Abstract

We use quantum mechanical analogy to introduce a new class of optical resonators with finite deflection profile mirrors that support a finite number of discrete confined transverse modes and a continuum of unconfined transverse modes. We develop theory of such resonators, experimentally demonstrate micro-optical resonators intrinsically confining only a single transverse mode, and demonstrate high finesse step-mirror-profile resonators. Such resonators have profound implications for optical resonator devices, such as lasers and interferometers.

© 2005 Optical Society of America

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References

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  1. B. E. A.  Saleh, M. C.  Teich, Fundamentals of Photonics (Wiley, New York, 1991).
    [CrossRef]
  2. A. E.  Siegman, Lasers (University Science Books, Mill Valley, CA, 1986).
  3. M.  Kuznetsov, F.  Hakimi, R.  Sprague, A.  Mooradian, “Design and characteristics of high power (>0.5 W cw) diode-pumped vertical-external-cavity surface-emitting semiconductor lasers with circular TEM00 beams,” IEEE J. Sel. Top. Quantum Electron. 5, 561–573 (1999).
    [CrossRef]
  4. A. G.  Fox, T.  Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–458 (1961).
  5. G. D.  Boyd, J. P.  Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Technol. J. 40, 489–508 (1961).
  6. G. D.  Boyd, H.  Kogelnik, “Generalized confocal resonator theory,” Bell Syst. Technol. J. 41, 1347–1369 (1962).
  7. H.  Kogelnik, T.  Li, “Laser beams and resonators,” Proceedings IEEE 54, 1312–1329 (1966) and Appl. Opt. 5, 1550–1567 (1966).
    [CrossRef]
  8. A. E.  Siegman, “Laser beams and resonators: the 1960s,” IEEE J. Sel. Top. Quantum Electron. 6, 1380–1388 (2000).
    [CrossRef]
  9. A. E.  Siegman, “Laser beams and resonators: beyond the 1960s,” IEEE J. Sel. Top. Quantum Electron. 6, 1389–1399 (2000).
    [CrossRef]
  10. D.  Marcuse, Light Transmission Optics, 2nd ed. (Van Nostrand Reinhold, New York, 1982).
  11. A. W.  Snyder, J. D.  Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).
  12. H. A.  Haus, Waves and Fields in Optoelectronics (Prentice Hall, Englewood Cliffs, NJ, 1984).
  13. E.  Merzbacher, Quantum Mechanics, 3rd ed. (Wiley, New York, 1997).
  14. Z. L.  Liau, D. E.  Mull, C. L.  Dennis, R. C.  Williamson, R. G.  Waarts, “Large-numerical-aperture microlens fabrication by one-step etching and mass-transport smoothing,” Appl. Phys. Lett. 64, 1484–1486 (1994).
    [CrossRef]
  15. D. A. B.  Miller, D. S.  Chemla, T. C.  Damen, A. C.  Gossard, W.  Wiegmann, T. H.  Wood, C. A.  Burrus, “Electric field dependence of optical absorption near the bandgap of quantum well structures,” Phys. Rev. B32, 1043–1060 (1985).
  16. J. P.  Koplow, D. A. V.  Kliner, L.  Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. 25, 442–444 (2000).
    [CrossRef]
  17. M. E.  Kuznetsov, “Optical resonators with mirror structure suppressing higher order transverse spatial modes,” US patent #6,810,062, issued October 26, 2004.

2000

A. E.  Siegman, “Laser beams and resonators: the 1960s,” IEEE J. Sel. Top. Quantum Electron. 6, 1380–1388 (2000).
[CrossRef]

A. E.  Siegman, “Laser beams and resonators: beyond the 1960s,” IEEE J. Sel. Top. Quantum Electron. 6, 1389–1399 (2000).
[CrossRef]

J. P.  Koplow, D. A. V.  Kliner, L.  Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. 25, 442–444 (2000).
[CrossRef]

1999

M.  Kuznetsov, F.  Hakimi, R.  Sprague, A.  Mooradian, “Design and characteristics of high power (>0.5 W cw) diode-pumped vertical-external-cavity surface-emitting semiconductor lasers with circular TEM00 beams,” IEEE J. Sel. Top. Quantum Electron. 5, 561–573 (1999).
[CrossRef]

1994

Z. L.  Liau, D. E.  Mull, C. L.  Dennis, R. C.  Williamson, R. G.  Waarts, “Large-numerical-aperture microlens fabrication by one-step etching and mass-transport smoothing,” Appl. Phys. Lett. 64, 1484–1486 (1994).
[CrossRef]

1985

D. A. B.  Miller, D. S.  Chemla, T. C.  Damen, A. C.  Gossard, W.  Wiegmann, T. H.  Wood, C. A.  Burrus, “Electric field dependence of optical absorption near the bandgap of quantum well structures,” Phys. Rev. B32, 1043–1060 (1985).

1962

G. D.  Boyd, H.  Kogelnik, “Generalized confocal resonator theory,” Bell Syst. Technol. J. 41, 1347–1369 (1962).

1961

A. G.  Fox, T.  Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–458 (1961).

G. D.  Boyd, J. P.  Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Technol. J. 40, 489–508 (1961).

Boyd, G. D.

G. D.  Boyd, H.  Kogelnik, “Generalized confocal resonator theory,” Bell Syst. Technol. J. 41, 1347–1369 (1962).

G. D.  Boyd, J. P.  Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Technol. J. 40, 489–508 (1961).

Burrus, C. A.

D. A. B.  Miller, D. S.  Chemla, T. C.  Damen, A. C.  Gossard, W.  Wiegmann, T. H.  Wood, C. A.  Burrus, “Electric field dependence of optical absorption near the bandgap of quantum well structures,” Phys. Rev. B32, 1043–1060 (1985).

Chemla, D. S.

D. A. B.  Miller, D. S.  Chemla, T. C.  Damen, A. C.  Gossard, W.  Wiegmann, T. H.  Wood, C. A.  Burrus, “Electric field dependence of optical absorption near the bandgap of quantum well structures,” Phys. Rev. B32, 1043–1060 (1985).

Damen, T. C.

D. A. B.  Miller, D. S.  Chemla, T. C.  Damen, A. C.  Gossard, W.  Wiegmann, T. H.  Wood, C. A.  Burrus, “Electric field dependence of optical absorption near the bandgap of quantum well structures,” Phys. Rev. B32, 1043–1060 (1985).

Dennis, C. L.

Z. L.  Liau, D. E.  Mull, C. L.  Dennis, R. C.  Williamson, R. G.  Waarts, “Large-numerical-aperture microlens fabrication by one-step etching and mass-transport smoothing,” Appl. Phys. Lett. 64, 1484–1486 (1994).
[CrossRef]

Fox, A. G.

A. G.  Fox, T.  Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–458 (1961).

Goldberg, L.

Gordon, J. P.

G. D.  Boyd, J. P.  Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Technol. J. 40, 489–508 (1961).

Gossard, A. C.

D. A. B.  Miller, D. S.  Chemla, T. C.  Damen, A. C.  Gossard, W.  Wiegmann, T. H.  Wood, C. A.  Burrus, “Electric field dependence of optical absorption near the bandgap of quantum well structures,” Phys. Rev. B32, 1043–1060 (1985).

Hakimi, F.

M.  Kuznetsov, F.  Hakimi, R.  Sprague, A.  Mooradian, “Design and characteristics of high power (>0.5 W cw) diode-pumped vertical-external-cavity surface-emitting semiconductor lasers with circular TEM00 beams,” IEEE J. Sel. Top. Quantum Electron. 5, 561–573 (1999).
[CrossRef]

Haus, H. A.

H. A.  Haus, Waves and Fields in Optoelectronics (Prentice Hall, Englewood Cliffs, NJ, 1984).

Kliner, D. A. V.

Kogelnik, H.

G. D.  Boyd, H.  Kogelnik, “Generalized confocal resonator theory,” Bell Syst. Technol. J. 41, 1347–1369 (1962).

H.  Kogelnik, T.  Li, “Laser beams and resonators,” Proceedings IEEE 54, 1312–1329 (1966) and Appl. Opt. 5, 1550–1567 (1966).
[CrossRef]

Koplow, J. P.

Kuznetsov, M.

M.  Kuznetsov, F.  Hakimi, R.  Sprague, A.  Mooradian, “Design and characteristics of high power (>0.5 W cw) diode-pumped vertical-external-cavity surface-emitting semiconductor lasers with circular TEM00 beams,” IEEE J. Sel. Top. Quantum Electron. 5, 561–573 (1999).
[CrossRef]

Kuznetsov, M. E.

M. E.  Kuznetsov, “Optical resonators with mirror structure suppressing higher order transverse spatial modes,” US patent #6,810,062, issued October 26, 2004.

Li, T.

A. G.  Fox, T.  Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–458 (1961).

H.  Kogelnik, T.  Li, “Laser beams and resonators,” Proceedings IEEE 54, 1312–1329 (1966) and Appl. Opt. 5, 1550–1567 (1966).
[CrossRef]

Liau, Z. L.

Z. L.  Liau, D. E.  Mull, C. L.  Dennis, R. C.  Williamson, R. G.  Waarts, “Large-numerical-aperture microlens fabrication by one-step etching and mass-transport smoothing,” Appl. Phys. Lett. 64, 1484–1486 (1994).
[CrossRef]

Love, J. D.

A. W.  Snyder, J. D.  Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

Marcuse, D.

D.  Marcuse, Light Transmission Optics, 2nd ed. (Van Nostrand Reinhold, New York, 1982).

Merzbacher, E.

E.  Merzbacher, Quantum Mechanics, 3rd ed. (Wiley, New York, 1997).

Miller, D. A. B.

D. A. B.  Miller, D. S.  Chemla, T. C.  Damen, A. C.  Gossard, W.  Wiegmann, T. H.  Wood, C. A.  Burrus, “Electric field dependence of optical absorption near the bandgap of quantum well structures,” Phys. Rev. B32, 1043–1060 (1985).

Mooradian, A.

M.  Kuznetsov, F.  Hakimi, R.  Sprague, A.  Mooradian, “Design and characteristics of high power (>0.5 W cw) diode-pumped vertical-external-cavity surface-emitting semiconductor lasers with circular TEM00 beams,” IEEE J. Sel. Top. Quantum Electron. 5, 561–573 (1999).
[CrossRef]

Mull, D. E.

Z. L.  Liau, D. E.  Mull, C. L.  Dennis, R. C.  Williamson, R. G.  Waarts, “Large-numerical-aperture microlens fabrication by one-step etching and mass-transport smoothing,” Appl. Phys. Lett. 64, 1484–1486 (1994).
[CrossRef]

Saleh, B. E. A.

B. E. A.  Saleh, M. C.  Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

Siegman, A. E.

A. E.  Siegman, “Laser beams and resonators: the 1960s,” IEEE J. Sel. Top. Quantum Electron. 6, 1380–1388 (2000).
[CrossRef]

A. E.  Siegman, “Laser beams and resonators: beyond the 1960s,” IEEE J. Sel. Top. Quantum Electron. 6, 1389–1399 (2000).
[CrossRef]

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

Snyder, A. W.

A. W.  Snyder, J. D.  Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

Sprague, R.

M.  Kuznetsov, F.  Hakimi, R.  Sprague, A.  Mooradian, “Design and characteristics of high power (>0.5 W cw) diode-pumped vertical-external-cavity surface-emitting semiconductor lasers with circular TEM00 beams,” IEEE J. Sel. Top. Quantum Electron. 5, 561–573 (1999).
[CrossRef]

Teich, M. C.

B. E. A.  Saleh, M. C.  Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

Waarts, R. G.

Z. L.  Liau, D. E.  Mull, C. L.  Dennis, R. C.  Williamson, R. G.  Waarts, “Large-numerical-aperture microlens fabrication by one-step etching and mass-transport smoothing,” Appl. Phys. Lett. 64, 1484–1486 (1994).
[CrossRef]

Wiegmann, W.

D. A. B.  Miller, D. S.  Chemla, T. C.  Damen, A. C.  Gossard, W.  Wiegmann, T. H.  Wood, C. A.  Burrus, “Electric field dependence of optical absorption near the bandgap of quantum well structures,” Phys. Rev. B32, 1043–1060 (1985).

Williamson, R. C.

Z. L.  Liau, D. E.  Mull, C. L.  Dennis, R. C.  Williamson, R. G.  Waarts, “Large-numerical-aperture microlens fabrication by one-step etching and mass-transport smoothing,” Appl. Phys. Lett. 64, 1484–1486 (1994).
[CrossRef]

Wood, T. H.

D. A. B.  Miller, D. S.  Chemla, T. C.  Damen, A. C.  Gossard, W.  Wiegmann, T. H.  Wood, C. A.  Burrus, “Electric field dependence of optical absorption near the bandgap of quantum well structures,” Phys. Rev. B32, 1043–1060 (1985).

Appl. Phys. Lett.

Z. L.  Liau, D. E.  Mull, C. L.  Dennis, R. C.  Williamson, R. G.  Waarts, “Large-numerical-aperture microlens fabrication by one-step etching and mass-transport smoothing,” Appl. Phys. Lett. 64, 1484–1486 (1994).
[CrossRef]

Bell Syst. Tech. J.

A. G.  Fox, T.  Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–458 (1961).

Bell Syst. Technol. J.

G. D.  Boyd, J. P.  Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Technol. J. 40, 489–508 (1961).

G. D.  Boyd, H.  Kogelnik, “Generalized confocal resonator theory,” Bell Syst. Technol. J. 41, 1347–1369 (1962).

IEEE J. Sel. Top. Quantum Electron.

A. E.  Siegman, “Laser beams and resonators: the 1960s,” IEEE J. Sel. Top. Quantum Electron. 6, 1380–1388 (2000).
[CrossRef]

A. E.  Siegman, “Laser beams and resonators: beyond the 1960s,” IEEE J. Sel. Top. Quantum Electron. 6, 1389–1399 (2000).
[CrossRef]

M.  Kuznetsov, F.  Hakimi, R.  Sprague, A.  Mooradian, “Design and characteristics of high power (>0.5 W cw) diode-pumped vertical-external-cavity surface-emitting semiconductor lasers with circular TEM00 beams,” IEEE J. Sel. Top. Quantum Electron. 5, 561–573 (1999).
[CrossRef]

Opt. Lett.

Phys. Rev.

D. A. B.  Miller, D. S.  Chemla, T. C.  Damen, A. C.  Gossard, W.  Wiegmann, T. H.  Wood, C. A.  Burrus, “Electric field dependence of optical absorption near the bandgap of quantum well structures,” Phys. Rev. B32, 1043–1060 (1985).

Other

M. E.  Kuznetsov, “Optical resonators with mirror structure suppressing higher order transverse spatial modes,” US patent #6,810,062, issued October 26, 2004.

B. E. A.  Saleh, M. C.  Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

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

D.  Marcuse, Light Transmission Optics, 2nd ed. (Van Nostrand Reinhold, New York, 1982).

A. W.  Snyder, J. D.  Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

H. A.  Haus, Waves and Fields in Optoelectronics (Prentice Hall, Englewood Cliffs, NJ, 1984).

E.  Merzbacher, Quantum Mechanics, 3rd ed. (Wiley, New York, 1997).

H.  Kogelnik, T.  Li, “Laser beams and resonators,” Proceedings IEEE 54, 1312–1329 (1966) and Appl. Opt. 5, 1550–1567 (1966).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Plane-curved optical resonator with effective lengths of its discrete transverse modes.

Fig. 2.
Fig. 2.

Physical analogies: particle in a potential well, an optical waveguide, and an optical resonator, with their discrete bound and continuum of unbound states.

Fig. 3.
Fig. 3.

Transverse mode profiles and mode deflections for a resonator with a finite depth secant hyperbolic mirror profile: (a) deeper mirror with three confined modes, (b) shallow mirror with a single confined mode.

Fig. 4.
Fig. 4.

(a) Calculated universal Λ-V diagram, with experimental data, for optical resonators with finite depth mirrors: mode deflection parameter Λ versus cavity V parameter for different transverse modes nrad,nazim ; (b) measured finesse of three confined transverse modes as a function of the cavity V parameter.

Fig. 5.
Fig. 5.

Measured transmission spectra of spherical mirror and single-transverse-mode optical resonators: resonator fundamental mode frequency, plotted on the horizontal axis, is swept past a single frequency laser source by tuning the resonator via cavity length change.

Fig. 6.
Fig. 6.

Wavelength mapping of confined and unconfined transverse modes of optical resonators.

Fig. 7.
Fig. 7.

Measured finesse of optical resonators with a cylindrical pillbox mirror profile as a function of the pillbox diameter.

Equations (10)

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

f m = m c 2 L c
f m , t m c 2 L m , t
Δ L m , t L c L m , t
Δ L m , t < d 0
d ( x ) = d 0 ( 1 sec h ( 2 r r 0 ) )
V π w λ d 0 L c < 2
V f = π w λ n core 2 n clad 2 < 2.405
Λ 1 Δ L m , t d 0
V < 1.5
λ m λ m , c = 2 d 0 m

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