Abstract

We conduct a detailed theoretical analysis of ultrashort pulse propagation through waveguide long-period grating (LPG) structures operating in the linear regime. We first consider the case of uniform LPGs, and we also investigate the effect of the typical grating nonuniformities, e.g., grating profile apodization, grating period chirping, and discrete phase shifts, on the spectral and temporal behavior of LPG structures. The two interacting modes are analyzed separately, and advanced representation tools, namely, space–wavelength and space–time diagrams (where space refers to the longitudinal grating dimension), are used to provide a deeper insight into the physics that determines the pulse evolution dynamics through the grating structures under analysis. In addition to its intrinsic physical interest, our study reveals the strong potential of LPG-based devices for optical pulse reshaping operations in the subpicosecond regime.

© 2005 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  34. F. Chollet, J.-P. Goedgebuer, G. Ramantoko, “Limitations imposed by birefringence uniformity on narrow-linewidth filters based on mode coupling,” Opt. Eng. 40, 2763–2770 (2001).
    [CrossRef]
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    [CrossRef]

2004 (4)

C. Martinez, B. Hoarau, L. Chirossel, O. Jacquin, C. Guidoux, “Advanced spectral filtering functionalities in ion-exchanged waveguides with artificial claddings,” Opt. Commun. 233, 97–106 (2004).
[CrossRef]

H. An, B. Ashton, S. Fleming, “Long-period-grating-assisted optical add–drop filter based on mismatched twin-core photosensitive cladding fiber,” Opt. Lett. 29, 343–345 (2004).
[CrossRef] [PubMed]

T.-J. Eom, S.-J. Kim, C.-S. Park, B. H. Lee, “Generation of high-repetition-rate optical pulse using cascaded long-period fiber gratings,” Electron. Lett. 40, 981–982 (2004).
[CrossRef]

M. Kulishov, J. Azaña, “Ultrashort pulse propagation in grating-assisted codirectional couplers,” Opt. Express 12, 2699–2709 (2004).
[CrossRef] [PubMed]

2003 (4)

2002 (1)

X. Yang, X. Guo, C. Lu, C. T. Hiang, “Apodized long-period grating with low insertion loss,” Microwave Opt. Technol. Lett. 35, 283–286 (2002).
[CrossRef]

2001 (5)

2000 (1)

1997 (4)

J. N. Kutz, B. J. Eggleton, J. B. Stark, R. E. Slusher, “Nonlinear pulse propagation in long-period fiber gratings: theory and experiment,” IEEE J. Sel. Top. Quantum Electron. 3, 1232–1245 (1997).
[CrossRef]

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[CrossRef]

L. R. Chen, S. D. Benjamin, W. E. Smith, J. E. Sipe, “Ultrashort pulse reflection from fiber gratings: a numerical investigation,” J. Lightwave Technol. 15, 1503–1512 (1997).
[CrossRef]

N. M. Litchinitser, B. J. Eggleton, D. B. Patterson, “Fiber Bragg gratings for dispersion compensation in transmission: theoretical model and design criteria for nearly ideal pulse recompression,” J. Lightwave Technol. 15, 1303–1312 (1997).
[CrossRef]

1996 (1)

1995 (1)

1994 (3)

J. E. Sipe, L. Poladian, C. Martijn de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A 11, 1307–1320 (1994).
[CrossRef]

K. Rottwitt, M. J. Guy, A. Boskovich, D. U. Noske, J. R. Taylor, R. Kashyap, “Interaction of uniform phase picosecond pulses with chirped and unchirped photosensitive fiber Bragg gratings,” Electron. Lett. 30, 995–996 (1994).
[CrossRef]

F. Ouellette, J.-F. Clishe, S. Gagnon, “All-fiber devices for chromatic dispersion compensation based on chirped distributed resonant coupling,” J. Lightwave Technol. 12, 1728–1738 (1994).
[CrossRef]

1993 (3)

Z.-M. Chuang, L. Coldern, “Design of widely tunable semiconductor lasers using grating assisted codirectional-coupler filters,” IEEE J. Quantum Electron. 29, 1071–1080 (1993).
[CrossRef]

D. Smith, A. d’Alessandro, J. Baran, “Source of sidelobe asymmetry in integrated acousto-optic filters,” Appl. Phys. Lett. 62, 814–816 (1993).
[CrossRef]

J. E. Roman, K. A. Winick, “Waveguide grating filters for dispersion compensation and pulse compression,” IEEE J. Quantum Electron. 29, 975–982 (1993).
[CrossRef]

1992 (1)

1991 (5)

R. R.A. Syms, S. Makrimichalou, A. S. Holmes, “High-speed optical signal processing potential of grating-coupled waveguide filters,” Appl. Opt. 30, 3762–3769 (1991).
[CrossRef] [PubMed]

Y. Chen, A. W. Snyder, “Grating-assisted couplers,” Opt. Lett. 16, 217–219 (1991).
[CrossRef] [PubMed]

G. Griffel, M. Itzkovitch, A. Hardy, “Coupled mode formulation for directional couplers with longitudinal perturbation,” IEEE J. Quantum Electron. 27, 985–994 (1991).
[CrossRef]

G. Griffel, A. Yariv, “Frequency response and tunability of grating assisted directional couplers,” IEEE J. Quantum Electron. 27, 1115–1118 (1991).
[CrossRef]

F. Ouellette, “All-fiber filter for efficient dispersion compensation,” Opt. Lett. 16, 303–305 (1991).
[CrossRef] [PubMed]

1987 (1)

Agogliati, B.

Ahn, S.-W.

S.-W. Ahn, S.-Y. Shin, “Grating assisted codirectional coupler filter using electro-optic and passive polymer waveguides,” Opt. Commun. 197, 289–293 (2001).
[CrossRef]

An, H.

Archambault, J.-L.

Ashton, B.

Azaña, J.

Baran, J.

D. Smith, A. d’Alessandro, J. Baran, “Source of sidelobe asymmetry in integrated acousto-optic filters,” Appl. Phys. Lett. 62, 814–816 (1993).
[CrossRef]

Belmonte, M.

Benjamin, S. D.

L. R. Chen, S. D. Benjamin, W. E. Smith, J. E. Sipe, “Ultrashort pulse reflection from fiber gratings: a numerical investigation,” J. Lightwave Technol. 15, 1503–1512 (1997).
[CrossRef]

Bergano, N. S.

Boskovich, A.

K. Rottwitt, M. J. Guy, A. Boskovich, D. U. Noske, J. R. Taylor, R. Kashyap, “Interaction of uniform phase picosecond pulses with chirped and unchirped photosensitive fiber Bragg gratings,” Electron. Lett. 30, 995–996 (1994).
[CrossRef]

Chen, L. R.

L. R. Chen, S. D. Benjamin, W. E. Smith, J. E. Sipe, “Ultrashort pulse reflection from fiber gratings: a numerical investigation,” J. Lightwave Technol. 15, 1503–1512 (1997).
[CrossRef]

Chen, Y.

Chiang, K. S.

Chirossel, L.

C. Martinez, B. Hoarau, L. Chirossel, O. Jacquin, C. Guidoux, “Advanced spectral filtering functionalities in ion-exchanged waveguides with artificial claddings,” Opt. Commun. 233, 97–106 (2004).
[CrossRef]

Chollet, F.

F. Chollet, J.-P. Goedgebuer, G. Ramantoko, “Limitations imposed by birefringence uniformity on narrow-linewidth filters based on mode coupling,” Opt. Eng. 40, 2763–2770 (2001).
[CrossRef]

Chuang, Z.-M.

Z.-M. Chuang, L. Coldern, “Design of widely tunable semiconductor lasers using grating assisted codirectional-coupler filters,” IEEE J. Quantum Electron. 29, 1071–1080 (1993).
[CrossRef]

Clishe, J.-F.

F. Ouellette, J.-F. Clishe, S. Gagnon, “All-fiber devices for chromatic dispersion compensation based on chirped distributed resonant coupling,” J. Lightwave Technol. 12, 1728–1738 (1994).
[CrossRef]

Coldern, L.

Z.-M. Chuang, L. Coldern, “Design of widely tunable semiconductor lasers using grating assisted codirectional-coupler filters,” IEEE J. Quantum Electron. 29, 1071–1080 (1993).
[CrossRef]

d’Alessandro, A.

D. Smith, A. d’Alessandro, J. Baran, “Source of sidelobe asymmetry in integrated acousto-optic filters,” Appl. Phys. Lett. 62, 814–816 (1993).
[CrossRef]

Davidson, C. R.

Daxhelet, X.

Eggleton, B. J.

N. M. Litchinitser, B. J. Eggleton, D. B. Patterson, “Fiber Bragg gratings for dispersion compensation in transmission: theoretical model and design criteria for nearly ideal pulse recompression,” J. Lightwave Technol. 15, 1303–1312 (1997).
[CrossRef]

J. N. Kutz, B. J. Eggleton, J. B. Stark, R. E. Slusher, “Nonlinear pulse propagation in long-period fiber gratings: theory and experiment,” IEEE J. Sel. Top. Quantum Electron. 3, 1232–1245 (1997).
[CrossRef]

Ellis, A. D.

Eom, T.-J.

T.-J. Eom, S.-J. Kim, C.-S. Park, B. H. Lee, “Generation of high-repetition-rate optical pulse using cascaded long-period fiber gratings,” Electron. Lett. 40, 981–982 (2004).
[CrossRef]

S.-J. Kim, T.-J. Eom, B. H. Lee, C.-S. Park, “Optical temporal encoding/decoding of short pulses using cascaded long-period fiber gratings,” Opt. Express 11, 3034–3040 (2003).
[CrossRef] [PubMed]

Erdogan, T.

Fleming, S.

Gagnon, S.

F. Ouellette, J.-F. Clishe, S. Gagnon, “All-fiber devices for chromatic dispersion compensation based on chirped distributed resonant coupling,” J. Lightwave Technol. 12, 1728–1738 (1994).
[CrossRef]

Goedgebuer, J.-P.

F. Chollet, J.-P. Goedgebuer, G. Ramantoko, “Limitations imposed by birefringence uniformity on narrow-linewidth filters based on mode coupling,” Opt. Eng. 40, 2763–2770 (2001).
[CrossRef]

Griffel, G.

G. Griffel, A. Yariv, “Frequency response and tunability of grating assisted directional couplers,” IEEE J. Quantum Electron. 27, 1115–1118 (1991).
[CrossRef]

G. Griffel, M. Itzkovitch, A. Hardy, “Coupled mode formulation for directional couplers with longitudinal perturbation,” IEEE J. Quantum Electron. 27, 985–994 (1991).
[CrossRef]

Guidoux, C.

C. Martinez, B. Hoarau, L. Chirossel, O. Jacquin, C. Guidoux, “Advanced spectral filtering functionalities in ion-exchanged waveguides with artificial claddings,” Opt. Commun. 233, 97–106 (2004).
[CrossRef]

Guo, X.

X. Yang, X. Guo, C. Lu, C. T. Hiang, “Apodized long-period grating with low insertion loss,” Microwave Opt. Technol. Lett. 35, 283–286 (2002).
[CrossRef]

Guy, M. J.

K. Rottwitt, M. J. Guy, A. Boskovich, D. U. Noske, J. R. Taylor, R. Kashyap, “Interaction of uniform phase picosecond pulses with chirped and unchirped photosensitive fiber Bragg gratings,” Electron. Lett. 30, 995–996 (1994).
[CrossRef]

Hardy, A.

G. Griffel, M. Itzkovitch, A. Hardy, “Coupled mode formulation for directional couplers with longitudinal perturbation,” IEEE J. Quantum Electron. 27, 985–994 (1991).
[CrossRef]

Hiang, C. T.

X. Yang, X. Guo, C. Lu, C. T. Hiang, “Apodized long-period grating with low insertion loss,” Microwave Opt. Technol. Lett. 35, 283–286 (2002).
[CrossRef]

Hoarau, B.

C. Martinez, B. Hoarau, L. Chirossel, O. Jacquin, C. Guidoux, “Advanced spectral filtering functionalities in ion-exchanged waveguides with artificial claddings,” Opt. Commun. 233, 97–106 (2004).
[CrossRef]

Holmes, A. S.

Ibsen, M.

Intrachat, K.

K. Intrachat, J. N. Kutz, “Theory and simulation of passive modelocking dynamics using a long-period fiber grating,” IEEE J. Quantum Electron. 39, 1572–1578 (2003).
[CrossRef]

Itzkovitch, M.

G. Griffel, M. Itzkovitch, A. Hardy, “Coupled mode formulation for directional couplers with longitudinal perturbation,” IEEE J. Quantum Electron. 27, 985–994 (1991).
[CrossRef]

Jacquin, O.

C. Martinez, B. Hoarau, L. Chirossel, O. Jacquin, C. Guidoux, “Advanced spectral filtering functionalities in ion-exchanged waveguides with artificial claddings,” Opt. Commun. 233, 97–106 (2004).
[CrossRef]

Judkins, J. B.

Kashyap, R.

K. Rottwitt, M. J. Guy, A. Boskovich, D. U. Noske, J. R. Taylor, R. Kashyap, “Interaction of uniform phase picosecond pulses with chirped and unchirped photosensitive fiber Bragg gratings,” Electron. Lett. 30, 995–996 (1994).
[CrossRef]

R. Kashyap, Fiber Bragg Gratings (Academic, 1999).

Kim, S.-J.

T.-J. Eom, S.-J. Kim, C.-S. Park, B. H. Lee, “Generation of high-repetition-rate optical pulse using cascaded long-period fiber gratings,” Electron. Lett. 40, 981–982 (2004).
[CrossRef]

S.-J. Kim, T.-J. Eom, B. H. Lee, C.-S. Park, “Optical temporal encoding/decoding of short pulses using cascaded long-period fiber gratings,” Opt. Express 11, 3034–3040 (2003).
[CrossRef] [PubMed]

Kulishov, M.

Kutz, J. N.

K. Intrachat, J. N. Kutz, “Theory and simulation of passive modelocking dynamics using a long-period fiber grating,” IEEE J. Quantum Electron. 39, 1572–1578 (2003).
[CrossRef]

J. N. Kutz, B. J. Eggleton, J. B. Stark, R. E. Slusher, “Nonlinear pulse propagation in long-period fiber gratings: theory and experiment,” IEEE J. Sel. Top. Quantum Electron. 3, 1232–1245 (1997).
[CrossRef]

Laporta, P.

Lee, B. H.

T.-J. Eom, S.-J. Kim, C.-S. Park, B. H. Lee, “Generation of high-repetition-rate optical pulse using cascaded long-period fiber gratings,” Electron. Lett. 40, 981–982 (2004).
[CrossRef]

S.-J. Kim, T.-J. Eom, B. H. Lee, C.-S. Park, “Optical temporal encoding/decoding of short pulses using cascaded long-period fiber gratings,” Opt. Express 11, 3034–3040 (2003).
[CrossRef] [PubMed]

Lemaire, P. J.

Litchinitser, N. M.

N. M. Litchinitser, B. J. Eggleton, D. B. Patterson, “Fiber Bragg gratings for dispersion compensation in transmission: theoretical model and design criteria for nearly ideal pulse recompression,” J. Lightwave Technol. 15, 1303–1312 (1997).
[CrossRef]

Liu, Q.

Longhi, S.

Lu, C.

X. Yang, X. Guo, C. Lu, C. T. Hiang, “Apodized long-period grating with low insertion loss,” Microwave Opt. Technol. Lett. 35, 283–286 (2002).
[CrossRef]

Makrimichalou, S.

Marano, M.

Martijn de Sterke, C.

Martinez, C.

C. Martinez, B. Hoarau, L. Chirossel, O. Jacquin, C. Guidoux, “Advanced spectral filtering functionalities in ion-exchanged waveguides with artificial claddings,” Opt. Commun. 233, 97–106 (2004).
[CrossRef]

Noske, D. U.

K. Rottwitt, M. J. Guy, A. Boskovich, D. U. Noske, J. R. Taylor, R. Kashyap, “Interaction of uniform phase picosecond pulses with chirped and unchirped photosensitive fiber Bragg gratings,” Electron. Lett. 30, 995–996 (1994).
[CrossRef]

Ouellette, F.

Park, C.-S.

T.-J. Eom, S.-J. Kim, C.-S. Park, B. H. Lee, “Generation of high-repetition-rate optical pulse using cascaded long-period fiber gratings,” Electron. Lett. 40, 981–982 (2004).
[CrossRef]

S.-J. Kim, T.-J. Eom, B. H. Lee, C.-S. Park, “Optical temporal encoding/decoding of short pulses using cascaded long-period fiber gratings,” Opt. Express 11, 3034–3040 (2003).
[CrossRef] [PubMed]

Patterson, D. B.

N. M. Litchinitser, B. J. Eggleton, D. B. Patterson, “Fiber Bragg gratings for dispersion compensation in transmission: theoretical model and design criteria for nearly ideal pulse recompression,” J. Lightwave Technol. 15, 1303–1312 (1997).
[CrossRef]

Payne, D. N.

Pedrazzani, J. R.

Petropoulos, P.

Poladian, L.

Ramantoko, G.

F. Chollet, J.-P. Goedgebuer, G. Ramantoko, “Limitations imposed by birefringence uniformity on narrow-linewidth filters based on mode coupling,” Opt. Eng. 40, 2763–2770 (2001).
[CrossRef]

Rastogi, V.

Reekie, L.

Richardson, D. J.

Roman, J. E.

J. E. Roman, K. A. Winick, “Waveguide grating filters for dispersion compensation and pulse compression,” IEEE J. Quantum Electron. 29, 975–982 (1993).
[CrossRef]

Rottwitt, K.

K. Rottwitt, M. J. Guy, A. Boskovich, D. U. Noske, J. R. Taylor, R. Kashyap, “Interaction of uniform phase picosecond pulses with chirped and unchirped photosensitive fiber Bragg gratings,” Electron. Lett. 30, 995–996 (1994).
[CrossRef]

Russel, P. St.J.

Shin, S.-Y.

S.-W. Ahn, S.-Y. Shin, “Grating assisted codirectional coupler filter using electro-optic and passive polymer waveguides,” Opt. Commun. 197, 289–293 (2001).
[CrossRef]

Sipe, J. E.

Slusher, R. E.

J. N. Kutz, B. J. Eggleton, J. B. Stark, R. E. Slusher, “Nonlinear pulse propagation in long-period fiber gratings: theory and experiment,” IEEE J. Sel. Top. Quantum Electron. 3, 1232–1245 (1997).
[CrossRef]

Smith, D.

D. Smith, A. d’Alessandro, J. Baran, “Source of sidelobe asymmetry in integrated acousto-optic filters,” Appl. Phys. Lett. 62, 814–816 (1993).
[CrossRef]

Smith, W. E.

L. R. Chen, S. D. Benjamin, W. E. Smith, J. E. Sipe, “Ultrashort pulse reflection from fiber gratings: a numerical investigation,” J. Lightwave Technol. 15, 1503–1512 (1997).
[CrossRef]

Snyder, A. W.

Stark, J. B.

J. N. Kutz, B. J. Eggleton, J. B. Stark, R. E. Slusher, “Nonlinear pulse propagation in long-period fiber gratings: theory and experiment,” IEEE J. Sel. Top. Quantum Electron. 3, 1232–1245 (1997).
[CrossRef]

Stegall, D. B.

Syms, R. R.A.

Taverner, D.

Taylor, J. R.

K. Rottwitt, M. J. Guy, A. Boskovich, D. U. Noske, J. R. Taylor, R. Kashyap, “Interaction of uniform phase picosecond pulses with chirped and unchirped photosensitive fiber Bragg gratings,” Electron. Lett. 30, 995–996 (1994).
[CrossRef]

Vengsarkar, A. M.

Winick, K. A.

J. E. Roman, K. A. Winick, “Waveguide grating filters for dispersion compensation and pulse compression,” IEEE J. Quantum Electron. 29, 975–982 (1993).
[CrossRef]

Yang, X.

X. Yang, X. Guo, C. Lu, C. T. Hiang, “Apodized long-period grating with low insertion loss,” Microwave Opt. Technol. Lett. 35, 283–286 (2002).
[CrossRef]

Yariv, A.

G. Griffel, A. Yariv, “Frequency response and tunability of grating assisted directional couplers,” IEEE J. Quantum Electron. 27, 1115–1118 (1991).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

D. Smith, A. d’Alessandro, J. Baran, “Source of sidelobe asymmetry in integrated acousto-optic filters,” Appl. Phys. Lett. 62, 814–816 (1993).
[CrossRef]

Electron. Lett. (2)

T.-J. Eom, S.-J. Kim, C.-S. Park, B. H. Lee, “Generation of high-repetition-rate optical pulse using cascaded long-period fiber gratings,” Electron. Lett. 40, 981–982 (2004).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Envelope of refractive-index distribution along fiber axis z and its piecewise approximation by uniform sections Δ z i with a pulse propagating through the Δ z n + 1 section.

Fig. 2
Fig. 2

Space–time (left) and space–wavelength (right) 2D diagrams of optical pulse propagation through the supplier mode of a GACC with a uniform grating. The corresponding temporal (left) and spectral (right) responses at three specific, representative locations along the grating length are also shown in the 1D plots (solid curves). For comparison, the input pulse is represented in these 1D plots as well (dashed curves).

Fig. 3
Fig. 3

Space–time (left) and space–wavelength (right) 2D diagrams of optical pulse propagation through the receptor mode of a GACC with a uniform grating. The corresponding temporal (left) and spectral (right) responses at three specific, representative locations along the grating length are also shown in the 1D plots (solid curves). For comparison, the input pulse is represented in these 1D plots as well (dashed curves).

Fig. 4
Fig. 4

Space–time (left) and space–wavelength (right) 2D diagrams of optical pulse propagation through the supplier mode of a GACC with an apodized grating. The corresponding temporal (left) and spectral (right) responses at three specific, representative locations along the grating length are also shown in the 1D plots.

Fig. 5
Fig. 5

Space–time (left) and space–wavelength (right) 2D diagrams of optical pulse propagation through the receptor mode of a GACC with an apodized grating. The corresponding temporal (left) and spectral (right) responses at three specific, representative locations along the grating length are also shown in the 1D plots.

Fig. 6
Fig. 6

Space–time (left) and space–wavelength (right) 2D diagrams of optical pulse propagation through the supplier mode of a GACC with a chirped grating. The corresponding temporal (left) and spectral (right) responses at three specific, representative locations along the grating length are also shown in the 1D plots.

Fig. 7
Fig. 7

Space–time (left) and space–wavelength (right) 2D diagrams of optical pulse propagation through the receptor mode of a GACC with a chirped grating. The corresponding temporal (left) and spectral (right) responses at three specific, representative locations along the grating length are also shown in the 1D plots.

Fig. 8
Fig. 8

Space–time (left) and space–wavelength (right) 2D diagrams of optical pulse propagation through the supplier mode of a GACC with a uniform grating having a π shift in its center. The corresponding temporal (left) and spectral (right) responses at three specific, representative locations along the grating length are also shown in the 1D plots.

Fig. 9
Fig. 9

Space–time (left) and space–wavelength (right) 2D diagrams of optical pulse propagation through the receptor mode of a GACC with a uniform grating having a π shift in its center. The corresponding temporal (left) and spectral (right) responses at three specific, representative locations along the grating length are also shown in the 1D plots.

Equations (18)

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h in ( t ) = exp [ t 2 2 τ 0 2 2 ln ( 2 ) ] exp ( i 2 π ν c t ) .
H in ( ν ) = ( π τ 0 2 2 ln ( 2 ) ) 1 2 exp [ π 2 τ 0 2 2 ln ( 2 ) ( ν ν c ) 2 ] ,
Δ ν FWHM = 2 ln ( 2 ) 1 2 π τ 0 ln [ π τ 0 2 ln ( 2 ) ] 1 2 .
δ n ( z ) = δ n 1 ( z ) cos [ 2 π Λ z + ϕ ( z ) ] ,
E ( z ) = j = 1 2 A j ( z ) E j ( x , y ) exp ( j β j z ) ,
d A 1 ( z ) d z = i κ A 2 ( z ) exp ( j 2 σ z ) ,
d A 2 ( z ) d z = i κ * A 1 ( z ) exp ( j 2 σ z ) ,
κ = ω 4 E 1 * ( x , y ) ϵ 0 Δ ϵ 1 E 2 ( x , y ) d x d y ,
( A 1 ( z , ν ) A 2 ( z , ν ) ) = [ T 11 ( z , ν ) T 12 ( z , ν ) T 21 ( z , ν ) T 22 ( z , ν ) ] ( A 1 ( 0 , ν ) A 2 ( 0 , ν ) ) .
h s ( z , t ) = J 1 [ H in ( ν ) T 11 ( z , ν ) ] ,
h r ( z , t ) = J 1 [ H in ( ν ) T 21 ( z , ν ) ]
T 11 ( z , ν ) = [ cos ( γ z ) + j σ γ sin ( γ z ) ] exp [ j ( β 1 σ ) z ] ,
T 12 ( z , ν ) = j κ γ sin ( γ z ) exp [ j ( β 1 σ ) z ] ,
T 21 ( z , ν ) = j κ * γ sin ( γ z ) exp [ j ( β 2 + σ ) z ] ,
T 22 ( z , ν ) = [ cos ( γ z ) j σ γ sin ( γ z ) ] exp [ j ( β 2 + σ ) z ] ,
T ( π ) ( z , ν ) = { T ( z , ν ) for 0 z L 2 T ( L 2 , ν ) I ( π 2 ) T ( z , ν ) for L 2 < z L } ,
I ( π 2 ) = [ exp ( j π 2 ) 0 0 exp ( j π 2 ) ] .
T Σ ( z , ν ) = [ i = 1 n + 1 T i ( Δ z i , ν ) ] for z Δ z n + 1 .

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