Abstract

We study the effect of nonparaxiality in a folded resonator by accurate measurements of the Gouy phase as a function of the mode number for mode numbers up to 1500. Our experimental method is based on tuning the resonator close to a frequency-degenerate point. The Gouy phase shows a nonparaxial behavior that is much stronger in the folding plane than in the perpendicular plane. Agreement with ray-tracing simulations is established, and a link with aberration theory is made.

© 2004 Optical Society of America

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References

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  1. J. Dingjan, E. Altewischer, M. P. van Exter, J. P. Woerdman, “Experimental observation of wave chaos in a conventional optical resonator,” Phys. Rev. Lett. 88, 064101 (2002).
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  6. H. Laabs, A. T. Friberg, “Nonparaxial eigenmodes of stable resonators,” IEEE J. Quantum Electron. 35, 198–207 (1999).
    [CrossRef]
  7. H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, UK, 1970).
  8. M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999).
  9. V. N. Mahajan, Optical Imaging and Aberrations (SPIE Press, Bellingham, Wash., 1998).
  10. D. L. Dickensheets, “Imaging performance of off-axis planar diffractive lenses,” J. Opt. Soc. Am. A 13, 1849–1858 (1996).
    [CrossRef]
  11. M. H. Dunn, A. I. Ferguson, “Coma compensation in off-axis laser resonators,” Opt. Commun. 20, 214–219 (1977).
    [CrossRef]
  12. M. Hercher, “The spherical Fabry–Perot interferometer,” in The Fabry–Perot Interferometer, J. M. Vaughan, ed. (Adam Hilger, Bristol, UK, 1989), pp. 185–213.

2002

J. Dingjan, E. Altewischer, M. P. van Exter, J. P. Woerdman, “Experimental observation of wave chaos in a conventional optical resonator,” Phys. Rev. Lett. 88, 064101 (2002).
[CrossRef] [PubMed]

1999

H. Laabs, A. T. Friberg, “Nonparaxial eigenmodes of stable resonators,” IEEE J. Quantum Electron. 35, 198–207 (1999).
[CrossRef]

1996

1995

1977

M. H. Dunn, A. I. Ferguson, “Coma compensation in off-axis laser resonators,” Opt. Commun. 20, 214–219 (1977).
[CrossRef]

1970

1964

Altewischer, E.

J. Dingjan, E. Altewischer, M. P. van Exter, J. P. Woerdman, “Experimental observation of wave chaos in a conventional optical resonator,” Phys. Rev. Lett. 88, 064101 (2002).
[CrossRef] [PubMed]

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999).

Buchdahl, H. A.

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, UK, 1970).

Degnan, J. J.

Dickensheets, D. L.

Dingjan, J.

J. Dingjan, E. Altewischer, M. P. van Exter, J. P. Woerdman, “Experimental observation of wave chaos in a conventional optical resonator,” Phys. Rev. Lett. 88, 064101 (2002).
[CrossRef] [PubMed]

Dunn, M. H.

M. H. Dunn, A. I. Ferguson, “Coma compensation in off-axis laser resonators,” Opt. Commun. 20, 214–219 (1977).
[CrossRef]

Ferguson, A. I.

M. H. Dunn, A. I. Ferguson, “Coma compensation in off-axis laser resonators,” Opt. Commun. 20, 214–219 (1977).
[CrossRef]

Friberg, A. T.

H. Laabs, A. T. Friberg, “Nonparaxial eigenmodes of stable resonators,” IEEE J. Quantum Electron. 35, 198–207 (1999).
[CrossRef]

Hercher, M.

M. Hercher, “The spherical Fabry–Perot interferometer,” in The Fabry–Perot Interferometer, J. M. Vaughan, ed. (Adam Hilger, Bristol, UK, 1989), pp. 185–213.

Herriot, D.

Kogelnik, H.

Kompfner, R.

Laabs, H.

H. Laabs, A. T. Friberg, “Nonparaxial eigenmodes of stable resonators,” IEEE J. Quantum Electron. 35, 198–207 (1999).
[CrossRef]

Mahajan, V. N.

V. N. Mahajan, Optical Imaging and Aberrations (SPIE Press, Bellingham, Wash., 1998).

Ramsay, I. A.

Siegman, A. E.

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

Ueda, K.

Uehara, N.

van Exter, M. P.

J. Dingjan, E. Altewischer, M. P. van Exter, J. P. Woerdman, “Experimental observation of wave chaos in a conventional optical resonator,” Phys. Rev. Lett. 88, 064101 (2002).
[CrossRef] [PubMed]

Woerdman, J. P.

J. Dingjan, E. Altewischer, M. P. van Exter, J. P. Woerdman, “Experimental observation of wave chaos in a conventional optical resonator,” Phys. Rev. Lett. 88, 064101 (2002).
[CrossRef] [PubMed]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999).

Appl. Opt.

IEEE J. Quantum Electron.

H. Laabs, A. T. Friberg, “Nonparaxial eigenmodes of stable resonators,” IEEE J. Quantum Electron. 35, 198–207 (1999).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

M. H. Dunn, A. I. Ferguson, “Coma compensation in off-axis laser resonators,” Opt. Commun. 20, 214–219 (1977).
[CrossRef]

Phys. Rev. Lett.

J. Dingjan, E. Altewischer, M. P. van Exter, J. P. Woerdman, “Experimental observation of wave chaos in a conventional optical resonator,” Phys. Rev. Lett. 88, 064101 (2002).
[CrossRef] [PubMed]

Other

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

M. Hercher, “The spherical Fabry–Perot interferometer,” in The Fabry–Perot Interferometer, J. M. Vaughan, ed. (Adam Hilger, Bristol, UK, 1989), pp. 185–213.

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, UK, 1970).

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999).

V. N. Mahajan, Optical Imaging and Aberrations (SPIE Press, Bellingham, Wash., 1998).

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Figures (8)

Fig. 1
Fig. 1

Folded three-mirror resonator and its orientation.

Fig. 2
Fig. 2

Overview of the setup, where the mirrors M1, MF, and M2 form the folded resonator: A1 and A2, lengths of resonator arms; PM, photomultiplier; L1 and L2, lenses; B1 and B2, beam splitters; M3 and M4, mirrors. The solid line indicates the fixed beam that excites the fundamental mode. The position of the other (dotted) beam on mirror M1 can be increased by rotating M3 to excite higher-order modes.

Fig. 3
Fig. 3

Spectrum for an almost eightfold degenerate cavity configuration. The modes collapse into eight clumps of peaks. The two highest peaks are due to the fundamental mode, which serves as a reference.

Fig. 4
Fig. 4

Spectral detail of the fundamental mode and the nearest clump of peaks. The almost eightfold degeneracy makes the difference in mode number between two subsequent modes equal to 8. Δνm is the distance between fundamental mode m=0 and mode m.

Fig. 5
Fig. 5

Δθm versus mode number m for various values of N. The top three curves are measured in the xz principal plane, the bottom three in the y principal plane. For easy comparison all curves have been vertically shifted by an arbitrary amount to bring them closer to each other; N- indicates an originally negative-valued Δθm.

Fig. 6
Fig. 6

Square root of the mode number versus off-axis distance of injection on mirror M1.

Fig. 7
Fig. 7

Ray-tracing calculations of Δθ versus square of the off-axis distance (xz principal plane) of injection for N=8 (triangles) and 9 (circles) and in the y principal plane for N=9 (squares).

Fig. 8
Fig. 8

Equivalent lens guide of a folded three-mirror resonator in the xz principal plane and the y principal plane for two round trips.

Equations (5)

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νq,nm=c2Lq+(m+n+1) θ02π,
νq,m0=c2LN [Nq+(m+1)],
θm=2πN+Δθm,
Δθm=2πνFSRΔνmm,
xn=Δr cos[nθ(Δr)],

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