Abstract

We show that the parity of the order of the harmonics generated by a bichromatic field with commensurate frequencies may be related to the invariance properties of the time-dependent Hamiltonian with respect to some suitable transformations. To simplify the mathematical treatment, we have used a simple model atom to perform calculations that show the effects of the relative phase of the driving fields. Harmonic generation has been described as a two-step process that involves replicas of the Stark levels. In agreement with other previous treatments in which other model atoms were used, we have found that a small contamination of the fundamental harmonic can produce significant modifications of the emission spectra.

© 1999 Optical Society of America

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References

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  1. For a review of HHG in the strong field see A. L’Huillier, L. A. Lompre, G. Mainfray, and C. Manus, Adv. At., Mol., Opt. Phys. 1, 139 (1992), and references therein.
  2. J. K. Krause, K. J. Schafer, and K. C. Kulander, Phys. Rev. Lett. 68, 3535 (1992).
    [CrossRef] [PubMed]
  3. A. E. Kaplan and P. L. Shkolnikov, Phys. Rev. A 49, 1275 (1994).
    [CrossRef] [PubMed]
  4. L. Plaja and L. Roso, J. Mod. Opt. 40, 793 (1993).
    [CrossRef]
  5. E. Conejero Jarque and L. Playa, J. Phys. B 31, 1687 (1988).
    [CrossRef]
  6. F. I. Gauthey, C. H. Keitel, P. L. Knight, and A. Maquet, Phys. Rev. A 52, 525 (1995).
    [CrossRef] [PubMed]
  7. F. I. Gauthey, B. M. Garraway, and P. L. Knight, Phys. Rev. A 56, 3093 (1997).
    [CrossRef]
  8. F. I. Gauthey, C. H. Keitel, P. L. Knight, and A. Maquet, Phys. Rev. A 55, 615 (1997), and references therein.
    [CrossRef]
  9. W. Becker, S. Lang, and J. K. McIver, Phys. Rev. A 50, 1540 (1994).
    [CrossRef] [PubMed]
  10. M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huiller, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994).
    [CrossRef] [PubMed]
  11. R. Burlon, G. Ferrante, C. Leone, P. A. Oleinikov, and V. T. Platonenko, J. Opt. Soc. Am. B 13, 162 (1996).
    [CrossRef]
  12. S. Watanabe, K. Kando, Y. Nabekawa, A. Sagisaka, and Y. Kolaysashi, Phys. Rev. Lett. 73, 2692 (1994).
    [CrossRef] [PubMed]
  13. M. D. Perry and J. K. Crane, Phys. Rev. A 48, 54051 (1993); H. Eichmann, A. Egbert, S. Nolte, C. Mornnea, B. Wellegehausen, W. Becker, S. Lang, and J. K. McIver, Phys. Rev. A 51, R3414 (1995).
    [CrossRef]
  14. J. Wang and S. I. Chu, Chem. Phys. Lett. 227, 663 (1994).
    [CrossRef]
  15. S. Lang, W. Becker, and J. K. McIver, Phys. Rev. A 52, 2262 (1995).
    [CrossRef]
  16. D. A. Telnov, J. Wang, and S.-I. Chu, Phys. Rev. A 52, 3988 (1995).
    [CrossRef] [PubMed]
  17. N. B. Delone, N. L. Manakov, and A. G. Fainshtein, Sov. Phys. JETP 59, 529 (1984).
  18. M.-Q. Bao and A. Starace, Phys. Rev. A 53, R3723 (1996).
    [CrossRef]
  19. W. Becker, A. Lohr, M. Kleber, and M. Lewenstein, Phys. Rev. A 56, 645 (1997).
    [CrossRef]
  20. S. Erhard and E. U. Gross, in Multiphoton Processes, eds., P. Lambropulos and M. Walther, Inst. Phys. Conf. Serv. 154, 37 (1996).

1997 (3)

F. I. Gauthey, B. M. Garraway, and P. L. Knight, Phys. Rev. A 56, 3093 (1997).
[CrossRef]

F. I. Gauthey, C. H. Keitel, P. L. Knight, and A. Maquet, Phys. Rev. A 55, 615 (1997), and references therein.
[CrossRef]

W. Becker, A. Lohr, M. Kleber, and M. Lewenstein, Phys. Rev. A 56, 645 (1997).
[CrossRef]

1996 (3)

S. Erhard and E. U. Gross, in Multiphoton Processes, eds., P. Lambropulos and M. Walther, Inst. Phys. Conf. Serv. 154, 37 (1996).

R. Burlon, G. Ferrante, C. Leone, P. A. Oleinikov, and V. T. Platonenko, J. Opt. Soc. Am. B 13, 162 (1996).
[CrossRef]

M.-Q. Bao and A. Starace, Phys. Rev. A 53, R3723 (1996).
[CrossRef]

1995 (3)

F. I. Gauthey, C. H. Keitel, P. L. Knight, and A. Maquet, Phys. Rev. A 52, 525 (1995).
[CrossRef] [PubMed]

S. Lang, W. Becker, and J. K. McIver, Phys. Rev. A 52, 2262 (1995).
[CrossRef]

D. A. Telnov, J. Wang, and S.-I. Chu, Phys. Rev. A 52, 3988 (1995).
[CrossRef] [PubMed]

1994 (5)

W. Becker, S. Lang, and J. K. McIver, Phys. Rev. A 50, 1540 (1994).
[CrossRef] [PubMed]

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huiller, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994).
[CrossRef] [PubMed]

S. Watanabe, K. Kando, Y. Nabekawa, A. Sagisaka, and Y. Kolaysashi, Phys. Rev. Lett. 73, 2692 (1994).
[CrossRef] [PubMed]

J. Wang and S. I. Chu, Chem. Phys. Lett. 227, 663 (1994).
[CrossRef]

A. E. Kaplan and P. L. Shkolnikov, Phys. Rev. A 49, 1275 (1994).
[CrossRef] [PubMed]

1993 (1)

L. Plaja and L. Roso, J. Mod. Opt. 40, 793 (1993).
[CrossRef]

1992 (2)

For a review of HHG in the strong field see A. L’Huillier, L. A. Lompre, G. Mainfray, and C. Manus, Adv. At., Mol., Opt. Phys. 1, 139 (1992), and references therein.

J. K. Krause, K. J. Schafer, and K. C. Kulander, Phys. Rev. Lett. 68, 3535 (1992).
[CrossRef] [PubMed]

1988 (1)

E. Conejero Jarque and L. Playa, J. Phys. B 31, 1687 (1988).
[CrossRef]

1984 (1)

N. B. Delone, N. L. Manakov, and A. G. Fainshtein, Sov. Phys. JETP 59, 529 (1984).

Balcou, Ph.

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huiller, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994).
[CrossRef] [PubMed]

Bao, M.-Q.

M.-Q. Bao and A. Starace, Phys. Rev. A 53, R3723 (1996).
[CrossRef]

Becker, W.

W. Becker, A. Lohr, M. Kleber, and M. Lewenstein, Phys. Rev. A 56, 645 (1997).
[CrossRef]

S. Lang, W. Becker, and J. K. McIver, Phys. Rev. A 52, 2262 (1995).
[CrossRef]

W. Becker, S. Lang, and J. K. McIver, Phys. Rev. A 50, 1540 (1994).
[CrossRef] [PubMed]

Burlon, R.

Chu, S. I.

J. Wang and S. I. Chu, Chem. Phys. Lett. 227, 663 (1994).
[CrossRef]

Chu, S.-I.

D. A. Telnov, J. Wang, and S.-I. Chu, Phys. Rev. A 52, 3988 (1995).
[CrossRef] [PubMed]

Conejero Jarque, E.

E. Conejero Jarque and L. Playa, J. Phys. B 31, 1687 (1988).
[CrossRef]

Corkum, P. B.

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huiller, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994).
[CrossRef] [PubMed]

Delone, N. B.

N. B. Delone, N. L. Manakov, and A. G. Fainshtein, Sov. Phys. JETP 59, 529 (1984).

Erhard, S.

S. Erhard and E. U. Gross, in Multiphoton Processes, eds., P. Lambropulos and M. Walther, Inst. Phys. Conf. Serv. 154, 37 (1996).

Fainshtein, A. G.

N. B. Delone, N. L. Manakov, and A. G. Fainshtein, Sov. Phys. JETP 59, 529 (1984).

Ferrante, G.

Garraway, B. M.

F. I. Gauthey, B. M. Garraway, and P. L. Knight, Phys. Rev. A 56, 3093 (1997).
[CrossRef]

Gauthey, F. I.

F. I. Gauthey, B. M. Garraway, and P. L. Knight, Phys. Rev. A 56, 3093 (1997).
[CrossRef]

F. I. Gauthey, C. H. Keitel, P. L. Knight, and A. Maquet, Phys. Rev. A 55, 615 (1997), and references therein.
[CrossRef]

F. I. Gauthey, C. H. Keitel, P. L. Knight, and A. Maquet, Phys. Rev. A 52, 525 (1995).
[CrossRef] [PubMed]

Gross, E. U.

S. Erhard and E. U. Gross, in Multiphoton Processes, eds., P. Lambropulos and M. Walther, Inst. Phys. Conf. Serv. 154, 37 (1996).

Ivanov, M. Yu.

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huiller, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994).
[CrossRef] [PubMed]

Kando, K.

S. Watanabe, K. Kando, Y. Nabekawa, A. Sagisaka, and Y. Kolaysashi, Phys. Rev. Lett. 73, 2692 (1994).
[CrossRef] [PubMed]

Kaplan, A. E.

A. E. Kaplan and P. L. Shkolnikov, Phys. Rev. A 49, 1275 (1994).
[CrossRef] [PubMed]

Keitel, C. H.

F. I. Gauthey, C. H. Keitel, P. L. Knight, and A. Maquet, Phys. Rev. A 55, 615 (1997), and references therein.
[CrossRef]

F. I. Gauthey, C. H. Keitel, P. L. Knight, and A. Maquet, Phys. Rev. A 52, 525 (1995).
[CrossRef] [PubMed]

Kleber, M.

W. Becker, A. Lohr, M. Kleber, and M. Lewenstein, Phys. Rev. A 56, 645 (1997).
[CrossRef]

Knight, P. L.

F. I. Gauthey, C. H. Keitel, P. L. Knight, and A. Maquet, Phys. Rev. A 55, 615 (1997), and references therein.
[CrossRef]

F. I. Gauthey, B. M. Garraway, and P. L. Knight, Phys. Rev. A 56, 3093 (1997).
[CrossRef]

F. I. Gauthey, C. H. Keitel, P. L. Knight, and A. Maquet, Phys. Rev. A 52, 525 (1995).
[CrossRef] [PubMed]

Kolaysashi, Y.

S. Watanabe, K. Kando, Y. Nabekawa, A. Sagisaka, and Y. Kolaysashi, Phys. Rev. Lett. 73, 2692 (1994).
[CrossRef] [PubMed]

Krause, J. K.

J. K. Krause, K. J. Schafer, and K. C. Kulander, Phys. Rev. Lett. 68, 3535 (1992).
[CrossRef] [PubMed]

Kulander, K. C.

J. K. Krause, K. J. Schafer, and K. C. Kulander, Phys. Rev. Lett. 68, 3535 (1992).
[CrossRef] [PubMed]

L’Huiller, A.

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huiller, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994).
[CrossRef] [PubMed]

L’Huillier, A.

For a review of HHG in the strong field see A. L’Huillier, L. A. Lompre, G. Mainfray, and C. Manus, Adv. At., Mol., Opt. Phys. 1, 139 (1992), and references therein.

Lang, S.

S. Lang, W. Becker, and J. K. McIver, Phys. Rev. A 52, 2262 (1995).
[CrossRef]

W. Becker, S. Lang, and J. K. McIver, Phys. Rev. A 50, 1540 (1994).
[CrossRef] [PubMed]

Leone, C.

Lewenstein, M.

W. Becker, A. Lohr, M. Kleber, and M. Lewenstein, Phys. Rev. A 56, 645 (1997).
[CrossRef]

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huiller, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994).
[CrossRef] [PubMed]

Lohr, A.

W. Becker, A. Lohr, M. Kleber, and M. Lewenstein, Phys. Rev. A 56, 645 (1997).
[CrossRef]

Lompre, L. A.

For a review of HHG in the strong field see A. L’Huillier, L. A. Lompre, G. Mainfray, and C. Manus, Adv. At., Mol., Opt. Phys. 1, 139 (1992), and references therein.

Mainfray, G.

For a review of HHG in the strong field see A. L’Huillier, L. A. Lompre, G. Mainfray, and C. Manus, Adv. At., Mol., Opt. Phys. 1, 139 (1992), and references therein.

Manakov, N. L.

N. B. Delone, N. L. Manakov, and A. G. Fainshtein, Sov. Phys. JETP 59, 529 (1984).

Manus, C.

For a review of HHG in the strong field see A. L’Huillier, L. A. Lompre, G. Mainfray, and C. Manus, Adv. At., Mol., Opt. Phys. 1, 139 (1992), and references therein.

Maquet, A.

F. I. Gauthey, C. H. Keitel, P. L. Knight, and A. Maquet, Phys. Rev. A 55, 615 (1997), and references therein.
[CrossRef]

F. I. Gauthey, C. H. Keitel, P. L. Knight, and A. Maquet, Phys. Rev. A 52, 525 (1995).
[CrossRef] [PubMed]

McIver, J. K.

S. Lang, W. Becker, and J. K. McIver, Phys. Rev. A 52, 2262 (1995).
[CrossRef]

W. Becker, S. Lang, and J. K. McIver, Phys. Rev. A 50, 1540 (1994).
[CrossRef] [PubMed]

Nabekawa, Y.

S. Watanabe, K. Kando, Y. Nabekawa, A. Sagisaka, and Y. Kolaysashi, Phys. Rev. Lett. 73, 2692 (1994).
[CrossRef] [PubMed]

Oleinikov, P. A.

Plaja, L.

L. Plaja and L. Roso, J. Mod. Opt. 40, 793 (1993).
[CrossRef]

Platonenko, V. T.

Playa, L.

E. Conejero Jarque and L. Playa, J. Phys. B 31, 1687 (1988).
[CrossRef]

Roso, L.

L. Plaja and L. Roso, J. Mod. Opt. 40, 793 (1993).
[CrossRef]

Sagisaka, A.

S. Watanabe, K. Kando, Y. Nabekawa, A. Sagisaka, and Y. Kolaysashi, Phys. Rev. Lett. 73, 2692 (1994).
[CrossRef] [PubMed]

Schafer, K. J.

J. K. Krause, K. J. Schafer, and K. C. Kulander, Phys. Rev. Lett. 68, 3535 (1992).
[CrossRef] [PubMed]

Shkolnikov, P. L.

A. E. Kaplan and P. L. Shkolnikov, Phys. Rev. A 49, 1275 (1994).
[CrossRef] [PubMed]

Starace, A.

M.-Q. Bao and A. Starace, Phys. Rev. A 53, R3723 (1996).
[CrossRef]

Telnov, D. A.

D. A. Telnov, J. Wang, and S.-I. Chu, Phys. Rev. A 52, 3988 (1995).
[CrossRef] [PubMed]

Wang, J.

D. A. Telnov, J. Wang, and S.-I. Chu, Phys. Rev. A 52, 3988 (1995).
[CrossRef] [PubMed]

J. Wang and S. I. Chu, Chem. Phys. Lett. 227, 663 (1994).
[CrossRef]

Watanabe, S.

S. Watanabe, K. Kando, Y. Nabekawa, A. Sagisaka, and Y. Kolaysashi, Phys. Rev. Lett. 73, 2692 (1994).
[CrossRef] [PubMed]

Adv. At., Mol., Opt. Phys. (1)

For a review of HHG in the strong field see A. L’Huillier, L. A. Lompre, G. Mainfray, and C. Manus, Adv. At., Mol., Opt. Phys. 1, 139 (1992), and references therein.

Chem. Phys. Lett. (1)

J. Wang and S. I. Chu, Chem. Phys. Lett. 227, 663 (1994).
[CrossRef]

Inst. Phys. Conf. Serv. (1)

S. Erhard and E. U. Gross, in Multiphoton Processes, eds., P. Lambropulos and M. Walther, Inst. Phys. Conf. Serv. 154, 37 (1996).

J. Mod. Opt. (1)

L. Plaja and L. Roso, J. Mod. Opt. 40, 793 (1993).
[CrossRef]

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

J. Phys. B (1)

E. Conejero Jarque and L. Playa, J. Phys. B 31, 1687 (1988).
[CrossRef]

Phys. Rev. A (10)

F. I. Gauthey, C. H. Keitel, P. L. Knight, and A. Maquet, Phys. Rev. A 52, 525 (1995).
[CrossRef] [PubMed]

F. I. Gauthey, B. M. Garraway, and P. L. Knight, Phys. Rev. A 56, 3093 (1997).
[CrossRef]

F. I. Gauthey, C. H. Keitel, P. L. Knight, and A. Maquet, Phys. Rev. A 55, 615 (1997), and references therein.
[CrossRef]

W. Becker, S. Lang, and J. K. McIver, Phys. Rev. A 50, 1540 (1994).
[CrossRef] [PubMed]

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huiller, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994).
[CrossRef] [PubMed]

S. Lang, W. Becker, and J. K. McIver, Phys. Rev. A 52, 2262 (1995).
[CrossRef]

D. A. Telnov, J. Wang, and S.-I. Chu, Phys. Rev. A 52, 3988 (1995).
[CrossRef] [PubMed]

M.-Q. Bao and A. Starace, Phys. Rev. A 53, R3723 (1996).
[CrossRef]

W. Becker, A. Lohr, M. Kleber, and M. Lewenstein, Phys. Rev. A 56, 645 (1997).
[CrossRef]

A. E. Kaplan and P. L. Shkolnikov, Phys. Rev. A 49, 1275 (1994).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

J. K. Krause, K. J. Schafer, and K. C. Kulander, Phys. Rev. Lett. 68, 3535 (1992).
[CrossRef] [PubMed]

S. Watanabe, K. Kando, Y. Nabekawa, A. Sagisaka, and Y. Kolaysashi, Phys. Rev. Lett. 73, 2692 (1994).
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

N. B. Delone, N. L. Manakov, and A. G. Fainshtein, Sov. Phys. JETP 59, 529 (1984).

Other (1)

M. D. Perry and J. K. Crane, Phys. Rev. A 48, 54051 (1993); H. Eichmann, A. Egbert, S. Nolte, C. Mornnea, B. Wellegehausen, W. Becker, S. Lang, and J. K. McIver, Phys. Rev. A 51, R3414 (1995).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

|dN|2 in arbitrary units versus harmonic order N calculated for ω2/ω1=3, ω10=25.5ω1, and Ω11=12ω1 at three values of relative phase φ. The ratio (F2/F1)2 between the intensities of the two fields has been taken equal to 1/100. Thick curve, |dN|2 when F2=0 and the intensity of the radiation field is kept fixed.

Fig. 2
Fig. 2

As in Fig. 1 but for (F2/F1)2=10-4.

Fig. 3
Fig. 3

Population of the Nth replica, |αN|2, versus N for two values of relative phase φ evaluated with the same parameters as for Fig. 1.

Fig. 4
Fig. 4

(a) |dN|2 in arbitrary units for N=5 versus the ratio of the field strengths (F2/F1) for two values of the relative phase φ evaluated at ω2/ω1=3, ω10=25.5ω1, and Ω11=12ω1. (b) As in (a) for N=25. (c) As in (a) for N=45.

Fig. 5
Fig. 5

As in Fig. 1, with ω2/ω1=2, for two values of relative phase φ.

Fig. 6
Fig. 6

|dN|2 in arbitrary units versus harmonic order N calculated for ω2/ω1=3 and ω2/ω1=2 with F2=F1, ω10=25.5ω1, Ω11=15/2ω1, and φ=0.

Equations (47)

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

F(t)=[F1 sin ω1t+F2 sin(ω2t+φ)]zˆ,
[H0(r)-D·F(t)]ψ(r, t)=itψ˙(r, t),
ψE(r, t)=exp(-iEt)χ(r, t),
χ(r, t)=nχn(r)exp(-inωt).
H0(r)-D·F(t)-itχ(r, t)=Eχ(r, t).
χ(-r, t+π/ω)=exp(iα)χ(r, t),
χ(r, t+2π/ω)=exp(iα)χ(-r, t+π/ω)=exp(2iα)χ(r, t).
χn(-r)=±(-1)nχn(r),
d(t)=ψE(r, t)|D|ψE(r, t)
d(t)=mnχn(r, t)|D|χm(r, t)exp[i(n-m)ωt],
H0(r)-D·F(t1, t2)-it1-it2χ(r, t1, t2)=Eχ(r, t1, t2),
F(t1, t2)=[F1 sin ω1t1+F2 sin(ω2t2+φ)]zˆ
χ(r, t1+2π/ω, t2+2π/ω)=χ(r, t1, t2).
ψ=exp(-iEt)χE(r, t1=t; t2=t).
χ(r, t1, t2)=n1,n2φn1,n2(r)exp(-in1ω1t1-in2ω2t2),
(-1)n1+n2φn1n2(-r)=±φn1n2(r).
d˜(t1, t2)=χE|D|χE=n1,n2φn1,n2(r)|D|φm1,m2(r)×exp{i[(n1-m1)ω1t+(n2-m2)ω2t]}
d(t)=χ(r, t1=t, t2=t)|D|χ(r, t1=t, t2=t),
d(t)=Ψ(r, t)|ez|Ψ(r, t),
Ψ(r, t)=j=02aj exp(-iEjt)Φj(r),
Φ0(r)=Ψ0(r),
Φ1,2(r)=[Ψ1(r)±Ψ2(r)]/(2)1/2,
a01,
aj(t)=-d0j*djjexp-iΩjjω1cos ω1t+γjjω2cos(ω2t+φ)×exp(iω10t)n1ω1+n2ω2n1ω1+n2ω2+ω10An1,n2(j)×(φ)exp[i(n1ω1+n2ω2)t],
Ωij=dijE1,
γij=dijE2,
dij=eΦj|z|Φi,
ω10=(1-0),
An1,n2(j)(φ)=in1+n2 exp(in2φ)Jn1-Ωjjω1Jn2-γjjω2,
d(t)=d(1)(t)+d(2)(t),
d(1)(t)=j=1|aj|2djj,
d(2)(t)=j=1ajd0j exp(-iω10t)+c.c.
|a1|2-|a2|2=|d01|2d112n1,n2NN n1ω1+n2ω2n1ω1+n2ω2+ω10×(n1-N1)ω1+(n2-N2)ω2(n1-N1)ω1+(n2-N2)ω2+ω10×An1-N1,n2-N2(1)*(φ)An1,n2(1)(φ)exp[i(N1ω1+N2ω2)t]×[1+(-1)N1+N2+1],
An1,n2(1)(φ)=(-1)n1+n2An1,n2(2)(φ).
A-n1,-n2(2)(φ)=An1n2(1)*(φ).
d(t)=N1,N2[dN1,N2(1)+dN1,N2(2)+d-N1,-N2(2)*]×exp[i(N1ω1+N2ω2)t+N2φ],
dN1,N2(1)=|d01|2|d11|n1,n2 n1ω1+n2ω2n1ω1+n2ω2+ω10×(n1-N1)ω1+(n2-N2)ω2(n1-N1)ω1+(n2-N2)ω2+ω10×An1-N1,n2-N2(1)*(φ)An1,n2(1)(φ)×[1+(-1)N1+N2+1,
dN1,N2(2)=-|d01|2|d11|[1+(-1)N1+N2+1]×n1,n2 n1ω1+n2ω2n1ω1+n2ω2+ω10×An1,n2(1)(φ)An1-N1,n2-N2(1)*(φ).
Φ1(r, t)=12[ψ1(r)+ψ2(r)]exp-i1t×s1,s2As1,s2(1)(φ)exp[i(s1ω1+s2ω2)t],
Φ2(r, t)=12[ψ1(r)-ψ2(r)]exp-i1t×s1,s2As1,s2(2)(φ)exp[i(s1ω1+s2ω2)t].
d(t)=dN(φ)exp(iNω1t),
dN(φ)=n2[dN-rn2,n2(1)(φ)+dN-rn2,n2(2)(φ)+d-N+rn2,n2(2)*(φ)]
αN(j)=s2AN-3s2,s2(j)(φ)=s2iN-2s2 exp(is2φ)JN-3s2-Ωjjω1Js2-γjjω2.
dN=N[CN(1)+CN(2)]αN(1)*αN+N(1)+CN(2)αN(1)αN-N(1)*,
CN(1)=(N+N)Nω12/{(ω01+Nω1)×[ω01+(N+N)ω1]}
CN(2)=-Nω1/(ω01-Nω1).
|d2K+1|2=a2K+12+b(φ)F2/F1+c(φ)(F2/F1)2,

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