Abstract

A transfer-matrix method based on perturbation expansion is proposed as an alternative way of simulating the transmission spectrum of a binary long-period grating (LPG). We first generalize the concept of transfer matrices for a heterojunction waveguide. For the couplings among copropagating modes, forward transfer matrices are used to describe the evolution of mode amplitudes along the grating. We show that these elements are related to the well-known coupling coefficients. The method is then used for the study of ideal two-mode grating couplers, and analytic solutions are obtained. We also use the matrix method to study multimode couplings in a LPG and compare the results with those obtained by using the coupled-mode theory. To further demonstrate its usefulness, we apply the method to a special quasi-periodic LPG, the Fibonacci grating. The results show that each cladding mode contributes to several transmission dips and that the dips of different cladding modes are grouped according to the special resonance conditions.

© 1999 Optical Society of America

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References

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  1. G. Meltz, W. W. Morey, W. H. Glenn, “Formation of Bragg gratings in optical fibers by a transverse holographic method,” Opt. Lett. 14, 823–825 (1989).
    [CrossRef] [PubMed]
  2. A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
    [CrossRef]
  3. A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, P. J. Lemaire, N. S. Bergano, C. R. Davidson, “Long-period fiber-grating-based gain equalizers,” Opt. Lett. 21, 336–338 (1996).
    [CrossRef] [PubMed]
  4. V. Bhatia, A. M. Vengsarkar, “Optical fiber long-period grating sensors,” Opt. Lett. 21, 692–694 (1996).
    [CrossRef] [PubMed]
  5. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, Boston, 1991).
  6. V. Mizrahi, J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. 11, 1513–1517 (1993).
    [CrossRef]
  7. T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14, 1760–1773 (1997).
    [CrossRef]
  8. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
  9. H. A. Macleod, Thin-Film Optical Filters (American Elsevier, New York, 1969).
  10. P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69, 742–756 (1979).
    [CrossRef]
  11. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1991), Secs. 31-1 and 31-8.
  12. A. W. Snyder, “Excitation and scattering of modes on a dielectric or optical fiber,” IEEE Trans. Microwave Theory Tech. MTT-17, 1138–1144 (1969).
    [CrossRef]
  13. D. Levine, P. J. Steinhardt, “Quasicrystals: a new class of ordered structures,” Phys. Rev. Lett. 53, 2477–2480 (1984).
    [CrossRef]
  14. R. Merlin, K. Bajema, R. Clarke, F.-Y. Juang, P. K. Bhattacharya, “Quasiperiodic GaAs-AlAs heterostructures,” Phys. Rev. Lett. 55, 1768–1771 (1985).
    [CrossRef] [PubMed]
  15. D. Levine, P. J. Steinhardt, “Quasicrystals. I. Definitions and structure,” Phys. Rev. B 34, 596–616 (1986).
    [CrossRef]

1997 (1)

1996 (3)

1993 (1)

V. Mizrahi, J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. 11, 1513–1517 (1993).
[CrossRef]

1989 (1)

1986 (1)

D. Levine, P. J. Steinhardt, “Quasicrystals. I. Definitions and structure,” Phys. Rev. B 34, 596–616 (1986).
[CrossRef]

1985 (1)

R. Merlin, K. Bajema, R. Clarke, F.-Y. Juang, P. K. Bhattacharya, “Quasiperiodic GaAs-AlAs heterostructures,” Phys. Rev. Lett. 55, 1768–1771 (1985).
[CrossRef] [PubMed]

1984 (1)

D. Levine, P. J. Steinhardt, “Quasicrystals: a new class of ordered structures,” Phys. Rev. Lett. 53, 2477–2480 (1984).
[CrossRef]

1979 (1)

1969 (1)

A. W. Snyder, “Excitation and scattering of modes on a dielectric or optical fiber,” IEEE Trans. Microwave Theory Tech. MTT-17, 1138–1144 (1969).
[CrossRef]

Bajema, K.

R. Merlin, K. Bajema, R. Clarke, F.-Y. Juang, P. K. Bhattacharya, “Quasiperiodic GaAs-AlAs heterostructures,” Phys. Rev. Lett. 55, 1768–1771 (1985).
[CrossRef] [PubMed]

Bergano, N. S.

Bhatia, V.

V. Bhatia, A. M. Vengsarkar, “Optical fiber long-period grating sensors,” Opt. Lett. 21, 692–694 (1996).
[CrossRef] [PubMed]

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

Bhattacharya, P. K.

R. Merlin, K. Bajema, R. Clarke, F.-Y. Juang, P. K. Bhattacharya, “Quasiperiodic GaAs-AlAs heterostructures,” Phys. Rev. Lett. 55, 1768–1771 (1985).
[CrossRef] [PubMed]

Clarke, R.

R. Merlin, K. Bajema, R. Clarke, F.-Y. Juang, P. K. Bhattacharya, “Quasiperiodic GaAs-AlAs heterostructures,” Phys. Rev. Lett. 55, 1768–1771 (1985).
[CrossRef] [PubMed]

Davidson, C. R.

Erdogan, T.

T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14, 1760–1773 (1997).
[CrossRef]

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

Glenn, W. H.

Juang, F.-Y.

R. Merlin, K. Bajema, R. Clarke, F.-Y. Juang, P. K. Bhattacharya, “Quasiperiodic GaAs-AlAs heterostructures,” Phys. Rev. Lett. 55, 1768–1771 (1985).
[CrossRef] [PubMed]

Judkins, J. B.

A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, P. J. Lemaire, N. S. Bergano, C. R. Davidson, “Long-period fiber-grating-based gain equalizers,” Opt. Lett. 21, 336–338 (1996).
[CrossRef] [PubMed]

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

Lemaire, P. J.

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, P. J. Lemaire, N. S. Bergano, C. R. Davidson, “Long-period fiber-grating-based gain equalizers,” Opt. Lett. 21, 336–338 (1996).
[CrossRef] [PubMed]

Levine, D.

D. Levine, P. J. Steinhardt, “Quasicrystals. I. Definitions and structure,” Phys. Rev. B 34, 596–616 (1986).
[CrossRef]

D. Levine, P. J. Steinhardt, “Quasicrystals: a new class of ordered structures,” Phys. Rev. Lett. 53, 2477–2480 (1984).
[CrossRef]

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1991), Secs. 31-1 and 31-8.

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters (American Elsevier, New York, 1969).

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, Boston, 1991).

Meltz, G.

Merlin, R.

R. Merlin, K. Bajema, R. Clarke, F.-Y. Juang, P. K. Bhattacharya, “Quasiperiodic GaAs-AlAs heterostructures,” Phys. Rev. Lett. 55, 1768–1771 (1985).
[CrossRef] [PubMed]

Mizrahi, V.

V. Mizrahi, J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. 11, 1513–1517 (1993).
[CrossRef]

Morey, W. W.

Pedrazzani, J. R.

Sipe, J. E.

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

V. Mizrahi, J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. 11, 1513–1517 (1993).
[CrossRef]

Snyder, A. W.

A. W. Snyder, “Excitation and scattering of modes on a dielectric or optical fiber,” IEEE Trans. Microwave Theory Tech. MTT-17, 1138–1144 (1969).
[CrossRef]

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1991), Secs. 31-1 and 31-8.

Steinhardt, P. J.

D. Levine, P. J. Steinhardt, “Quasicrystals. I. Definitions and structure,” Phys. Rev. B 34, 596–616 (1986).
[CrossRef]

D. Levine, P. J. Steinhardt, “Quasicrystals: a new class of ordered structures,” Phys. Rev. Lett. 53, 2477–2480 (1984).
[CrossRef]

Vengsarkar, A. M.

Yeh, P.

IEEE Trans. Microwave Theory Tech. (1)

A. W. Snyder, “Excitation and scattering of modes on a dielectric or optical fiber,” IEEE Trans. Microwave Theory Tech. MTT-17, 1138–1144 (1969).
[CrossRef]

J. Lightwave Technol. (2)

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

V. Mizrahi, J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. 11, 1513–1517 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Lett. (3)

Phys. Rev. B (1)

D. Levine, P. J. Steinhardt, “Quasicrystals. I. Definitions and structure,” Phys. Rev. B 34, 596–616 (1986).
[CrossRef]

Phys. Rev. Lett. (2)

D. Levine, P. J. Steinhardt, “Quasicrystals: a new class of ordered structures,” Phys. Rev. Lett. 53, 2477–2480 (1984).
[CrossRef]

R. Merlin, K. Bajema, R. Clarke, F.-Y. Juang, P. K. Bhattacharya, “Quasiperiodic GaAs-AlAs heterostructures,” Phys. Rev. Lett. 55, 1768–1771 (1985).
[CrossRef] [PubMed]

Other (4)

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

H. A. Macleod, Thin-Film Optical Filters (American Elsevier, New York, 1969).

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1991), Secs. 31-1 and 31-8.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, Boston, 1991).

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Figures (15)

Fig. 1
Fig. 1

Schematic diagram of a waveguide heterojunction. The two waveguide structures are slightly different, and the same number is used to label the modes of each waveguide.

Fig. 2
Fig. 2

Incidence of one waveguide mode k on the interface from region 0 to region 1. Region 1 is formed by UV exposure, and its waveguide structure is assumed to be slightly perturbed from that of region 0. The incident mode will partially transmit and partially reflect.

Fig. 3
Fig. 3

Incidence of mode k from region 1 to region 0, opposite to that shown in Fig. 2.

Fig. 4
Fig. 4

Schematic diagram showing the distributed coupling between two waveguide modes that occurs in each interface of the 0 and 1 regions in a binary LPG. The interface acts as a splitter with the amplitude splitting ratios shown in the diagram.

Fig. 5
Fig. 5

Schematic diagram showing the mechanism of coupling between two modes in a binary LPG. The couplings occur at the interfaces of regions 1 and 0 with well-defined splitting ratios. The propagation of complex amplitudes of the modes can be described by using transfer matrices. The transfer matrix of a 01 unit cell is composed of four elementary matrices: F(0|1), P(1), F(1|0), and P(0), which are defined in the text.

Fig. 6
Fig. 6

Comparison of transmission spectra for binary LPG’s with number of cells equal to 25, 30, and 35.

Fig. 7
Fig. 7

Transmission spectra of a binary and a uniform sinusoidal LPG. Each has a grating length of (a) 15 mm, (b) 27.5 mm, and (c) 55 mm.

Fig. 8
Fig. 8

Comparison of transmission spectra for binary LPG’s with different duty cycles.

Fig. 9
Fig. 9

Dispersion curves of effective indices of LP01 core mode and HE1ν cladding modes with ν=1, 3, 5, 7, 9.

Fig. 10
Fig. 10

Dispersion curves of coupling coefficients κ011νcocl of cladding modes with ν=1, 3, 5, 7, 9.

Fig. 11
Fig. 11

Transmission spectra of a binary and a uniform sinusoidal LPG. The duty cycle of the binary LPG is set to (a) unity, (b) 1/4, and (c) 4.

Fig. 12
Fig. 12

Schematic diagram of a quasi-periodic binary grating. The point set {z0, z1, z2 ,} forms the Bravis lattice of the quasi-periodic lattice. Each lattice point is then placed in a region 1, which acts as a diffractor.

Fig. 13
Fig. 13

Transmission spectrum of a quasi-periodic Fibonacci LPG. The transmission dips are grouped according to the resonance conditions (m, n). The first group corresponds to (1, 2), the second corresponds to (1, 1), and the third corresponds to (0, 1).

Fig. 14
Fig. 14

Transmission spectrum of the ν=3 cladding mode in a quasi-periodic Fibonacci LPG. The labels above the main peaks indicate the corresponding resonance conditions.

Fig. 15
Fig. 15

Transmission spectrum of the ν=9 cladding mode in a quasi-periodic Fibonacci LPG. The labels above the main peaks indicate the corresponding resonance conditions.

Equations (139)

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Et(r)=j[Aj(r)etj(r) exp(iβjz)+Bj(r)etj(r) exp(-iβjz)],
Ht(r)=j[Aj(r)htj(r) exp(iβjz)-Bj(r)htj(r) exp(-iβjz)]
(r=a, b).
12 A[ei(r)×hj(r)*]z dA
=12 A[ei(r)*×hj(r)]z dA=Nj(r)δij,
Ak(a)=j12 (Ikj+Jkj)Aj(b)+12 (Ikj-Jkj)Bj(b),
Bk(a)=j12 (Ikj-Jkj)Aj(b)+12 (Ikj+Jkj)Bj(b),
Ikj=12Nk(a) A[etj(b)×htk(a)*]z dA,
Jkj=12Nk(a) A[etk(a)*×htj(b)]z dA.
Ak(a)Bk(a)=12(Ikj+Jkj)12(Ikj-Jkj)12(Ikj-Jkj)12(Ikj+Jkj) Aj(b)Bj(b).
e¯j=ej(0),h¯j=hj(0),
ej=ej(0)+μδej,hj=hj(0)+μδhj.
(E0+B¯k)e¯tk+jkB¯je¯tj=Ak(e¯tk+μδetk)+jkAj(e¯tj+μδetj),
(E0-B¯k)h¯tk-jkB¯jh¯tj=Ak(h¯tk+μδhtk)+jkAj(h¯tj+μδhtj).
E0+B¯k=(1+μαkk)Ak+μjkαjkAj,
E0-B¯k=(1+μβkk)Ak+μjkβjkAj,
B¯j=(1+μαjj)Aj+μαkjAk+μij,kαijAi
(jk),
-B¯j=(1+μβjj)Aj+μβkjAk+μij,kBijAi
(jk),
αjk=12N¯k A(δetj×h¯tk*)z dA,
βjk=12N¯k A(e¯tk*×δhtj)z dA.
B¯j=B¯j(0)+μB¯j(1)+μ2B¯j(2)+ ,
Aj=Aj(0)+μAj(1)+μ2Aj(2)+ .
Ak(0)=E0,B¯k(0)=0,
Aj(0)=B¯j(0)=0.
Ak(1)=-12(αkk+βkk)E0,
B¯k(1)=12(αkk-βkk)E0,
Aj(1)=-12(αkj+βkj)E0(jk),
B¯j(1)=12(αkj-βkj)E0(jk).
αkj=akj+bkj-δkj,βkj=akj-bkj-δkj,
Ak=1-μ22 DkE0,
B¯k=μbkk(1)E0,
Aj=-μakj(1)E0(jk),
B¯j=μbkj(1)E0(jk),
E02-|B¯k|2-jk|B¯j|2 = |Ak|2+jk|Aj|2.
A¯k=1-μ22 DkE1,
Bk=-μbkk(1)E1,
A¯j=μakj(1)E1(jk),
Bj=-μbkj(1)E1(jk).
κkj=κkjt+κkjz,κkj=κkjt-κkjz,
κkjt=k04 0μ01/2A(n2-n¯2)etk(0)*  etj(0) dA,
κkjz=k04 0μ01/2 n¯2n2 A(n2-n¯2)ezk(0)*ezj(0) dA.
ajk(1)=1βj(0)-βk(0) (κkjt+κkjz)=κkjβj(0)-βk(0),
bjk(1)=-1βj(0)+βk(0) (κkjt-κkjz)=-κkjβj(0)+βk(0).
A¯kB¯kA¯jB¯j=F11F12F13F14F21F22F23F24F31F32F33F34F41F42F43F44 AkBkAjBj.
Ak(L)Bk(L)Aj(L)Bj(L)
=exp(iβkL)0000exp(-iβkL)0000exp(iβjL)0000exp(-iβjL)
×Ak(0)Bk(0)Aj(0)Bj(0),
AkAj=δkγ-γ*δj A¯kA¯jF(1|0)A¯kA¯j;
A¯kA¯j=δk-γγ*δj AkAjF(0|1)AkAj.
δk1-μ2Dk/2,
δj1-μ2Dj/2,
γμ κkjβk(0)-βj(0)=μajk(1).
P(0)=exp[iβk(0)Λ(0)]00exp[iβj(0)Λ(0)]exp[iθk(0)]00exp[iθj(0)],
P(1)=exp{i[βk(0)+μβk(1)]Λ(1)}00exp{i[βj(0)+μβj(1)]Λ(1)}exp[iθk(1)]00exp[iθj(1)].
F=P(0)F(0|1)P(1)F(1|0).
Ak(n+1)Aj(n+1)=FAk(n)Aj(n),
F=exp(iθk){δk2+|γ|2 exp[iΔ(1)]}exp(iθk)γ{δk-δj exp[iΔ(1)]}-exp(iθj)γ*{δj-δk exp[-iΔ(1)]}exp(iθj){δk2+|γ|2 exp[-iΔ(1)]}.
θk=θk(0)+θk(1),θj=θj(0)+θj(1),
Δ(1)=θj(1)-θk(1),Δ(0)=θj(0)-θk(0),
Fkj(1|0)=δj=1-μ2Dj/2(k=j)γkj=μ κkjβk(0)-βj(0)(kj).
Fkj(0|1)=δj=1-μ2Dj/2(k=j)-γkj=μ -κkjβk(0)-βj(0)(kj).
Pkj(0)=δkj exp[iβj(0)Λ(0)],
Pkj(1)=δkj exp{i[βj(0)+μβj(1)]Λ(1)}.
δk=δj=1-γ2δ.
F=exp(iθ¯)expi -Δ2{δ2+γ2 exp[iΔ(1)]}expi -Δ2γδ{1-exp[iΔ(1)]}-expi Δ2γδ{1-exp[-iΔ(1)]}expi Δ2{δ2+γ2 exp[-iΔ(1)]}exp(iθ¯)ABCD,
F+F=I.
FN=exp(iNθ¯)A sin(NKΛ)-sin[(N-1)KΛ]sin(KΛ)B sin(NKΛ)sin(KΛ)C sin(NKΛ)sin(KΛ)D sin(NKΛ)-sin[(N-1)KΛ]sin(KΛ),
cos(KΛ)=12 (A+D)=(1-γ2)cosΔ(1)+Δ(0)2+γ2 cosΔ(1)-Δ(0)2.
Δ(1)+Δ(0)±2mπ.
βk(0)-βj(0)-2πΛ+μ[βk(1)-βj(1)] Λ(1)Λ=0,
κkjω0n12 ΔngAco(et01co*et1νcl)dA=κ011νcocl,
βk(1)ω0n12 ΔngAco(et01co*et01co)dA=κ0101coco,
Δnsin(z)=Δng 12 [1+sin(Kz)]=Δng12+14i exp(iKz)-14i exp(-iKz),
Δnbin(z)=Δngn[u(z-nΛ)-u(z-nΛ-Λ(1))]=Δng12+1πi exp(iKz)-1πi exp(-iKz)+13πi exp(i3Kz)-13πi exp(-i3Kz)+,
λ0=(n01co-n1νcl)Λ/1-κ0101cocoΛ2π R1+R.
AcoA1νcl=1-D01co2κ011νcoclβ01co-β1νclκ1ν01clcoβ1νcl-β01co1-D1νcl2 A¯coA¯1νclF(1|0)A¯coA¯1νcl;
A¯coA¯1νcl=1-D01co2-κ011νcoclβ01co-β1νcl-κ1ν01clcoβ1νcl-β01co1-D1νcl2 AcoA1νclF(0|1)AcoA1νcl,
P(0)=exp[iβ01coΛ(0)]00exp[iβ1νclΛ(0)],
P(1)=exp[i(β01co+κ0101coco)Λ(1)]00exp[i(β1νcl+κ1ν1νclcl)Λ(1)].
Aco(n+1)A1cl(n+1)A2cl(n+1)=FAco(n)A1cl(n)A2cl(n).
A1νclAco(0)exp(iβ1νclzN)ηn=0N exp{i[(β01co-β1νcl)zn+nκ0101cocoΛ(1)]}ηn=1N exp[iQzn+nκ0101cocoΛ(1)]ηS(Q),
zn=ΛBn+1τ nτ,
S(Q)=n expiQn(1+α)+1τ nτΛB.
S(Q)p,q sin Xp,qXp,q exp(iXp,q)δ(Q-Qp,q),
Qp,q=τ(q+pτ)1+ττ(1+α) 2πΛB
β01co-β1νcl-(m+nτ) 2πΛ¯=0,
βj=βj(0)+k00μ01/2 μA(n2-n¯2)ej  ej(0)* dAA[ej×hj(0)*+ej(0)*×hj] z dA;
etj=k(ajk+bjk)etk(0),
htj=k(ajk-bjk)htk(0),
ezj=n¯2n2 k(ajk-bjk)ezk(0),
hzj=k(ajk+bjk)hzk(0),
ajk=k04Nk(0)[βj-βk(0)] 0μ01/2μA(n2-n¯2)ej  ek(0)* dA,
bjk=-k04Nk(0)[βj+βk(0)] 0μ01/2μA(n2-n¯2)ej  e-k(0)* dA,
βj=βj(0)+μβj(1)+μ2βj(2)+ ,
ej=ej(0)+μej(1)+μ2ej(2)+ ,
hj=hj(0)+μhj(1)+μ2hj(2)+ .
βj(1)=k04Nj(0) 0μ01/2A(n2-n¯2)ej(0)  ej(0)* dA.
ajk=μajk(1)+μ2ajk(2)+ ,ajk(0)=0,
ajk(1)=k04Nk(0)[βj(0)-βk(0)] 0μ01/2×A(n2-n¯2)ej(0)  ek(0)* dA,
bjk=μbjk(1)+μ2bjk(2)+ ,bjk(0)=0,
bjk(1)=-k04Nk(0)[βj(0)+βk(0)] 0μ01/2×A(n2-n¯2)ej(0)  e-k(0)* dA.
bjj(0)=0,
bjj(1)=-k04Nj(0)2βj(0) 0μ01/2A(n2-n¯2)ej(0)  e-j(0)* dA.
ajj(0)=1.
Nj(1)=12 A{[etj(0)+μetj(1)+μ2etj(2)]×[htj(0)+μhtj(1)+μ2htj(2)]} z dA=[1+2μajj(1)+2μ2ajj(2)]Nj(0)+μ2k[|ajk(1)|2-|bjk(1)|2]Nk(0).
Zj=1+2μajj(1)+2μ2ajj(2)+μ2k[|ajk(1)|2-|bjk(1)|2].
ajj(0)=1/Zj1-12 2μajj(1)+2μ2ajj(2)+μ2k[|ajk(1)|2-|bjk(1)|2].
1=|ajj(0)|2+μ2|bjj(1)|2+kj[|ajk(1)|2+|bjk(1)|2].
ajj(1)=0,ajj(2)=k|bjk(1)|2.
ajj1-12 μ2Dj=1-12 μ2|bjj(1)|2+kj[|ajk(1)|2+|bjk(1)|2].
1μbkk(1)0μbkj(1)=[F]1-μ22 Dk0-μakj(1)0.
0μbjk(1)1μbjj(1)=[F]-μajk(1)01-μ22 Dj0.
1=F111-μ22 Dk+F13[-μakj(1)],
0=F11[-μajk(1)]+F131-μ22 Dj.
F11=1+μ2κkjκjk[βj(0)-βk(0)]2+12 Dk,
F12=μ -κkk2βk(0)+μ2 -κkjκjk[βj(0)-βk(0)][βj(0)+βk(0)],
F13=μ κkjβj(0)-βk(0),
F14=μ -κkjβj(0)+βk(0)+μ2 -κkjκjj2βj(0)[βj(0)-βk(0)],
F21=μ -κkk2βk(0)+μ2 -κkjκjk[βj(0)-βk(0)][βj(0)+βk(0)],
F22=1+μ2κkk2[2βk(0)]2+κkjκjk[βj(0)+βk(0)]2-12 Dk,
F23=μ -κkjβj(0)+βk(0)+μ2 -κkkκkj2βk(0)[βj(0)-βk(0)],
F24=μ κkjβj(0)-βk(0)+μ2κkkκkj2βk(0)[βj(0)+βk(0)]+κkjκjj2βj(0)[βj(0)+βk(0)],
F31=μ κjkβk(0)-βj(0),
F32=μ -κjkβk(0)+βj(0)+μ2 -κjkκkk2βk(0)[βk(0)-βj(0)],
F33=1+μ2κjkκkj[βk(0)-βj(0)]2+12 Dj,
F34=μ -κjj2βj(0)+μ2 -κjkκkj[βk(0)-βj(0)][βk(0)+βj(0)],
F41=μ -κjkβk(0)+βj(0)+μ2 -κjjκjk2βj(0)[βk(0)+βj(0)],
F42=μ κjkβk(0)-βj(0)+μ2κjkκkk2βk(0)[βk(0)+βj(0)]+κjjκjk2βj(0)[βk(0)+βj(0)],
F43=μ -κjj2βj(0)+μ2 -κjkκkj[βk(0)-βj(0)][βk(0)+βj(0)],
F44=1+μ2κjj2[2βj(0)]2+κjkκkj[βk(0)+βj(0)]2-12 Dj.
dAkdz=iβk(1)Ak+jiκkjAj exp(-i2δkjz),
δkj=-δjk12 βk(0)-βj(0)-2πΛ.
Ak(Lg)Aj(Lg)
=cos(kLg)+i δk sin(kLg)i κk sin(kLg)i κ*k sin(kLg)cos(kLg)+i δk sin(kLg)
×Ak(0)Aj(0),
dAcodz=iκ0101cocoAco+iν m2 κ1ν01clcoAνcl exp(-i2δ1ν01clcoz),
νdAνcldz=+i m2 κ1ν01clcoAco exp(+i2δ1ν01clcoz),

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