Abstract

Much of nonlinear optics relates to the coupling of waves and changes in frequency components in the weakly nonlinear regime. The total field of a nonlinear optical waveguide can be expanded in terms of the modal fields of the linear waveguide, with the nonlinearity acting to couple power between the modes. For lossless systems there are at least two constants of the motion, one always being the conserved total power. The second constant has been constructed in various ways in specific problems and has sometimes been identified as a Hamiltonian. We show that a second constant can always be constructed by deriving a general formula for it in terms of the electromagnetic-field variables. Further, the second constant can then be used to write the coupled amplitude equations in Hamiltonian form. Specific examples are given.

[Optical Society of America ]

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  1. J. A. Armstrong , N. Bloembergen , J. Ducuing , and P. S. Pershan , Interactions between light waves in a nonlinear dielectric , Phys. Rev. PHRVAO 127 , 1918 1939 ( 1962
    [CrossRef]
  2. R. H. Stolen , J. Botineau , and A. Ashkin , Intensity discrimination of optical pulses with birefringent fibers , Opt. Lett. OPLEDP 7 , 512 514 ( 1982
    [CrossRef] [PubMed]
  3. R. H. Stolen and J. E. Bjorkholm , Parametric amplification and frequency conversion in optical fibers , IEEE J. Quantum Electron. IEJQA7 QE-18 , 1062 1072 ( 1982
    [CrossRef]
  4. S. M. Jensen , The nonlinear coherent coupler , IEEE J. Quantum Electron. IEJQA7 QE-18 , 1580 1583 ( 1982
    [CrossRef]
  5. B. Daino , G. Gregori , and S. Wabnitz , Stability analysis of nonlinear coherent coupling , J. Appl. Phys. JAPIAU 58 , 4512 4514 ( 1985
    [CrossRef]
  6. S. Trillo and S. Wabnitz , Nonlinear nonreciprocity in a coherent mismatched directional coupler , Appl. Phys. Lett. APPLAB 49 , 752 754 ( 1986
    [CrossRef]
  7. E. Caglioti , S. Trillo , S. Wabnitz , B. Daino , and G. I. Stegeman , Power-dependent switching in a coherent nonlinear directional coupler in the presence of saturation , Appl. Phys. Lett. APPLAB 51 , 293 295 ( 1987
    [CrossRef]
  8. Y. Chen , Four-wave mixing in optical fibers: exact solution , J. Opt. Soc. Am. B JOBPDE 6 , 1986 1993 ( 1989
    [CrossRef]
  9. G. Capellini and S. Trillo , Third-order three-wave mixing in single-mode fibers: exact solutions and spatial instability effects , J. Opt. Soc. Am. B JOBPDE 8 , 824 838 ( 1991
    [CrossRef]
  10. A. W. Snyder and D. J. Mitchell , Description of nonlinear couplers by power conservation , Opt. Lett. OPLEDP 14 , 1146 1148 ( 1989
    [CrossRef] [PubMed]
  11. A. W. Snyder , D. J. Mitchell , L. Poladian , D. R. Rowland , and Y. Chen , Physics of nonlinear fiber couplers , J. Opt. Soc. Am. B JOBPDE 8 , 2102 2118 ( 1991
    [CrossRef]
  12. D. R. Rowland , All-optical devices using nonlinear fiber couplers , J. Lightwave Technol. JLTEDG 9 , 1074 1082 ( 1991
    [CrossRef]
  13. Y. Chen , Mismatched nonlinear couplers with saturable nonlinearity , J. Opt. Soc. Am. B JOBPDE 8 , 986 992 ( 1991
    [CrossRef]
  14. W. Samir , S. J. Garth , and C. Pask , Theory of fused-tapered nonlinear optical fiber couplers , Appl. Opt. APOPAI 32 , 4513 4516 ( 1993
    [CrossRef] [PubMed]
  15. E. Caglioti , S. Trillo , S. Wabnitz , and G. I. Stegeman , Limitations to all-optical switching using nonlinear couplers in the presence of linear and nonlinear absorption and saturation , J. Opt. Soc. Am. B JOBPDE 5 , 472 482 ( 1988
    [CrossRef]
  16. W. Samir , S. J. Garth , and C. Pask , Interplay of grating and nonlinearity in mode coupling , J. Opt. Soc. Am. B JOBPDE 11 , 64 71 ( 1994
    [CrossRef]
  17. S. J. Garth and C. Pask , Polarization behavior in lossy nonlinear birefringent optical fibres , Opt. Quantum Electron. OQELDI 22 , 37 53 ( 1990
    [CrossRef]
  18. J. E. Sipe and G. I. Stegeman , Comparison of normal mode and total field analysis techniques in planar integrated optics , J. Opt. Soc. Am. JOSAAH 69 , 1676 1683 ( 1979
    [CrossRef]
  19. B. Crosignani , P. Di Porto , and A. Yariv , Coupled-mode theory and slowly-varying approximation in guided-wave optics , Opt. Commun. OPCOB8 78 , 237 239 ( 1990
    [CrossRef]
  20. B. Crosignani , P. Di Porto , and A. Yariv , Slowly-varying approximation and coupled-mode equations in guiding structures , Opt. Commun. OPCOB8 91 , 341 342 ( 1992
    [CrossRef]
  21. R. A. Betts , T. Tjugiarto , Y. L. Xue , and P. L. Chu , Nonlinear refractive index in erbium doped optical fiber: theory and experiment , IEEE J. Quantum Electron. IEJQA7 27 , 908 913 ( 1991
    [CrossRef]
  22. S. Trillo , S. Wabnitz , R. Chisari , and G. Capellini , Two-wave mixing in a quadratic nonlinear medium: bifurcations, spatial instabilities, and chaos , Opt. Lett. OPLEDP 17 , 637 639 ( 1992
    [CrossRef] [PubMed]
  23. A. Ankiewicz , A. W. Snyder , and X.-H. Zheng , Coupling between parallel optical fiber cores: critical examination , J. Lightwave Technol. JLTEDG LT-4 , 1317 1323 ( 1986
    [CrossRef]
  24. S. Trillo , S. Wabnitz , R. H. Stolen , G. Assanto , C. T. Seaton , and G. I. Stegeman , Experimental observation of polarization instability in a birefringent optical fiber , Appl. Phys. Lett. APPLAB 49 , 1224 1226 ( 1986
    [CrossRef]
  25. B. Crosignani and P. Di Porto , Intensity-induced rotation of the polarization ellipse in low-birefringence, single-mode optical fibers , Opt. Acta OPACAT 32 , 1251 1258 ( 1985
  26. P. A. Franken , A. E. Hill , C. W. Peters , and G. Weinreich , Generation of optical harmonics , Phys. Rev. Lett. PRLTAO 7 , 118 119 ( 1961
    [CrossRef]
  27. J. H. Marburger and J. F. Lam , Nonlinear theory of degenerate four-wave mixing , Appl. Phys. Lett. APPLAB 34 , 389 391 ( 1979
    [CrossRef]
  28. B. Kryzhanovsky , A. Karapetyan , and B. Glushko , Theory of energy exchange and conversion via four-wave mixing in a nondissipative (3) material , Phys. Rev. A PLRAAN 44 , 6036 6042 ( 1991
    [CrossRef] [PubMed]
  29. J. Brown , Electromagnetic momentum associated with waveguide modes , Proc. IEEE IEEPAD 113 , 27 34 ( 1966
  30. H. A. Haus and H. Kogelnik , Electromagnetic momentum and momentum flow in dielectric waveguides , J. Opt. Soc. Am. A JOAOD6 66 , 320 327 ( 1976
    [CrossRef]
  31. P. S. Pershan , Nonlinear optical properties of solids: energy considerations , Phys. Rev. PHRVAO 130 , 919 929 ( 1963
    [CrossRef]
  32. M. V. Tratnik and J. E. Sipe , Nonlinear polarization dynamics. I. The single pulse equation , Phys. Rev. A PLRAAN 35 , 2965 2975 ( 1987
    [CrossRef] [PubMed]
  33. C. M. de Sterke and J. E. Sipe , Extensions and generalizations of an envelope-function approach for the electrodynamics of nonlinear periodic structures , Phys. Rev. A PLRAAN 39 , 5163 5178 ( 1989
    [CrossRef] [PubMed]
  34. F. N. H. Robinson , Electromagnetic stress and momentum in matter , Phys. Rep. PRPLCM 16 , 313 354 ( 1975
    [CrossRef]

Capellini, G

Karapetyan, A

B. Kryzhanovsky , A. Karapetyan , and B. Glushko , Theory of energy exchange and conversion via four-wave mixing in a nondissipative (3) material , Phys. Rev. A PLRAAN 44 , 6036 6042 ( 1991
[CrossRef] [PubMed]

Other (34)

J. A. Armstrong , N. Bloembergen , J. Ducuing , and P. S. Pershan , Interactions between light waves in a nonlinear dielectric , Phys. Rev. PHRVAO 127 , 1918 1939 ( 1962
[CrossRef]

R. H. Stolen , J. Botineau , and A. Ashkin , Intensity discrimination of optical pulses with birefringent fibers , Opt. Lett. OPLEDP 7 , 512 514 ( 1982
[CrossRef] [PubMed]

R. H. Stolen and J. E. Bjorkholm , Parametric amplification and frequency conversion in optical fibers , IEEE J. Quantum Electron. IEJQA7 QE-18 , 1062 1072 ( 1982
[CrossRef]

S. M. Jensen , The nonlinear coherent coupler , IEEE J. Quantum Electron. IEJQA7 QE-18 , 1580 1583 ( 1982
[CrossRef]

B. Daino , G. Gregori , and S. Wabnitz , Stability analysis of nonlinear coherent coupling , J. Appl. Phys. JAPIAU 58 , 4512 4514 ( 1985
[CrossRef]

S. Trillo and S. Wabnitz , Nonlinear nonreciprocity in a coherent mismatched directional coupler , Appl. Phys. Lett. APPLAB 49 , 752 754 ( 1986
[CrossRef]

E. Caglioti , S. Trillo , S. Wabnitz , B. Daino , and G. I. Stegeman , Power-dependent switching in a coherent nonlinear directional coupler in the presence of saturation , Appl. Phys. Lett. APPLAB 51 , 293 295 ( 1987
[CrossRef]

Y. Chen , Four-wave mixing in optical fibers: exact solution , J. Opt. Soc. Am. B JOBPDE 6 , 1986 1993 ( 1989
[CrossRef]

G. Capellini and S. Trillo , Third-order three-wave mixing in single-mode fibers: exact solutions and spatial instability effects , J. Opt. Soc. Am. B JOBPDE 8 , 824 838 ( 1991
[CrossRef]

A. W. Snyder and D. J. Mitchell , Description of nonlinear couplers by power conservation , Opt. Lett. OPLEDP 14 , 1146 1148 ( 1989
[CrossRef] [PubMed]

A. W. Snyder , D. J. Mitchell , L. Poladian , D. R. Rowland , and Y. Chen , Physics of nonlinear fiber couplers , J. Opt. Soc. Am. B JOBPDE 8 , 2102 2118 ( 1991
[CrossRef]

D. R. Rowland , All-optical devices using nonlinear fiber couplers , J. Lightwave Technol. JLTEDG 9 , 1074 1082 ( 1991
[CrossRef]

Y. Chen , Mismatched nonlinear couplers with saturable nonlinearity , J. Opt. Soc. Am. B JOBPDE 8 , 986 992 ( 1991
[CrossRef]

W. Samir , S. J. Garth , and C. Pask , Theory of fused-tapered nonlinear optical fiber couplers , Appl. Opt. APOPAI 32 , 4513 4516 ( 1993
[CrossRef] [PubMed]

E. Caglioti , S. Trillo , S. Wabnitz , and G. I. Stegeman , Limitations to all-optical switching using nonlinear couplers in the presence of linear and nonlinear absorption and saturation , J. Opt. Soc. Am. B JOBPDE 5 , 472 482 ( 1988
[CrossRef]

W. Samir , S. J. Garth , and C. Pask , Interplay of grating and nonlinearity in mode coupling , J. Opt. Soc. Am. B JOBPDE 11 , 64 71 ( 1994
[CrossRef]

S. J. Garth and C. Pask , Polarization behavior in lossy nonlinear birefringent optical fibres , Opt. Quantum Electron. OQELDI 22 , 37 53 ( 1990
[CrossRef]

J. E. Sipe and G. I. Stegeman , Comparison of normal mode and total field analysis techniques in planar integrated optics , J. Opt. Soc. Am. JOSAAH 69 , 1676 1683 ( 1979
[CrossRef]

B. Crosignani , P. Di Porto , and A. Yariv , Coupled-mode theory and slowly-varying approximation in guided-wave optics , Opt. Commun. OPCOB8 78 , 237 239 ( 1990
[CrossRef]

B. Crosignani , P. Di Porto , and A. Yariv , Slowly-varying approximation and coupled-mode equations in guiding structures , Opt. Commun. OPCOB8 91 , 341 342 ( 1992
[CrossRef]

R. A. Betts , T. Tjugiarto , Y. L. Xue , and P. L. Chu , Nonlinear refractive index in erbium doped optical fiber: theory and experiment , IEEE J. Quantum Electron. IEJQA7 27 , 908 913 ( 1991
[CrossRef]

S. Trillo , S. Wabnitz , R. Chisari , and G. Capellini , Two-wave mixing in a quadratic nonlinear medium: bifurcations, spatial instabilities, and chaos , Opt. Lett. OPLEDP 17 , 637 639 ( 1992
[CrossRef] [PubMed]

A. Ankiewicz , A. W. Snyder , and X.-H. Zheng , Coupling between parallel optical fiber cores: critical examination , J. Lightwave Technol. JLTEDG LT-4 , 1317 1323 ( 1986
[CrossRef]

S. Trillo , S. Wabnitz , R. H. Stolen , G. Assanto , C. T. Seaton , and G. I. Stegeman , Experimental observation of polarization instability in a birefringent optical fiber , Appl. Phys. Lett. APPLAB 49 , 1224 1226 ( 1986
[CrossRef]

B. Crosignani and P. Di Porto , Intensity-induced rotation of the polarization ellipse in low-birefringence, single-mode optical fibers , Opt. Acta OPACAT 32 , 1251 1258 ( 1985

P. A. Franken , A. E. Hill , C. W. Peters , and G. Weinreich , Generation of optical harmonics , Phys. Rev. Lett. PRLTAO 7 , 118 119 ( 1961
[CrossRef]

J. H. Marburger and J. F. Lam , Nonlinear theory of degenerate four-wave mixing , Appl. Phys. Lett. APPLAB 34 , 389 391 ( 1979
[CrossRef]

B. Kryzhanovsky , A. Karapetyan , and B. Glushko , Theory of energy exchange and conversion via four-wave mixing in a nondissipative (3) material , Phys. Rev. A PLRAAN 44 , 6036 6042 ( 1991
[CrossRef] [PubMed]

J. Brown , Electromagnetic momentum associated with waveguide modes , Proc. IEEE IEEPAD 113 , 27 34 ( 1966

H. A. Haus and H. Kogelnik , Electromagnetic momentum and momentum flow in dielectric waveguides , J. Opt. Soc. Am. A JOAOD6 66 , 320 327 ( 1976
[CrossRef]

P. S. Pershan , Nonlinear optical properties of solids: energy considerations , Phys. Rev. PHRVAO 130 , 919 929 ( 1963
[CrossRef]

M. V. Tratnik and J. E. Sipe , Nonlinear polarization dynamics. I. The single pulse equation , Phys. Rev. A PLRAAN 35 , 2965 2975 ( 1987
[CrossRef] [PubMed]

C. M. de Sterke and J. E. Sipe , Extensions and generalizations of an envelope-function approach for the electrodynamics of nonlinear periodic structures , Phys. Rev. A PLRAAN 39 , 5163 5178 ( 1989
[CrossRef] [PubMed]

F. N. H. Robinson , Electromagnetic stress and momentum in matter , Phys. Rep. PRPLCM 16 , 313 354 ( 1975
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

Overview of the analysis.

Equations (71)

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Γ=m=1Lk=1NmβmkΠmk+1n+1 APtNLEt*+PtNL*EtdA,
dΠmkdz=-HBmk,
dBmkdz=HΠmk,
×E=-Bt,
×H=Dt,
D=0,
B=0,
B=μ0H,
D=E+PNL,
Et=m 12 [am(z)exp(iβmz)+bm(z)exp(-iβmz)]e¯t,m(x, y),
Ht=m 12 [am(z)exp(iβmz)-bm(z)exp(-iβmz)]h¯t,m(x, y),
z A(Et×H¯t*+E¯t*×Ht)zˆdA
=iωAE¯*PNLdA,
A(e¯t,m×h¯t,k*)zˆdA=A(e¯t,m*×h¯t,k)zˆdA=2skδm,k,
dakdz=iω2sk Ae¯k*PNLdA exp(-iβkz).
ak* dakdz=-ωskβk A 12 ak*(PtNLe¯t,mk*)×dA ddz [exp(-iβkz)].
Et,k*=12 ak*e¯t,k* exp(-iβkz).
dEt,k*dz=12 ak*e¯t,k* ddz [exp(-iβkz)]+12 e¯t,k* exp(-iβkz) dak*dz.
12 ak*(PtNLe¯t,k*) ddz [exp(-iβkz)]
=PtNL dEt,k*dz-12 (PtNLe¯t,k*)exp(-iβkz) dak*dz.
βkω skak* dakdz=-APtNL dEt,k*dzdA+dak*dz×A 12 (PtNLe¯t,k*)exp(-iβkz)dA.
βkω skak* dakdz=-APtNL dEt,m*dzdA-iω sk dak*dz dakdz.
βkω sk d|ak|2dz=-APtNL dEt,k*dz+PtNL* dEt,kdzdA.
ddz k=1N βkPkω=-APtNL dEt*dz+PtNL*dEtdzdA.
m=1Lk=1Nm βmkωm dPmkdz
=-m=1LAPmNL dEtm*dz+PmNL* dEtmdzdA
=-APNL dEt*dz+PNL* dEtdzdA,
P(3)=0χ(3)EEE,
P(3)=P(3)+P(3)*
E=E+E*.
ddz (P(3)E)=ddz (E0χ(3)EEE),
dP(3)dz E=34 ddz (P(3)E).
dP(3)dz E*+dP(3)*dz E=34 ddz P(3)E*+P(3)*E.
k=1N βkPkω+14 APt(3)Et*+Pt(3)*EtdA=Γ(3),
k=1N βkPkω+1n+1 APt(n)Et*+Pt(n)*EtdA
=Γ(n).
PNL=P(3)+P(5)+ .
Γsat=k=1N βkPkω+n 1n+1 APt(n)Et*+Pt(n)*EtdA,n=3, 5,  .
β1P1ω+β2P22ω+13 APt(2)Et*+Pt(2)*EtdA
=Γ(2),
skak=Pk exp(iϕk),
Pk=sk|ak|2.
sk d|ak|2dz=iωak*2 APtNLe¯t,k*dA exp(-iβkz)-ak2 APtNL*e¯t,kdA exp(iβkz).
dPkdz=iωA[PtNLEt,k*-PtNL*Et,k]dA.
dϕkdz=ω2Pk A[PtNLEt,k*+PtNL*Et,k]dA.
Bk=βkz+ϕk,
dBkdz=βk+ω2Pk A[PtNLEt,k*+PtNL*Et,k]dA.
d(Pk/ω)dz=-HBk,
dBkdz=H(Pk/ω),
h¯t,k*n00czˆ×e¯t,k*,
sk12 n00cAe¯t,k2dA,
E(x, y, z)=12 P1(z)s1 e¯1(x, y)exp(B1)+12 P2(z)s2 e¯2(x, y)exp(B2),
P(3)(ω)=0χ(3)[2(EE*)E+(EE)E*],
H=Γ=β1Π1+β2Π2+ω4 [Q11Π12+Q22Π22+4Q12Π1Π2+2Q12Π1Π2 cos 2(B1-B2)],
Qij=3ω0χ(3)8sisj Ae¯i2e¯j2dA,i, j=1, 2,
Ae¯i3e¯jdA=0,ij,
Γs=S2+Q2(β1-β2) S12,
S1=2P1P2 cos(B1-B2),
S2=P1-P2.
ωH=(β1-β2)2 Γs+Q4 P2+(β1+β2)2 P,
dΠxdz=-dΠydz=ωRΠxΠy sin 2θ,
dBxdz=βx+3ωR2 Πx+ωR2 Πy(2+cos 2θ),
dBydz=βy+3ωR2 Πy+ωR2 Πx(2+cos 2θ),
H=βxΠx+βyΠy+3ωR4 (Πx2+Πy2)+ωR2 ΠxΠy(2+cos 2θ).
E=12 a1(z)e¯1(x, y)exp(iβ1z-iω1t)+12 a2(z)e¯2(x, y)exp(iβ2z-iω2t)+c.c.
H=β1Π1+β2Π2+Rω1Π1ω2Π2 cos θ,
E=j=14 12 aj(z)e¯j(x, y)exp(iβjz-iωjt)+c.c.
H=j=14βjΠj+2c1ω1ω2ω3ω4Π1Π2Π3Π4×cos θ+c22 j=14k=14IjkωjωkΠjΠk,
Pn=-FEn*.
PnNL=-FNLEn*.
Γ=m,kβmkΠmk-AFNLdA.

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