Abstract

We show that we can efficiently achieve surface-emitting second-harmonic generation in vertical and horizontal cavities. The fundamental beam is coupled into the waveguide, which consists of III–V or II–VI semiconductor multilayers or asymmetric quantum-well domain structures. The generated second-harmonic radiation propagates along the growth direction of these layers (which is normal to the propagation direction of the fundamental beam). The quasi-phase matching is achieved when second-order susceptibility is modulated along the growth direction in these structures. By the proper design of these structures, the frequency doublers based on these structures together can cover the spectrum of 0.8–2.0 μm. If the pump power density is sufficiently large, the conversion efficiency approaches saturation. The saturation power per unit waveguide width is between ~0.9 and ~435 mW/μm. At such a power density, 72% conversion efficiency can be achieved. In addition, the proposed frequency doublers are, in principle, broadband.

© 1995 Optical Society of America

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References

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  1. N. Bloembergen, U.S. Patent 3,384,433 (May 21, 1968).
  2. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992).
    [CrossRef]
  3. R. Normandin, R. L. Williams, and F. Chatenoud, Electron. Lett. 26, 2088 (1990); R. Normandin, H. Dai, S. Janz, A. Delage, J. Brown, and F. Chatenoud, Appl. Phys. Lett. 62, 118 (1993); D. Vakhshoori, R. J. Fischer, M. Hong, D. L. Sivco, G. J. Zydzik, G. N. S. Chu, and A. Y. Cho, Appl. Phys. Lett. 59, 896 (1991).
    [CrossRef]
  4. R. Lodenkamper, M. L. Bortz, M. M. Fejer, K. Bacher, and J. S. Harris, Jr., Opt. Lett. 18, 1798 (1993).
    [CrossRef] [PubMed]
  5. J. Khurgin, Appl. Phys. Lett. 21, 2100 (1987); Phys. Rev. B38, 4056 (1988); J. Appl. Phys. 64, 5026 (1988); J. Opt. Soc. Am. B 6, 1673 (1989).
    [CrossRef]
  6. S. Janz, F. Chatenoud, and R. Normandin, Opt. Lett. 19, 622 (1994).
    [CrossRef] [PubMed]
  7. J. B. Khurgin, S. J. Lee, and Y. J. Ding, in Proceedings of NAECON′94 (IEEE, New York, 1994), p. 520.
  8. R. Normandin, S. Letourneau, F. Chatenoud, and R. L. Williams, IEEE J. Quantum Electron. 27, 1520 (1991); D. Vakhshoori and S. Wang, J. Lightwave Technol. 9, 906 (1991).
    [CrossRef]
  9. S. Janz, E. Frlan, H. Dai, F. Chatenoud, and R. Normandin, in Nonlinear Optics: Materials, Fundamentals, and Applications, Vol. 18 of OSA 1992 Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 263.
  10. R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. I. Stegeman, E. W. Van Stryland, and H. Vanhertzeele, Opt. Lett. 17, 28 (1992); G. I. Stegeman, M. Sheik-Bahae, E. W. Van Stryland, and G. Assanto, Opt. Lett. 18, 13 (1993); A. E. Kaplan, Opt. Lett. 18, 1223 (1993).
    [CrossRef] [PubMed]
  11. J. B. Khurgin and Y. J. Ding, Opt. Lett. 19, 1016 (1994).
    [CrossRef] [PubMed]
  12. Y. J. Ding, J. B. Khurgin, and S. J. Lee, in CCAST-WL Workshop Series: Vol. 38, Ultrafast Phenomena, K. Shum, Y. J. Ding, and X. C. Zhang, eds. (Gordon & Breach, Beijing, 1994), p. 60.
  13. S. J. Lee, J. B. Khurgin, and Y. J. Ding, J. Opt. Soc. Amer. B 12, 275 (1995).
    [CrossRef]
  14. A. Yariv, Quantum Electronics (Wiley, New York, 1989), p. 500.
  15. S. Li and J. Khurgin, Appl. Phys. Lett. 62, 1727 (1993); Z. Chen, M. Li, D. Cui, H. Lu, and G. Yang, Appl. Phys. Lett. 62, 1502 (1993).
    [CrossRef]
  16. A. Villeneuve, C. C. Yang, G. I. Stegeman, C. H. Lin, and H. H. Lin, Appl. Phys. Lett. 62, 2465 (1993).
    [CrossRef]
  17. T. S. Moss, Optical Properties of Semiconductors (Butter-worths, London, 1959), p. 48.
  18. K. W. Böer, Survey of Semiconductor Physics (Van Nostrand, New York, 1990), p. 235.

Bacher, K.

R. Lodenkamper, M. L. Bortz, M. M. Fejer, K. Bacher, and J. S. Harris, Jr., Opt. Lett. 18, 1798 (1993).
[CrossRef] [PubMed]

Bloembergen, N.

N. Bloembergen, U.S. Patent 3,384,433 (May 21, 1968).

Böer, K. W.

K. W. Böer, Survey of Semiconductor Physics (Van Nostrand, New York, 1990), p. 235.

Bortz, M. L.

R. Lodenkamper, M. L. Bortz, M. M. Fejer, K. Bacher, and J. S. Harris, Jr., Opt. Lett. 18, 1798 (1993).
[CrossRef] [PubMed]

Byer, R. L.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992).
[CrossRef]

Chatenoud, F.

R. Normandin, R. L. Williams, and F. Chatenoud, Electron. Lett. 26, 2088 (1990); R. Normandin, H. Dai, S. Janz, A. Delage, J. Brown, and F. Chatenoud, Appl. Phys. Lett. 62, 118 (1993); D. Vakhshoori, R. J. Fischer, M. Hong, D. L. Sivco, G. J. Zydzik, G. N. S. Chu, and A. Y. Cho, Appl. Phys. Lett. 59, 896 (1991).
[CrossRef]

S. Janz, F. Chatenoud, and R. Normandin, Opt. Lett. 19, 622 (1994).
[CrossRef] [PubMed]

R. Normandin, S. Letourneau, F. Chatenoud, and R. L. Williams, IEEE J. Quantum Electron. 27, 1520 (1991); D. Vakhshoori and S. Wang, J. Lightwave Technol. 9, 906 (1991).
[CrossRef]

S. Janz, E. Frlan, H. Dai, F. Chatenoud, and R. Normandin, in Nonlinear Optics: Materials, Fundamentals, and Applications, Vol. 18 of OSA 1992 Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 263.

Dai, H.

S. Janz, E. Frlan, H. Dai, F. Chatenoud, and R. Normandin, in Nonlinear Optics: Materials, Fundamentals, and Applications, Vol. 18 of OSA 1992 Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 263.

DeSalvo, R.

R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. I. Stegeman, E. W. Van Stryland, and H. Vanhertzeele, Opt. Lett. 17, 28 (1992); G. I. Stegeman, M. Sheik-Bahae, E. W. Van Stryland, and G. Assanto, Opt. Lett. 18, 13 (1993); A. E. Kaplan, Opt. Lett. 18, 1223 (1993).
[CrossRef] [PubMed]

Ding, Y. J.

J. B. Khurgin and Y. J. Ding, Opt. Lett. 19, 1016 (1994).
[CrossRef] [PubMed]

Y. J. Ding, J. B. Khurgin, and S. J. Lee, in CCAST-WL Workshop Series: Vol. 38, Ultrafast Phenomena, K. Shum, Y. J. Ding, and X. C. Zhang, eds. (Gordon & Breach, Beijing, 1994), p. 60.

J. B. Khurgin, S. J. Lee, and Y. J. Ding, in Proceedings of NAECON′94 (IEEE, New York, 1994), p. 520.

S. J. Lee, J. B. Khurgin, and Y. J. Ding, J. Opt. Soc. Amer. B 12, 275 (1995).
[CrossRef]

Fejer, M. M.

R. Lodenkamper, M. L. Bortz, M. M. Fejer, K. Bacher, and J. S. Harris, Jr., Opt. Lett. 18, 1798 (1993).
[CrossRef] [PubMed]

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992).
[CrossRef]

Frlan, E.

S. Janz, E. Frlan, H. Dai, F. Chatenoud, and R. Normandin, in Nonlinear Optics: Materials, Fundamentals, and Applications, Vol. 18 of OSA 1992 Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 263.

Hagan, D. J.

R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. I. Stegeman, E. W. Van Stryland, and H. Vanhertzeele, Opt. Lett. 17, 28 (1992); G. I. Stegeman, M. Sheik-Bahae, E. W. Van Stryland, and G. Assanto, Opt. Lett. 18, 13 (1993); A. E. Kaplan, Opt. Lett. 18, 1223 (1993).
[CrossRef] [PubMed]

Harris, J. S.

R. Lodenkamper, M. L. Bortz, M. M. Fejer, K. Bacher, and J. S. Harris, Jr., Opt. Lett. 18, 1798 (1993).
[CrossRef] [PubMed]

Janz, S.

S. Janz, F. Chatenoud, and R. Normandin, Opt. Lett. 19, 622 (1994).
[CrossRef] [PubMed]

S. Janz, E. Frlan, H. Dai, F. Chatenoud, and R. Normandin, in Nonlinear Optics: Materials, Fundamentals, and Applications, Vol. 18 of OSA 1992 Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 263.

Jundt, D. H.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992).
[CrossRef]

Khurgin, J.

J. Khurgin, Appl. Phys. Lett. 21, 2100 (1987); Phys. Rev. B38, 4056 (1988); J. Appl. Phys. 64, 5026 (1988); J. Opt. Soc. Am. B 6, 1673 (1989).
[CrossRef]

S. Li and J. Khurgin, Appl. Phys. Lett. 62, 1727 (1993); Z. Chen, M. Li, D. Cui, H. Lu, and G. Yang, Appl. Phys. Lett. 62, 1502 (1993).
[CrossRef]

Khurgin, J. B.

J. B. Khurgin and Y. J. Ding, Opt. Lett. 19, 1016 (1994).
[CrossRef] [PubMed]

S. J. Lee, J. B. Khurgin, and Y. J. Ding, J. Opt. Soc. Amer. B 12, 275 (1995).
[CrossRef]

J. B. Khurgin, S. J. Lee, and Y. J. Ding, in Proceedings of NAECON′94 (IEEE, New York, 1994), p. 520.

Y. J. Ding, J. B. Khurgin, and S. J. Lee, in CCAST-WL Workshop Series: Vol. 38, Ultrafast Phenomena, K. Shum, Y. J. Ding, and X. C. Zhang, eds. (Gordon & Breach, Beijing, 1994), p. 60.

Lee, S. J.

Y. J. Ding, J. B. Khurgin, and S. J. Lee, in CCAST-WL Workshop Series: Vol. 38, Ultrafast Phenomena, K. Shum, Y. J. Ding, and X. C. Zhang, eds. (Gordon & Breach, Beijing, 1994), p. 60.

S. J. Lee, J. B. Khurgin, and Y. J. Ding, J. Opt. Soc. Amer. B 12, 275 (1995).
[CrossRef]

J. B. Khurgin, S. J. Lee, and Y. J. Ding, in Proceedings of NAECON′94 (IEEE, New York, 1994), p. 520.

Letourneau, S.

R. Normandin, S. Letourneau, F. Chatenoud, and R. L. Williams, IEEE J. Quantum Electron. 27, 1520 (1991); D. Vakhshoori and S. Wang, J. Lightwave Technol. 9, 906 (1991).
[CrossRef]

Li, S.

S. Li and J. Khurgin, Appl. Phys. Lett. 62, 1727 (1993); Z. Chen, M. Li, D. Cui, H. Lu, and G. Yang, Appl. Phys. Lett. 62, 1502 (1993).
[CrossRef]

Lin, C. H.

A. Villeneuve, C. C. Yang, G. I. Stegeman, C. H. Lin, and H. H. Lin, Appl. Phys. Lett. 62, 2465 (1993).
[CrossRef]

Lin, H. H.

A. Villeneuve, C. C. Yang, G. I. Stegeman, C. H. Lin, and H. H. Lin, Appl. Phys. Lett. 62, 2465 (1993).
[CrossRef]

Lodenkamper, R.

R. Lodenkamper, M. L. Bortz, M. M. Fejer, K. Bacher, and J. S. Harris, Jr., Opt. Lett. 18, 1798 (1993).
[CrossRef] [PubMed]

Magel, G. A.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992).
[CrossRef]

Moss, T. S.

T. S. Moss, Optical Properties of Semiconductors (Butter-worths, London, 1959), p. 48.

Normandin, R.

S. Janz, E. Frlan, H. Dai, F. Chatenoud, and R. Normandin, in Nonlinear Optics: Materials, Fundamentals, and Applications, Vol. 18 of OSA 1992 Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 263.

R. Normandin, R. L. Williams, and F. Chatenoud, Electron. Lett. 26, 2088 (1990); R. Normandin, H. Dai, S. Janz, A. Delage, J. Brown, and F. Chatenoud, Appl. Phys. Lett. 62, 118 (1993); D. Vakhshoori, R. J. Fischer, M. Hong, D. L. Sivco, G. J. Zydzik, G. N. S. Chu, and A. Y. Cho, Appl. Phys. Lett. 59, 896 (1991).
[CrossRef]

S. Janz, F. Chatenoud, and R. Normandin, Opt. Lett. 19, 622 (1994).
[CrossRef] [PubMed]

R. Normandin, S. Letourneau, F. Chatenoud, and R. L. Williams, IEEE J. Quantum Electron. 27, 1520 (1991); D. Vakhshoori and S. Wang, J. Lightwave Technol. 9, 906 (1991).
[CrossRef]

Sheik-Bahae, M.

R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. I. Stegeman, E. W. Van Stryland, and H. Vanhertzeele, Opt. Lett. 17, 28 (1992); G. I. Stegeman, M. Sheik-Bahae, E. W. Van Stryland, and G. Assanto, Opt. Lett. 18, 13 (1993); A. E. Kaplan, Opt. Lett. 18, 1223 (1993).
[CrossRef] [PubMed]

Stegeman, G. I.

R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. I. Stegeman, E. W. Van Stryland, and H. Vanhertzeele, Opt. Lett. 17, 28 (1992); G. I. Stegeman, M. Sheik-Bahae, E. W. Van Stryland, and G. Assanto, Opt. Lett. 18, 13 (1993); A. E. Kaplan, Opt. Lett. 18, 1223 (1993).
[CrossRef] [PubMed]

A. Villeneuve, C. C. Yang, G. I. Stegeman, C. H. Lin, and H. H. Lin, Appl. Phys. Lett. 62, 2465 (1993).
[CrossRef]

Stryland, E. W. Van

R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. I. Stegeman, E. W. Van Stryland, and H. Vanhertzeele, Opt. Lett. 17, 28 (1992); G. I. Stegeman, M. Sheik-Bahae, E. W. Van Stryland, and G. Assanto, Opt. Lett. 18, 13 (1993); A. E. Kaplan, Opt. Lett. 18, 1223 (1993).
[CrossRef] [PubMed]

Vanhertzeele, H.

R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. I. Stegeman, E. W. Van Stryland, and H. Vanhertzeele, Opt. Lett. 17, 28 (1992); G. I. Stegeman, M. Sheik-Bahae, E. W. Van Stryland, and G. Assanto, Opt. Lett. 18, 13 (1993); A. E. Kaplan, Opt. Lett. 18, 1223 (1993).
[CrossRef] [PubMed]

Villeneuve, A.

A. Villeneuve, C. C. Yang, G. I. Stegeman, C. H. Lin, and H. H. Lin, Appl. Phys. Lett. 62, 2465 (1993).
[CrossRef]

Williams, R. L.

R. Normandin, S. Letourneau, F. Chatenoud, and R. L. Williams, IEEE J. Quantum Electron. 27, 1520 (1991); D. Vakhshoori and S. Wang, J. Lightwave Technol. 9, 906 (1991).
[CrossRef]

R. Normandin, R. L. Williams, and F. Chatenoud, Electron. Lett. 26, 2088 (1990); R. Normandin, H. Dai, S. Janz, A. Delage, J. Brown, and F. Chatenoud, Appl. Phys. Lett. 62, 118 (1993); D. Vakhshoori, R. J. Fischer, M. Hong, D. L. Sivco, G. J. Zydzik, G. N. S. Chu, and A. Y. Cho, Appl. Phys. Lett. 59, 896 (1991).
[CrossRef]

Yang, C. C.

A. Villeneuve, C. C. Yang, G. I. Stegeman, C. H. Lin, and H. H. Lin, Appl. Phys. Lett. 62, 2465 (1993).
[CrossRef]

Yariv, A.

A. Yariv, Quantum Electronics (Wiley, New York, 1989), p. 500.

Other

N. Bloembergen, U.S. Patent 3,384,433 (May 21, 1968).

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992).
[CrossRef]

R. Normandin, R. L. Williams, and F. Chatenoud, Electron. Lett. 26, 2088 (1990); R. Normandin, H. Dai, S. Janz, A. Delage, J. Brown, and F. Chatenoud, Appl. Phys. Lett. 62, 118 (1993); D. Vakhshoori, R. J. Fischer, M. Hong, D. L. Sivco, G. J. Zydzik, G. N. S. Chu, and A. Y. Cho, Appl. Phys. Lett. 59, 896 (1991).
[CrossRef]

R. Lodenkamper, M. L. Bortz, M. M. Fejer, K. Bacher, and J. S. Harris, Jr., Opt. Lett. 18, 1798 (1993).
[CrossRef] [PubMed]

J. Khurgin, Appl. Phys. Lett. 21, 2100 (1987); Phys. Rev. B38, 4056 (1988); J. Appl. Phys. 64, 5026 (1988); J. Opt. Soc. Am. B 6, 1673 (1989).
[CrossRef]

S. Janz, F. Chatenoud, and R. Normandin, Opt. Lett. 19, 622 (1994).
[CrossRef] [PubMed]

J. B. Khurgin, S. J. Lee, and Y. J. Ding, in Proceedings of NAECON′94 (IEEE, New York, 1994), p. 520.

R. Normandin, S. Letourneau, F. Chatenoud, and R. L. Williams, IEEE J. Quantum Electron. 27, 1520 (1991); D. Vakhshoori and S. Wang, J. Lightwave Technol. 9, 906 (1991).
[CrossRef]

S. Janz, E. Frlan, H. Dai, F. Chatenoud, and R. Normandin, in Nonlinear Optics: Materials, Fundamentals, and Applications, Vol. 18 of OSA 1992 Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 263.

R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. I. Stegeman, E. W. Van Stryland, and H. Vanhertzeele, Opt. Lett. 17, 28 (1992); G. I. Stegeman, M. Sheik-Bahae, E. W. Van Stryland, and G. Assanto, Opt. Lett. 18, 13 (1993); A. E. Kaplan, Opt. Lett. 18, 1223 (1993).
[CrossRef] [PubMed]

J. B. Khurgin and Y. J. Ding, Opt. Lett. 19, 1016 (1994).
[CrossRef] [PubMed]

Y. J. Ding, J. B. Khurgin, and S. J. Lee, in CCAST-WL Workshop Series: Vol. 38, Ultrafast Phenomena, K. Shum, Y. J. Ding, and X. C. Zhang, eds. (Gordon & Breach, Beijing, 1994), p. 60.

S. J. Lee, J. B. Khurgin, and Y. J. Ding, J. Opt. Soc. Amer. B 12, 275 (1995).
[CrossRef]

A. Yariv, Quantum Electronics (Wiley, New York, 1989), p. 500.

S. Li and J. Khurgin, Appl. Phys. Lett. 62, 1727 (1993); Z. Chen, M. Li, D. Cui, H. Lu, and G. Yang, Appl. Phys. Lett. 62, 1502 (1993).
[CrossRef]

A. Villeneuve, C. C. Yang, G. I. Stegeman, C. H. Lin, and H. H. Lin, Appl. Phys. Lett. 62, 2465 (1993).
[CrossRef]

T. S. Moss, Optical Properties of Semiconductors (Butter-worths, London, 1959), p. 48.

K. W. Böer, Survey of Semiconductor Physics (Van Nostrand, New York, 1990), p. 235.

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Figures (6)

Fig. 1
Fig. 1

Configuration for surface-emitting SHG. The middle plot is the band diagram of the large-scale ACQW domains. The bottom plot shows the sinusoidal component of the second-order susceptibility (solid curve) from the Fourier expansion of the periodically modulated second-order susceptibility in the large-scale ACQW domains (dashed line) for achieving quasi-phase matching.

Fig. 2
Fig. 2

Conversion efficiency versus the normalized pump power density without the horizontal cavity.

Fig. 3
Fig. 3

Normalized input and reflected power densities (P1/Ps and P2/Ps) versus the propagation distance z/L: line 1, P0/Ps ≈ 0.05; curve 2, P0/Ps ≈ 1; curve 3, P0/Ps ≈ 5; inset; the cavity structure.

Fig. 4
Fig. 4

Comparison between the conversion efficiencies versus the normalized pump power densities with (solid curve, top) and without (solid line, bottom) the horizontal cavity.

Fig. 5
Fig. 5

Normalized input and reflected power densities (P1/Ps and P2/Ps) versus the propagation distance (z/L): curve 1, P0/Ps ≈ 1.1 × 10−4; curve 2, P0/Ps ≈ 1.7 × 10−3, curve 3, P0/Ps ≈ 0.02; inset, the cavity structure.

Fig. 6
Fig. 6

Side-view structure of the one-dimensional array of the optical frequency doublers. The plane is (a) parallel to and (b) normal to the propagation direction of the input beam.

Tables (1)

Tables Icon

Table 1 Structure Parameters for Frequency Doubling the Laser Light in the Range of 0.98–1.8 μm

Equations (50)

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

2 E 1 ( 2 ) - n ω 2 c 2 2 E 1 ( 2 ) t 2 = 1 c 2 2 t 2 [ χ ( 2 ) E 2 ( 1 ) E 2 ω ] ,
2 E 2 ω - n 2 ω 2 c 2 2 E 2 ω t 2 - μ 0 σ E 2 ω t = 1 c 2 2 t 2 [ χ ( 2 ) E 1 E 2 ] ,
E 1 ( 2 ) = A 1 ( 2 ) ( z ) ψ 1 ( 2 ) ( x ) exp [ i ( ± β z - ω t ) ] + c . c . ,
E 2 ω = A 2 ω ( t ) ψ 2 ω ( x ) exp ( - 2 i ω t ) + c . c . ,
2 i β d A 1 ( 2 ) d z ψ 1 ( 2 ) ( x ) = χ ( 2 ) ( x ) × ω 2 c 2 A 2 ω A 2 ( 1 ) * ψ 2 ( 1 ) ( x ) ψ 2 ω ( x ) ,
4 i ω n 2 ω 2 c 2 A 2 ω t ψ 2 ω ( x ) + 2 i ω μ 0 σ A 2 ω ψ 2 ω ( x ) = - 4 ω 2 c 2 χ ( 2 ) ( x ) A 1 A 2 ψ 1 ( x ) ψ 2 ( x ) .
d A 1 ( 2 ) d z = ± i ω 2 n ω c χ 0 ( 2 ) d eff 1 / 2 A 2 ω A 2 ( 1 ) * ,
d A 2 ω d z + A 2 ω 2 τ = i ω n 2 ω 2 χ 0 ( 2 ) d eff 1 / 2 A 1 A 2 ,
d eff = [ χ 0 ( 2 ) ] 2 / [ - χ ( 2 ) ( x ) ψ 1 ( x ) ψ 2 ( x ) ψ 2 ω ( x ) d x ] 2 .
A 2 ω = 2 i ω τ n 2 ω 2 χ 0 ( 2 ) d eff 1 / 2 A 1 A 2 .
d A 1 d z = - 2 ω 2 τ c n ω n 2 ω 2 [ χ 0 ( 2 ) ] 2 d eff A 1 A 2 2 ,
d A 2 d z = 2 ω 2 τ c n ω n 2 ω 2 [ χ 0 ( 2 ) ] 2 d eff A 2 A 1 2 .
d P 1 / d z = - κ P 1 P 2 ,
d P 2 / d z = κ P 1 P 2 ,
κ = ω 2 τ η 0 c n ω 2 n 2 ω 2 [ χ 0 ( 2 ) ] 2 d eff ,
P s = ( κ L ) - 1 = λ 2 n ω 2 n 2 ω ( 1 - R 1 R 2 ) 8 π 2 L η 0 [ χ 0 ( 2 ) ] 2 d eff d ,
d u 1 / d ζ = - u 1 u 2
d u 2 / d ζ = u 1 u 2 .
u 1 + u 2 = const .
u 1 = a cos 2 B ,             u 2 = a sin 2 B ,
d B d ζ = a cos B sin B 2 .
tan B = c 1 exp ( a ζ / 2 ) .
u 2 ( 1 ) = R 4 u 1 ( 1 ) .
c 1 = R 4 exp ( - a / 2 ) .
tan B = R 4 exp [ a ( ζ - 1 ) / 2 ] .
u 1 = a 1 + R 4 exp [ a ( ζ - 1 ) ] , u 2 = a R 4 exp [ a ( ζ - 1 ) ] 1 + R 4 exp [ a ( ζ - 1 ) ] .
P 3 P s 2 0 1 u 1 u 2 d ζ = 2 a [ 1 - exp ( - a ) ] R 4 ( 1 + R 4 ) [ 1 + R 4 exp ( - a ) ] .
η = P 3 P s u 0 = 2 a [ 1 - exp ( - a ) ] R 4 ( 1 + R 4 ) [ 1 + R 4 exp ( - a ) ] u 0 .
u 1 ( 0 ) = u 0 .
u 0 = a 1 + R 4 exp ( - a ) .
η = 2 R 4 1 + R 4 [ 1 - exp ( - a ) ] .
u 0 a / ( 1 + R 4 )
η 2 R 4 u 0 = 2 R 4 P 0 / P s
u 0 a
η 2 R 4 1 + R 4 [ 1 - ( 1 + R 4 ) exp ( - u 0 ) ] 2 R 4 1 + R 4 .
u 0 = u 1 ( 0 ) - R 3 u 2 ( 0 ) .
u 0 = a [ 1 - R 3 R 4 exp ( - a ) ] 1 + R 4 exp ( - a ) .
η = 2 R 4 1 + R 4 [ 1 - exp ( - a ) ] [ 1 - R 3 R 4 exp ( - a ) ] .
u 0 a
η 2 R 4 1 + R 4 [ 1 - ( 1 + R 4 ) exp ( - u 0 ) ] 2 R 4 1 + R 4 .
u 0 a ( 1 - R 3 R 4 ) / ( 1 + R 4 ) .
η 2 R 4 ( 1 - R 3 R 4 ) 2 u 0 .
d eff ( pm ) 2 d .
d eff ( npm ) [ 2 π n 2 ω d / λ 2 ω sin ( 2 π n 2 ω d / λ 2 ω ) ] 2 d 2 .
d eff ( npm ) d eff ( pm ) π 2 n 2 ω 2 d 2 λ 2 ω 2 .
d eff ( λ 2 ω ) = d eff ( λ 0 ) { 2 π ( λ 2 ω - 1 - λ 0 - 1 ) d n 2 ω sin [ 2 π ( λ 2 ω - 1 - λ 0 - 1 ) d n 2 ω ] } 2 .
d P s d T = 2 N P s [ 1 n 2 ω ( ZnS ) d n 2 ω ( ZnS ) d T - 1 n 2 ω ( ZnSe ) d n 2 ω ( ZnSe ) d T ] ,
d P s d T = - 1.9 × 10 - 5 N P s .
Δ η 4.8 × 10 - 4 η Δ T .
Δ L L m + 1 - L m = λ / 2 n ω ,

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