Abstract

We show that the eigenmodes of unstable-cavity lasers have fractal structure, in contrast with the well-known stable-cavity eigenmodes. As with all fractals, the dynamic range over which self-similarity holds is limited; in this case the range is set by diffraction, i.e., by the Fresnel number of the resonator. We determine the fractal dimension of the mode profiles and show that it is related to the aperture shape.

© 1998 Optical Society of America

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References

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    [Crossref] [PubMed]
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  5. To calculate the eigenmodes we used the software package paraxia, distributed by Sciopt Enterprises, San Jose, Calif. 95120. The package employs virtual-source theory.1,2
  6. D. Stoyan and H. Stoyan, Fractals, Random Shapes and Point Fields (Wiley, New York, 1992), Sec. 5.3.
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]

1997 (1)

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, Opt. Commun. 137, 303 (1997).
[Crossref]

1996 (4)

M. V. Berry, J. Phys. A 29, 6617 (1996).
[Crossref]

M. V. Berry and S. Klein, J. Mod. Opt. 43, 2139 (1996).
[Crossref]

Y. J. Cheng, C. G. Fanning, and A. E. Siegman, Phys. Rev. Lett. 77, 627 (1996).
[Crossref] [PubMed]

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, Phys. Rev. Lett. 77, 4314 (1996).
[Crossref] [PubMed]

1995 (1)

G. H. C. New, J. Mod. Opt. 42, 799 (1995).
[Crossref]

1986 (1)

1981 (1)

1973 (1)

Berry, M. V.

M. V. Berry, J. Phys. A 29, 6617 (1996).
[Crossref]

M. V. Berry and S. Klein, J. Mod. Opt. 43, 2139 (1996).
[Crossref]

Cheng, Y. J.

Y. J. Cheng, C. G. Fanning, and A. E. Siegman, Phys. Rev. Lett. 77, 627 (1996).
[Crossref] [PubMed]

Fanning, C. G.

Y. J. Cheng, C. G. Fanning, and A. E. Siegman, Phys. Rev. Lett. 77, 627 (1996).
[Crossref] [PubMed]

Horwitz, P.

Klein, S.

M. V. Berry and S. Klein, J. Mod. Opt. 43, 2139 (1996).
[Crossref]

Lindberg, Å. M.

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, Opt. Commun. 137, 303 (1997).
[Crossref]

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, Phys. Rev. Lett. 77, 4314 (1996).
[Crossref] [PubMed]

New, G. H. C.

G. H. C. New, J. Mod. Opt. 42, 799 (1995).
[Crossref]

Siegman, A. E.

Y. J. Cheng, C. G. Fanning, and A. E. Siegman, Phys. Rev. Lett. 77, 627 (1996).
[Crossref] [PubMed]

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

Southwell, W. H.

Stoyan, D.

D. Stoyan and H. Stoyan, Fractals, Random Shapes and Point Fields (Wiley, New York, 1992), Sec. 5.3.

Stoyan, H.

D. Stoyan and H. Stoyan, Fractals, Random Shapes and Point Fields (Wiley, New York, 1992), Sec. 5.3.

Thijssen, M. S.

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, Opt. Commun. 137, 303 (1997).
[Crossref]

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, Phys. Rev. Lett. 77, 4314 (1996).
[Crossref] [PubMed]

van Eijkelenborg, M. A.

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, Opt. Commun. 137, 303 (1997).
[Crossref]

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, Phys. Rev. Lett. 77, 4314 (1996).
[Crossref] [PubMed]

Woerdman, J. P.

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, Opt. Commun. 137, 303 (1997).
[Crossref]

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, Phys. Rev. Lett. 77, 4314 (1996).
[Crossref] [PubMed]

J. Mod. Opt. (2)

M. V. Berry and S. Klein, J. Mod. Opt. 43, 2139 (1996).
[Crossref]

G. H. C. New, J. Mod. Opt. 42, 799 (1995).
[Crossref]

J. Opt. Soc. Am. (1)

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

J. Phys. A (1)

M. V. Berry, J. Phys. A 29, 6617 (1996).
[Crossref]

Opt. Commun. (1)

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, Opt. Commun. 137, 303 (1997).
[Crossref]

Opt. Lett. (1)

Phys. Rev. Lett. (2)

Y. J. Cheng, C. G. Fanning, and A. E. Siegman, Phys. Rev. Lett. 77, 627 (1996).
[Crossref] [PubMed]

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, Phys. Rev. Lett. 77, 4314 (1996).
[Crossref] [PubMed]

Other (3)

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

To calculate the eigenmodes we used the software package paraxia, distributed by Sciopt Enterprises, San Jose, Calif. 95120. The package employs virtual-source theory.1,2

D. Stoyan and H. Stoyan, Fractals, Random Shapes and Point Fields (Wiley, New York, 1992), Sec. 5.3.

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Figures (4)

Fig. 1
Fig. 1

Geometry of a confocal unstable cavity; the two mirrors share a common focus. We calculated the mode profiles in the aperture plane indicated by the dashed line.

Fig. 2
Fig. 2

Intensity profiles for a strip resonator for different Fresnel numbers N. The resonator has a magnification M=2. (a) N=10, (b) N=100, and (c) N=1000. Curve (b) is shifted downward by an amount 0.35 and (c) by 0.6 to avoid overlap with curve (a).

Fig. 3
Fig. 3

Application of the box-counting method to a strip resonator, showing the number of boxes of size d needed to cover curve (c) in Fig. 2. The straight line is a fit with a slope of -1.58±0.03.

Fig. 4
Fig. 4

Application of the box-counting method to an unstable resonator with a circular aperture, M=2, and N=1000, showing the number of boxes of size d needed to cover the intensity mode profile of the lowest-loss mode. The straight line is a fit with a slope of -1.26±0.03.

Equations (2)

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N=12M-1a2λL,
D=limd0lognlog1/d.

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