Abstract

We present a detailed theoretical analysis for the determination of the total internal loss in Fabry–Perot resonators based on Fourier analysis of the emission or transmission spectrum. The observation of higher-order harmonics and their relative height in the Fourier-transformed spectrum allow us to quantify the total resonator loss. Because this new method considers both contrast and shape of the Fabry–Perot fringes it is especially well suited for the evaluation of high-finesse laser resonators such as those of vertical cavity surface-emitting lasers in terms of propagation loss/gain.

© 1997 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1984), p. 256.
  2. B. W. Hakki and T. L. Paoli, J. Appl. Phys. 44, 4113 (1973).
    [CrossRef]
  3. B. W. Hakki and T. L. Paoli, J. Appl. Phys. 46, 1299 (1975).
    [CrossRef]
  4. E. Hecht and A. Zajac, Optics, 3rd ed. (Addison-Wesley, Boston, Mass., 1976), p. 307.
  5. V. G. Cooper, Appl. Opt. 10, 525 (1971).
    [CrossRef] [PubMed]
  6. W. H. Steel, Interferometry, 2nd ed. (Cambridge U.  Press, Cambridge, 1983), pp. 141–149.
  7. B. D. Patterson, J. E. Epler, B. Graf, H. W. Lehmann, and H. C. Sigg, IEEE J. Quantum Electron. 30, 703 (1994).
    [CrossRef]
  8. K. Takada, I. Yokohama, K. Chida, and J. Noda, Appl. Opt. 26, 1603 (1987).
    [CrossRef] [PubMed]
  9. R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1986).

1994 (1)

B. D. Patterson, J. E. Epler, B. Graf, H. W. Lehmann, and H. C. Sigg, IEEE J. Quantum Electron. 30, 703 (1994).
[CrossRef]

1987 (1)

1975 (1)

B. W. Hakki and T. L. Paoli, J. Appl. Phys. 46, 1299 (1975).
[CrossRef]

1973 (1)

B. W. Hakki and T. L. Paoli, J. Appl. Phys. 44, 4113 (1973).
[CrossRef]

1971 (1)

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1984), p. 256.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1986).

Chida, K.

Cooper, V. G.

Epler, J. E.

B. D. Patterson, J. E. Epler, B. Graf, H. W. Lehmann, and H. C. Sigg, IEEE J. Quantum Electron. 30, 703 (1994).
[CrossRef]

Graf, B.

B. D. Patterson, J. E. Epler, B. Graf, H. W. Lehmann, and H. C. Sigg, IEEE J. Quantum Electron. 30, 703 (1994).
[CrossRef]

Hakki, B. W.

B. W. Hakki and T. L. Paoli, J. Appl. Phys. 46, 1299 (1975).
[CrossRef]

B. W. Hakki and T. L. Paoli, J. Appl. Phys. 44, 4113 (1973).
[CrossRef]

Hecht, E.

E. Hecht and A. Zajac, Optics, 3rd ed. (Addison-Wesley, Boston, Mass., 1976), p. 307.

Lehmann, H. W.

B. D. Patterson, J. E. Epler, B. Graf, H. W. Lehmann, and H. C. Sigg, IEEE J. Quantum Electron. 30, 703 (1994).
[CrossRef]

Noda, J.

Paoli, T. L.

B. W. Hakki and T. L. Paoli, J. Appl. Phys. 46, 1299 (1975).
[CrossRef]

B. W. Hakki and T. L. Paoli, J. Appl. Phys. 44, 4113 (1973).
[CrossRef]

Patterson, B. D.

B. D. Patterson, J. E. Epler, B. Graf, H. W. Lehmann, and H. C. Sigg, IEEE J. Quantum Electron. 30, 703 (1994).
[CrossRef]

Sigg, H. C.

B. D. Patterson, J. E. Epler, B. Graf, H. W. Lehmann, and H. C. Sigg, IEEE J. Quantum Electron. 30, 703 (1994).
[CrossRef]

Steel, W. H.

W. H. Steel, Interferometry, 2nd ed. (Cambridge U.  Press, Cambridge, 1983), pp. 141–149.

Takada, K.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1984), p. 256.

Yokohama, I.

Zajac, A.

E. Hecht and A. Zajac, Optics, 3rd ed. (Addison-Wesley, Boston, Mass., 1976), p. 307.

Appl. Opt. (2)

IEEE J. Quantum Electron. (1)

B. D. Patterson, J. E. Epler, B. Graf, H. W. Lehmann, and H. C. Sigg, IEEE J. Quantum Electron. 30, 703 (1994).
[CrossRef]

J. Appl. Phys. (2)

B. W. Hakki and T. L. Paoli, J. Appl. Phys. 44, 4113 (1973).
[CrossRef]

B. W. Hakki and T. L. Paoli, J. Appl. Phys. 46, 1299 (1975).
[CrossRef]

Other (4)

E. Hecht and A. Zajac, Optics, 3rd ed. (Addison-Wesley, Boston, Mass., 1976), p. 307.

W. H. Steel, Interferometry, 2nd ed. (Cambridge U.  Press, Cambridge, 1983), pp. 141–149.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1984), p. 256.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1986).

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Figures (4)

Fig. 1
Fig. 1

Analytically calculated transmission spectrum of a FP resonator with 30% reflective mirrors. The numerical Fourier transform of the spectrum is shown in the inset.

Fig. 2
Fig. 2

Analytically calculated transmission spectrum of a FP resonator with 70% reflective mirrors. The numerical Fourier transform of the spectrum is shown in the inset.

Fig. 3
Fig. 3

Absorption loss in an active resonator for the values of mirror reflectance R shown and mirror reflectance for absorption loss α as a function of HAR.

Fig. 4
Fig. 4

Emission spectrum of a AlGalnN double heterostructure. The Fourier transform of the spectrum is shown in the inset.

Equations (7)

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Aβ=1-R exp2iψm=0R exp2iψm×exp-2kLm+1/2βexp2inLmβ.
Iβ=Aβ2=AβA*β=1-R2 exp-2kLβ+4 sin2 ψ1-R exp-2kLβ2+4R exp-2kLβsin2ψ+nLβ.
Ad¯=1-R exp2iψm=00βmaxR exp2iψm×exp-2kLm+1/2+iπd-nLmβdβ.
Ad¯=1-R exp2iψ×m=0R exp2iψm2kLm+1/2+iπd-nLm×1-exp-2kLm+1/2+iπd-nLmβmax.
Id¯=1-R exp2iψ2m=0l=0Rl+m exp-2iψl-m×-1-exp-2πix-x1βmax1-exp-2πix2-xβmax2πi2x-x1x2-xdx,
x1=-kLm+1/2+inLmiπ, x2=kLl+1/2+iπd+nLliπ
Id¯=1-R exp2iψ2×m=0l=0Rl+m exp-2iψl-mπix2-x1=1-R exp2iψ2×m=0l=0Rl+m exp-2iψl-m{kLl+m+1+i[πd+nL(l-m]},

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