. G. Reznikov and A. I. Khizhnyak, “Properties of resonators with a wavefront-reversing mirror,” Sov. J. Quantum Electron. 10, 633–634 (1980).
[CrossRef]

M. Bel’dyugin and E. M. Zemskov, “Calculation of the field in a laser resonator with a wavefront-reversing mirror,” Sov. J. Quantum Electron. 10, 764–765 (1980).
[CrossRef]

J. F. Lam and W. P. Brown, “Optical resonators with phase-conjugate mirrors,” Opt. Lett. 5, 61–63 (1980).
[CrossRef] [PubMed]

1979 (2)

I. M. Bel’dyugin, M. G. Galushin, and E. M. Zemskov, “Properties of resonators with wavefront-reversing mirrors,” Sov. J. Quantum Electron. 9, 20–23 (1979).
[CrossRef]

J. Auyeung, D. Fekete, D. M. Pepper, and A. Yariv, “A theoretical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
[CrossRef]

1977 (1)

1975 (1)

A. Yariv and P. Yeh, “Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers,” Opt. Commun. 13, 370–374 (1975).
[CrossRef]

1974 (1)

1972 (1)

B. Ya. Zel’dovich, V. I. Popvichev, V. V. Ragul’skii, and F. S. Faizullov, “Connection between the wavefronts of the reflected and exciting light in stimulated Mandel’shtam Brillouin scattering,” JETP Lett. 15, 109–113 (1972).

1966 (1)

Auyeung, J.

J. Auyeung, D. Fekete, D. M. Pepper, and A. Yariv, “A theoretical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
[CrossRef]

Bel’dyugin, I. M.

I. M. Bel’dyugin, M. V. Zolotarev, S. E. Kireev, and A. I. Odintsov, “Copper vapor laser with a self-pumped wavefront reversing mirror,” Sov. J. Quantum Electron. 16, 535–537 (1986).
[CrossRef]

I. M. Bel’dyugin, M. G. Galushin, and E. M. Zemskov, “Properties of resonators with wavefront-reversing mirrors,” Sov. J. Quantum Electron. 9, 20–23 (1979).
[CrossRef]

Bel’dyugin, M.

M. Bel’dyugin and E. M. Zemskov, “Calculation of the field in a laser resonator with a wavefront-reversing mirror,” Sov. J. Quantum Electron. 10, 764–765 (1980).
[CrossRef]

Brignon, A.

P. Sillard, A. Brignon, and J.-P. Huignard, “Nd:YAG loop resonator with a Cr4+:YAG self-pumped phase-conjugate mirror,” IEEE J. Quantum Electron. QE-33, 483 (1997).
[CrossRef]

Brown, W. P.

Crofts, G. J.

M. J. Damzen, G. J. Crofts, and R. P. M. Green, “Spatial characteristics of a laser oscillator formed by optically-written holographic gain gratings,” Opt. Commun. 110, 152–156 (1994).
[CrossRef]

Cronin-Golomb, M.

J. O. White, M. Cronin-Golomb, B. Fisher, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450–452 (1982).
[CrossRef]

Damzen, M. J.

M. J. Damzen, R. P. M. Green, and K. S. Syed, “Self-adaptive solid-state laser oscillator formed by dynamic gain-grating holograms,” Opt. Lett. 20, 1704–1706 (1995).
[CrossRef] [PubMed]

M. J. Damzen, G. J. Crofts, and R. P. M. Green, “Spatial characteristics of a laser oscillator formed by optically-written holographic gain gratings,” Opt. Commun. 110, 152–156 (1994).
[CrossRef]

Faizullov, F. S.

B. Ya. Zel’dovich, V. I. Popvichev, V. V. Ragul’skii, and F. S. Faizullov, “Connection between the wavefronts of the reflected and exciting light in stimulated Mandel’shtam Brillouin scattering,” JETP Lett. 15, 109–113 (1972).

Fekete, D.

J. Auyeung, D. Fekete, D. M. Pepper, and A. Yariv, “A theoretical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
[CrossRef]

Fisher, B.

J. O. White, M. Cronin-Golomb, B. Fisher, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450–452 (1982).
[CrossRef]

Galushin, M. G.

I. M. Bel’dyugin, M. G. Galushin, and E. M. Zemskov, “Properties of resonators with wavefront-reversing mirrors,” Sov. J. Quantum Electron. 9, 20–23 (1979).
[CrossRef]

Green, R. P. M.

M. J. Damzen, R. P. M. Green, and K. S. Syed, “Self-adaptive solid-state laser oscillator formed by dynamic gain-grating holograms,” Opt. Lett. 20, 1704–1706 (1995).
[CrossRef] [PubMed]

M. J. Damzen, G. J. Crofts, and R. P. M. Green, “Spatial characteristics of a laser oscillator formed by optically-written holographic gain gratings,” Opt. Commun. 110, 152–156 (1994).
[CrossRef]

Hellwarth, R. W.

Huignard, J.-P.

P. Sillard, A. Brignon, and J.-P. Huignard, “Nd:YAG loop resonator with a Cr4+:YAG self-pumped phase-conjugate mirror,” IEEE J. Quantum Electron. QE-33, 483 (1997).
[CrossRef]

Khizhnyak, A. I.

M. G. Reznikov and A. I. Khizhnyak, “Properties of resonators with a wavefront-reversing mirror,” Sov. J. Quantum Electron. 10, 633–634 (1980).
[CrossRef]

Kireev, S. E.

I. M. Bel’dyugin, M. V. Zolotarev, S. E. Kireev, and A. I. Odintsov, “Copper vapor laser with a self-pumped wavefront reversing mirror,” Sov. J. Quantum Electron. 16, 535–537 (1986).
[CrossRef]

Kogelnik, H.

Lam, J. F.

Li, T.

Odintsov, A. I.

I. M. Bel’dyugin, M. V. Zolotarev, S. E. Kireev, and A. I. Odintsov, “Copper vapor laser with a self-pumped wavefront reversing mirror,” Sov. J. Quantum Electron. 16, 535–537 (1986).
[CrossRef]

Pepper, D. M.

J. Auyeung, D. Fekete, D. M. Pepper, and A. Yariv, “A theoretical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
[CrossRef]

Popvichev, V. I.

B. Ya. Zel’dovich, V. I. Popvichev, V. V. Ragul’skii, and F. S. Faizullov, “Connection between the wavefronts of the reflected and exciting light in stimulated Mandel’shtam Brillouin scattering,” JETP Lett. 15, 109–113 (1972).

Ragul’skii, V. V.

B. Ya. Zel’dovich, V. I. Popvichev, V. V. Ragul’skii, and F. S. Faizullov, “Connection between the wavefronts of the reflected and exciting light in stimulated Mandel’shtam Brillouin scattering,” JETP Lett. 15, 109–113 (1972).

Reznikov, M. G.

M. G. Reznikov and A. I. Khizhnyak, “Properties of resonators with a wavefront-reversing mirror,” Sov. J. Quantum Electron. 10, 633–634 (1980).
[CrossRef]

Siegman, A. E.

Sillard, P.

P. Sillard, A. Brignon, and J.-P. Huignard, “Nd:YAG loop resonator with a Cr4+:YAG self-pumped phase-conjugate mirror,” IEEE J. Quantum Electron. QE-33, 483 (1997).
[CrossRef]

Syed, K. S.

White, J. O.

J. O. White, M. Cronin-Golomb, B. Fisher, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450–452 (1982).
[CrossRef]

Yariv, A.

J. O. White, M. Cronin-Golomb, B. Fisher, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450–452 (1982).
[CrossRef]

J. Auyeung, D. Fekete, D. M. Pepper, and A. Yariv, “A theoretical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
[CrossRef]

A. Yariv and P. Yeh, “Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers,” Opt. Commun. 13, 370–374 (1975).
[CrossRef]

Yeh, P.

A. Yariv and P. Yeh, “Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers,” Opt. Commun. 13, 370–374 (1975).
[CrossRef]

Zel’dovich, B. Ya.

B. Ya. Zel’dovich, V. I. Popvichev, V. V. Ragul’skii, and F. S. Faizullov, “Connection between the wavefronts of the reflected and exciting light in stimulated Mandel’shtam Brillouin scattering,” JETP Lett. 15, 109–113 (1972).

Zemskov, E. M.

M. Bel’dyugin and E. M. Zemskov, “Calculation of the field in a laser resonator with a wavefront-reversing mirror,” Sov. J. Quantum Electron. 10, 764–765 (1980).
[CrossRef]

I. M. Bel’dyugin, M. G. Galushin, and E. M. Zemskov, “Properties of resonators with wavefront-reversing mirrors,” Sov. J. Quantum Electron. 9, 20–23 (1979).
[CrossRef]

Zolotarev, M. V.

I. M. Bel’dyugin, M. V. Zolotarev, S. E. Kireev, and A. I. Odintsov, “Copper vapor laser with a self-pumped wavefront reversing mirror,” Sov. J. Quantum Electron. 16, 535–537 (1986).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

J. O. White, M. Cronin-Golomb, B. Fisher, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450–452 (1982).
[CrossRef]

IEEE J. Quantum Electron. (2)

J. Auyeung, D. Fekete, D. M. Pepper, and A. Yariv, “A theoretical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
[CrossRef]

P. Sillard, A. Brignon, and J.-P. Huignard, “Nd:YAG loop resonator with a Cr4+:YAG self-pumped phase-conjugate mirror,” IEEE J. Quantum Electron. QE-33, 483 (1997).
[CrossRef]

J. Opt. Soc. Am. (1)

JETP Lett. (1)

B. Ya. Zel’dovich, V. I. Popvichev, V. V. Ragul’skii, and F. S. Faizullov, “Connection between the wavefronts of the reflected and exciting light in stimulated Mandel’shtam Brillouin scattering,” JETP Lett. 15, 109–113 (1972).

Opt. Commun. (2)

A. Yariv and P. Yeh, “Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers,” Opt. Commun. 13, 370–374 (1975).
[CrossRef]

M. J. Damzen, G. J. Crofts, and R. P. M. Green, “Spatial characteristics of a laser oscillator formed by optically-written holographic gain gratings,” Opt. Commun. 110, 152–156 (1994).
[CrossRef]

Opt. Lett. (2)

Sov. J. Quantum Electron. (4)

I. M. Bel’dyugin, M. V. Zolotarev, S. E. Kireev, and A. I. Odintsov, “Copper vapor laser with a self-pumped wavefront reversing mirror,” Sov. J. Quantum Electron. 16, 535–537 (1986).
[CrossRef]

M. Bel’dyugin and E. M. Zemskov, “Calculation of the field in a laser resonator with a wavefront-reversing mirror,” Sov. J. Quantum Electron. 10, 764–765 (1980).
[CrossRef]

I. M. Bel’dyugin, M. G. Galushin, and E. M. Zemskov, “Properties of resonators with wavefront-reversing mirrors,” Sov. J. Quantum Electron. 9, 20–23 (1979).
[CrossRef]

M. G. Reznikov and A. I. Khizhnyak, “Properties of resonators with a wavefront-reversing mirror,” Sov. J. Quantum Electron. 10, 633–634 (1980).
[CrossRef]

Other (3)

P. A. Bélanger, A. Hardy, and A. E. Siegman, “Resonant modes of optical cavities with phase conjugate mirrors,” Appl. Opt. 19, 602–609 (1980); “Resonant modes of optical cavities with phase conjugate mirrors: higher order modes,” Appl. Opt. 19, 479–480 (1980).
[CrossRef]

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

W. Koechner, Solid-State Laser Engineering (Springer-Verlag, Berlin, 1992).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Schematic of an adaptive laser resonator, where beam E4 is generated by the FWM interaction of beams E1, E2, and E3.

Fig. 2
Fig. 2

Simplified schematic of an adaptive laser resonator, of loop length L and distance from the output coupler to the FWM interaction d. The dashed box represents additional (optional) optical elements.

Fig. 3
Fig. 3

(a) Spot size and (b) curvature (reciprocal of ROC) for a self-consistent mode solution of the simplified resonator as a function of the beam position in the cavity. The clockwise propagation of A1 to A3 is represented by the solid lines. The anticlockwise propagation (A4 to A2) is represented by the dotted lines.

Fig. 4
Fig. 4

Output spot size as a function of the cavity parameter g=1-L/f for the simple ALO with an intracavity lens.

Fig. 5
Fig. 5

(a) Spot size and (b) curvature of the self-consistent mode as the beam propagates around the cavity, for a simple ALO with an intracavity lens and cavity parameter g=+0.22.

Fig. 6
Fig. 6

Schematic of the experimental ALO system using Nd:YAG as the FWM medium (AMP 1) and amplifier (AMP 2). The output, probe, and conjugate beam sizes are monitored by CCD cameras 1, 2, and 3, respectively.

Fig. 7
Fig. 7

Simple schematic of the experimental ALO used in the numerical model, which uses lenses and Gaussian apertures to simulate optical elements.

Fig. 8
Fig. 8

Spatial profiles of (a) the experimental output beam, (b) the experimental probe beam, (c) the theoretical output beam, and (d) the theoretical probe beam. Beam sizes shown are the full width at the 1/e2 intensity point.

Fig. 9
Fig. 9

(a) Experimental (with fitted Gaussian beam) and (b) theoretical probe and conjugate beam sizes measured as a function of distance from the FWM amplifier (AMP1).

Tables (1)

Tables Icon

Table 1 Examples of ABCD Ray Transfer Matrices

Equations (32)

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

Ej=½A˜j exp[i(ωt-kz)]+complexconjugate,
A˜j(r, z)exp[-ikr2/2q˜j(z)],
1q˜j(z)=1Rj(z)-i λπwj2(z),j=14,
A˜4A˜1A˜2A˜3*.
1q˜4=1q˜1+1q˜2-1q˜3*,
1R4=1R1+1R2-1R3,
1w42=1w12+1w22+1w32.
q˜=Aq˜+BCq˜+D,
q˜0=q˜2+d,
q˜1=q˜0+d,
q˜2=q˜4+L,
q˜3=q˜1+L,
q˜0=L(2+i),
w0=(5Lλ/π)1/2,
R0=5L/2.
q˜2=(q˜4+L)ff-(q˜4+L),
q˜3=q˜1ff-q˜1+L.
q˜0=L{2g+i[(4g2-4g4+1)]1/2/g},
w0=λLπ1/2(4g2+1)2g2(4g2-4g4+1)1/4,
R0=L4g2+12g3.
|g|1+221/2(-1.098g1.098).
AMP1,2=1-iλ/πa201 10z1 1-1/fth01 10z1 ×1-iλ/πa201,
NRTE=1-iλ/πa201 10z1 1-iλ/πa201,
ACBDFWM=1-2/RFWM-iλ/πwFWM201=11/Q˜FWM01,
2RFWM=2R3-1R1+1R2,
1wFWM2=1w12+1w22.
11/Q˜FWM01=1-2/R301.
10L1 11/Q˜FWM01 10L1
=1+L/Q˜FWM1/Q˜FWM2L+L2/Q˜FWM1+L/Q˜FWM,
m˜=1+L/Q˜FWM
1q˜0=[(A˜+D˜)2/4-1]1/2B˜=1L (m˜2-1)1/2m˜+1.
λ˜=m˜-(m˜2-1)1/2=(2+i)/5.

Metrics