Abstract

We detail the oscillation properties of cw type-II optical parametric oscillators (signal and idler modes with orthogonal polarizations). When the signal and idler frequencies are very close, they are shown to have characteristics that are quite different from the well-known type-I optical parametric oscillators. We determine in particular the cavity-length values for which the oscillation occurs and how the frequencies of the output fields vary when one changes this length, the crystal angle, or the temperature. Finally, we determine the influence of the mirror phase shifts on the oscillation characteristics of a linear-cavity OPO.

© 1993 Optical Society of America

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  1. J. A. Giordmaine and R. C. Miller, Phys. Rev. Lett. 14, 973 (1965).
    [CrossRef]
  2. A. Akhmanov, A. I. Kovrigin, A. I. Kolosov, A. Piskarkas, V. V. Fadeev, and R. V. Khokhlov, Sov. Phys. JETP 3, 241 (1966).
  3. S. E. Harris, Proc. IEEE 57, 2096 (1969).
    [CrossRef]
  4. R. L. Byer, in Treatise in Quantum Electronics, H. Rabin and C. L. Tang, eds. (Academic, N.Y.1975), p. 587.
  5. W. Brunner and H. Paul, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1977), p. 15.
  6. J. Falk, IEEE J. Quantum Electron. QE-7, 230 (1971).
    [CrossRef]
  7. R. G. Smith, IEEE J. Quantum Electron. QE-9, 530 (1973).
    [CrossRef]
  8. W. J. Kozlovsky, E. R. Gustafson, R. C. Eckardt, and R. L. Byer, Opt. Lett. 13, 1102 (1988).
    [CrossRef] [PubMed]
  9. L. Wu, H. J. Kimble, J. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
    [CrossRef] [PubMed]
  10. A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, Phys. Rev. Lett. 59, 2555 (1987).
    [CrossRef] [PubMed]
  11. C. D. Nabors and R. M. Shelby, Phys. Rev. A 42, 556 (1990).
    [CrossRef] [PubMed]
  12. J. Mertz, T. Debuisschert, A. Heidmann, C. Fabre, and E. Giacobino, Opt. Lett. 16, 1234 (1991).
    [CrossRef] [PubMed]
  13. R. C. Eckardt, C. D. Nabors, W. J. Kozlovsky, and R. L. Byer, J. Opt. Soc. Am. B 8, 646 (1991).
    [CrossRef]
  14. C. M. Savage and D. F. Walls, J. Opt. Soc. Am. B 4, 1514 (1987).
    [CrossRef]
  15. L. Lugiato, C. Oldano, C. Fabre, E. Giacobino, and R. Horowicz, Nuovo Cimento D 10, 959 (1988).
    [CrossRef]
  16. N. C. Wong, Phys. Rev. A 45, 3176 (1992).
    [CrossRef] [PubMed]
  17. J. Snyder, E. Giacobino, C. Fabre, A. Heidmann, and M. Ducloy, J. Opt. Soc. Am. B 7, 2132 (1990).
    [CrossRef]
  18. J. Q. Yao and T. S. Fahlen, J. Appl. Phys. 55, 65 (1984).
    [CrossRef]
  19. D. Lee and N. C. Wong, Opt. Lett. 17, 13 (1992).
    [CrossRef] [PubMed]
  20. H. J. Kimble and J. Hall, in Quantum Optics IV, J. Harvey, ed. (Springer-Verlag, Heidelberg, 1987), p. 58.
  21. P. Kurz, R. Paschotta, K. Fiedler, A. Sizmann, G. Leuchs, and J. Mlynek, Appl. Phys. B 55, 216 (1992).
    [CrossRef]
  22. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).
    [CrossRef]
  23. G. D. Boyd and D. A. Kleinman, J. Appl. Phys. 39, 3597 (1968).
    [CrossRef]

1992 (3)

N. C. Wong, Phys. Rev. A 45, 3176 (1992).
[CrossRef] [PubMed]

P. Kurz, R. Paschotta, K. Fiedler, A. Sizmann, G. Leuchs, and J. Mlynek, Appl. Phys. B 55, 216 (1992).
[CrossRef]

D. Lee and N. C. Wong, Opt. Lett. 17, 13 (1992).
[CrossRef] [PubMed]

1991 (2)

1990 (2)

1988 (2)

L. Lugiato, C. Oldano, C. Fabre, E. Giacobino, and R. Horowicz, Nuovo Cimento D 10, 959 (1988).
[CrossRef]

W. J. Kozlovsky, E. R. Gustafson, R. C. Eckardt, and R. L. Byer, Opt. Lett. 13, 1102 (1988).
[CrossRef] [PubMed]

1987 (2)

C. M. Savage and D. F. Walls, J. Opt. Soc. Am. B 4, 1514 (1987).
[CrossRef]

A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, Phys. Rev. Lett. 59, 2555 (1987).
[CrossRef] [PubMed]

1986 (1)

L. Wu, H. J. Kimble, J. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

1984 (1)

J. Q. Yao and T. S. Fahlen, J. Appl. Phys. 55, 65 (1984).
[CrossRef]

1973 (1)

R. G. Smith, IEEE J. Quantum Electron. QE-9, 530 (1973).
[CrossRef]

1971 (1)

J. Falk, IEEE J. Quantum Electron. QE-7, 230 (1971).
[CrossRef]

1969 (1)

S. E. Harris, Proc. IEEE 57, 2096 (1969).
[CrossRef]

1968 (1)

G. D. Boyd and D. A. Kleinman, J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

1966 (1)

A. Akhmanov, A. I. Kovrigin, A. I. Kolosov, A. Piskarkas, V. V. Fadeev, and R. V. Khokhlov, Sov. Phys. JETP 3, 241 (1966).

1965 (1)

J. A. Giordmaine and R. C. Miller, Phys. Rev. Lett. 14, 973 (1965).
[CrossRef]

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).
[CrossRef]

Akhmanov, A.

A. Akhmanov, A. I. Kovrigin, A. I. Kolosov, A. Piskarkas, V. V. Fadeev, and R. V. Khokhlov, Sov. Phys. JETP 3, 241 (1966).

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).
[CrossRef]

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).
[CrossRef]

Boyd, G. D.

G. D. Boyd and D. A. Kleinman, J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

Brunner, W.

W. Brunner and H. Paul, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1977), p. 15.

Byer, R. L.

Camy, G.

A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, Phys. Rev. Lett. 59, 2555 (1987).
[CrossRef] [PubMed]

Debuisschert, T.

Ducloy, M.

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).
[CrossRef]

Eckardt, R. C.

Fabre, C.

J. Mertz, T. Debuisschert, A. Heidmann, C. Fabre, and E. Giacobino, Opt. Lett. 16, 1234 (1991).
[CrossRef] [PubMed]

J. Snyder, E. Giacobino, C. Fabre, A. Heidmann, and M. Ducloy, J. Opt. Soc. Am. B 7, 2132 (1990).
[CrossRef]

L. Lugiato, C. Oldano, C. Fabre, E. Giacobino, and R. Horowicz, Nuovo Cimento D 10, 959 (1988).
[CrossRef]

A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, Phys. Rev. Lett. 59, 2555 (1987).
[CrossRef] [PubMed]

Fadeev, V. V.

A. Akhmanov, A. I. Kovrigin, A. I. Kolosov, A. Piskarkas, V. V. Fadeev, and R. V. Khokhlov, Sov. Phys. JETP 3, 241 (1966).

Fahlen, T. S.

J. Q. Yao and T. S. Fahlen, J. Appl. Phys. 55, 65 (1984).
[CrossRef]

Falk, J.

J. Falk, IEEE J. Quantum Electron. QE-7, 230 (1971).
[CrossRef]

Fiedler, K.

P. Kurz, R. Paschotta, K. Fiedler, A. Sizmann, G. Leuchs, and J. Mlynek, Appl. Phys. B 55, 216 (1992).
[CrossRef]

Giacobino, E.

J. Mertz, T. Debuisschert, A. Heidmann, C. Fabre, and E. Giacobino, Opt. Lett. 16, 1234 (1991).
[CrossRef] [PubMed]

J. Snyder, E. Giacobino, C. Fabre, A. Heidmann, and M. Ducloy, J. Opt. Soc. Am. B 7, 2132 (1990).
[CrossRef]

L. Lugiato, C. Oldano, C. Fabre, E. Giacobino, and R. Horowicz, Nuovo Cimento D 10, 959 (1988).
[CrossRef]

A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, Phys. Rev. Lett. 59, 2555 (1987).
[CrossRef] [PubMed]

Giordmaine, J. A.

J. A. Giordmaine and R. C. Miller, Phys. Rev. Lett. 14, 973 (1965).
[CrossRef]

Gustafson, E. R.

Hall, J.

L. Wu, H. J. Kimble, J. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

H. J. Kimble and J. Hall, in Quantum Optics IV, J. Harvey, ed. (Springer-Verlag, Heidelberg, 1987), p. 58.

Harris, S. E.

S. E. Harris, Proc. IEEE 57, 2096 (1969).
[CrossRef]

Heidmann, A.

Horowicz, R.

L. Lugiato, C. Oldano, C. Fabre, E. Giacobino, and R. Horowicz, Nuovo Cimento D 10, 959 (1988).
[CrossRef]

Horowicz, R. J.

A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, Phys. Rev. Lett. 59, 2555 (1987).
[CrossRef] [PubMed]

Khokhlov, R. V.

A. Akhmanov, A. I. Kovrigin, A. I. Kolosov, A. Piskarkas, V. V. Fadeev, and R. V. Khokhlov, Sov. Phys. JETP 3, 241 (1966).

Kimble, H. J.

L. Wu, H. J. Kimble, J. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

H. J. Kimble and J. Hall, in Quantum Optics IV, J. Harvey, ed. (Springer-Verlag, Heidelberg, 1987), p. 58.

Kleinman, D. A.

G. D. Boyd and D. A. Kleinman, J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

Kolosov, A. I.

A. Akhmanov, A. I. Kovrigin, A. I. Kolosov, A. Piskarkas, V. V. Fadeev, and R. V. Khokhlov, Sov. Phys. JETP 3, 241 (1966).

Kovrigin, A. I.

A. Akhmanov, A. I. Kovrigin, A. I. Kolosov, A. Piskarkas, V. V. Fadeev, and R. V. Khokhlov, Sov. Phys. JETP 3, 241 (1966).

Kozlovsky, W. J.

Kurz, P.

P. Kurz, R. Paschotta, K. Fiedler, A. Sizmann, G. Leuchs, and J. Mlynek, Appl. Phys. B 55, 216 (1992).
[CrossRef]

Lee, D.

Leuchs, G.

P. Kurz, R. Paschotta, K. Fiedler, A. Sizmann, G. Leuchs, and J. Mlynek, Appl. Phys. B 55, 216 (1992).
[CrossRef]

Lugiato, L.

L. Lugiato, C. Oldano, C. Fabre, E. Giacobino, and R. Horowicz, Nuovo Cimento D 10, 959 (1988).
[CrossRef]

Mertz, J.

Miller, R. C.

J. A. Giordmaine and R. C. Miller, Phys. Rev. Lett. 14, 973 (1965).
[CrossRef]

Mlynek, J.

P. Kurz, R. Paschotta, K. Fiedler, A. Sizmann, G. Leuchs, and J. Mlynek, Appl. Phys. B 55, 216 (1992).
[CrossRef]

Nabors, C. D.

Oldano, C.

L. Lugiato, C. Oldano, C. Fabre, E. Giacobino, and R. Horowicz, Nuovo Cimento D 10, 959 (1988).
[CrossRef]

Paschotta, R.

P. Kurz, R. Paschotta, K. Fiedler, A. Sizmann, G. Leuchs, and J. Mlynek, Appl. Phys. B 55, 216 (1992).
[CrossRef]

Paul, H.

W. Brunner and H. Paul, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1977), p. 15.

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).
[CrossRef]

Piskarkas, A.

A. Akhmanov, A. I. Kovrigin, A. I. Kolosov, A. Piskarkas, V. V. Fadeev, and R. V. Khokhlov, Sov. Phys. JETP 3, 241 (1966).

Reynaud, S.

A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, Phys. Rev. Lett. 59, 2555 (1987).
[CrossRef] [PubMed]

Savage, C. M.

Shelby, R. M.

C. D. Nabors and R. M. Shelby, Phys. Rev. A 42, 556 (1990).
[CrossRef] [PubMed]

Sizmann, A.

P. Kurz, R. Paschotta, K. Fiedler, A. Sizmann, G. Leuchs, and J. Mlynek, Appl. Phys. B 55, 216 (1992).
[CrossRef]

Smith, R. G.

R. G. Smith, IEEE J. Quantum Electron. QE-9, 530 (1973).
[CrossRef]

Snyder, J.

Walls, D. F.

Wong, N. C.

Wu, H.

L. Wu, H. J. Kimble, J. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

Wu, L.

L. Wu, H. J. Kimble, J. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

Yao, J. Q.

J. Q. Yao and T. S. Fahlen, J. Appl. Phys. 55, 65 (1984).
[CrossRef]

Appl. Phys. B (1)

P. Kurz, R. Paschotta, K. Fiedler, A. Sizmann, G. Leuchs, and J. Mlynek, Appl. Phys. B 55, 216 (1992).
[CrossRef]

IEEE J. Quantum Electron. (2)

J. Falk, IEEE J. Quantum Electron. QE-7, 230 (1971).
[CrossRef]

R. G. Smith, IEEE J. Quantum Electron. QE-9, 530 (1973).
[CrossRef]

J. Appl. Phys. (2)

J. Q. Yao and T. S. Fahlen, J. Appl. Phys. 55, 65 (1984).
[CrossRef]

G. D. Boyd and D. A. Kleinman, J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

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

Nuovo Cimento D (1)

L. Lugiato, C. Oldano, C. Fabre, E. Giacobino, and R. Horowicz, Nuovo Cimento D 10, 959 (1988).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).
[CrossRef]

Phys. Rev. A (2)

C. D. Nabors and R. M. Shelby, Phys. Rev. A 42, 556 (1990).
[CrossRef] [PubMed]

N. C. Wong, Phys. Rev. A 45, 3176 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett. (3)

L. Wu, H. J. Kimble, J. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, Phys. Rev. Lett. 59, 2555 (1987).
[CrossRef] [PubMed]

J. A. Giordmaine and R. C. Miller, Phys. Rev. Lett. 14, 973 (1965).
[CrossRef]

Proc. IEEE (1)

S. E. Harris, Proc. IEEE 57, 2096 (1969).
[CrossRef]

Sov. Phys. JETP (1)

A. Akhmanov, A. I. Kovrigin, A. I. Kolosov, A. Piskarkas, V. V. Fadeev, and R. V. Khokhlov, Sov. Phys. JETP 3, 241 (1966).

Other (3)

R. L. Byer, in Treatise in Quantum Electronics, H. Rabin and C. L. Tang, eds. (Academic, N.Y.1975), p. 587.

W. Brunner and H. Paul, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1977), p. 15.

H. J. Kimble and J. Hall, in Quantum Optics IV, J. Harvey, ed. (Springer-Verlag, Heidelberg, 1987), p. 58.

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Figures (10)

Fig. 1
Fig. 1

Schematic of a TRO, consisting of a crystal of length l located in a ring cavity made of two perfectly reflecting mirrors and one coupling mirror. The reflection coefficients for the pump, signal, and idler fields are denoted r0, r1, r2, respectively. The incoming pump field α0in produces one output signal field α1out and one idler field α2out. A part of the pump field, α0out, is reflected from the TRO cavity.

Fig. 2
Fig. 2

Permitted signal and idler modes for two adjacent resonances of the cavity in the case when δnl0 is an integer. The idler and signal frequencies of a given mode (denoted by letters a, b, etc.) are located symmetrically with respect to the degeneracy frequency (ν0/2). Two adjacent signal or idler frequencies are separated by the FSR of the ring cavity d FSR = c / ( L ¯ + n ¯ l ).

Fig. 3
Fig. 3

Selection of beat-note frequencies resulting from pump resonance when the birefringence term δn is nonzero. The two schemes correspond to two adjacent resonance lengths of the cavity. In each one, regularly spaced groups of beat notes are allowed that are separated by Dpr. The width of each groupjs Wpr. Two consecutive beat notes in a group are separated by 2 D ¯.

Fig. 4
Fig. 4

Selection of beat-note frequencies resulting from phase matching. The beat note Δνk corresponding to perfect phase matching does not necessarily match a group. In the case of a low pumping level, only the group closest to Δνk is selected for each pump resonance of the cavity.

Fig. 5
Fig. 5

Experimental curves obtained with a linear-cavity OPO by use of a KTP crystal when sweeping its length. The top curve is the intensity of the pump beam reflected by the cavity. The bottom curve is the intensity in the signal and idler beams emerging from the cavity. Oscillation occurs only for cavity lengths in the vicinity of each minimum of the reflected pump intensity, i.e, of a resonance of the cavity with the pump field. The bottom curve shows maximum intensity peaks that correspond to cavity lengths that are resonant for both signal and idler fields. The peaks are narrow enough that the peaks are well separated. The two displayed groups of peaks are different because the allowed signal and idler frequencies are different for two adjacent pump resonances.

Fig. 6
Fig. 6

Beat-note frequencies when phase matching is obtained in the vicinity of degeneracy. Fine tuning is necessary for exact degeneracy.

Fig. 7
Fig. 7

Typical dependence of the phase-matched beat note Δνk when an external parameter x is tuned. xd is the value of x that gives degeneracy. When x is tuned near this value, Δνk experiences (a) a linear dependence in the case of type-II phase matching and (b) a quadratic dependence in the case of type-I phase matching.

Fig. 8
Fig. 8

Beat-note frequencies Δνmi corresponding to values of m that are allowed for a given pump resonance. They experience an identical linear dependence when an external parameter x is tuned. The phase-matched beat-note frequencies Δνk have a linear dependence as a function of x with a slope that is more important that that of the lines Δνmi. The OPO oscillates on the curve Δνmi that is closest to the exact phase-matching curve Δνk. Thus, as x is tuned, the beat-note frequency experiences a succession of mode hops.

Fig. 9
Fig. 9

Sketch of a semimonolithic linear-cavity TRO. The crystal length is l, and the cavity length is l + l. Mirror M is directly coated on one end of the crystal.

Fig. 10
Fig. 10

Variation of the normalized oscillation threshold as a function of phase-matching coefficient Δkl/2 for different values of the mirror-phase-shift parameter θ. Dotted curve, θ = 0; short-dashed curves, θ = π/2; long-dashed curves, θ = π; solid curves, θ = 3π/2. When θ = π, the curve has two minima, located at Δkl/2 = ±1.16, which are equal to 1.92 times the θ = 0 threshold.

Equations (134)

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L = L cav l .
r j = 1 γ j ,
t j = 2 γ j ,
α 0 ( l ) = α 0 ( 0 ) 2 χ * α 2 ( 0 ) α 1 ( 0 ) ,
α 1 ( l ) = α 1 ( 0 ) + 2 χ α 0 ( 0 ) α 2 * ( 0 ) ,
α 2 ( l ) = α 2 ( 0 ) + 2 χ α 0 ( 0 ) α 1 * ( 0 ) ,
χ = χ eff l w 0 w 1 w 2 w 0 2 w 1 2 + w 0 2 w 2 2 + w 2 2 w 1 2 ( ћ ω 0 ω 1 ω 2 π 0 c 3 n 0 n 1 n 2 ) 1 / 2 × sinc ( Δ k l 2 ) exp [ ( i Δ k l ) / 2 ] ,
Δ k = k 0 k 1 k 2 ,
φ j = ω j c ( n j l + L ) .
α 0 [ 1 r 0 exp ( i φ 0 ) ] = 2 χ * α 1 α 2 r 0 exp ( i φ 0 ) + t 0 α 0 in ,
α 1 [ 1 r 1 exp ( i φ 1 ) ] = 2 χ α 0 α 2 * r 1 exp ( i φ 1 ) ,
α 2 [ 1 r 2 exp ( i φ 2 ) ] = 2 χ α 0 α 1 * r 2 exp ( i φ 2 ) ,
φ j = 2 p j π + δ φ j , p j an integer .
| δ φ j | 2 π .
α 0 ( γ 0 i δ φ 0 ) = 2 χ * α 1 α 2 + 2 γ 0 α 0 in ,
α 1 ( γ 1 i δ φ 1 ) = 2 χ α 0 α 2 * ,
α 2 ( γ 2 i δ φ 2 ) = 2 χ α 0 α 1 * ,
γ j = γ j + μ j .
Δ j = δ φ j γ j ,
α 0 γ 0 ( 1 i Δ 0 ) = 2 χ * α 1 α 2 + 2 γ 0 α 0 in ,
α 1 γ 1 ( 1 i Δ 1 ) = 2 χ α 0 α 2 * ,
α 2 γ 2 ( 1 i Δ 2 ) = 2 χ α 0 α 1 * ,
α 1 = α 2 = 0.
γ 1 γ 2 ( 1 i Δ 1 ) ( 1 + i Δ 2 ) = 4 | χ | 2 | α 0 | 2 .
Δ 1 = Δ 2 = Δ .
| α j out | 2 = 2 γ j | α j | 2 .
| α 1 out | 2 ( 1 i Δ 1 ) = | α 2 out | 2 ( 1 i Δ 2 ) .
I j = | α j out | 2 = 2 γ j | α j | 2 , j = 1 , 2.
I 1 I 2 = γ 1 | α 1 | 2 γ 2 | α 2 | 2 = γ 1 γ 2 γ 2 γ 1 .
| α 0 | 2 = γ 1 γ 2 ( 1 + Δ 2 ) 4 | χ | 2 .
γ 0 2 | α 0 | 2 ( 1 + Δ 0 2 ) = 2 γ 0 | α 0 in | th 2 .
| α 0 in | th 2 = γ 0 2 γ 1 γ 2 8 | χ | 2 γ 0 ( 1 + Δ 2 ) ( 1 + Δ 0 2 ) .
σ = | α 0 in | 2 | α 0 in | res 2 .
σ th = ( 1 + Δ 2 ) ( 1 + Δ 0 2 ) = ( Δ 0 + Δ ) 2 + ( 1 Δ 0 Δ ) 2 .
[ ( 1 i Δ 0 ) ( 1 i Δ ) + 4 | χ | 2 | α 1 | 2 γ 0 γ 2 ] α 0 = ( 1 i Δ ) 2 γ 0 γ 0 α 0 in .
( 1 Δ Δ 0 + 4 | χ | 2 | α 1 | 2 γ 2 γ 0 ) 2 + ( Δ + Δ 0 ) 2 = σ .
| α j | 2 = γ k γ 0 4 | χ | 2 { [ σ ( Δ 0 + Δ ) 1 / 2 + Δ 0 Δ 1 ] } , j , k = 1 , 2 , j k .
| α j | 2 = γ k γ 0 4 | χ | 2 [ σ 1 ] , j , k = 1 , 2 , j k .
Δ ν = ν 1 ν 2 .
ν 1 = ν 0 2 + Δ ν 2 ,
ν 2 = ν 0 2 Δ ν 2 .
φ 1 = 2 π ν 1 c [ L + n 1 ( ν 1 ) l ] ,
φ 2 = 2 π ν 2 c [ L + n 2 ( ν 2 ) l ] .
γ 1 = γ 2 .
φ 2 = φ 1 + 2 m π , m an integer .
Δ ν = ν 0 δ n ( ν 0 , ν 1 ) l L + n ¯ ( ν 0 , ν 1 ) l c m L + n ¯ ( ν 0 , ν 1 ) l ,
n ¯ ( ν 0 , ν 1 ) = n 1 ( ν 1 ) + n 2 ( ν 0 ν 1 ) 2 ,
δ n ( ν 0 , ν 1 ) = n 1 ( ν 1 ) n 2 ( ν 0 ν 1 ) 2 .
n 1 ( ν 1 ) = n 1 ( ν 0 2 ) ,
n 2 ( ν 2 ) = n 2 ( ν 0 2 ) .
n ¯ ( ν 0 , ν 1 ) = n ¯ = n 1 ( ν 0 / 2 ) + n 2 ( ν 0 / 2 ) 2 ,
δ n ( ν 0 , ν 1 ) = δ n = n 1 ( ν 0 / 2 ) n 2 ( ν 0 / 2 ) 2 .
Δ ν = ν 0 δ n l + c m L + n ¯ l .
D = c L + n ¯ l .
d FSR = D 2 = c 2 ( L + n ¯ l ) ,
δ φ 1 = δ φ 2 = δ φ = 0
φ 1 = 2 p π ,
φ 2 = 2 ( p + m ) π , p a positive integer , m an integer .
Δ ν = 2 c p ν 0 [ L + ( n ¯ + δ n ) l ] L + ( n ¯ + δ n ) l .
L + n ¯ l = λ 0 ( 2 p + m ) δ n l Δ ν ν 0 .
L + n ¯ l = λ 0 ( 2 p + m ) + δ n l ν 0 ( ν 0 δ n l + c m L + n ¯ l ) .
L + n ¯ l = λ 0 [ ( 2 p + m ) + δ n l ( L ¯ + n ¯ l ) ( m + δ n l λ 0 ) ] .
D ¯ = c L ¯ + n ¯ l ,
Δ ν 0 = ν 0 δ n l L ¯ + n ¯ l = δ n l λ 0 D ¯ .
ν 1 = ν 0 2 + Δ ν 0 2 m D ¯ , ν 2 = ν 0 2 Δ ν 0 2 + m D ¯ , m an integer .
ν 1 = ν 0 2 + Δ ν 0 2 ( m + 1 2 ) D ¯ , ν 2 = ν 0 2 Δ ν 0 2 + ( m + 1 2 ) D ¯ .
Δ L 1 λ 0 = 2 δ n l L ¯ + n ¯ l .
Δ L 2 λ 0 = γ π σ 1 .
Δ L 1 λ 0 = 2 × 10 2 .
Δ L 2 λ 0 10 3 .
Δ ν = ν 0 δ n l + c m L ¯ + n ¯ l .
L + n ¯ l = λ 0 s = λ 0 ( 2 p + m ) δ n l Δ ν s ν 0 ,
Δ ν s = c δ n l ( 2 p + m s ) .
W pr = c δ n l γ 0 2 π σ 1 .
D pr = c δ n l .
Δ k = 2 π c ( n 0 ν 0 n 1 ν 1 n 2 ν 2 ) .
Δ k = 2 π c [ ( n 0 n ¯ ) ν 0 δ n Δ ν ] .
Δ ν k = ν 0 n 0 n ¯ δ n .
| α 0 in | res 2 = γ 0 2 γ 1 γ 2 8 κ 2 l 2 γ 0 1 sin c 2 ( Δ k l 2 ) .
σ = | α 0 in | 2 | α 0 in | pm 2 .
| Δ k l | = 2 3 ( σ 1 ) .
D pm = 2 c π δ n l 3 ( σ 1 ) .
n 0 = n ¯ .
D ¯ 2 = c 2 ( L ¯ + n ¯ l ) .
( Δ ν k ) x = ( n 0 x n ¯ x ) ν 0 δ n .
( Δ ν m ) x = ν 0 l L ¯ + n ¯ l ( δ n ) x .
n 0 φ = n 1 φ = n φ .
( δ n ) φ = n ¯ φ = 1 2 n φ .
[ ( Δ ν m ) ] / φ [ ( Δ ν k ) ] / φ = δ n l L ¯ + n ¯ l .
n 0 T n ¯ T n 0 T n 1 T n 1 T n 2 T 2 ( δ n ) T .
[ ( Δ ν m ) ] / T [ ( Δ ν k ) ] / T δ n l 2 ( L ¯ + n ¯ l ) .
α 0 [ 1 r 0 exp ( i φ 0 ) ] = 2 χ * α 1 α 2 exp ( i φ 0 ) [ r 0 + r 1 r 2 exp ( i θ i Δ k l ) ] + t 0 α 0 in , α 1 [ 1 r 1 exp ( i φ 1 ) ] = 2 χ α 0 α 2 * exp ( i φ 1 ) [ r 1 + r 0 r 2 exp ( i θ + i Δ k l ) ] , α 2 [ 1 r 2 exp ( i φ 2 ) ] = 2 χ α 0 α 1 * exp ( i φ 2 ) [ r 2 + r 0 r 1 exp ( i θ + i Δ k l ) ] ,
φ j = 2 ( n j l + l ) ( ω j / c ) + δ j + δ j ,
θ = δ 1 + δ 2 δ 0 .
α 0 ( γ 0 i δ φ 0 ) = 2 χ * α 1 α 2 × [ 1 + exp ( i θ i Δ k l ) ] + 2 γ 0 α 0 in , α 1 ( γ 1 i δ φ 1 ) = 2 χ α 0 α 2 * × [ 1 + exp ( i θ + i Δ k l ) ] , α 2 ( γ 2 i δ φ 2 ) = 2 χ α 0 α 1 * × [ 1 + exp ( i θ + i Δ k l ) ] .
χ = κ l sinc ( Δ k l 2 ) exp ( i Δ k l / 2 ) [ 1 + exp ( i Δ k l i θ ) ] .
| α 0 in | th 2 = γ 0 2 γ 1 γ 2 32 κ 2 l 2 γ 0 ( Δ k l / 2 ) 2 sin 2 ( Δ k l / 2 ) cos 2 ( Δ k l / 2 θ / 2 ) .
θ = δ 0 δ 1 δ 2 ,
E j ( z , r , t ) = A j ( z ) exp ( r 2 / w j 2 ) exp [ i ( ω j t k j z ) ] , j = 0 , 1 , 2.
P 0 ( r , z ) = 0 χ eff A 2 ( z ) A 1 ( z ) exp ( r 2 / w ¯ 0 2 ) exp ( i Δ k z ) ,
P 1 ( r , z ) = 0 χ eff A 0 ( z ) A 2 * ( z ) exp ( r 2 / w ¯ 1 2 ) exp ( i Δ k z ) ,
P 2 ( r , z ) = 0 χ eff A 0 ( z ) A 1 * ( z ) exp ( r 2 / w ¯ 2 2 ) exp ( i Δ k z ) .
Δ k = k 0 k 1 k 2 .
1 w ¯ j 2 = 1 w l 2 + 1 w k 2 , 1 j , k j , l k .
d α 0 d z = 2 κ α 2 ( z ) α 1 ( z ) exp ( i Δ k z ) ,
d α 1 d z = 2 κ α 0 ( z ) α 2 * ( z ) exp ( i Δ k z ) ,
d α 2 d z = 2 κ α 0 ( z ) α 1 * ( z ) exp ( i Δ k z ) ,
α j ( z ) = ( n j c 0 π w j 2 4 ћ ω j ) 1 / 2 A j ( z ) , j = 1 , 2 ; α 0 = i ( n 0 c 0 π w 0 2 4 ћ ω 0 ) 1 / 2 A 0 ( z )
κ = χ eff w 0 w 1 w 2 w 0 2 w 1 2 + w 0 2 w 2 2 + w 1 2 w 2 2 ( ћ ω 0 ω 1 ω 2 π 0 c 3 n 0 n 1 n 2 ) 1 / 2 .
α 0 ( l ) = α 0 ( 0 ) 2 χ * α 2 ( 0 ) α 1 ( 0 ) ,
α 1 ( l ) = α 1 ( 0 ) + 2 χ α 0 ( 0 ) α 2 * ( 0 ) ,
α 2 ( l ) = α 2 ( 0 ) + 2 χ α 0 ( 0 ) α 1 * ( 0 ) ,
χ = κ l sinc ( Δ k l 2 ) exp ( i Δ k l / 2 ) .
σ = σ th = ( 1 + Δ 2 ) ( 1 + Δ 0 2 ) .
| Δ | σ 1 .
| δ L λ 0 | γ π σ 1 ,
Δ L 1 λ 0 = γ π σ 1 .
Δ L 0 λ 0 = γ 0 2 π σ 1 .
ν 1 = ν 0 2 + Δ ν 2 ,
ν 2 = ν 0 2 Δ ν 2 ,
n 1 ( ν 1 ) = n 1 ( ν 0 2 + Δ ν 2 ) = n 1 ( ν 0 2 ) + Δ ν 2 d n 1 d ν ( ν 0 2 ) .
n 2 ( ν 2 ) = n 2 ( ν 0 2 Δ ν 2 ) = n 2 ( ν 0 2 ) Δ ν 2 d n 2 d ν ( ν 0 2 ) .
n ¯ ( ν 0 , ν 1 ) = n 1 ( ν 0 / 2 ) + n 2 ( ν 0 / 2 ) 2 + Δ ν 4 [ d n 1 d ν ( ν 0 2 ) d n 2 d ν ( ν 0 2 ) ] ,
δ n ( ν 0 , ν 1 ) = n 1 ( ν 0 / 2 ) n 2 ( ν 0 / 2 ) 2 + Δ ν 4 [ d n 1 d ν ( ν 0 2 ) + d n 2 d ν ( ν 0 2 ) ] .
n ¯ ( ν 0 , ν 1 ) = n ¯ + B Δ ν ,
δ n ( ν 0 , ν 1 ) = δ n + A Δ ν ,
n ¯ = n 1 ( ν 0 / 2 ) + n 2 ( ν 0 / 2 ) 2 ,
δ n = n 1 ( ν 0 / 2 ) n 2 ( ν 0 / 2 ) 2 ,
A = 1 4 [ d n 1 d ν ( ν 0 2 ) + d n 2 d ν ( ν 0 2 ) ] ,
B = 1 4 [ d n 1 d ν ( ν 0 2 ) d n 2 d ν ( ν 0 2 ) ] .
Δ ν = ν 0 δ n l c m L + n ¯ l + ν 0 A l .
Δ ν = 2 p c ν 0 [ L + ( n ¯ + δ n ) l ] L + ( n ¯ + δ n ) l + ( A + B ) l ν 0 .
ν 0 ( L + n ¯ l ) + ( 2 p + m ) c = Δ ν ( δ n l + B l ν 0 ) .
δ n = δ n + B ν 0 .

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