Abstract

We develop a detailed theory of the transverse instabilities that can occur as two laser beams intersect in a nonlinear Kerr medium. Our analysis of the interaction includes all the various sources of nonlinear phase shifts of the interacting fields as well as the mutual interaction of the two pump waves. In general, the interaction gives rise to a four-sidemode process. The couplings among the sidemodes arise from three distinct interactions of modulational instability, two-beam-excited (TBE) conical emission, and nonlinear Bragg diffraction. Modulational instability and TBE conical emission are shown to exhibit exponential spatial gain. The value of this gain is higher for the case in which both processes contribute than in the cases in which the two processes occur singly. Nonlinear Bragg diffraction is shown to be a spatially stable process by itself. However, in the presence of the other two processes this process provides an additional nonlinear phase shift that changes the direction of maximum growth of the instability. In none of the cases does the maximum growth occur for perfect linear phase matching.

© 1993 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. See, for example, R. W. Boyd, Nonlinear Optics (Academic, San Diego, 1992).
  2. V. I. Bespalov and V. I. Talanov, Sov. Phys. JETP Lett. 3, 307 (1966); R. Y. Chiao, P. L. Kelley, and E. Garmire, Phys. Rev. Lett. 17, 1158 (1966); S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, Sov. Phys. JETP 23, 1025 (1966); R. G. Brewer, J. R. Lifsitz, E. Garmire, R. Y. Chiao, and C. H. Townes, Phys. Rev. 166, 326 (1968).
    [CrossRef]
  3. L. A. Ostrovskii, Sov. Phys. JETP 24, 797 (1967); see, for example, G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, 1992), Chap. 5.
  4. For a tutorial introduction to modulational instabilities of collinear waves, see C. J. McKinstrie and G. G. Luther, Phys. Scr. T-30, 31 (1990).
    [CrossRef]
  5. C. J. McKinstrie and R. Bingham, Phys. Fluids B 1, 230 (1989); Phys. Fluids B 2, 3215 (1990).
    [CrossRef]
  6. G. G. Luther and C. J. McKinstrie, J. Opt. Soc. Am. B 7, 1125 (1990).
    [CrossRef]
  7. See, for example, G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, 1992), Chap. 7.
  8. G. P. Agrawal, Phys. Rev. Lett. 64, 2487 (1990); A. J. Stentz, M. Kauranen, J. J. Maki, G. P. Agrawal, and R. W. Boyd, Opt. Lett. 17, 19 (1992).
    [CrossRef] [PubMed]
  9. A. C. Tam, Phys. Rev. A 19, 1971 (1979); D. J. Harter, P. Narum, M. G. Raymer, and R. W. Boyd, Phys. Rev. Lett. 46, 1192 (1981); D. J. Harter and R. W. Boyd, Opt. Lett. 7, 491 (1982); R. W. Boyd and D. J. Harter, Appl. Phys. B 29, 163 (1982); J. Pender and L. Hesselink, IEEE J. Quantum Electron. 25, 395 (1989); J. Opt. Soc. Am. B 7, 1361 (1990); J. F. Valley, G. Khitrova, H. M. Gibbs, J. Grantham, and Xu Jiajin, Phys. Rev. Lett. 64, 2362 (1990).
    [CrossRef] [PubMed]
  10. S. N. Vlasov and E. V. Sheinina, Radiophys. Quantum Electron. 26, 15 (1983); W. J. Firth and C. Paré, Opt. Lett. 13, 1096 (1988); G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, S. Le Boiteux, and M. Ducloy, Opt. Commun. 67, 363 (1988); G. Grynberg and J. Paye, Europhys. Lett. 8, 29 (1989).
    [CrossRef] [PubMed]
  11. E. K. Kirilenko, S. A. Lesnik, V. B. Markov, and A. I. Khyzniak, Opt. Commun. 60, 9 (1986).
    [CrossRef]
  12. M. Kauranen, J. J. Maki, A. L. Gaeta, and R. W. Boyd, Opt. Lett. 16, 943 (1991).
    [CrossRef] [PubMed]
  13. Note that the form of Eqs. (11) and (12) contains an implicit assumption that the angle between the wave vectors k1and k2is large enough that the phase mismatch of four-wave mixing terms such as 3χA12A2*exp[i(2k1− k2) · r] is sufficiently large that their contributions can be neglected. The requirement for this is that the nonlinear phase shift experienced by the waves be smaller than the linear phase mismatch of these terms. This can be shown to be true for 6πkχ|A1,2|2≪ kx2/k.
  14. Note that Eqs. (13) and (14) and the expression for β contain an assumption that the fields A1and A2propagate almost parallel to the z axis, i.e., |kx| ≪ |kz|. We make this assumption only for convenience and notational simplicity. It is a straightforward procedure to generalize the results of this paper for arbitrary crossing angles of the two pump beams.
  15. B. Ya. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Optical Phase Conjugation (Springer-Verlag, Berlin, 1985), Chap. 2.
    [CrossRef]

1991 (1)

1990 (3)

For a tutorial introduction to modulational instabilities of collinear waves, see C. J. McKinstrie and G. G. Luther, Phys. Scr. T-30, 31 (1990).
[CrossRef]

G. G. Luther and C. J. McKinstrie, J. Opt. Soc. Am. B 7, 1125 (1990).
[CrossRef]

G. P. Agrawal, Phys. Rev. Lett. 64, 2487 (1990); A. J. Stentz, M. Kauranen, J. J. Maki, G. P. Agrawal, and R. W. Boyd, Opt. Lett. 17, 19 (1992).
[CrossRef] [PubMed]

1989 (1)

C. J. McKinstrie and R. Bingham, Phys. Fluids B 1, 230 (1989); Phys. Fluids B 2, 3215 (1990).
[CrossRef]

1986 (1)

E. K. Kirilenko, S. A. Lesnik, V. B. Markov, and A. I. Khyzniak, Opt. Commun. 60, 9 (1986).
[CrossRef]

1983 (1)

S. N. Vlasov and E. V. Sheinina, Radiophys. Quantum Electron. 26, 15 (1983); W. J. Firth and C. Paré, Opt. Lett. 13, 1096 (1988); G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, S. Le Boiteux, and M. Ducloy, Opt. Commun. 67, 363 (1988); G. Grynberg and J. Paye, Europhys. Lett. 8, 29 (1989).
[CrossRef] [PubMed]

1979 (1)

A. C. Tam, Phys. Rev. A 19, 1971 (1979); D. J. Harter, P. Narum, M. G. Raymer, and R. W. Boyd, Phys. Rev. Lett. 46, 1192 (1981); D. J. Harter and R. W. Boyd, Opt. Lett. 7, 491 (1982); R. W. Boyd and D. J. Harter, Appl. Phys. B 29, 163 (1982); J. Pender and L. Hesselink, IEEE J. Quantum Electron. 25, 395 (1989); J. Opt. Soc. Am. B 7, 1361 (1990); J. F. Valley, G. Khitrova, H. M. Gibbs, J. Grantham, and Xu Jiajin, Phys. Rev. Lett. 64, 2362 (1990).
[CrossRef] [PubMed]

1967 (1)

L. A. Ostrovskii, Sov. Phys. JETP 24, 797 (1967); see, for example, G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, 1992), Chap. 5.

1966 (1)

V. I. Bespalov and V. I. Talanov, Sov. Phys. JETP Lett. 3, 307 (1966); R. Y. Chiao, P. L. Kelley, and E. Garmire, Phys. Rev. Lett. 17, 1158 (1966); S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, Sov. Phys. JETP 23, 1025 (1966); R. G. Brewer, J. R. Lifsitz, E. Garmire, R. Y. Chiao, and C. H. Townes, Phys. Rev. 166, 326 (1968).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Phys. Rev. Lett. 64, 2487 (1990); A. J. Stentz, M. Kauranen, J. J. Maki, G. P. Agrawal, and R. W. Boyd, Opt. Lett. 17, 19 (1992).
[CrossRef] [PubMed]

See, for example, G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, 1992), Chap. 7.

Bespalov, V. I.

V. I. Bespalov and V. I. Talanov, Sov. Phys. JETP Lett. 3, 307 (1966); R. Y. Chiao, P. L. Kelley, and E. Garmire, Phys. Rev. Lett. 17, 1158 (1966); S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, Sov. Phys. JETP 23, 1025 (1966); R. G. Brewer, J. R. Lifsitz, E. Garmire, R. Y. Chiao, and C. H. Townes, Phys. Rev. 166, 326 (1968).
[CrossRef]

Bingham, R.

C. J. McKinstrie and R. Bingham, Phys. Fluids B 1, 230 (1989); Phys. Fluids B 2, 3215 (1990).
[CrossRef]

Boyd, R. W.

M. Kauranen, J. J. Maki, A. L. Gaeta, and R. W. Boyd, Opt. Lett. 16, 943 (1991).
[CrossRef] [PubMed]

See, for example, R. W. Boyd, Nonlinear Optics (Academic, San Diego, 1992).

Gaeta, A. L.

Kauranen, M.

Khyzniak, A. I.

E. K. Kirilenko, S. A. Lesnik, V. B. Markov, and A. I. Khyzniak, Opt. Commun. 60, 9 (1986).
[CrossRef]

Kirilenko, E. K.

E. K. Kirilenko, S. A. Lesnik, V. B. Markov, and A. I. Khyzniak, Opt. Commun. 60, 9 (1986).
[CrossRef]

Lesnik, S. A.

E. K. Kirilenko, S. A. Lesnik, V. B. Markov, and A. I. Khyzniak, Opt. Commun. 60, 9 (1986).
[CrossRef]

Luther, G. G.

G. G. Luther and C. J. McKinstrie, J. Opt. Soc. Am. B 7, 1125 (1990).
[CrossRef]

For a tutorial introduction to modulational instabilities of collinear waves, see C. J. McKinstrie and G. G. Luther, Phys. Scr. T-30, 31 (1990).
[CrossRef]

Maki, J. J.

Markov, V. B.

E. K. Kirilenko, S. A. Lesnik, V. B. Markov, and A. I. Khyzniak, Opt. Commun. 60, 9 (1986).
[CrossRef]

McKinstrie, C. J.

For a tutorial introduction to modulational instabilities of collinear waves, see C. J. McKinstrie and G. G. Luther, Phys. Scr. T-30, 31 (1990).
[CrossRef]

G. G. Luther and C. J. McKinstrie, J. Opt. Soc. Am. B 7, 1125 (1990).
[CrossRef]

C. J. McKinstrie and R. Bingham, Phys. Fluids B 1, 230 (1989); Phys. Fluids B 2, 3215 (1990).
[CrossRef]

Ostrovskii, L. A.

L. A. Ostrovskii, Sov. Phys. JETP 24, 797 (1967); see, for example, G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, 1992), Chap. 5.

Pilipetsky, N. F.

B. Ya. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Optical Phase Conjugation (Springer-Verlag, Berlin, 1985), Chap. 2.
[CrossRef]

Sheinina, E. V.

S. N. Vlasov and E. V. Sheinina, Radiophys. Quantum Electron. 26, 15 (1983); W. J. Firth and C. Paré, Opt. Lett. 13, 1096 (1988); G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, S. Le Boiteux, and M. Ducloy, Opt. Commun. 67, 363 (1988); G. Grynberg and J. Paye, Europhys. Lett. 8, 29 (1989).
[CrossRef] [PubMed]

Shkunov, V. V.

B. Ya. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Optical Phase Conjugation (Springer-Verlag, Berlin, 1985), Chap. 2.
[CrossRef]

Talanov, V. I.

V. I. Bespalov and V. I. Talanov, Sov. Phys. JETP Lett. 3, 307 (1966); R. Y. Chiao, P. L. Kelley, and E. Garmire, Phys. Rev. Lett. 17, 1158 (1966); S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, Sov. Phys. JETP 23, 1025 (1966); R. G. Brewer, J. R. Lifsitz, E. Garmire, R. Y. Chiao, and C. H. Townes, Phys. Rev. 166, 326 (1968).
[CrossRef]

Tam, A. C.

A. C. Tam, Phys. Rev. A 19, 1971 (1979); D. J. Harter, P. Narum, M. G. Raymer, and R. W. Boyd, Phys. Rev. Lett. 46, 1192 (1981); D. J. Harter and R. W. Boyd, Opt. Lett. 7, 491 (1982); R. W. Boyd and D. J. Harter, Appl. Phys. B 29, 163 (1982); J. Pender and L. Hesselink, IEEE J. Quantum Electron. 25, 395 (1989); J. Opt. Soc. Am. B 7, 1361 (1990); J. F. Valley, G. Khitrova, H. M. Gibbs, J. Grantham, and Xu Jiajin, Phys. Rev. Lett. 64, 2362 (1990).
[CrossRef] [PubMed]

Vlasov, S. N.

S. N. Vlasov and E. V. Sheinina, Radiophys. Quantum Electron. 26, 15 (1983); W. J. Firth and C. Paré, Opt. Lett. 13, 1096 (1988); G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, S. Le Boiteux, and M. Ducloy, Opt. Commun. 67, 363 (1988); G. Grynberg and J. Paye, Europhys. Lett. 8, 29 (1989).
[CrossRef] [PubMed]

Zel’dovich, B. Ya.

B. Ya. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Optical Phase Conjugation (Springer-Verlag, Berlin, 1985), Chap. 2.
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

E. K. Kirilenko, S. A. Lesnik, V. B. Markov, and A. I. Khyzniak, Opt. Commun. 60, 9 (1986).
[CrossRef]

Opt. Lett. (1)

Phys. Fluids B (1)

C. J. McKinstrie and R. Bingham, Phys. Fluids B 1, 230 (1989); Phys. Fluids B 2, 3215 (1990).
[CrossRef]

Phys. Rev. A (1)

A. C. Tam, Phys. Rev. A 19, 1971 (1979); D. J. Harter, P. Narum, M. G. Raymer, and R. W. Boyd, Phys. Rev. Lett. 46, 1192 (1981); D. J. Harter and R. W. Boyd, Opt. Lett. 7, 491 (1982); R. W. Boyd and D. J. Harter, Appl. Phys. B 29, 163 (1982); J. Pender and L. Hesselink, IEEE J. Quantum Electron. 25, 395 (1989); J. Opt. Soc. Am. B 7, 1361 (1990); J. F. Valley, G. Khitrova, H. M. Gibbs, J. Grantham, and Xu Jiajin, Phys. Rev. Lett. 64, 2362 (1990).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

G. P. Agrawal, Phys. Rev. Lett. 64, 2487 (1990); A. J. Stentz, M. Kauranen, J. J. Maki, G. P. Agrawal, and R. W. Boyd, Opt. Lett. 17, 19 (1992).
[CrossRef] [PubMed]

Phys. Scr. (1)

For a tutorial introduction to modulational instabilities of collinear waves, see C. J. McKinstrie and G. G. Luther, Phys. Scr. T-30, 31 (1990).
[CrossRef]

Radiophys. Quantum Electron. (1)

S. N. Vlasov and E. V. Sheinina, Radiophys. Quantum Electron. 26, 15 (1983); W. J. Firth and C. Paré, Opt. Lett. 13, 1096 (1988); G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, S. Le Boiteux, and M. Ducloy, Opt. Commun. 67, 363 (1988); G. Grynberg and J. Paye, Europhys. Lett. 8, 29 (1989).
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

L. A. Ostrovskii, Sov. Phys. JETP 24, 797 (1967); see, for example, G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, 1992), Chap. 5.

Sov. Phys. JETP Lett. (1)

V. I. Bespalov and V. I. Talanov, Sov. Phys. JETP Lett. 3, 307 (1966); R. Y. Chiao, P. L. Kelley, and E. Garmire, Phys. Rev. Lett. 17, 1158 (1966); S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, Sov. Phys. JETP 23, 1025 (1966); R. G. Brewer, J. R. Lifsitz, E. Garmire, R. Y. Chiao, and C. H. Townes, Phys. Rev. 166, 326 (1968).
[CrossRef]

Other (5)

See, for example, R. W. Boyd, Nonlinear Optics (Academic, San Diego, 1992).

See, for example, G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, 1992), Chap. 7.

Note that the form of Eqs. (11) and (12) contains an implicit assumption that the angle between the wave vectors k1and k2is large enough that the phase mismatch of four-wave mixing terms such as 3χA12A2*exp[i(2k1− k2) · r] is sufficiently large that their contributions can be neglected. The requirement for this is that the nonlinear phase shift experienced by the waves be smaller than the linear phase mismatch of these terms. This can be shown to be true for 6πkχ|A1,2|2≪ kx2/k.

Note that Eqs. (13) and (14) and the expression for β contain an assumption that the fields A1and A2propagate almost parallel to the z axis, i.e., |kx| ≪ |kz|. We make this assumption only for convenience and notational simplicity. It is a straightforward procedure to generalize the results of this paper for arbitrary crossing angles of the two pump beams.

B. Ya. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Optical Phase Conjugation (Springer-Verlag, Berlin, 1985), Chap. 2.
[CrossRef]

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 (5)

Fig. 1
Fig. 1

Geometry of the interaction. Two strong pump waves intersect in a nonlinear Kerr medium. The transmitted pump waves and the weak sidemodes that grow from noise are viewed on a screen placed after the nonlinear medium.

Fig. 2
Fig. 2

Appearance of various nonlinear processes on the screen placed after the Kerr medium: a, nonlinear Bragg diffraction; b, modulational instability; c, TBE conical emission. The large spots represent pump beams, and the small spots represent side-modes. The couplings between the sidemodes by the nonlinear processes are indicated by lines with arrows.

Fig. 3
Fig. 3

Labeling of the sidemodes on the screen placed after the Kerr medium. The large spots represent the transmitted pump beams, and the small spots represent sidemodes.

Fig. 4
Fig. 4

Optimum location of the sidemodes on the screen for the case of modulational instability and TBE conical emission occurring together as well as for the case of all three processes occurring together.

Fig. 5
Fig. 5

Joint growth rate of modulational instability, TBE conical emission, and nonlinear Bragg diffraction as a function of the transverse wave-vector components qx and qy. The parameters describing the interaction are k/kx = 25 and kx/κ = 1000. The feature labeled MI corresponds to modulational instability with the maximum growth rate of κ, and the feature labeled CE corresponds to TBE conical emission with the maximum growth rate of 2κ. The spot with the maximum growth rate of 3κ, labeled all, corresponds to the joint action of all three processes.

Equations (57)

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

E ( r , t ) = E ( r ) exp ( i ω t ) ,
E ( r ) = j A j ( r ) exp ( i k j · r ) ,
P NL ( r , t ) = P NL ( r ) exp ( i ω t ) ,
P NL ( r ) = j P j ( r ) exp ( i k j · r ) .
d A j d z = i 2 π k j P j = i 2 π k P j ,
P NL ( r ) = 3 χ | E ( r ) | 2 E ( r ) ,
E ( r ) = E p ( r ) + E w ( r ) .
P NL = 3 χ | E p | 2 E p + 6 χ | E p | 2 E w + 3 χ E p 2 E w * ,
E p ( r ) = A 1 ( r ) exp ( i k 1 · r ) + A 2 ( r ) exp ( i k 2 · r ) .
k 1 = k x x ˆ + k z z ˆ , k 2 = k x x ˆ + k z z ˆ ,
P 1 exp ( i k 1 · r ) = 3 χ ( | A 1 | 2 + 2 | A 2 | 2 ) A 1 exp ( i k 1 · r ) ,
P 2 exp ( i k 2 · r ) = 3 χ ( 2 | A 1 | 2 + | A 2 | 2 ) A 2 exp ( i k 2 · r ) .
A 1 ( z ) = A 1 ( 0 ) exp [ i β ( | A 1 | 2 + 2 | A 2 | 2 ) z ] ,
A 2 ( z ) = A 2 ( 0 ) exp [ i β ( 2 | A 1 | 2 + | A 2 | 2 ) z ] ,
E w ( r ) = A w ( r ) exp ( i k w · r ) .
j 1 , 2 P j exp ( i k j · r ) = 6 χ ( | A 1 | 2 + | A 2 | 2 ) A w exp ( k k w · r ) + 6 χ A 1 A 2 * A w exp [ i ( k 1 k 2 + k w ) · r ] + 6 χ A 1 * A 2 A w exp [ i ( k 1 + k 2 + k w ) · r ] + 3 χ A 1 2 A w * exp [ i ( 2 k 1 k w ) · r ] + 3 χ A 2 2 A w * exp [ i ( 2 k 2 k w ) · r ] + 6 χ A 1 A 2 A w * exp [ i ( k 1 + k 2 k w ) · r ] .
k w = k w , x x ˆ + k w , y y ˆ + k w , z z ˆ ,
k j = k j , x x ˆ + k j , y y ˆ + k j , z z ˆ .
E w ( r ) = A 3 ( r ) exp ( i k 3 · r ) + A 4 ( r ) exp ( i k 4 · r ) + A 5 ( r ) exp ( i k 5 · r ) + A 6 ( r ) exp ( i k 6 · r ) ,
k 3 = q x x ˆ + q y y ˆ + Q 1 z ˆ , k 4 = ( 2 k x q x ) x ˆ + q y y ˆ + Q 2 z ˆ , k 5 = ( 2 k x q x ) x ˆ q y y ˆ + Q 2 z ˆ , k 6 = q x x ˆ q y y ˆ + Q 1 z ˆ ,
Q 1 = ( k 2 q x 2 q y 2 ) 1 / 2 , Q 2 = [ k 2 ( 2 k x q x ) 2 q y 2 ] 1 / 2 .
P 3 = 6 χ ( | A 1 | 2 + | A 2 | 2 ) A 3 + 6 χ A 1 A 2 * A 4 exp [ i ( k 1 k 2 + k 4 k 3 ) · r ] + 3 χ A 1 2 A 5 * exp [ i ( 2 k 1 k 5 k 3 ) · r ] + 6 χ A 1 A 2 A 6 * exp [ i ( k 1 + k 2 k 6 k 3 ) · r ] ,
Δ 1 = k z Q 1 , Δ 2 = k z Q 2
P 3 = 6 χ ( | A 1 | 2 + | A 2 | 2 ) A 3 + 6 χ A 1 A 2 * A 4 exp [ i ( Δ 1 Δ 2 ) z ] + 3 χ A 1 2 A 5 * exp [ i ( Δ 1 + Δ 2 ) z ] + 6 χ A 1 A 2 A 6 * exp ( i 2 Δ 1 z ) .
d A 3 d z = i 2 β ( | A 1 | 2 + | A 2 | 2 ) A 3 + i 2 β A 1 A 2 * A 4 exp [ i ( Δ 1 Δ 2 ) z ] + i β A 1 2 A 5 * exp [ i ( Δ 1 + Δ 2 ) z ] + i 2 β A 1 A 2 A 6 * exp ( i 2 Δ 1 z ) .
κ 1 = β | A 1 ( 0 ) | 2 = β | A 1 ( z ) | 2 , κ 2 = β | A 2 ( 0 ) | 2 = β | A 2 ( z ) | 2 , κ 11 = β A 1 2 ( 0 ) , κ 22 = β A 2 2 ( 0 ) , κ 12 = β A 1 ( 0 ) A 2 ( 0 ) , τ 12 = β A 1 ( 0 ) A 2 * ( 0 ) .
β A 1 2 = β A 1 2 ( z ) = κ 11 exp [ i 2 ( κ 1 + 2 κ 2 ) z ] , β A 2 2 = β A 2 2 ( z ) = κ 22 exp [ i 2 ( 2 κ 1 + κ 2 ) z ] , β A 1 A 2 = β A 1 ( z ) A 2 ( z ) = κ 12 exp [ i 3 ( κ 1 + κ 2 ) z ] , β A 1 A 2 * = β A 1 ( z ) A 2 * ( z ) = τ 12 exp [ i ( κ 2 κ 1 ) z ] ,
d A 3 d z = i 2 ( κ 1 + κ 2 ) A 3 + i 2 τ 12 exp [ i ( κ 2 κ 1 ) z ] exp [ i ( Δ 1 Δ 2 ) z ] A 4 + i κ 11 exp [ i 2 ( κ 1 + 2 κ 2 ) z ] exp [ i ( Δ 1 + Δ 2 ) z ] A 5 * + i 2 κ 12 exp [ i 3 ( κ 1 + κ 2 ) z ] exp ( i 2 Δ 1 z ) A 6 * .
A j ( z ) = B j ( z ) exp ( i ϕ j z ) ,
ϕ 3 = κ 1 + 2 κ 2 + Δ 1 , ϕ 4 = 2 κ 1 + κ 2 + Δ 2 ϕ 5 = κ 1 + 2 κ 2 + Δ 2 , ϕ 6 = 2 κ 1 + κ 2 + Δ 1 .
d B 3 d z = i ( κ 1 Δ 1 ) B 3 + i 2 τ 12 B 4 + i κ 11 B 5 * + i 2 κ 12 B 6 * .
d B d z = M · B ,
M = [ i ( κ 1 Δ 1 ) i 2 τ 12 i κ 11 i 2 κ 12 i 2 τ 12 * i ( κ 2 Δ 2 ) i 2 κ 12 i κ 22 i κ 11 * i 2 κ 12 * i ( κ 1 Δ 2 ) i 2 τ 12 * i 2 κ 12 * i κ 22 * i 2 τ 12 i ( κ 2 Δ 1 ) ] .
κ 11 = κ 1 , κ 22 = κ 2 , κ 12 = τ 12 = κ 1 κ 2 .
M BD = [ i ( κ 1 Δ 1 ) i 2 τ 12 i 2 τ 12 * i ( κ 2 Δ 2 ) ] = [ i ( κ 1 Δ 1 ) i 2 κ 1 κ 2 i 2 κ 1 κ 2 i ( κ 2 Δ 2 ) ] ,
λ = i 2 ( 2 κ Δ 1 Δ 2 ) ± i 2 [ 16 κ 2 + ( Δ 1 Δ 2 ) 2 ] 1 / 2 ,
M MI = [ i ( κ 1 Δ 1 ) i κ 11 i κ 11 * i ( κ 1 Δ 2 ) ] = [ i ( κ 1 Δ 1 ) i κ 1 i κ 1 i ( κ 1 Δ 2 ) ] .
λ = i 2 ( Δ 2 Δ 1 ) ± 1 2 [ 4 κ ( Δ 1 + Δ 2 ) ( Δ 1 + Δ 2 ) 2 ] 1 / 2 ,
λ real = 1 2 [ 4 κ ( Δ 1 + Δ 2 ) ( Δ 1 + Δ 2 ) 2 ] 1 / 2 .
Δ 1 + Δ 2 = δ x 2 + δ y 2 k z .
G MI = 2 λ max = 2 κ = 24 π 2 k χ c I , ( Δ 1 + Δ 2 ) MI = 2 κ ,
I = c 2 π | A | 2 = c 2 π | A 1 | 2 = c 2 π | A 2 | 2 .
M CE = [ i ( κ 1 Δ 1 ) i 2 κ 12 i 2 κ 12 * i ( κ 2 Δ 1 ) ] = [ i ( κ 1 Δ 1 ) i 2 κ 1 κ 2 i 2 κ 1 κ 2 i ( κ 2 Δ 1 ) ] .
λ = ± [ 4 κ 2 ( κ Δ 1 ) 2 ] 1 / 2 ,
q x 2 + q y 2 = k x 2 + 2 k z κ κ 2 .
G CE = 4 κ = 48 π 2 k χ c I , Δ 1 , CE = κ .
λ = ± ( a ± b ) 1 / 2 2 ,
a = Δ 1 2 Δ 2 2 + 2 ( Δ 1 + Δ 2 ) κ + 8 κ 2 , b = ( Δ 2 Δ 1 ) 2 ( Δ 1 + Δ 2 ) 2 4 ( Δ 2 Δ 1 ) 2 ( Δ 1 + Δ 2 ) κ + 64 κ 4 .
λ real = ( a + b ) 1 / 2 2 ,
λ real = ( 8 κ 2 + 2 Δ 1 κ Δ 1 2 ) 1 / 2 = [ 9 κ 2 ( κ Δ 1 ) ] 1 / 2 ,
G MI&CE = 6 κ = 72 π 2 k χ c I , Δ 1 , MI&CE = Δ 2 , MI&CE = κ .
a = Δ 1 2 Δ 2 2 + 2 ( Δ 1 + Δ 2 ) κ , b = ( Δ 2 Δ 1 ) 2 ( Δ 1 + Δ 2 ) 2 4 ( Δ 2 Δ 1 ) ( Δ 1 + Δ 2 ) κ + 16 ( Δ 1 + Δ 2 ) 2 κ 2 .
λ real = ( 6 Δ 1 κ Δ 1 2 ) 1 / 2 = [ 9 κ 2 ( 3 κ Δ 1 ) 2 ] 1 / 2 .
G all = 6 κ = 72 π 2 k χ c I , Δ 1 , all = Δ 2 , all = 3 κ .
cos θ 1 Δ 1 / k .
cos θ 1 3 κ / k = 1 36 π 2 χ c I = 0.99997 ,
λ real = [ Δ 1 ( κ 1 + κ 2 ) Δ 1 2 + Δ 1 ( κ 1 2 + κ 2 2 + 14 κ 1 κ 2 ) 1 / 2 ] 1 / 2 .

Metrics