Abstract

We measure and model parametric gain and oscillation for two crystals arranged for walkoff compensation. We show how the orientation of the crystals determines the relative sign of the nonlinear mixing coefficient in the two crystals. This sign dramatically influences small signal gain and oscillator performance, and we show how to determine the correct crystal orientation from parametric-gain measurements. The performance of two-crystal oscillators is examined with particular attention to beam tilts, conversion efficiency, and beam quality. We find reduced efficiency and increased oscillation threshold when the coefficients have opposite signs in a two-crystal ring oscillator. Sign reversal seems to have little influence on spectral purity or far-field beam profiles when the oscillator is seeded.

© 1997 Optical Society of America

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References

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  1. V. D. Volosov, A. G. Kalintsev, and V. N. Krylov, “Degenerate parametric processes in three-wave interactions in tandem crystals,” Sov. Tech. Phys. Lett. 2, 32–34 (1976).
  2. V. D. Volosov, A. G. Kalintsev, and V. N. Krylov, “Suppression of degenerate parametric processes limiting frequency-doubling efficiency of crystals,” Sov. J. Quantum Electron. 6, 1163–1167 (1976).
    [Crossref]
  3. V. D. Volosov and A. G. Kalintsev, “Optimum optical second-harmonic generation in tandem crystals,” Sov. Tech. Phys. Lett. 2, 373–375 (1976).
  4. V. D. Volosov, A. G. Kalintsev, and V. N. Krylov, “Phase effects in a double-pass frequency doubler,” Sov. Tech. Phys. Lett. 5, 5–7 (1979).
  5. R. B. Andreev, K. V. Vetrov, V. D. Volosov, and A. G. Kalintsev, “Three-wave parametric processes in multicrystal nonlinear frequency converters,” Opt. Spectrosc. 65, 90–93 (1988).
  6. B. Ya. Zel’dovich, Yu. E. Kapitskii, and A. N. Chudinov, “Interference between second harmonics generated in two different KTP crystals,” Sov. J. Quantum Electron. 20, 1120–1121 (1990).
    [Crossref]
  7. M. Watanabe, K. Hayasaka, H. Imajo, and S. Urabe, “Continuous-wave sum-frequency generation near 194 nm with a collinear double enhancement cavity,” Opt. Commun. 97, 225–227 (1993).
    [Crossref]
  8. L. K. Samanta, T. Yanagawa, and Y. Yamamoto, “Technique for enhanced second harmonic output power,” Opt. Commun. 76, 250–252 (1990).
    [Crossref]
  9. W. R. Bosenberg, W. S. Pelouch, and C. L. Tang, “High-efficiency and narrow-linewidth operation of a two-crystal β-BaB2O4 optical parametric oscillator,” Appl. Phys. Lett. 55, 1952–1954 (1989).
    [Crossref]
  10. J. Zondy, M. Abed, and S. Khodja, “Twin-crystal walk-off-compensated type-II second-harmonic generation: single-pass and cavity-enhanced experiments in KTiOPO4,” J. Opt. Soc. Am. B 11, 2368–2379 (1994).
    [Crossref]
  11. J.-J. Zondy, “Experimental investigation of single and twin AgGaSe2 crystals for cw 10.2 µm SHG,” Opt. Commun. 119, 320–326 (1995).
    [Crossref]
  12. M. A. Norton, D. Eimerl, C. A. Ebbers, S. P. Velsko, and C. S. Petty, “KD*P frequency doubler for high average power applications,” Proc. SPIE 1223, 75–83 (1990).
    [Crossref]
  13. S. E. Harris, “Method to lock an optical parametric oscillator to an atomic transition,” Appl. Phys. Lett. 14, 335–337 (1969).
    [Crossref]
  14. A. Yariv, Introduction to Optical Electronics (Holt, Rinehart, & Winston, New York, 1976).
  15. R. W. Boyd, Nonlinear Optics (Academic, Boston, Mass., 1992).
  16. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).
  17. F. Zernike and J. E. Midwinter, Applied Nonlinear Optics (Wiley, New York, 1973).
  18. V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, New York, 1991).
  19. D. J. Armstrong, W. J. Alford, T. D. Raymond, and A. V. Smith, “Absolute measurement of the effective nonlinearities of KTP and BBO crystals by optical parametric amplification,” Appl. Opt. 35, 2032–2040 (1996).
    [Crossref] [PubMed]
  20. T. D. Raymond, W. J. Alford, A. V. Smith, and M. S. Bowers, “Frequency shifts in injection-seeded optical parametric oscillators with phase mismatch,” Opt. Lett. 19, 1520–1522 (1994).
    [Crossref] [PubMed]
  21. A. V. Smith, W. J. Alford, T. D. Raymond, and M. S. Bowers, “Comparison of a numerical model with measured performance of a seeded, nanosecond KTP optical parametric oscillator,” J. Opt. Soc. Am. B 12, 2253–2267 (1995).
    [Crossref]
  22. A. V. Smith and M. S. Bowers, “Phase distortions in sum- and difference-frequency mixing in crystals,” J. Opt. Soc. Am. B 12, 49–57 (1995).
    [Crossref]

1996 (1)

1995 (3)

1994 (2)

1993 (1)

M. Watanabe, K. Hayasaka, H. Imajo, and S. Urabe, “Continuous-wave sum-frequency generation near 194 nm with a collinear double enhancement cavity,” Opt. Commun. 97, 225–227 (1993).
[Crossref]

1990 (3)

L. K. Samanta, T. Yanagawa, and Y. Yamamoto, “Technique for enhanced second harmonic output power,” Opt. Commun. 76, 250–252 (1990).
[Crossref]

B. Ya. Zel’dovich, Yu. E. Kapitskii, and A. N. Chudinov, “Interference between second harmonics generated in two different KTP crystals,” Sov. J. Quantum Electron. 20, 1120–1121 (1990).
[Crossref]

M. A. Norton, D. Eimerl, C. A. Ebbers, S. P. Velsko, and C. S. Petty, “KD*P frequency doubler for high average power applications,” Proc. SPIE 1223, 75–83 (1990).
[Crossref]

1989 (1)

W. R. Bosenberg, W. S. Pelouch, and C. L. Tang, “High-efficiency and narrow-linewidth operation of a two-crystal β-BaB2O4 optical parametric oscillator,” Appl. Phys. Lett. 55, 1952–1954 (1989).
[Crossref]

1988 (1)

R. B. Andreev, K. V. Vetrov, V. D. Volosov, and A. G. Kalintsev, “Three-wave parametric processes in multicrystal nonlinear frequency converters,” Opt. Spectrosc. 65, 90–93 (1988).

1979 (1)

V. D. Volosov, A. G. Kalintsev, and V. N. Krylov, “Phase effects in a double-pass frequency doubler,” Sov. Tech. Phys. Lett. 5, 5–7 (1979).

1976 (3)

V. D. Volosov, A. G. Kalintsev, and V. N. Krylov, “Degenerate parametric processes in three-wave interactions in tandem crystals,” Sov. Tech. Phys. Lett. 2, 32–34 (1976).

V. D. Volosov, A. G. Kalintsev, and V. N. Krylov, “Suppression of degenerate parametric processes limiting frequency-doubling efficiency of crystals,” Sov. J. Quantum Electron. 6, 1163–1167 (1976).
[Crossref]

V. D. Volosov and A. G. Kalintsev, “Optimum optical second-harmonic generation in tandem crystals,” Sov. Tech. Phys. Lett. 2, 373–375 (1976).

1969 (1)

S. E. Harris, “Method to lock an optical parametric oscillator to an atomic transition,” Appl. Phys. Lett. 14, 335–337 (1969).
[Crossref]

Abed, M.

Alford, W. J.

Andreev, R. B.

R. B. Andreev, K. V. Vetrov, V. D. Volosov, and A. G. Kalintsev, “Three-wave parametric processes in multicrystal nonlinear frequency converters,” Opt. Spectrosc. 65, 90–93 (1988).

Armstrong, D. J.

Bosenberg, W. R.

W. R. Bosenberg, W. S. Pelouch, and C. L. Tang, “High-efficiency and narrow-linewidth operation of a two-crystal β-BaB2O4 optical parametric oscillator,” Appl. Phys. Lett. 55, 1952–1954 (1989).
[Crossref]

Bowers, M. S.

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, Boston, Mass., 1992).

Chudinov, A. N.

B. Ya. Zel’dovich, Yu. E. Kapitskii, and A. N. Chudinov, “Interference between second harmonics generated in two different KTP crystals,” Sov. J. Quantum Electron. 20, 1120–1121 (1990).
[Crossref]

Dmitriev, V. G.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, New York, 1991).

Ebbers, C. A.

M. A. Norton, D. Eimerl, C. A. Ebbers, S. P. Velsko, and C. S. Petty, “KD*P frequency doubler for high average power applications,” Proc. SPIE 1223, 75–83 (1990).
[Crossref]

Eimerl, D.

M. A. Norton, D. Eimerl, C. A. Ebbers, S. P. Velsko, and C. S. Petty, “KD*P frequency doubler for high average power applications,” Proc. SPIE 1223, 75–83 (1990).
[Crossref]

Gurzadyan, G. G.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, New York, 1991).

Harris, S. E.

S. E. Harris, “Method to lock an optical parametric oscillator to an atomic transition,” Appl. Phys. Lett. 14, 335–337 (1969).
[Crossref]

Hayasaka, K.

M. Watanabe, K. Hayasaka, H. Imajo, and S. Urabe, “Continuous-wave sum-frequency generation near 194 nm with a collinear double enhancement cavity,” Opt. Commun. 97, 225–227 (1993).
[Crossref]

Imajo, H.

M. Watanabe, K. Hayasaka, H. Imajo, and S. Urabe, “Continuous-wave sum-frequency generation near 194 nm with a collinear double enhancement cavity,” Opt. Commun. 97, 225–227 (1993).
[Crossref]

Kalintsev, A. G.

R. B. Andreev, K. V. Vetrov, V. D. Volosov, and A. G. Kalintsev, “Three-wave parametric processes in multicrystal nonlinear frequency converters,” Opt. Spectrosc. 65, 90–93 (1988).

V. D. Volosov, A. G. Kalintsev, and V. N. Krylov, “Phase effects in a double-pass frequency doubler,” Sov. Tech. Phys. Lett. 5, 5–7 (1979).

V. D. Volosov and A. G. Kalintsev, “Optimum optical second-harmonic generation in tandem crystals,” Sov. Tech. Phys. Lett. 2, 373–375 (1976).

V. D. Volosov, A. G. Kalintsev, and V. N. Krylov, “Degenerate parametric processes in three-wave interactions in tandem crystals,” Sov. Tech. Phys. Lett. 2, 32–34 (1976).

V. D. Volosov, A. G. Kalintsev, and V. N. Krylov, “Suppression of degenerate parametric processes limiting frequency-doubling efficiency of crystals,” Sov. J. Quantum Electron. 6, 1163–1167 (1976).
[Crossref]

Kapitskii, Yu. E.

B. Ya. Zel’dovich, Yu. E. Kapitskii, and A. N. Chudinov, “Interference between second harmonics generated in two different KTP crystals,” Sov. J. Quantum Electron. 20, 1120–1121 (1990).
[Crossref]

Khodja, S.

Krylov, V. N.

V. D. Volosov, A. G. Kalintsev, and V. N. Krylov, “Phase effects in a double-pass frequency doubler,” Sov. Tech. Phys. Lett. 5, 5–7 (1979).

V. D. Volosov, A. G. Kalintsev, and V. N. Krylov, “Suppression of degenerate parametric processes limiting frequency-doubling efficiency of crystals,” Sov. J. Quantum Electron. 6, 1163–1167 (1976).
[Crossref]

V. D. Volosov, A. G. Kalintsev, and V. N. Krylov, “Degenerate parametric processes in three-wave interactions in tandem crystals,” Sov. Tech. Phys. Lett. 2, 32–34 (1976).

Midwinter, J. E.

F. Zernike and J. E. Midwinter, Applied Nonlinear Optics (Wiley, New York, 1973).

Nikogosyan, D. N.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, New York, 1991).

Norton, M. A.

M. A. Norton, D. Eimerl, C. A. Ebbers, S. P. Velsko, and C. S. Petty, “KD*P frequency doubler for high average power applications,” Proc. SPIE 1223, 75–83 (1990).
[Crossref]

Pelouch, W. S.

W. R. Bosenberg, W. S. Pelouch, and C. L. Tang, “High-efficiency and narrow-linewidth operation of a two-crystal β-BaB2O4 optical parametric oscillator,” Appl. Phys. Lett. 55, 1952–1954 (1989).
[Crossref]

Petty, C. S.

M. A. Norton, D. Eimerl, C. A. Ebbers, S. P. Velsko, and C. S. Petty, “KD*P frequency doubler for high average power applications,” Proc. SPIE 1223, 75–83 (1990).
[Crossref]

Raymond, T. D.

Samanta, L. K.

L. K. Samanta, T. Yanagawa, and Y. Yamamoto, “Technique for enhanced second harmonic output power,” Opt. Commun. 76, 250–252 (1990).
[Crossref]

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

Smith, A. V.

Tang, C. L.

W. R. Bosenberg, W. S. Pelouch, and C. L. Tang, “High-efficiency and narrow-linewidth operation of a two-crystal β-BaB2O4 optical parametric oscillator,” Appl. Phys. Lett. 55, 1952–1954 (1989).
[Crossref]

Urabe, S.

M. Watanabe, K. Hayasaka, H. Imajo, and S. Urabe, “Continuous-wave sum-frequency generation near 194 nm with a collinear double enhancement cavity,” Opt. Commun. 97, 225–227 (1993).
[Crossref]

Velsko, S. P.

M. A. Norton, D. Eimerl, C. A. Ebbers, S. P. Velsko, and C. S. Petty, “KD*P frequency doubler for high average power applications,” Proc. SPIE 1223, 75–83 (1990).
[Crossref]

Vetrov, K. V.

R. B. Andreev, K. V. Vetrov, V. D. Volosov, and A. G. Kalintsev, “Three-wave parametric processes in multicrystal nonlinear frequency converters,” Opt. Spectrosc. 65, 90–93 (1988).

Volosov, V. D.

R. B. Andreev, K. V. Vetrov, V. D. Volosov, and A. G. Kalintsev, “Three-wave parametric processes in multicrystal nonlinear frequency converters,” Opt. Spectrosc. 65, 90–93 (1988).

V. D. Volosov, A. G. Kalintsev, and V. N. Krylov, “Phase effects in a double-pass frequency doubler,” Sov. Tech. Phys. Lett. 5, 5–7 (1979).

V. D. Volosov, A. G. Kalintsev, and V. N. Krylov, “Suppression of degenerate parametric processes limiting frequency-doubling efficiency of crystals,” Sov. J. Quantum Electron. 6, 1163–1167 (1976).
[Crossref]

V. D. Volosov and A. G. Kalintsev, “Optimum optical second-harmonic generation in tandem crystals,” Sov. Tech. Phys. Lett. 2, 373–375 (1976).

V. D. Volosov, A. G. Kalintsev, and V. N. Krylov, “Degenerate parametric processes in three-wave interactions in tandem crystals,” Sov. Tech. Phys. Lett. 2, 32–34 (1976).

Watanabe, M.

M. Watanabe, K. Hayasaka, H. Imajo, and S. Urabe, “Continuous-wave sum-frequency generation near 194 nm with a collinear double enhancement cavity,” Opt. Commun. 97, 225–227 (1993).
[Crossref]

Yamamoto, Y.

L. K. Samanta, T. Yanagawa, and Y. Yamamoto, “Technique for enhanced second harmonic output power,” Opt. Commun. 76, 250–252 (1990).
[Crossref]

Yanagawa, T.

L. K. Samanta, T. Yanagawa, and Y. Yamamoto, “Technique for enhanced second harmonic output power,” Opt. Commun. 76, 250–252 (1990).
[Crossref]

Yariv, A.

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart, & Winston, New York, 1976).

Zel’dovich, B. Ya.

B. Ya. Zel’dovich, Yu. E. Kapitskii, and A. N. Chudinov, “Interference between second harmonics generated in two different KTP crystals,” Sov. J. Quantum Electron. 20, 1120–1121 (1990).
[Crossref]

Zernike, F.

F. Zernike and J. E. Midwinter, Applied Nonlinear Optics (Wiley, New York, 1973).

Zondy, J.

Zondy, J.-J.

J.-J. Zondy, “Experimental investigation of single and twin AgGaSe2 crystals for cw 10.2 µm SHG,” Opt. Commun. 119, 320–326 (1995).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

S. E. Harris, “Method to lock an optical parametric oscillator to an atomic transition,” Appl. Phys. Lett. 14, 335–337 (1969).
[Crossref]

W. R. Bosenberg, W. S. Pelouch, and C. L. Tang, “High-efficiency and narrow-linewidth operation of a two-crystal β-BaB2O4 optical parametric oscillator,” Appl. Phys. Lett. 55, 1952–1954 (1989).
[Crossref]

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

Opt. Commun. (3)

J.-J. Zondy, “Experimental investigation of single and twin AgGaSe2 crystals for cw 10.2 µm SHG,” Opt. Commun. 119, 320–326 (1995).
[Crossref]

M. Watanabe, K. Hayasaka, H. Imajo, and S. Urabe, “Continuous-wave sum-frequency generation near 194 nm with a collinear double enhancement cavity,” Opt. Commun. 97, 225–227 (1993).
[Crossref]

L. K. Samanta, T. Yanagawa, and Y. Yamamoto, “Technique for enhanced second harmonic output power,” Opt. Commun. 76, 250–252 (1990).
[Crossref]

Opt. Lett. (1)

Opt. Spectrosc. (1)

R. B. Andreev, K. V. Vetrov, V. D. Volosov, and A. G. Kalintsev, “Three-wave parametric processes in multicrystal nonlinear frequency converters,” Opt. Spectrosc. 65, 90–93 (1988).

Proc. SPIE (1)

M. A. Norton, D. Eimerl, C. A. Ebbers, S. P. Velsko, and C. S. Petty, “KD*P frequency doubler for high average power applications,” Proc. SPIE 1223, 75–83 (1990).
[Crossref]

Sov. J. Quantum Electron. (2)

B. Ya. Zel’dovich, Yu. E. Kapitskii, and A. N. Chudinov, “Interference between second harmonics generated in two different KTP crystals,” Sov. J. Quantum Electron. 20, 1120–1121 (1990).
[Crossref]

V. D. Volosov, A. G. Kalintsev, and V. N. Krylov, “Suppression of degenerate parametric processes limiting frequency-doubling efficiency of crystals,” Sov. J. Quantum Electron. 6, 1163–1167 (1976).
[Crossref]

Sov. Tech. Phys. Lett. (3)

V. D. Volosov and A. G. Kalintsev, “Optimum optical second-harmonic generation in tandem crystals,” Sov. Tech. Phys. Lett. 2, 373–375 (1976).

V. D. Volosov, A. G. Kalintsev, and V. N. Krylov, “Phase effects in a double-pass frequency doubler,” Sov. Tech. Phys. Lett. 5, 5–7 (1979).

V. D. Volosov, A. G. Kalintsev, and V. N. Krylov, “Degenerate parametric processes in three-wave interactions in tandem crystals,” Sov. Tech. Phys. Lett. 2, 32–34 (1976).

Other (5)

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart, & Winston, New York, 1976).

R. W. Boyd, Nonlinear Optics (Academic, Boston, Mass., 1992).

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

F. Zernike and J. E. Midwinter, Applied Nonlinear Optics (Wiley, New York, 1973).

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, New York, 1991).

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Figures (13)

Fig. 1
Fig. 1

Two-crystal, single-pass parametric gain calculated from Eq. (16) for identical crystals with εi0=0 and A1B1=A2B2=7.47×103m-2 with deff values of (a) the same sign and (b) opposite signs.

Fig. 2
Fig. 2

Same as Fig. 1 but with A1B1=A2B2=2.99×104m-2.

Fig. 3
Fig. 3

Diagrams used to deduce relative signs of the deff values for various crystal orientations. The orientation labeled (a) represents the baseline case. The dashed arrows in (b), (c), and (d) indicate axes about which the crystal is rotated by 180°. The arrows to the left of the crystals represent the directions of the applied fields and the induced polarization. The diagonal lines on top of the crystals represent the orientation of the optic axes.

Fig. 4
Fig. 4

(a) Contour plot of a two-crystal gain surface in the low gain limit with Gs=0.1. (b) Cuts through the gain surface along the lines of slope ±1. Line styles in (b) correspond to the axes shown in (a).

Fig. 5
Fig. 5

Experimental arrangement for two-crystal, single-pass parametric-gain measurements. HR, highly reflective.

Fig. 6
Fig. 6

Two-crystal, single-pass parametric gain. (a) Measured and (b) calculated with Eq. (16), the crystals are oriented so the signs of deff are the same. (c) For the measured gain the signs of deff are opposite, but a phase-correction plate is inserted between the crystals. Pump fluence and the resulting gain in (c) are larger than in (a).

Fig. 7
Fig. 7

Two-crystal, single-pass parametric gain with crystals oriented so the signs of deff are opposite. (a) Measured and (b) calculated with Eq. (16).

Fig. 8
Fig. 8

Experimental arrangement for two-crystal optical parametric oscillator measurements. PZT, piezoelectric transducer.

Fig. 9
Fig. 9

Contour plots of calculated two-crystal, single-pass parametric gain. Crystal rotations for “Tune” and “Tilt” behavior exhibited by the two-crystal walkoff-compensated oscillator are indicated on the diagonal axes. (a) Crystals are oriented so the signs of deff are the same. (b) Crystals are oriented so the signs of deff are opposite.

Fig. 10
Fig. 10

Measured far-field signal-fluence profiles for increasing values of Δk1L/π, Δk2L/π plotted against the critical and the noncritical divergence angles ϴC and ϴNC with the crystals oriented so the signs of deff are the same. The peak fluence for each profile is normalized to 1. In the left column the range of Δk1L/π, Δk2L/π is from 0.32, -0.32 to 1.62, -1.62 (Δk1, Δk2=1, -1 to 5, -5 cm-1). In the right column the range is the same, but the signs of Δk1L/π, Δk2L/π are reversed.

Fig. 11
Fig. 11

Path taken by a tilted e-polarized wave in the two-crystal walkoff-compensated three-mirror ring OPO cavity. Lc is the cavity length, and α is the tilt angle.

Fig. 12
Fig. 12

Signal energy versus pump energy for (a) the correct orientation with Δk1=Δk2=0 and (b) the incorrect orientation with Δk1L/π=Δk2L/π-0.45.

Fig. 13
Fig. 13

Far-field signal-fluence profiles with 6 mJ of pump energy plotted against the critical and the noncritical divergence angles ϴC and ϴNC. The peak fluence for each profile is normalized to 1. (a) Measured, seeded, correct orientation, Δk1=Δk2=0. (b) Calculated, same as (a). (c) Measured, seeded, incorrect orientation, Δk1L/π=Δk2L/π-0.45. (d) Calculated, same as (c). (e) Same as (c) but unseeded.

Equations (21)

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

Pj(ω3)=χj,k,l(2)(-ω3; ω1ω2)Ek(ω1)El(ω2),
P(ω3)=2odeffE(ω1)E(ω2).
dεsdz=ideffcωsnsεpεi* exp(iΔkz),
dεidz=ideffcωiniεpεs* exp(iΔkz),
dεpdz=ideffcωpnpεsεi exp(-iΔkz),
Eω=12{εω exp[-i(ωt-kωz)]+εω* exp[i(ωt-kωz)]},
Δk=kp-ks-ki,
ωp=ωs+ωi.
εs(z)=εs0cosh γz-iΔk2γsinh γzexpiΔkz2+iAγεi*0sinh γz expiΔkz2,
εi*z=εi*0cosh γz+iΔk2γsinh γzexp-iΔkz2-iBγεs0sinh γz exp-iΔkz2,
A=deffωsεpcns,
B=deffωiεp*cni,
γ=124AB-Δk2.
εsz=εs0cosh γz-iΔk2γsinh γzexpiΔkz2.
Gs=εsLεs02-1.
εs=εs0expiΔk1L1/2+iΔk2L2/2×cosh γ1L1 cosh γ2L2-Δk1Δk24γ1γ2sinh γ1L1 sinh γ2L2+A2B1γ1γ2sinh γ1L1 sinh γ2L2 expiθ-iΔk12γ1sinh γ1L1 cosh γ2L2-iΔk22γ2cosh γ1L1 sinh γ2L2+εi*0expiΔk1L1/2+iΔk2L2/2×A1Δk22γ1γ2sinh γ1L1 sinh γ2L2-A2Δk12γ1γ2sinh γ1L1 sinh γ2L2 expiθ+iA1γ1sinh γ1L1 cosh γ2L2+iA2γ2cosh γ1L1 sinh γ2L2 expiθ,
E2ω=ωdeffncEω2LsinΔkL/2ΔkL/2expiΔkL/2,
E2ω=ωdeffncEω2L2sinΔkL/4ΔkL/4expiΔkL/4exp-iΔkL/2
E2ω=ωdeffncEω2L2sinΔkL/4ΔkL/4exp-iΔkL/4.
E2ω=ωdeffncEω2LsinΔkL/4ΔkL/4exp-iΔkL/4.
E2ω=ωdeffncEω2LsinΔkL/2NΔkL/2Nexp-iΔkL/2N,

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