Abstract

The theory required for description of the effect of the temperature rise that is due to self-induced heating on second-harmonic generation is developed and applied to the biaxial molecular crystal (-)2-α-(methylbenzylamino)-5-nitropyridine (MBANP). It is shown that for the range of parameters considered there is an optimum input power of the fundamental beam for highest second-harmonic conversion efficiency. A maximum efficiency of 61% was calculated for the optimum input power of 19.6 kW with a beam radius of 0.25 mm and an interaction length of 1.6 cm.

© 1999 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S.-C. Sheng and A. E. Siegman, “Nonlinear-optical calculations using fast-transform methods: second-harmonic generation with depletion and diffraction,” Phys. Rev. A 21, 599–606 (1980).
    [CrossRef]
  2. M. Nieto-Vesperinas and G. Lera, “Solution to non-linear optical frequency mixing equations with depletion and diffraction: difference frequency generation,” Opt. Commun. 69, 329–333 (1989).
    [CrossRef]
  3. M. A. Dreger and J. K. McIver, “Second-harmonic generation in a nonlinear, anisotropic medium with diffraction and depletion,” J. Opt. Soc. Am. B 7, 776–784 (1990).
    [CrossRef]
  4. G. Arisholm, “General numerical methods for simulating second-order nonlinear interactions in birefringent media,” J. Opt. Soc. Am. B 14, 2543–2549 (1997).
    [CrossRef]
  5. M. Okada and S. Ieiri, “Influence of self-induced thermal effects on second-harmonic generation,” IEEE J. Quantum Electron. QE-7, 469–470 (1971).
    [CrossRef]
  6. 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]
  7. R. T. Bailey, F. R. Cruickshank, D. Pugh, and J. N. Sherwood, “Growth, perfection and properties of organic nonlinear materials,” Acta Crystallogr., Sect. A: Found. Crystallogr. 47, 145–155 (1991).
    [CrossRef]
  8. R. J. Twieg and C. W. Dirk, “Molecular and crystal structure of the nonlinear optical material (-)2-(α-methylbenzylamino)-5-nitropyridine,” Research Rep. RJ 5237 (54077) (IBM Almaden Research Center, San Jose, Calif., July 1968).
  9. R. T. Bailey, F. R. Cruickshank, D. Pugh, J. N. Sherwood, G. S. Simpson, and S. Wilkie, “The linear optical properties of the organic molecular crystal (-)2-(α-methylbenzylamino)-5-nitropyridine,” Mol. Cryst. Liq. Cryst. 231, 223–229 (1993).
    [CrossRef]
  10. P. Kerkoc, R. T. Bailey, F. R. Cruickshank, and D. Pugh, “Second-harmonic generation in biaxial crystals for a focused fundamental Gaussian beam: application to (-)2-(α-methylbenzylamino)-5-nitropyridine single crystals,” J. Opt. Soc. Am. B 15, 438–445 (1998).
    [CrossRef]
  11. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 14, pp. 665–678.
  12. S. A. Akhmanov, A. I. Kovrygin, and A. P. Sukhorukov, “Optical harmonic generation and optical frequency multipliers,” in Quantum Electronics: a Treatise, H. Rabin and C. L. Tang, eds. (Academic, New York, 1975), Vol. 1, Chap. 8, pp. 511–513.
  13. R. T. Bailey, F. R. Cruickshank, P. Kerkoc, D. Pugh, and J. N. Sherwood, “Thermal conductivity of the molecular crystal (-)2-α-(methylbenzylamino)-5-nitropyridine,” J. Appl. Phys. 74, 3047–3051 (1993).
    [CrossRef]
  14. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in FORTRAN 77: the Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992), Chap. 12, pp. 515–519.
  15. Ref. 14, Chap. 16, pp. 735–739.
  16. Ref. 14, Chap. 17, pp. 749–751.
  17. P. Kerkoc, R. T. Bailey, F. R. Cruickshank, D. Pugh, and J. N. Sherwood, “Low values of the tensor components of the non-resonant optical absorption in the molecular nonlinear optical single crystal (-)2-α-(methylbenzylamino)-5-nitropyridine,” Opt. Commun. 132, 484–488 (1996).
    [CrossRef]
  18. G. N. Ramachandran and S. Ramaseshan, “Crystal Optics,” in Handbuch der Physik 25, S. Flügge, ed. (Springer-Verlag, Berlin, 1961), Vol. 25, Chap. 1, pp. 87–89.
  19. 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]
  20. G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
    [CrossRef]
  21. P. Kerkoc, M. Zgonik, K. Sutter, Ch. Bosshard, and P. Günter, “4-(N, N-dimethylamino)-3-acetamidonitrobenzene single crystals for nonlinear optical applications,” J. Opt. Soc. Am. B 7, 313–319 (1990).
    [CrossRef]
  22. H. S. Nalwa, T. Watanabe, and S. Miyata, “Organic materials for second-order nonlinear optics,” in Nonlinear Optics of Organic Molecules and Polymers, H. S. Nalwa and S. Miyata, eds. (CRC Press, Boca Raton, Fla., 1997), Chap. 4, pp. 136–137.

1998

1997

1996

P. Kerkoc, R. T. Bailey, F. R. Cruickshank, D. Pugh, and J. N. Sherwood, “Low values of the tensor components of the non-resonant optical absorption in the molecular nonlinear optical single crystal (-)2-α-(methylbenzylamino)-5-nitropyridine,” Opt. Commun. 132, 484–488 (1996).
[CrossRef]

1995

1993

R. T. Bailey, F. R. Cruickshank, D. Pugh, J. N. Sherwood, G. S. Simpson, and S. Wilkie, “The linear optical properties of the organic molecular crystal (-)2-(α-methylbenzylamino)-5-nitropyridine,” Mol. Cryst. Liq. Cryst. 231, 223–229 (1993).
[CrossRef]

R. T. Bailey, F. R. Cruickshank, P. Kerkoc, D. Pugh, and J. N. Sherwood, “Thermal conductivity of the molecular crystal (-)2-α-(methylbenzylamino)-5-nitropyridine,” J. Appl. Phys. 74, 3047–3051 (1993).
[CrossRef]

1991

R. T. Bailey, F. R. Cruickshank, D. Pugh, and J. N. Sherwood, “Growth, perfection and properties of organic nonlinear materials,” Acta Crystallogr., Sect. A: Found. Crystallogr. 47, 145–155 (1991).
[CrossRef]

1990

1989

M. Nieto-Vesperinas and G. Lera, “Solution to non-linear optical frequency mixing equations with depletion and diffraction: difference frequency generation,” Opt. Commun. 69, 329–333 (1989).
[CrossRef]

1980

S.-C. Sheng and A. E. Siegman, “Nonlinear-optical calculations using fast-transform methods: second-harmonic generation with depletion and diffraction,” Phys. Rev. A 21, 599–606 (1980).
[CrossRef]

1971

M. Okada and S. Ieiri, “Influence of self-induced thermal effects on second-harmonic generation,” IEEE J. Quantum Electron. QE-7, 469–470 (1971).
[CrossRef]

1968

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

Alford, W. J.

Arisholm, G.

Bailey, R. T.

P. Kerkoc, R. T. Bailey, F. R. Cruickshank, and D. Pugh, “Second-harmonic generation in biaxial crystals for a focused fundamental Gaussian beam: application to (-)2-(α-methylbenzylamino)-5-nitropyridine single crystals,” J. Opt. Soc. Am. B 15, 438–445 (1998).
[CrossRef]

P. Kerkoc, R. T. Bailey, F. R. Cruickshank, D. Pugh, and J. N. Sherwood, “Low values of the tensor components of the non-resonant optical absorption in the molecular nonlinear optical single crystal (-)2-α-(methylbenzylamino)-5-nitropyridine,” Opt. Commun. 132, 484–488 (1996).
[CrossRef]

R. T. Bailey, F. R. Cruickshank, P. Kerkoc, D. Pugh, and J. N. Sherwood, “Thermal conductivity of the molecular crystal (-)2-α-(methylbenzylamino)-5-nitropyridine,” J. Appl. Phys. 74, 3047–3051 (1993).
[CrossRef]

R. T. Bailey, F. R. Cruickshank, D. Pugh, J. N. Sherwood, G. S. Simpson, and S. Wilkie, “The linear optical properties of the organic molecular crystal (-)2-(α-methylbenzylamino)-5-nitropyridine,” Mol. Cryst. Liq. Cryst. 231, 223–229 (1993).
[CrossRef]

R. T. Bailey, F. R. Cruickshank, D. Pugh, and J. N. Sherwood, “Growth, perfection and properties of organic nonlinear materials,” Acta Crystallogr., Sect. A: Found. Crystallogr. 47, 145–155 (1991).
[CrossRef]

Bosshard, Ch.

Bowers, M. S.

Boyd, G. D.

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

Cruickshank, F. R.

P. Kerkoc, R. T. Bailey, F. R. Cruickshank, and D. Pugh, “Second-harmonic generation in biaxial crystals for a focused fundamental Gaussian beam: application to (-)2-(α-methylbenzylamino)-5-nitropyridine single crystals,” J. Opt. Soc. Am. B 15, 438–445 (1998).
[CrossRef]

P. Kerkoc, R. T. Bailey, F. R. Cruickshank, D. Pugh, and J. N. Sherwood, “Low values of the tensor components of the non-resonant optical absorption in the molecular nonlinear optical single crystal (-)2-α-(methylbenzylamino)-5-nitropyridine,” Opt. Commun. 132, 484–488 (1996).
[CrossRef]

R. T. Bailey, F. R. Cruickshank, P. Kerkoc, D. Pugh, and J. N. Sherwood, “Thermal conductivity of the molecular crystal (-)2-α-(methylbenzylamino)-5-nitropyridine,” J. Appl. Phys. 74, 3047–3051 (1993).
[CrossRef]

R. T. Bailey, F. R. Cruickshank, D. Pugh, J. N. Sherwood, G. S. Simpson, and S. Wilkie, “The linear optical properties of the organic molecular crystal (-)2-(α-methylbenzylamino)-5-nitropyridine,” Mol. Cryst. Liq. Cryst. 231, 223–229 (1993).
[CrossRef]

R. T. Bailey, F. R. Cruickshank, D. Pugh, and J. N. Sherwood, “Growth, perfection and properties of organic nonlinear materials,” Acta Crystallogr., Sect. A: Found. Crystallogr. 47, 145–155 (1991).
[CrossRef]

Dreger, M. A.

Günter, P.

Ieiri, S.

M. Okada and S. Ieiri, “Influence of self-induced thermal effects on second-harmonic generation,” IEEE J. Quantum Electron. QE-7, 469–470 (1971).
[CrossRef]

Kerkoc, P.

P. Kerkoc, R. T. Bailey, F. R. Cruickshank, and D. Pugh, “Second-harmonic generation in biaxial crystals for a focused fundamental Gaussian beam: application to (-)2-(α-methylbenzylamino)-5-nitropyridine single crystals,” J. Opt. Soc. Am. B 15, 438–445 (1998).
[CrossRef]

P. Kerkoc, R. T. Bailey, F. R. Cruickshank, D. Pugh, and J. N. Sherwood, “Low values of the tensor components of the non-resonant optical absorption in the molecular nonlinear optical single crystal (-)2-α-(methylbenzylamino)-5-nitropyridine,” Opt. Commun. 132, 484–488 (1996).
[CrossRef]

R. T. Bailey, F. R. Cruickshank, P. Kerkoc, D. Pugh, and J. N. Sherwood, “Thermal conductivity of the molecular crystal (-)2-α-(methylbenzylamino)-5-nitropyridine,” J. Appl. Phys. 74, 3047–3051 (1993).
[CrossRef]

P. Kerkoc, M. Zgonik, K. Sutter, Ch. Bosshard, and P. Günter, “4-(N, N-dimethylamino)-3-acetamidonitrobenzene single crystals for nonlinear optical applications,” J. Opt. Soc. Am. B 7, 313–319 (1990).
[CrossRef]

Kleinman, D. A.

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

Lera, G.

M. Nieto-Vesperinas and G. Lera, “Solution to non-linear optical frequency mixing equations with depletion and diffraction: difference frequency generation,” Opt. Commun. 69, 329–333 (1989).
[CrossRef]

McIver, J. K.

Nieto-Vesperinas, M.

M. Nieto-Vesperinas and G. Lera, “Solution to non-linear optical frequency mixing equations with depletion and diffraction: difference frequency generation,” Opt. Commun. 69, 329–333 (1989).
[CrossRef]

Okada, M.

M. Okada and S. Ieiri, “Influence of self-induced thermal effects on second-harmonic generation,” IEEE J. Quantum Electron. QE-7, 469–470 (1971).
[CrossRef]

Pugh, D.

P. Kerkoc, R. T. Bailey, F. R. Cruickshank, and D. Pugh, “Second-harmonic generation in biaxial crystals for a focused fundamental Gaussian beam: application to (-)2-(α-methylbenzylamino)-5-nitropyridine single crystals,” J. Opt. Soc. Am. B 15, 438–445 (1998).
[CrossRef]

P. Kerkoc, R. T. Bailey, F. R. Cruickshank, D. Pugh, and J. N. Sherwood, “Low values of the tensor components of the non-resonant optical absorption in the molecular nonlinear optical single crystal (-)2-α-(methylbenzylamino)-5-nitropyridine,” Opt. Commun. 132, 484–488 (1996).
[CrossRef]

R. T. Bailey, F. R. Cruickshank, P. Kerkoc, D. Pugh, and J. N. Sherwood, “Thermal conductivity of the molecular crystal (-)2-α-(methylbenzylamino)-5-nitropyridine,” J. Appl. Phys. 74, 3047–3051 (1993).
[CrossRef]

R. T. Bailey, F. R. Cruickshank, D. Pugh, J. N. Sherwood, G. S. Simpson, and S. Wilkie, “The linear optical properties of the organic molecular crystal (-)2-(α-methylbenzylamino)-5-nitropyridine,” Mol. Cryst. Liq. Cryst. 231, 223–229 (1993).
[CrossRef]

R. T. Bailey, F. R. Cruickshank, D. Pugh, and J. N. Sherwood, “Growth, perfection and properties of organic nonlinear materials,” Acta Crystallogr., Sect. A: Found. Crystallogr. 47, 145–155 (1991).
[CrossRef]

Raymond, T. D.

Sheng, S.-C.

S.-C. Sheng and A. E. Siegman, “Nonlinear-optical calculations using fast-transform methods: second-harmonic generation with depletion and diffraction,” Phys. Rev. A 21, 599–606 (1980).
[CrossRef]

Sherwood, J. N.

P. Kerkoc, R. T. Bailey, F. R. Cruickshank, D. Pugh, and J. N. Sherwood, “Low values of the tensor components of the non-resonant optical absorption in the molecular nonlinear optical single crystal (-)2-α-(methylbenzylamino)-5-nitropyridine,” Opt. Commun. 132, 484–488 (1996).
[CrossRef]

R. T. Bailey, F. R. Cruickshank, P. Kerkoc, D. Pugh, and J. N. Sherwood, “Thermal conductivity of the molecular crystal (-)2-α-(methylbenzylamino)-5-nitropyridine,” J. Appl. Phys. 74, 3047–3051 (1993).
[CrossRef]

R. T. Bailey, F. R. Cruickshank, D. Pugh, J. N. Sherwood, G. S. Simpson, and S. Wilkie, “The linear optical properties of the organic molecular crystal (-)2-(α-methylbenzylamino)-5-nitropyridine,” Mol. Cryst. Liq. Cryst. 231, 223–229 (1993).
[CrossRef]

R. T. Bailey, F. R. Cruickshank, D. Pugh, and J. N. Sherwood, “Growth, perfection and properties of organic nonlinear materials,” Acta Crystallogr., Sect. A: Found. Crystallogr. 47, 145–155 (1991).
[CrossRef]

Siegman, A. E.

S.-C. Sheng and A. E. Siegman, “Nonlinear-optical calculations using fast-transform methods: second-harmonic generation with depletion and diffraction,” Phys. Rev. A 21, 599–606 (1980).
[CrossRef]

Simpson, G. S.

R. T. Bailey, F. R. Cruickshank, D. Pugh, J. N. Sherwood, G. S. Simpson, and S. Wilkie, “The linear optical properties of the organic molecular crystal (-)2-(α-methylbenzylamino)-5-nitropyridine,” Mol. Cryst. Liq. Cryst. 231, 223–229 (1993).
[CrossRef]

Smith, A. V.

Sutter, K.

Wilkie, S.

R. T. Bailey, F. R. Cruickshank, D. Pugh, J. N. Sherwood, G. S. Simpson, and S. Wilkie, “The linear optical properties of the organic molecular crystal (-)2-(α-methylbenzylamino)-5-nitropyridine,” Mol. Cryst. Liq. Cryst. 231, 223–229 (1993).
[CrossRef]

Zgonik, M.

Acta Crystallogr., Sect. A: Found. Crystallogr.

R. T. Bailey, F. R. Cruickshank, D. Pugh, and J. N. Sherwood, “Growth, perfection and properties of organic nonlinear materials,” Acta Crystallogr., Sect. A: Found. Crystallogr. 47, 145–155 (1991).
[CrossRef]

IEEE J. Quantum Electron.

M. Okada and S. Ieiri, “Influence of self-induced thermal effects on second-harmonic generation,” IEEE J. Quantum Electron. QE-7, 469–470 (1971).
[CrossRef]

J. Appl. Phys.

R. T. Bailey, F. R. Cruickshank, P. Kerkoc, D. Pugh, and J. N. Sherwood, “Thermal conductivity of the molecular crystal (-)2-α-(methylbenzylamino)-5-nitropyridine,” J. Appl. Phys. 74, 3047–3051 (1993).
[CrossRef]

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

J. Opt. Soc. Am. B

Mol. Cryst. Liq. Cryst.

R. T. Bailey, F. R. Cruickshank, D. Pugh, J. N. Sherwood, G. S. Simpson, and S. Wilkie, “The linear optical properties of the organic molecular crystal (-)2-(α-methylbenzylamino)-5-nitropyridine,” Mol. Cryst. Liq. Cryst. 231, 223–229 (1993).
[CrossRef]

Opt. Commun.

M. Nieto-Vesperinas and G. Lera, “Solution to non-linear optical frequency mixing equations with depletion and diffraction: difference frequency generation,” Opt. Commun. 69, 329–333 (1989).
[CrossRef]

P. Kerkoc, R. T. Bailey, F. R. Cruickshank, D. Pugh, and J. N. Sherwood, “Low values of the tensor components of the non-resonant optical absorption in the molecular nonlinear optical single crystal (-)2-α-(methylbenzylamino)-5-nitropyridine,” Opt. Commun. 132, 484–488 (1996).
[CrossRef]

Phys. Rev. A

S.-C. Sheng and A. E. Siegman, “Nonlinear-optical calculations using fast-transform methods: second-harmonic generation with depletion and diffraction,” Phys. Rev. A 21, 599–606 (1980).
[CrossRef]

Other

R. J. Twieg and C. W. Dirk, “Molecular and crystal structure of the nonlinear optical material (-)2-(α-methylbenzylamino)-5-nitropyridine,” Research Rep. RJ 5237 (54077) (IBM Almaden Research Center, San Jose, Calif., July 1968).

G. N. Ramachandran and S. Ramaseshan, “Crystal Optics,” in Handbuch der Physik 25, S. Flügge, ed. (Springer-Verlag, Berlin, 1961), Vol. 25, Chap. 1, pp. 87–89.

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 14, pp. 665–678.

S. A. Akhmanov, A. I. Kovrygin, and A. P. Sukhorukov, “Optical harmonic generation and optical frequency multipliers,” in Quantum Electronics: a Treatise, H. Rabin and C. L. Tang, eds. (Academic, New York, 1975), Vol. 1, Chap. 8, pp. 511–513.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in FORTRAN 77: the Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992), Chap. 12, pp. 515–519.

Ref. 14, Chap. 16, pp. 735–739.

Ref. 14, Chap. 17, pp. 749–751.

H. S. Nalwa, T. Watanabe, and S. Miyata, “Organic materials for second-order nonlinear optics,” in Nonlinear Optics of Organic Molecules and Polymers, H. S. Nalwa and S. Miyata, eds. (CRC Press, Boca Raton, Fla., 1997), Chap. 4, pp. 136–137.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Orthogonal phase-matching system xDω, yD2ω, and zs. The propagation of the waves at both frequencies is in the z direction. The angle ρω is the beam walk-off of the fundamental beam, with the Poynting vector tω propagating in the (x, z) plane. The direction of the second-harmonic Poynting vector is denoted t2ω, and its beam walk-off angle is ρ2ω. The projections of this angle onto planes (y, z) and (x, z) are denoted ρx 2ω and ρy 2ω, respectively. For clarity, the angles shown are exaggerated.

Fig. 2
Fig. 2

Dependence of the second-harmonic efficiency on the interaction length z. The values of the input powers of the fundamental beam are A, 19.6 kW; B, 104 kW; C, 234 kW (pulselength, 25 ns). The dependence for DAN is superimposed for comparison (input power of 19.6 kW). The input peak intensity has a constant value of 20 MW/cm2.

Fig. 3
Fig. 3

Temperature rise along the phase-matching direction in the center of the fundamental and second-harmonic waves, i.e., x=0, y=0. The levels of the input powers of the fundamental beam are the same as given in Fig. 2: A, 19.6 kW, B, 104 kW; C, 234 kW.

Fig. 4
Fig. 4

Projections of the second-harmonic fluence distribution at the exit of the crystal under the conditions for the highest efficiency on (a) the (y, z) plane and (b) the (x, z) plane. For a definition of the planes, see Fig. 1.

Tables (1)

Tables Icon

Table 1 Values of the Parameters of MBANP Crystal Used in the Computationa

Equations (19)

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

cos ρΩ=tΩ·s.
cos ρx2ω=cos ρ2ωsin δx,
cos ρy2ω=cos ρ2ωsin δy,
Aωz+ρωAωx-12ikω2Aωx2+2Aωy2=H1,
A2ωz+ρy2ωA2ωx+ρx2ωA2ωy
-12ik2ω2A2ωx2+2A2ωy2=H2,
H1=-iωdeffcnωA2ωAω* exp(-iΔkz)+ikωnωdnωdTTAω+αω2Aω,
H2=-iωdeffcn2ωAω2 exp(iΔkz)+ik2ωn2ωdn2ωdTTA2ω+α2ω2A2ω,
k112Tx2+k222Ty2+k332Tz2+2k132Txz
+2k232Tyz+2k122Txy
=-14πln 21/2fτ0c(nωαω|Aω|2+n2ωα2ω|A2ω|2),
k33Tz+k32Ty+k31Tx=±hT,
dBωdz-i2πρωp+2π2kω(p2+q2)Bω=F(H1),
dB2ωdz-i2π(ρy2ωp+ρx2ωq)+2π2k2ω(p2+q2)B2ω
=F(H2),
k33d2Θdz2-4πi(k13p+k23q)dΘdz
-4π2(k11p2+k22q2+2k12pq)Θ
=-14πln 21/2fτ0c(nωαωF|Aω|2+n2ωα2ωF|A2ω|2),
k33dΘdz-2πi(k32q+k31p)Θ=±hΘ

Metrics