Abstract

We present a new approach for the calculation of resonator mirror shapes based on a simulated annealing optimization algorithm. Compared with the possibilities of conventional resonators with spherical mirrors, the method presented permits a large mode volume of the stationary field distribution of the oscillating lowest-loss mode and an increase in its loss discrimination against the remaining ensemble of other modes at the same time. We present results for circular symmetric graded-phase mirrors that yield a difference of 30% between the diffraction losses of the fundamental mode and of other evaluated higher modes.

© 1996 Optical Society of America

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References

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  1. M. Lax, C. E. Greninger, W. H. Louisell, W. B. McKnight, J. Opt. Soc. Am. 65, 642 (1975).
    [CrossRef]
  2. C. Paré, L. Gagnon, P. A. Bélanger, Phys. Rev. A 46, 4150 (1992).
    [CrossRef] [PubMed]
  3. R. Van Neste, C. Paré, R. L. Lachance, P. A. Bélanger, IEEE J. Quantum Electron. 30, 2663 (1994).
    [CrossRef]
  4. J. R. Leger, D. Chen, Z. Wang, Opt. Lett. 19, 108 (1994).
    [CrossRef] [PubMed]
  5. J. R. Leger, D. Chen, G. Mowry, Appl. Opt. 34, 2498 (1995).
    [CrossRef] [PubMed]
  6. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 22, p. 858.
  7. F. Laeri, J. Opt. Soc. Am. B 7, 2169 (1990).
    [CrossRef]
  8. S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, Science 220, 671 (1983).
    [CrossRef] [PubMed]
  9. Ch. W. Groetsch, Inverse Problems in the Mathematical Sciences (Vieweg, Braunschweig, 1993).
  10. A. E. Siegman, E. A. Sziklas, Appl. Opt. 13, 2775 (1974).
    [CrossRef] [PubMed]
  11. C. Palma, V. Bagini, Opt. Commun. 111, 6 (1994).
    [CrossRef]

1995 (1)

1994 (3)

R. Van Neste, C. Paré, R. L. Lachance, P. A. Bélanger, IEEE J. Quantum Electron. 30, 2663 (1994).
[CrossRef]

J. R. Leger, D. Chen, Z. Wang, Opt. Lett. 19, 108 (1994).
[CrossRef] [PubMed]

C. Palma, V. Bagini, Opt. Commun. 111, 6 (1994).
[CrossRef]

1992 (1)

C. Paré, L. Gagnon, P. A. Bélanger, Phys. Rev. A 46, 4150 (1992).
[CrossRef] [PubMed]

1990 (1)

1983 (1)

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, Science 220, 671 (1983).
[CrossRef] [PubMed]

1975 (1)

1974 (1)

Bagini, V.

C. Palma, V. Bagini, Opt. Commun. 111, 6 (1994).
[CrossRef]

Bélanger, P. A.

R. Van Neste, C. Paré, R. L. Lachance, P. A. Bélanger, IEEE J. Quantum Electron. 30, 2663 (1994).
[CrossRef]

C. Paré, L. Gagnon, P. A. Bélanger, Phys. Rev. A 46, 4150 (1992).
[CrossRef] [PubMed]

Chen, D.

Gagnon, L.

C. Paré, L. Gagnon, P. A. Bélanger, Phys. Rev. A 46, 4150 (1992).
[CrossRef] [PubMed]

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, Science 220, 671 (1983).
[CrossRef] [PubMed]

Greninger, C. E.

Groetsch, Ch. W.

Ch. W. Groetsch, Inverse Problems in the Mathematical Sciences (Vieweg, Braunschweig, 1993).

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, Science 220, 671 (1983).
[CrossRef] [PubMed]

Lachance, R. L.

R. Van Neste, C. Paré, R. L. Lachance, P. A. Bélanger, IEEE J. Quantum Electron. 30, 2663 (1994).
[CrossRef]

Laeri, F.

Lax, M.

Leger, J. R.

Louisell, W. H.

McKnight, W. B.

Mowry, G.

Palma, C.

C. Palma, V. Bagini, Opt. Commun. 111, 6 (1994).
[CrossRef]

Paré, C.

R. Van Neste, C. Paré, R. L. Lachance, P. A. Bélanger, IEEE J. Quantum Electron. 30, 2663 (1994).
[CrossRef]

C. Paré, L. Gagnon, P. A. Bélanger, Phys. Rev. A 46, 4150 (1992).
[CrossRef] [PubMed]

Siegman, A. E.

A. E. Siegman, E. A. Sziklas, Appl. Opt. 13, 2775 (1974).
[CrossRef] [PubMed]

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 22, p. 858.

Sziklas, E. A.

Van Neste, R.

R. Van Neste, C. Paré, R. L. Lachance, P. A. Bélanger, IEEE J. Quantum Electron. 30, 2663 (1994).
[CrossRef]

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, Science 220, 671 (1983).
[CrossRef] [PubMed]

Wang, Z.

Appl. Opt. (2)

IEEE J. Quantum Electron. (1)

R. Van Neste, C. Paré, R. L. Lachance, P. A. Bélanger, IEEE J. Quantum Electron. 30, 2663 (1994).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (1)

C. Palma, V. Bagini, Opt. Commun. 111, 6 (1994).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

C. Paré, L. Gagnon, P. A. Bélanger, Phys. Rev. A 46, 4150 (1992).
[CrossRef] [PubMed]

Science (1)

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, Science 220, 671 (1983).
[CrossRef] [PubMed]

Other (2)

Ch. W. Groetsch, Inverse Problems in the Mathematical Sciences (Vieweg, Braunschweig, 1993).

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 22, p. 858.

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Figures (4)

Fig. 1
Fig. 1

Comparison of the mirror profiles obtained by the PC propagation method (dotted curves) and by SA optimization (solid curves). The upper plot shows the radial profiles of the graded-phase rear mirror, and the lower plot is the power density distribution in the plane of the output coupler.

Fig. 2
Fig. 2

Results with a resonator with a 20-m radius-of-curvature output coupler, a 1-in. (2.54-cm) mirror aperture, and a 5.8 Fresnel number. The upper plot is the radial profile of the graded-phase rear mirror, and the lower plot is the power density distribution in the plane of the output coupler. SA optimization with enforced loss increase for all modes except the fundamental is shown by the solid curves, and the PC method is shown by the dotted curves.

Fig. 3
Fig. 3

Dependence of the loss factor (squared eigenvalue modulus) of the evaluated (0, 0) mode (solid curves), the (0, 1)–(0, 19) modes (dotted curves), and the (1, 1)–(1, 19) modes on additional refractive power (which in the annealing simulation was lumped into a mirror). The resonator is the same as in Fig. 2.

Fig. 4
Fig. 4

Same as Fig. 2, except with a resonator with a 30-m radius-of-curvature output coupler, a 1-in. mirror aperture, and a 5.8 Fresnel number.

Equations (6)

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γ p l ψ p l ( r , θ ) = K p l ( r , θ , r , θ ) ψ p l ( r , θ ) r d θ d r .
η p l = 1 γ p l 2 ,
ɛ f = α ψ 00 β ψ ref ,
ɛ d ( 1 ) = c 00 ( 1 γ 00 2 ) + c 10 γ 10 2 + + c P L γ P L 2 ,
ɛ d ( 2 ) = c 00 γ 00 2 + c 10 1 γ 10 2 + + c P L 1 γ P L 2 ,
ɛ d ( 3 ) = c 00 cos ( π 2 γ 00 2 ) + c 10 sin ( π 2 γ 10 2 ) + + c P L sin ( π 2 γ P L 2 ) .

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