Abstract

We report the first calculations of mode patterns of unstable-cavity lasers with truly two-dimensional transverse geometries. A detailed account of numerical techniques, incorporating a nonorthogonal beam-propagation method, and results for cavities with a range of transverse symmetries, such as regular polygonal and rhomboid, are presented. In view of the beautiful complexity of the eigenmodes predicted, a novel kaleidoscope laser is proposed.

© 2000 Optical Society of America

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References

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  1. A. E. Siegman, “Unstable optical resonators,” Appl. Opt. 13, 353–367 (1974).
    [CrossRef] [PubMed]
  2. J. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).
  3. A. E. Siegman, Lasers (Oxford University, London, 1986).
  4. P. Horwitz, “Asymptotic theory of unstable resonator modes,” J. Opt. Soc. Am. 63, 1528–1543 (1973).
    [CrossRef]
  5. M. A. Lauder and G. H. C. New, “Biorthogonality properties and excess noise factors of unstable optical resonators,” Opt. Commun. 31, 343–348 (1979).
  6. W. H. Southwell, “Unstable-resonator-mode derivation using virtual-source theory,” J. Opt. Soc. Am. A 3, 1885–1891 (1986).
    [CrossRef]
  7. G. H. C. New, “The origin of excess noise,” J. Mod. Opt. 42, 799–810 (1995).
    [CrossRef]
  8. M. A. Rippin and G. H. C. New, “Excess noise factors in circular unstable resonators,” J. Mod. Opt. 43, 993–1008 (1996).
    [CrossRef]
  9. M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, “Resonance of quantum noise in an unstable cavity laser,” Phys. Rev. Lett. 77, 4314–4317 (1996).
    [CrossRef] [PubMed]
  10. M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, “Unstable-resonator diffraction losses and the excess-noise factor,” Phys. Rev. A 55, 4556–4562 (1997).
    [CrossRef]
  11. M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, “Higher-order transverse modes of an unstable-cavity laser,” IEEE J. Quantum Electron. 34, 955–965 (1998).
    [CrossRef]
  12. G. S. McDonald, J. P. Woerdman, and G. H. C. New, “Excess noise in low Fresnel number unstable resonators,” Opt. Commun. 164, 285–295 (1999).
    [CrossRef]
  13. Sir Isaac Newton, Opticks, 4th ed. (London, 1730, reprinted by Dover, New York, 1952).
  14. M. P. Silverman and W. Strange, “The Newton two-knife experiment: intricacies of wedge diffraction,” Am. J. Phys. 64, 773–787 (1996).
    [CrossRef]
  15. G. S. McDonald and W. J. Firth, “Spatial grid symmetries and reduced models in the simulation of beam counterpropagation in a nonlinear medium,” J. Mod. Opt. 40, 23–32 (1993).
    [CrossRef]
  16. Y. J. Cheng, C. G. Fanning, and A. E. Siegman, “Experimental observation of a large excess quantum noise factor in the linewidth of a laser oscillator having nonorthogonal modes,” Phys. Rev. Lett. 77, 627–630 (1996).
    [CrossRef] [PubMed]
  17. G. P. Karman, G. S. McDonald, J. P. Woerdman, and G. H. C. New, “Excess-noise dependence on intra-cavity aperture shape,” Appl. Opt. 38, 6874–6878 (1999).
    [CrossRef]
  18. G. P. Karman and J. P. Woerdman, “Fractal structure of eigenmodes of unstable-cavity lasers,” Opt. Lett. 23, 1909–1911 (1998).
    [CrossRef]
  19. J. Walker, “The amateur scientist,” Sci. Am. 253, 124–130 (1985).
    [CrossRef]

1999 (2)

G. S. McDonald, J. P. Woerdman, and G. H. C. New, “Excess noise in low Fresnel number unstable resonators,” Opt. Commun. 164, 285–295 (1999).
[CrossRef]

G. P. Karman, G. S. McDonald, J. P. Woerdman, and G. H. C. New, “Excess-noise dependence on intra-cavity aperture shape,” Appl. Opt. 38, 6874–6878 (1999).
[CrossRef]

1998 (2)

G. P. Karman and J. P. Woerdman, “Fractal structure of eigenmodes of unstable-cavity lasers,” Opt. Lett. 23, 1909–1911 (1998).
[CrossRef]

M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, “Higher-order transverse modes of an unstable-cavity laser,” IEEE J. Quantum Electron. 34, 955–965 (1998).
[CrossRef]

1997 (1)

M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, “Unstable-resonator diffraction losses and the excess-noise factor,” Phys. Rev. A 55, 4556–4562 (1997).
[CrossRef]

1996 (4)

M. A. Rippin and G. H. C. New, “Excess noise factors in circular unstable resonators,” J. Mod. Opt. 43, 993–1008 (1996).
[CrossRef]

M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, “Resonance of quantum noise in an unstable cavity laser,” Phys. Rev. Lett. 77, 4314–4317 (1996).
[CrossRef] [PubMed]

M. P. Silverman and W. Strange, “The Newton two-knife experiment: intricacies of wedge diffraction,” Am. J. Phys. 64, 773–787 (1996).
[CrossRef]

Y. J. Cheng, C. G. Fanning, and A. E. Siegman, “Experimental observation of a large excess quantum noise factor in the linewidth of a laser oscillator having nonorthogonal modes,” Phys. Rev. Lett. 77, 627–630 (1996).
[CrossRef] [PubMed]

1995 (1)

G. H. C. New, “The origin of excess noise,” J. Mod. Opt. 42, 799–810 (1995).
[CrossRef]

1993 (1)

G. S. McDonald and W. J. Firth, “Spatial grid symmetries and reduced models in the simulation of beam counterpropagation in a nonlinear medium,” J. Mod. Opt. 40, 23–32 (1993).
[CrossRef]

1986 (1)

1985 (1)

J. Walker, “The amateur scientist,” Sci. Am. 253, 124–130 (1985).
[CrossRef]

1979 (1)

M. A. Lauder and G. H. C. New, “Biorthogonality properties and excess noise factors of unstable optical resonators,” Opt. Commun. 31, 343–348 (1979).

1974 (1)

1973 (1)

Cheng, Y. J.

Y. J. Cheng, C. G. Fanning, and A. E. Siegman, “Experimental observation of a large excess quantum noise factor in the linewidth of a laser oscillator having nonorthogonal modes,” Phys. Rev. Lett. 77, 627–630 (1996).
[CrossRef] [PubMed]

Fanning, C. G.

Y. J. Cheng, C. G. Fanning, and A. E. Siegman, “Experimental observation of a large excess quantum noise factor in the linewidth of a laser oscillator having nonorthogonal modes,” Phys. Rev. Lett. 77, 627–630 (1996).
[CrossRef] [PubMed]

Firth, W. J.

G. S. McDonald and W. J. Firth, “Spatial grid symmetries and reduced models in the simulation of beam counterpropagation in a nonlinear medium,” J. Mod. Opt. 40, 23–32 (1993).
[CrossRef]

Horwitz, P.

Karman, G. P.

Lauder, M. A.

M. A. Lauder and G. H. C. New, “Biorthogonality properties and excess noise factors of unstable optical resonators,” Opt. Commun. 31, 343–348 (1979).

Lindberg, A. M.

M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, “Higher-order transverse modes of an unstable-cavity laser,” IEEE J. Quantum Electron. 34, 955–965 (1998).
[CrossRef]

M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, “Unstable-resonator diffraction losses and the excess-noise factor,” Phys. Rev. A 55, 4556–4562 (1997).
[CrossRef]

M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, “Resonance of quantum noise in an unstable cavity laser,” Phys. Rev. Lett. 77, 4314–4317 (1996).
[CrossRef] [PubMed]

McDonald, G. S.

G. P. Karman, G. S. McDonald, J. P. Woerdman, and G. H. C. New, “Excess-noise dependence on intra-cavity aperture shape,” Appl. Opt. 38, 6874–6878 (1999).
[CrossRef]

G. S. McDonald, J. P. Woerdman, and G. H. C. New, “Excess noise in low Fresnel number unstable resonators,” Opt. Commun. 164, 285–295 (1999).
[CrossRef]

G. S. McDonald and W. J. Firth, “Spatial grid symmetries and reduced models in the simulation of beam counterpropagation in a nonlinear medium,” J. Mod. Opt. 40, 23–32 (1993).
[CrossRef]

New, G. H. C.

G. S. McDonald, J. P. Woerdman, and G. H. C. New, “Excess noise in low Fresnel number unstable resonators,” Opt. Commun. 164, 285–295 (1999).
[CrossRef]

G. P. Karman, G. S. McDonald, J. P. Woerdman, and G. H. C. New, “Excess-noise dependence on intra-cavity aperture shape,” Appl. Opt. 38, 6874–6878 (1999).
[CrossRef]

M. A. Rippin and G. H. C. New, “Excess noise factors in circular unstable resonators,” J. Mod. Opt. 43, 993–1008 (1996).
[CrossRef]

G. H. C. New, “The origin of excess noise,” J. Mod. Opt. 42, 799–810 (1995).
[CrossRef]

M. A. Lauder and G. H. C. New, “Biorthogonality properties and excess noise factors of unstable optical resonators,” Opt. Commun. 31, 343–348 (1979).

Rippin, M. A.

M. A. Rippin and G. H. C. New, “Excess noise factors in circular unstable resonators,” J. Mod. Opt. 43, 993–1008 (1996).
[CrossRef]

Siegman, A. E.

Y. J. Cheng, C. G. Fanning, and A. E. Siegman, “Experimental observation of a large excess quantum noise factor in the linewidth of a laser oscillator having nonorthogonal modes,” Phys. Rev. Lett. 77, 627–630 (1996).
[CrossRef] [PubMed]

A. E. Siegman, “Unstable optical resonators,” Appl. Opt. 13, 353–367 (1974).
[CrossRef] [PubMed]

Silverman, M. P.

M. P. Silverman and W. Strange, “The Newton two-knife experiment: intricacies of wedge diffraction,” Am. J. Phys. 64, 773–787 (1996).
[CrossRef]

Southwell, W. H.

Strange, W.

M. P. Silverman and W. Strange, “The Newton two-knife experiment: intricacies of wedge diffraction,” Am. J. Phys. 64, 773–787 (1996).
[CrossRef]

Thijssen, M. S.

M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, “Higher-order transverse modes of an unstable-cavity laser,” IEEE J. Quantum Electron. 34, 955–965 (1998).
[CrossRef]

M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, “Unstable-resonator diffraction losses and the excess-noise factor,” Phys. Rev. A 55, 4556–4562 (1997).
[CrossRef]

M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, “Resonance of quantum noise in an unstable cavity laser,” Phys. Rev. Lett. 77, 4314–4317 (1996).
[CrossRef] [PubMed]

van Eijkelenborg, M. A.

M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, “Higher-order transverse modes of an unstable-cavity laser,” IEEE J. Quantum Electron. 34, 955–965 (1998).
[CrossRef]

M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, “Unstable-resonator diffraction losses and the excess-noise factor,” Phys. Rev. A 55, 4556–4562 (1997).
[CrossRef]

M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, “Resonance of quantum noise in an unstable cavity laser,” Phys. Rev. Lett. 77, 4314–4317 (1996).
[CrossRef] [PubMed]

Walker, J.

J. Walker, “The amateur scientist,” Sci. Am. 253, 124–130 (1985).
[CrossRef]

Woerdman, J. P.

G. P. Karman, G. S. McDonald, J. P. Woerdman, and G. H. C. New, “Excess-noise dependence on intra-cavity aperture shape,” Appl. Opt. 38, 6874–6878 (1999).
[CrossRef]

G. S. McDonald, J. P. Woerdman, and G. H. C. New, “Excess noise in low Fresnel number unstable resonators,” Opt. Commun. 164, 285–295 (1999).
[CrossRef]

G. P. Karman and J. P. Woerdman, “Fractal structure of eigenmodes of unstable-cavity lasers,” Opt. Lett. 23, 1909–1911 (1998).
[CrossRef]

M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, “Higher-order transverse modes of an unstable-cavity laser,” IEEE J. Quantum Electron. 34, 955–965 (1998).
[CrossRef]

M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, “Unstable-resonator diffraction losses and the excess-noise factor,” Phys. Rev. A 55, 4556–4562 (1997).
[CrossRef]

M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, “Resonance of quantum noise in an unstable cavity laser,” Phys. Rev. Lett. 77, 4314–4317 (1996).
[CrossRef] [PubMed]

Am. J. Phys. (1)

M. P. Silverman and W. Strange, “The Newton two-knife experiment: intricacies of wedge diffraction,” Am. J. Phys. 64, 773–787 (1996).
[CrossRef]

Appl. Opt. (2)

IEEE J. Quantum Electron. (1)

M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, “Higher-order transverse modes of an unstable-cavity laser,” IEEE J. Quantum Electron. 34, 955–965 (1998).
[CrossRef]

J. Mod. Opt. (3)

G. S. McDonald and W. J. Firth, “Spatial grid symmetries and reduced models in the simulation of beam counterpropagation in a nonlinear medium,” J. Mod. Opt. 40, 23–32 (1993).
[CrossRef]

G. H. C. New, “The origin of excess noise,” J. Mod. Opt. 42, 799–810 (1995).
[CrossRef]

M. A. Rippin and G. H. C. New, “Excess noise factors in circular unstable resonators,” J. Mod. Opt. 43, 993–1008 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (2)

M. A. Lauder and G. H. C. New, “Biorthogonality properties and excess noise factors of unstable optical resonators,” Opt. Commun. 31, 343–348 (1979).

G. S. McDonald, J. P. Woerdman, and G. H. C. New, “Excess noise in low Fresnel number unstable resonators,” Opt. Commun. 164, 285–295 (1999).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, “Unstable-resonator diffraction losses and the excess-noise factor,” Phys. Rev. A 55, 4556–4562 (1997).
[CrossRef]

Phys. Rev. Lett. (2)

M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, “Resonance of quantum noise in an unstable cavity laser,” Phys. Rev. Lett. 77, 4314–4317 (1996).
[CrossRef] [PubMed]

Y. J. Cheng, C. G. Fanning, and A. E. Siegman, “Experimental observation of a large excess quantum noise factor in the linewidth of a laser oscillator having nonorthogonal modes,” Phys. Rev. Lett. 77, 627–630 (1996).
[CrossRef] [PubMed]

Sci. Am. (1)

J. Walker, “The amateur scientist,” Sci. Am. 253, 124–130 (1985).
[CrossRef]

Other (3)

Sir Isaac Newton, Opticks, 4th ed. (London, 1730, reprinted by Dover, New York, 1952).

J. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).

A. E. Siegman, Lasers (Oxford University, London, 1986).

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Figures (3)

Fig. 1
Fig. 1

Bare cavity configuration of the kaleidoscope laser. (a) Schematic picture of the unstable cavity, consisting of a convex mirror (focal length f) and a concave mirror (focal length fv) separated by a distance L. The circulating field is apertured on each round trip by one of the mirrors or, equivalently, by an aperture placed directly against that mirror. (b) The transverse shapes of the apertures considered are plotted. (c) In the modeling we transform the above system to the equivalent lens guide sketched, which is a series of diverging lenses (focal length fe and a distance Le apart) and apertures adjacent to each of these lenses.

Fig. 2
Fig. 2

Orthogonal (xy) and nonorthogonal (pq) axes in the transverse plane. An arbitrary point in this space, which is labeled 1, has two pairs of coordinates, (x1, y1) and (p1, q1).

Fig. 3
Fig. 3

Cross section of the transverse-mode profiles of the kaleidoscope laser. The defining aperture of the cavity has a different shape in each column of this figure. The profiles generally exhibit increasing complexity as the equivalent Fresnel number is increased (moving downward through each column). In column 1 the aperture has the shape of an equilateral triangle (Neq=0.5, 1.5, 3.0, and 4.0). Columns 2–5 are for rhomboid (Neq=0.5, 1.0, 1.5, and 3.0), pentagonal (Neq=0.5, 1.5, 2.0, and 4.0), hexagonal (Neq=0.5, 1.5, 3.0, and 4.0), and octagonal aperturing (Neq=0.5, 1.0, 3.0, and 4.0), respectively. The colors yellow, red, green, blue, and black are used to represent different (decreasing) levels of light intensity.

Equations (10)

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

Ue1(p, q)=U(p, q)×exp[-iπβ(p2+q2+2pq cos Φ)/a2],
Ue2(p, q)=Ue1(p, q) exp[-iπβ(M-1)×(p2+q2+2pq cos Φ)/a2]
U(x, y)z=i2k2x2+2y2U(x, y),
U(p, q)z=i2k sin2 Φ×2p2+2q2-2 cos Φ pqU(p, q).
U(p, q)=--U˜(K1, K2)×exp[i(K1p+K2q)]dK1dK2,
U˜e3(K1, K2)
=U˜e2(K1, K2)
×exp-iLe2k sin2 Φ(K12+K22-2 cos Φ K1K2).
U(p)=-U˜(K)exp(iKp)dK,
U(x)=-U˜(K)exp(iKq cos Φ)exp(iKp)dK.

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