Abstract

Two compact analytical descriptions of Fresnel diffraction patterns from polygonal apertures under uniform illumination are detailed. In particular, a simple expression for the diffracted field from constituent edges is derived. These results have fundamental importance as well as specific applications, and they promise new physical insights into diffraction-related phenomena. The usefulness of the formulations is illuminated in the context of a virtual source theory that accounts for two transverse dimensions. This application permits calculation of fractal unstable-resonator modes of arbitrary order and unprecedented accuracy.

© 2006 Optical Society of America

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References

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    [CrossRef]
  2. J. Komrska, "Simple derivation of formulas for Fraunhofer diffraction at polygonal apertures," J. Opt. Soc. Am. 72, 1382-l384 (1982).
    [CrossRef]
  3. S. Ganci, "Simple derivation of formulas for Fraunhofer diffraction at polygonal apertures from Maggi--Rubinowicz transformation," J. Opt. Soc. Am. A 1, 559-561 (1984).
    [CrossRef]
  4. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980).
  5. E. Hecht, Optics, 4th ed. (Addison Wesley, 2002).
  6. G. Brooker, Modern Classical Optics (Oxford University, 2003).
  7. F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 1981).
  8. G. P. Karman and J. P. Woerdman, "Fractal structure of eigenmodes of unstable cavity lasers," Opt. Lett. 23, 1909-1911 (1998).
    [CrossRef]
  9. G. P. Karman, G. S. McDonald, G. H. C. New, and J. P. Woerdman, "Laser optics--fractal modes in unstable resonators," Nature 402, 138 (1999).
    [CrossRef]
  10. G. S. McDonald, G. P. Karman, G. H. C. New, and J. P. Woerdman, "Kaleidoscope laser," J. Opt. Soc. Am. B 17, 524-529 (2000).
    [CrossRef]
  11. G. H. C. New, M. A. Yates, J. P. Woerdman, and G. S. McDonald, "Diffractive origin of fractal resonator modes," Opt. Commun. 193, 261-266 (2001).
    [CrossRef]
  12. M. A. Yates and G. H. C. New, "Fractal dimension of unstable resonator modes," Opt. Commun. 208, 377-380 (2002).
    [CrossRef]
  13. C. M. G. Watterson, M. J. Padgett, and J. Courtial, "Classic-fractal eigenmodes of unstable canonical resonators," Opt. Commun. 223, 17-23 (2003).
    [CrossRef]
  14. G. P. Karman, G. S. McDonald, J. P. Woerdman, and G. H. C. New, "Excess-noise dependence on intracavity aperture shape," Appl. Opt. 38, 6874-6878 (1999).
    [CrossRef]
  15. G. S. McDonald, G. H. C. New, and J. P. Woerdman, "Excess noise in low Fresnel number unstable resonators," Opt. Commun. 164, 285-295 (1999).
    [CrossRef]
  16. M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, "Influence of transverse resonator symmetry on excess noise," Opt. Commun. 137, 303-307 (1997).
    [CrossRef]
  17. M. V. Berry, "Mode degeneracies and the Petermann excess-noise factor for unstable lasers," J. Mod. Opt. 50, 63-81 (2003).
  18. M. P. Silverman and W. Strange, "The Newton two-knife experiment: Intricacies of wedge diffraction," Am. J. Phys. 64, 773-787 (1996).
    [CrossRef]
  19. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Wiley-Interscience, l972).
  20. J. H. Hannay, "Fresnel diffraction as an aperture edge integral," J. Mod. Opt. 47, 121-124 (2000).
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    [CrossRef]
  22. S. M. Wang, Q. Lin, L. P. Yu, and X. L. Xu, "Fresnel number of a regular polygon and slit," Appl. Opt. 39, 3453-3455 (2000).
    [CrossRef]
  23. A. E. Siegman, Lasers (University Science Books, 1986).
  24. K. D. Mielenz, "Computation of Fresnel integrals," J. Res. Natl. Inst. Stand. Technol. 102, 363-365 (1997).
  25. K. D. Mielenz, "Computation of Fresnel integrals. II," J. Res. Natl. Inst. Stand. Technol. 105, 589-590 (2000).
  26. G. S. McDonald, G. H. C. New, and J. P. Woerdman, "The two-dimensional virtual source method," presented at the 14th National Quantum Electronics Conference, Manchester, UK, September 6-9, 1999.
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    [CrossRef] [PubMed]
  28. W. H. Southwell, "Unstable-resonator-mode derivation using virtual-source theory," J. Opt. Soc. Am. A 3, 1885-1891 (1986).
    [CrossRef]
  29. M. Berry, C. Storm, and W. van Saarloos, "Theory of unstable laser modes: edge waves and fractality," Opt. Commun. 197, 393-402 (2001).
    [CrossRef]
  30. M. A. Yates, G. H. C. New, and T. Albaho, "Calculating higher-order modes of one-dimensional unstable laser resonators," J. Mod. Opt. 51, 657-667 (2004).
  31. A. G. Fox and T. Li, "Resonator modes in a maser interferometer," Bell Syst. Tech. J. 40, 453-488 (1961).
  32. M. V. Berry, "Fractal modes of unstable lasers with polygonal and circular mirrors," Opt. Commun. 200, 321-330 (2001).
    [CrossRef]
  33. J. A. Loaiza, E. R. Eliel, and J. P. Woerdman, "Experimental observation of fractal modes in unstable optical resonators," arXiv:physics/0304046 vl, 11 Apr, 2003, (http://arxiv.org/PSlowbarcache/physics/pdf/0304/0304046.pdf).
  34. K. Petermann, "Calculated spontaneous emission factor for double-heterostructure injection-lasers with gain-induced waveguiding," IEEE J. Quantum Electron. QE-15, 566-570 (1979).
    [CrossRef]
  35. G. H. C. New, "The origin of excess noise," J. Mod. Opt. 42, 799-810 (1995).
    [CrossRef]

2004 (1)

M. A. Yates, G. H. C. New, and T. Albaho, "Calculating higher-order modes of one-dimensional unstable laser resonators," J. Mod. Opt. 51, 657-667 (2004).

2003 (2)

C. M. G. Watterson, M. J. Padgett, and J. Courtial, "Classic-fractal eigenmodes of unstable canonical resonators," Opt. Commun. 223, 17-23 (2003).
[CrossRef]

M. V. Berry, "Mode degeneracies and the Petermann excess-noise factor for unstable lasers," J. Mod. Opt. 50, 63-81 (2003).

2002 (1)

M. A. Yates and G. H. C. New, "Fractal dimension of unstable resonator modes," Opt. Commun. 208, 377-380 (2002).
[CrossRef]

2001 (3)

G. H. C. New, M. A. Yates, J. P. Woerdman, and G. S. McDonald, "Diffractive origin of fractal resonator modes," Opt. Commun. 193, 261-266 (2001).
[CrossRef]

M. Berry, C. Storm, and W. van Saarloos, "Theory of unstable laser modes: edge waves and fractality," Opt. Commun. 197, 393-402 (2001).
[CrossRef]

M. V. Berry, "Fractal modes of unstable lasers with polygonal and circular mirrors," Opt. Commun. 200, 321-330 (2001).
[CrossRef]

2000 (4)

G. S. McDonald, G. P. Karman, G. H. C. New, and J. P. Woerdman, "Kaleidoscope laser," J. Opt. Soc. Am. B 17, 524-529 (2000).
[CrossRef]

K. D. Mielenz, "Computation of Fresnel integrals. II," J. Res. Natl. Inst. Stand. Technol. 105, 589-590 (2000).

J. H. Hannay, "Fresnel diffraction as an aperture edge integral," J. Mod. Opt. 47, 121-124 (2000).

S. M. Wang, Q. Lin, L. P. Yu, and X. L. Xu, "Fresnel number of a regular polygon and slit," Appl. Opt. 39, 3453-3455 (2000).
[CrossRef]

1999 (3)

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

G. P. Karman, G. S. McDonald, G. H. C. New, and J. P. Woerdman, "Laser optics--fractal modes in unstable resonators," Nature 402, 138 (1999).
[CrossRef]

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

1998 (1)

1997 (2)

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, "Influence of transverse resonator symmetry on excess noise," Opt. Commun. 137, 303-307 (1997).
[CrossRef]

K. D. Mielenz, "Computation of Fresnel integrals," J. Res. Natl. Inst. Stand. Technol. 102, 363-365 (1997).

1996 (1)

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

1995 (1)

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

1986 (1)

1985 (1)

1984 (1)

1982 (1)

1981 (1)

1979 (1)

K. Petermann, "Calculated spontaneous emission factor for double-heterostructure injection-lasers with gain-induced waveguiding," IEEE J. Quantum Electron. QE-15, 566-570 (1979).
[CrossRef]

1974 (1)

1961 (1)

A. G. Fox and T. Li, "Resonator modes in a maser interferometer," Bell Syst. Tech. J. 40, 453-488 (1961).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Wiley-Interscience, l972).

Albaho, T.

M. A. Yates, G. H. C. New, and T. Albaho, "Calculating higher-order modes of one-dimensional unstable laser resonators," J. Mod. Opt. 51, 657-667 (2004).

Asvestas, J. S.

Berry, M.

M. Berry, C. Storm, and W. van Saarloos, "Theory of unstable laser modes: edge waves and fractality," Opt. Commun. 197, 393-402 (2001).
[CrossRef]

Berry, M. V.

M. V. Berry, "Mode degeneracies and the Petermann excess-noise factor for unstable lasers," J. Mod. Opt. 50, 63-81 (2003).

M. V. Berry, "Fractal modes of unstable lasers with polygonal and circular mirrors," Opt. Commun. 200, 321-330 (2001).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980).

Brooker, G.

G. Brooker, Modern Classical Optics (Oxford University, 2003).

Courtial, J.

C. M. G. Watterson, M. J. Padgett, and J. Courtial, "Classic-fractal eigenmodes of unstable canonical resonators," Opt. Commun. 223, 17-23 (2003).
[CrossRef]

Eliel, E. R.

J. A. Loaiza, E. R. Eliel, and J. P. Woerdman, "Experimental observation of fractal modes in unstable optical resonators," arXiv:physics/0304046 vl, 11 Apr, 2003, (http://arxiv.org/PSlowbarcache/physics/pdf/0304/0304046.pdf).

Fox, A. G.

A. G. Fox and T. Li, "Resonator modes in a maser interferometer," Bell Syst. Tech. J. 40, 453-488 (1961).

Ganci, S.

Hannay, J. H.

J. H. Hannay, "Fresnel diffraction as an aperture edge integral," J. Mod. Opt. 47, 121-124 (2000).

Hecht, E.

E. Hecht, Optics, 4th ed. (Addison Wesley, 2002).

Jenkins, F. A.

F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 1981).

Karman, G. P.

Komrska, J.

Li, T.

A. G. Fox and T. Li, "Resonator modes in a maser interferometer," Bell Syst. Tech. J. 40, 453-488 (1961).

Lin, Q.

Lindberg, Å. M.

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, "Influence of transverse resonator symmetry on excess noise," Opt. Commun. 137, 303-307 (1997).
[CrossRef]

Loaiza, J. A.

J. A. Loaiza, E. R. Eliel, and J. P. Woerdman, "Experimental observation of fractal modes in unstable optical resonators," arXiv:physics/0304046 vl, 11 Apr, 2003, (http://arxiv.org/PSlowbarcache/physics/pdf/0304/0304046.pdf).

Marsh, J. S.

McDonald, G. S.

G. H. C. New, M. A. Yates, J. P. Woerdman, and G. S. McDonald, "Diffractive origin of fractal resonator modes," Opt. Commun. 193, 261-266 (2001).
[CrossRef]

G. S. McDonald, G. P. Karman, G. H. C. New, and J. P. Woerdman, "Kaleidoscope laser," J. Opt. Soc. Am. B 17, 524-529 (2000).
[CrossRef]

G. S. McDonald, G. H. C. New, and J. P. Woerdman, "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 intracavity aperture shape," Appl. Opt. 38, 6874-6878 (1999).
[CrossRef]

G. P. Karman, G. S. McDonald, G. H. C. New, and J. P. Woerdman, "Laser optics--fractal modes in unstable resonators," Nature 402, 138 (1999).
[CrossRef]

G. S. McDonald, G. H. C. New, and J. P. Woerdman, "The two-dimensional virtual source method," presented at the 14th National Quantum Electronics Conference, Manchester, UK, September 6-9, 1999.

Mielenz, K. D.

K. D. Mielenz, "Computation of Fresnel integrals. II," J. Res. Natl. Inst. Stand. Technol. 105, 589-590 (2000).

K. D. Mielenz, "Computation of Fresnel integrals," J. Res. Natl. Inst. Stand. Technol. 102, 363-365 (1997).

New, G. H. C.

M. A. Yates, G. H. C. New, and T. Albaho, "Calculating higher-order modes of one-dimensional unstable laser resonators," J. Mod. Opt. 51, 657-667 (2004).

M. A. Yates and G. H. C. New, "Fractal dimension of unstable resonator modes," Opt. Commun. 208, 377-380 (2002).
[CrossRef]

G. H. C. New, M. A. Yates, J. P. Woerdman, and G. S. McDonald, "Diffractive origin of fractal resonator modes," Opt. Commun. 193, 261-266 (2001).
[CrossRef]

G. S. McDonald, G. P. Karman, G. H. C. New, and J. P. Woerdman, "Kaleidoscope laser," J. Opt. Soc. Am. B 17, 524-529 (2000).
[CrossRef]

G. S. McDonald, G. H. C. New, and J. P. Woerdman, "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 intracavity aperture shape," Appl. Opt. 38, 6874-6878 (1999).
[CrossRef]

G. P. Karman, G. S. McDonald, G. H. C. New, and J. P. Woerdman, "Laser optics--fractal modes in unstable resonators," Nature 402, 138 (1999).
[CrossRef]

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

G. S. McDonald, G. H. C. New, and J. P. Woerdman, "The two-dimensional virtual source method," presented at the 14th National Quantum Electronics Conference, Manchester, UK, September 6-9, 1999.

Padgett, M. J.

C. M. G. Watterson, M. J. Padgett, and J. Courtial, "Classic-fractal eigenmodes of unstable canonical resonators," Opt. Commun. 223, 17-23 (2003).
[CrossRef]

Petermann, K.

K. Petermann, "Calculated spontaneous emission factor for double-heterostructure injection-lasers with gain-induced waveguiding," IEEE J. Quantum Electron. QE-15, 566-570 (1979).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

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]

Smith, R. C.

Southwell, W. H.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Wiley-Interscience, l972).

Storm, C.

M. Berry, C. Storm, and W. van Saarloos, "Theory of unstable laser modes: edge waves and fractality," Opt. Commun. 197, 393-402 (2001).
[CrossRef]

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, Å. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, "Influence of transverse resonator symmetry on excess noise," Opt. Commun. 137, 303-307 (1997).
[CrossRef]

van Eijkelenborg, M. A.

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, "Influence of transverse resonator symmetry on excess noise," Opt. Commun. 137, 303-307 (1997).
[CrossRef]

van Saarloos, W.

M. Berry, C. Storm, and W. van Saarloos, "Theory of unstable laser modes: edge waves and fractality," Opt. Commun. 197, 393-402 (2001).
[CrossRef]

Wang, S. M.

Watterson, C. M. G.

C. M. G. Watterson, M. J. Padgett, and J. Courtial, "Classic-fractal eigenmodes of unstable canonical resonators," Opt. Commun. 223, 17-23 (2003).
[CrossRef]

White, H. E.

F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 1981).

Woerdman, J. P.

G. H. C. New, M. A. Yates, J. P. Woerdman, and G. S. McDonald, "Diffractive origin of fractal resonator modes," Opt. Commun. 193, 261-266 (2001).
[CrossRef]

G. S. McDonald, G. P. Karman, G. H. C. New, and J. P. Woerdman, "Kaleidoscope laser," J. Opt. Soc. Am. B 17, 524-529 (2000).
[CrossRef]

G. S. McDonald, G. H. C. New, and J. P. Woerdman, "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 intracavity aperture shape," Appl. Opt. 38, 6874-6878 (1999).
[CrossRef]

G. P. Karman, G. S. McDonald, G. H. C. New, and J. P. Woerdman, "Laser optics--fractal modes in unstable resonators," Nature 402, 138 (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, Å. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, "Influence of transverse resonator symmetry on excess noise," Opt. Commun. 137, 303-307 (1997).
[CrossRef]

G. S. McDonald, G. H. C. New, and J. P. Woerdman, "The two-dimensional virtual source method," presented at the 14th National Quantum Electronics Conference, Manchester, UK, September 6-9, 1999.

J. A. Loaiza, E. R. Eliel, and J. P. Woerdman, "Experimental observation of fractal modes in unstable optical resonators," arXiv:physics/0304046 vl, 11 Apr, 2003, (http://arxiv.org/PSlowbarcache/physics/pdf/0304/0304046.pdf).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980).

Xu, X. L.

Yates, M. A.

M. A. Yates, G. H. C. New, and T. Albaho, "Calculating higher-order modes of one-dimensional unstable laser resonators," J. Mod. Opt. 51, 657-667 (2004).

M. A. Yates and G. H. C. New, "Fractal dimension of unstable resonator modes," Opt. Commun. 208, 377-380 (2002).
[CrossRef]

G. H. C. New, M. A. Yates, J. P. Woerdman, and G. S. McDonald, "Diffractive origin of fractal resonator modes," Opt. Commun. 193, 261-266 (2001).
[CrossRef]

Yu, L. P.

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)

Bell Syst. Tech. J. (1)

A. G. Fox and T. Li, "Resonator modes in a maser interferometer," Bell Syst. Tech. J. 40, 453-488 (1961).

IEEE J. Quantum Electron. (1)

K. Petermann, "Calculated spontaneous emission factor for double-heterostructure injection-lasers with gain-induced waveguiding," IEEE J. Quantum Electron. QE-15, 566-570 (1979).
[CrossRef]

J. Mod. Opt. (4)

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

M. A. Yates, G. H. C. New, and T. Albaho, "Calculating higher-order modes of one-dimensional unstable laser resonators," J. Mod. Opt. 51, 657-667 (2004).

J. H. Hannay, "Fresnel diffraction as an aperture edge integral," J. Mod. Opt. 47, 121-124 (2000).

M. V. Berry, "Mode degeneracies and the Petermann excess-noise factor for unstable lasers," J. Mod. Opt. 50, 63-81 (2003).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

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

J. Res. Natl. Inst. Stand. Technol. (2)

K. D. Mielenz, "Computation of Fresnel integrals," J. Res. Natl. Inst. Stand. Technol. 102, 363-365 (1997).

K. D. Mielenz, "Computation of Fresnel integrals. II," J. Res. Natl. Inst. Stand. Technol. 105, 589-590 (2000).

Nature (1)

G. P. Karman, G. S. McDonald, G. H. C. New, and J. P. Woerdman, "Laser optics--fractal modes in unstable resonators," Nature 402, 138 (1999).
[CrossRef]

Opt. Commun. (7)

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

M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, "Influence of transverse resonator symmetry on excess noise," Opt. Commun. 137, 303-307 (1997).
[CrossRef]

M. Berry, C. Storm, and W. van Saarloos, "Theory of unstable laser modes: edge waves and fractality," Opt. Commun. 197, 393-402 (2001).
[CrossRef]

G. H. C. New, M. A. Yates, J. P. Woerdman, and G. S. McDonald, "Diffractive origin of fractal resonator modes," Opt. Commun. 193, 261-266 (2001).
[CrossRef]

M. A. Yates and G. H. C. New, "Fractal dimension of unstable resonator modes," Opt. Commun. 208, 377-380 (2002).
[CrossRef]

C. M. G. Watterson, M. J. Padgett, and J. Courtial, "Classic-fractal eigenmodes of unstable canonical resonators," Opt. Commun. 223, 17-23 (2003).
[CrossRef]

M. V. Berry, "Fractal modes of unstable lasers with polygonal and circular mirrors," Opt. Commun. 200, 321-330 (2001).
[CrossRef]

Opt. Lett. (2)

Other (8)

J. A. Loaiza, E. R. Eliel, and J. P. Woerdman, "Experimental observation of fractal modes in unstable optical resonators," arXiv:physics/0304046 vl, 11 Apr, 2003, (http://arxiv.org/PSlowbarcache/physics/pdf/0304/0304046.pdf).

A. E. Siegman, Lasers (University Science Books, 1986).

G. S. McDonald, G. H. C. New, and J. P. Woerdman, "The two-dimensional virtual source method," presented at the 14th National Quantum Electronics Conference, Manchester, UK, September 6-9, 1999.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Wiley-Interscience, l972).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980).

E. Hecht, Optics, 4th ed. (Addison Wesley, 2002).

G. Brooker, Modern Classical Optics (Oxford University, 2003).

F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 1981).

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram illustrating the coordinate system used to describe diffraction patterns from an aperture Ω illuminated by a plane wave of amplitude U 0 .

Fig. 2
Fig. 2

Schematic diagram of an aperture of arbitrary shape in the relative coordinates ( u , v ) .

Fig. 3
Fig. 3

Two schemes for dividing a regular polygon into subapertures: (a) a triangle ensemble, and (b) a combination of other shapes.

Fig. 4
Fig. 4

Geometrical constructs used in the formulation of the line-integral method.

Fig. 5
Fig. 5

Fresnel patterns (normalized intensity distribution, defined by U ( x , y ) 2 U 0 2 ) from (a) triangle, (b) pentagon, (c) hexagon, and (d) decahedron apertures.

Fig. 6
Fig. 6

Fresnel patterns from a triangular aperture with different Fresnel number F e f f . (a) F e f f = 3.06 , L = 200 mm ; (b) F e f f = 3.89 , L = 150 mm ; (c) F e f f = 7.23 , L = 75 mm ; (d) F e f f = 10.56 , L = 50 mm . (In each case, R = 1 mm and λ = 0.5 μ m ).

Fig. 7
Fig. 7

Lowest-order mode of an unstable resonator with triangular aperture ( M = 4 , N e q = 7.4604 , R = 1 mm ).

Fig. 8
Fig. 8

Lowest-order modes of unstable resonators ( M = 1.5 , R = 1 mm ) with (a) triangular ( N e q = 12.5 ) , (b) pentagonal ( N e q = 32.725 ) , (c) hexagonal ( N e q = 37.5 ) , and (d) decahedral ( N e q = 45.225 ) apertures. Note that a redefinition of a to be the radius of the circumcircle of each polygon yields a value of 50 for N e q in each of these four configurations.

Equations (37)

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U ( x , y ) = k U 0 2 π i L Ω d ξ d η exp { i k 2 L [ ( x ξ ) 2 + ( y η ) 2 ] } ,
U ( x , y ) = ε U 0 + E ( x , y ) .
U ( x , y ) = U 0 ( 1 + i ) 2 Ω d u d v exp [ i π 2 ( u 2 + v 2 ) ] .
S ( χ 1 , χ 2 ) = 1 + 1 ( 1 + i ) χ 2 χ 1 d v exp ( i π 2 v 2 ) .
S ( χ 1 , χ 2 ) = D ( χ 1 ) + D ( χ 2 ) ,
D ( χ j ) = i 1 + i ϕ * ( χ j ) exp ( i π 2 χ j 2 ) ,
ϕ ( χ j ) f ( χ j ) + i g ( χ j ) .
c ( χ j ) = 1 2 + f ( χ j ) sin ( π 2 χ j 2 ) g ( χ j ) cos ( π 2 χ j 2 ) ,
s ( χ j ) = 1 2 f ( χ j ) cos ( π 2 χ j 2 ) g ( χ j ) sin ( π 2 χ j 2 ) ,
f ( χ j ) = 1 + 0.926 χ j 2 + 1.792 χ j + 3.104 χ j 2 ,
g ( χ j ) = 1 2 + 4.142 χ j + 3.492 χ j 2 + 6.67 χ j 3 .
f ( χ j ) = f ( χ j ) 2 [ c ( χ j ) sin ( π 2 χ j 2 ) s ( χ j ) cos ( π 2 χ j 2 ) ] ,
g ( χ j ) = g ( χ j ) + 2 [ c ( χ j ) cos ( π 2 χ j 2 ) + s ( χ j ) sin ( π 2 χ j 2 ) ] ,
U ( x , y ) = U 0 [ 1 + S ( u 2 , u 1 ) + 1 1 + i u 1 u 2 d u S [ v 1 ( u ) , v 2 ( u ) ] exp ( i π 2 u 2 ) ] ,
U ( x , y ) = U 0 ε + E ( x , y ) ,
E ( x , y ) j = 1 N E j ( x , y ) ,
E ( x , y ) = U 0 [ [ 1 ε + S ( x , x A ) ] + 1 1 + i x A x d u S ( w 1 , w 2 ) exp ( i π 2 u 2 ) ] ,
w 1 ( u , x , y , θ ) = y ( u x ) tan θ ,
w 2 ( u , x , y , θ ) = y + ( u x ) tan θ ,
U ( p ) = k U 0 2 π i L Ω d 2 q exp ( i k 2 L p q 2 ) .
U ( p ) = U 0 [ ε 1 2 π σ exp ( i k 2 L q p 2 ) q p q p 2 n d l ] ,
U ( p ) = U 0 [ ε 1 2 π j = 1 N I j ( q j , t j ) ] ,
I j ( q j , t j ) = 0 L j exp ( i k 2 L q j + t j l p 2 ) q j + t j l p q j + t j l p 2 n j d l .
I j ( q j , t j ) = ( q j p ) n j exp ( i k 2 L ( q j p ) n j 2 ) ( q j p ) t j L j + ( q j p ) t j d l l 2 + ( q j p ) n j 2 exp ( i k 2 L l 2 ) .
E ( p ) = U 0 2 π j = 1 N I j ( q j , t j ) .
q j = R cos ( φ 0 + 2 π j N ) + i R sin ( φ 0 + 2 π j N ) ,
p m = r cos ( θ 0 + 2 π m N ) + i r sin ( θ 0 + 2 π m N ) ,
( q j p m ) n j = ( q j + 1 p m + 1 ) n j + 1 ,
( q j p m ) t j = ( q j + 1 p m + 1 ) t j + 1 ,
U ( p m ) = U ( p m + 1 ) ,
F e f f = a 2 λ L + 1 f ( N ) ,
f ( N ) = 0.30618 N 2 0.10533 N 0.68095 .
h k h ( ξ M k , η M k ) = 0 ,
Z k = B M M 2 k 1 M 2 1 ,
V ( x , y ) = e 0 [ E N s + 1 ( x c , y c ) α N s ( α 1 ) + k = 1 N s α k E k ( x , y ) ] ,
α N s + 1 + k = 0 N s [ E k ( x c , y c ) E k + 1 ( x c , y c ) ] α N s k = 0 ,
N e q = ( a 2 λ B ) ( M 2 1 2 M )

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