Abstract

We develop a general procedure that allows the determination of the spectral transmittance and reflectance at normal incidence for arbitrary one-dimensional continuous materials as well as the analysis of the time-domain propagation of pulses through them. This procedure consists of a generalization of Fresnel equations, and it is supported by an iterative algorithm also developed here: the polynomial fixed-point algorithm (PFPA). We apply these theoretical results to some concrete examples, such as determining the transmittance and reflectance for an absorptionless photonic crystal, an optical rugate filter, and a photonic crystal with periodic absorption. We also analyze the time-domain propagation of ultrashort Gaussian pulses through different structures.

© 2007 Optical Society of America

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2005

2004

R. Letiel, O. Stenzel, S. Wildbrandt, D. Gäbler, V. Janicki, and N. Kaiser, "Optical and non-optical characterization of Nb2O5-SiO2 compositional graded-index layers and rugate structures," Thin Solid Films 497, 135-141 (2004).
[CrossRef]

K. Huang, E. L. E. X. Jiang, J. Joannopoulos, K. Nelson, P. Bienstman, and S. Fan, "Nature of lossy Bloch states in polaritonic photonic crystals," Phys. Rev. B 69, 195111 (2004).
[CrossRef]

G. Morozov, D. Sprung, and J. Martorell, "Semiclassical coupled-wave theory and its application to TE waves in one-dimensional photonic crystals," Phys. Rev. E 69, 016612 (2004).
[CrossRef]

G. Morozov, D. Sprung, and J. Martorell, "Semiclassical coupled-wave theory and its application to TM waves in one-dimensional photonic crystals," Phys. Rev. E 70, 016606 (2004).
[CrossRef]

2001

2000

G. Burlak, S. Koshevaya, J. Sanchez-Mondragon, and V. Grimalsky, "Electromagnetic oscillations in a multilayer spherical stack," Opt. Commun. 180, 49-58 (2000).
[CrossRef]

C. Chen, P. Berini, D. Feng, S. Tanev, and V. Tzolov, "Efficient and accurate numerical analysis of multilayer planar optical waveguides in lossy anisotropic media," Opt. Express 7, 260-272 (2000).
[CrossRef] [PubMed]

1999

1995

1990

1989

1988

1987

1982

1981

1969

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909-2945 (1969).

Acebal, P.

Belendez, A.

Berini, P.

Bienstman, P.

K. Huang, E. L. E. X. Jiang, J. Joannopoulos, K. Nelson, P. Bienstman, and S. Fan, "Nature of lossy Bloch states in polaritonic photonic crystals," Phys. Rev. B 69, 195111 (2004).
[CrossRef]

Blaya, S.

Born, M.

M. Born and E. Wolf, Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon, 1984).
[PubMed]

Bovard, B. G.

Burden, R.

R. Burden and J. Faires, Numerical Analysis (Brooks Cole, 2000).

Burlak, G.

G. Burlak, S. Koshevaya, J. Sanchez-Mondragon, and V. Grimalsky, "Electromagnetic oscillations in a multilayer spherical stack," Opt. Commun. 180, 49-58 (2000).
[CrossRef]

Carretero, L.

Chen, C.

Dorf, M. C.

Faires, J.

R. Burden and J. Faires, Numerical Analysis (Brooks Cole, 2000).

Fan, S.

Feng, D.

Fimia, A.

Gäbler, D.

R. Letiel, O. Stenzel, S. Wildbrandt, D. Gäbler, V. Janicki, and N. Kaiser, "Optical and non-optical characterization of Nb2O5-SiO2 compositional graded-index layers and rugate structures," Thin Solid Films 497, 135-141 (2004).
[CrossRef]

Gaylord, T.

Gluck, N.

Grimalsky, V.

G. Burlak, S. Koshevaya, J. Sanchez-Mondragon, and V. Grimalsky, "Electromagnetic oscillations in a multilayer spherical stack," Opt. Commun. 180, 49-58 (2000).
[CrossRef]

Gunning, W. J.

Hall, R.

Huang, K.

K. Huang, E. L. E. X. Jiang, J. Joannopoulos, K. Nelson, P. Bienstman, and S. Fan, "Nature of lossy Bloch states in polaritonic photonic crystals," Phys. Rev. B 69, 195111 (2004).
[CrossRef]

Ippen, E.

Jameson, G.

G. Jameson, Topology and Normed Spaces (Chapman and Hall, 1974).

Janicki, V.

R. Letiel, O. Stenzel, S. Wildbrandt, D. Gäbler, V. Janicki, and N. Kaiser, "Optical and non-optical characterization of Nb2O5-SiO2 compositional graded-index layers and rugate structures," Thin Solid Films 497, 135-141 (2004).
[CrossRef]

Jiang, E. L. E. X.

K. Huang, E. L. E. X. Jiang, J. Joannopoulos, K. Nelson, P. Bienstman, and S. Fan, "Nature of lossy Bloch states in polaritonic photonic crystals," Phys. Rev. B 69, 195111 (2004).
[CrossRef]

Joannopoulos, J.

Kaiser, N.

R. Letiel, O. Stenzel, S. Wildbrandt, D. Gäbler, V. Janicki, and N. Kaiser, "Optical and non-optical characterization of Nb2O5-SiO2 compositional graded-index layers and rugate structures," Thin Solid Films 497, 135-141 (2004).
[CrossRef]

Kogelnik, H.

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909-2945 (1969).

Kolodziejski, L.

Koshevaya, S.

G. Burlak, S. Koshevaya, J. Sanchez-Mondragon, and V. Grimalsky, "Electromagnetic oscillations in a multilayer spherical stack," Opt. Commun. 180, 49-58 (2000).
[CrossRef]

Lekner, J.

Letiel, R.

R. Letiel, O. Stenzel, S. Wildbrandt, D. Gäbler, V. Janicki, and N. Kaiser, "Optical and non-optical characterization of Nb2O5-SiO2 compositional graded-index layers and rugate structures," Thin Solid Films 497, 135-141 (2004).
[CrossRef]

Lim, K.

Madrigal, R.

Martorell, J.

G. Morozov, D. Sprung, and J. Martorell, "Semiclassical coupled-wave theory and its application to TM waves in one-dimensional photonic crystals," Phys. Rev. E 70, 016606 (2004).
[CrossRef]

G. Morozov, D. Sprung, and J. Martorell, "Semiclassical coupled-wave theory and its application to TE waves in one-dimensional photonic crystals," Phys. Rev. E 69, 016612 (2004).
[CrossRef]

Matloub, S.

A. Rostami and S. Matloub, "Exactly solvable inhomogeneous Fibonacci-class quasi-periodic structures (optical filtering)," Opt. Commun. 247, 247-256 (2005).
[CrossRef]

Moharam, M.

Morozov, G.

G. Morozov, D. Sprung, and J. Martorell, "Semiclassical coupled-wave theory and its application to TM waves in one-dimensional photonic crystals," Phys. Rev. E 70, 016606 (2004).
[CrossRef]

G. Morozov, D. Sprung, and J. Martorell, "Semiclassical coupled-wave theory and its application to TE waves in one-dimensional photonic crystals," Phys. Rev. E 69, 016612 (2004).
[CrossRef]

Morris, G.

Nelson, K.

K. Huang, E. L. E. X. Jiang, J. Joannopoulos, K. Nelson, P. Bienstman, and S. Fan, "Nature of lossy Bloch states in polaritonic photonic crystals," Phys. Rev. B 69, 195111 (2004).
[CrossRef]

Peng, S.

Pérez-Molina, M.

Petrich, G.

Ripin, D.

Rostami, A.

A. Rostami and S. Matloub, "Exactly solvable inhomogeneous Fibonacci-class quasi-periodic structures (optical filtering)," Opt. Commun. 247, 247-256 (2005).
[CrossRef]

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001).

Sanchez-Mondragon, J.

G. Burlak, S. Koshevaya, J. Sanchez-Mondragon, and V. Grimalsky, "Electromagnetic oscillations in a multilayer spherical stack," Opt. Commun. 180, 49-58 (2000).
[CrossRef]

Southwell, W. H.

Sprung, D.

G. Morozov, D. Sprung, and J. Martorell, "Semiclassical coupled-wave theory and its application to TE waves in one-dimensional photonic crystals," Phys. Rev. E 69, 016612 (2004).
[CrossRef]

G. Morozov, D. Sprung, and J. Martorell, "Semiclassical coupled-wave theory and its application to TM waves in one-dimensional photonic crystals," Phys. Rev. E 70, 016606 (2004).
[CrossRef]

Stenzel, O.

R. Letiel, O. Stenzel, S. Wildbrandt, D. Gäbler, V. Janicki, and N. Kaiser, "Optical and non-optical characterization of Nb2O5-SiO2 compositional graded-index layers and rugate structures," Thin Solid Films 497, 135-141 (2004).
[CrossRef]

Tanev, S.

Thoen, E.

Tzolov, V.

Villeneuve, P.

Wildbrandt, S.

R. Letiel, O. Stenzel, S. Wildbrandt, D. Gäbler, V. Janicki, and N. Kaiser, "Optical and non-optical characterization of Nb2O5-SiO2 compositional graded-index layers and rugate structures," Thin Solid Films 497, 135-141 (2004).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon, 1984).
[PubMed]

Woodberry, J.

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, 2005).

Appl. Opt.

Bell Syst. Tech. J.

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909-2945 (1969).

J. Lightwave Technol.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

G. Burlak, S. Koshevaya, J. Sanchez-Mondragon, and V. Grimalsky, "Electromagnetic oscillations in a multilayer spherical stack," Opt. Commun. 180, 49-58 (2000).
[CrossRef]

A. Rostami and S. Matloub, "Exactly solvable inhomogeneous Fibonacci-class quasi-periodic structures (optical filtering)," Opt. Commun. 247, 247-256 (2005).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. B

K. Huang, E. L. E. X. Jiang, J. Joannopoulos, K. Nelson, P. Bienstman, and S. Fan, "Nature of lossy Bloch states in polaritonic photonic crystals," Phys. Rev. B 69, 195111 (2004).
[CrossRef]

Phys. Rev. E

G. Morozov, D. Sprung, and J. Martorell, "Semiclassical coupled-wave theory and its application to TE waves in one-dimensional photonic crystals," Phys. Rev. E 69, 016612 (2004).
[CrossRef]

G. Morozov, D. Sprung, and J. Martorell, "Semiclassical coupled-wave theory and its application to TM waves in one-dimensional photonic crystals," Phys. Rev. E 70, 016606 (2004).
[CrossRef]

Thin Solid Films

R. Letiel, O. Stenzel, S. Wildbrandt, D. Gäbler, V. Janicki, and N. Kaiser, "Optical and non-optical characterization of Nb2O5-SiO2 compositional graded-index layers and rugate structures," Thin Solid Films 497, 135-141 (2004).
[CrossRef]

Other

M. Born and E. Wolf, Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon, 1984).
[PubMed]

K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001).

P. Yeh, Optical Waves in Layered Media (Wiley, 2005).

G. Jameson, Topology and Normed Spaces (Chapman and Hall, 1974).

R. Burden and J. Faires, Numerical Analysis (Brooks Cole, 2000).

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Figures (7)

Fig. 1
Fig. 1

Three-layer structure.

Fig. 2
Fig. 2

Photonic crystal spectral response. Bandgaps centered at λ 1 = 480 nm and λ 2 = 640 nm .

Fig. 3
Fig. 3

Optical rugate filter spectral response.

Fig. 4
Fig. 4

Lossy photonic crystal spectral response.

Fig. 5
Fig. 5

Distortionless Gaussian pulse echoes, with multiple reflections at z = 0 and z = L .

Fig. 6
Fig. 6

Reflected ( z = 0 ) and transmitted ( z = L ) pulses are both broadened.

Fig. 7
Fig. 7

Reflected and transmitted pulses are compressed in the frequency domain, and therefore they are spread in the time domain.

Equations (81)

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E i ( z ) = e i [ ( 2 π ε 1 λ ) ] z , E r ( z ) = ρ ( λ ) e i [ ( 2 π ε 1 λ ) ] z , z < 0 ,
E t ( z ) = τ ( λ ) e i [ ( 2 π ε 3 λ ) ] ( z L ) , z > L .
2 E ( z ) z 2 + ( 2 π λ ) 2 ε 2 ( z ) E ( z ) = 0 .
E ( L ) = τ ( λ ) , E z z = L = i 2 π ε 3 λ τ ( λ ) ,
E ( 0 ) = 1 + ρ ( λ ) , E z z = 0 = i 2 π ε 1 λ [ 1 ρ ( λ ) ] .
E i ( z ) E ( L z ) , ε 2 i ( z ) ε 2 ( L z ) .
2 E i ( z ) z 2 + ( 2 π λ ) 2 ε 2 i ( z ) E i ( z ) = 0 ,
E i ( 0 ) = τ ( λ ) , E i z z = 0 = i 2 π ε 3 λ τ ( λ ) ,
E i ( L ) = 1 + ρ ( λ ) , E i z z = L = i 2 π ε 1 λ [ ρ ( λ ) 1 ] .
2 E p , λ i z 2 + ( 2 π λ ) 2 ε 2 i ( z ) E p , λ i = 0 ; E p , λ i ( 0 ) = 1 ,
E p , λ i z z = 0 = i 2 π ε 3 λ , z [ 0 , L ] .
{ ρ ( λ ) + 1 = τ ( λ ) E p , λ i ( L ) ρ ( λ ) 1 = i λ τ ( λ ) 2 π ε 1 E p , λ i z z = L } .
ρ ( λ ) = 2 π i ε 1 E p , λ i ( L ) + λ E p , λ i z z = L 2 π i ε 1 E p , λ i ( L ) λ E p , λ i z z = L ,
τ ( λ ) = 4 π i ε 1 2 π i ε 1 E p , λ i ( L ) λ E p , λ i z z = L .
T ( λ ) = 100 ε 3 ε 1 τ ( λ ) 2 ( % ) , R ( λ ) = 100 ρ ( λ ) 2 ( % ) .
y λ ( z ) h λ ( z ) y λ ( z ) = 0 ; { y λ ( 0 ) = e 1 λ y λ ( 0 ) = e 2 λ } ,
z [ 0 , L ] , λ S .
{ h λ , k ( z ) = h λ ( z + k L d ) y λ , k ( z ) = y λ ( z + k L d ) } , z [ 0 , L d ] , k S K , λ S .
y λ , k ( z ) h λ , k ( z ) y λ , k ( z ) = 0 ; { y λ , k ( 0 ) = y λ ( k L d ) y λ , k ( 0 ) = y λ ( k L d ) } .
h λ , k ( z ) g λ , k ( z ) = p = 0 m k c k , p λ z p ; z [ 0 , L d ] .
f λ , k ( z ) g λ , k ( z ) f λ , k ( z ) = 0 ; { f λ , k ( 0 ) = f λ , k 1 ( L d ) f λ , k ( 0 ) = f λ , k 1 ( L d ) } .
( y λ ( L ) y λ ( L ) ) ( f λ , K 1 ( L d ) f λ , K 1 ( L d ) ) = [ k = 0 K 1 ( f λ , 1 , K 1 k ( L d ) f λ , 2 , K 1 k ( L d ) f λ , 1 , K 1 k ( L d ) f λ , 2 , K 1 k ( L d ) ) ] . ( e 1 λ e 2 λ ) .
f λ , i , k , n + 1 ( z ) = z i 1 + 0 z ( 0 x g λ , k ( τ ) f λ , i , k , n ( τ ) d τ ) d x ,
z [ 0 , L d ] , n N .
f λ , i , k , n ( z ) = p a i , k , n , p λ z p .
f λ , i , k , n + 1 ( z ) = z i 1 + 0 z ( 0 x [ p = 0 m k c k , p λ τ p ] [ p a i , k , n , p λ τ p ] d τ ) d x = z i 1 + p e i , k , n , p λ z p ,
e i , k , n , 0 λ = e i , k , n , 1 λ = 0 , e i , k , n , p λ = 1 p ( p 1 ) d i , k , n , p 2 λ for p 2 ,
{ d i , k , n , p λ } = { c k , p λ } { a i , k , n , p λ } = r , p r 0 c k , r λ a i , k , n , p τ λ for each p N
a i , k , n + 1 , p λ = { 1 p ( p 1 ) r , p r 2 0 c k , r λ a i , k , n , p r 2 λ if p 2 δ ( p , i 1 ) if p { 0 , 1 } } ,
max z [ 0 , L d ] f λ , i , k , n ( z ) f λ , i , k ( z ) Γ λ 2 n ( 2 n ) ! L d i 1 cosh ( Γ λ ) ;
k S K , i { 1 , 2 } , λ S .
( y λ ( L ) y λ ( L ) ) [ k = 0 K 1 ( p a 1 , K 1 k , N , p λ L d p p a 2 , K 1 k , N , p λ L d p p p a 1 , K 1 k , N , p λ L d p 1 p p a 2 , K 1 k , N , p λ L d p 1 ) ] . ( e 1 λ e 2 λ ) .
f λ , i , k , n ( z ) f λ , i , k + Φ , n ( z ) .
ε 2 ( z ) = ε p + ε m 1 cos ( 2 π Λ 1 z ) + ε m 2 cos ( 2 π Λ 2 z ) ; z [ 0 , L ] .
ε 1 = ε 2 = ε p = 2.56 , ε m 1 = ε m 2 = 0.096 ,
Λ 1 = 0.15 μ m , Λ 2 = 4 2 Λ 1 , L = 7.8 μ m .
Λ = 0.6 μ m .
N = 13 , max z [ 0 , 0.12 ] f λ , i , k , 13 ( z ) f λ , i , k ( z ) < ϵ ; i { 1 , 2 } ,
k S 5 , λ S .
( E p , λ i ( 7.8 ) E p , λ i z z = 7.8 ) [ k = 0 4 ( p a 1 , 4 k , 13 , p λ ( 0.12 ) p p a 2 , 4 k , 13 , p λ ( 0.12 ) p o p a 1 , 4 k , 13 , p λ ( 0.12 ) p 1 p p a 2 , 4 k , 13 , p λ ( 0.12 ) p 1 ) ] 13 ( 1 i 2 π 1.6 λ ) .
ε 2 ( z ) = ε p + sinc ( 2 z L 1 ) [ ε m 1 cos ( 2 π Λ 1 z ) + ε m 2 cos ( 2 π Λ 2 π ) ] , with z [ 0 , L ] .
ε m 1 = ε m 2 = 0.4 .
N = 16 , max z [ 0 , 0.15 ] f λ , i , k , 16 ( z ) f λ , i , k ( z ) < ϵ , i { 1 , 2 } ,
k S 52 , λ S .
( E p , λ i ( 7.8 ) E p , λ i z z = 7.8 ) [ k = 0 51 ( p a 1 , 51 k , 16 , p λ ( 0.15 ) p p a 2 , 51 k , 16 , p λ ( 0.15 ) p p p a 1 , 51 k , 16 , p λ ( 0.15 ) p 1 p p a 2 , 51 k , 16 , p λ ( 0.15 ) p 1 ) ] ( 1 i 2 π 1.6 λ ) .
ε 2 ( z ) = ε p + ε m 1 cos ( 2 π Λ 1 z ) + ε m 2 cos ( 2 π Λ 2 z ) + i λ c 2 π μ 0 σ ( z ) ,
σ ( z ) = σ p + σ m cos ( 2 π Λ 2 z ) , z [ 0 , L ] .
σ p = σ m = 3 × 10 10 m Ω .
N = 13 , max z [ 0 , 0.12 ] f λ , i , k , 13 ( z ) f λ , i , k ( z ) < ϵ ,
i { 1 , 2 } , k S 5 , λ S .
( E p , λ i ( 7.8 ) E p , λ i z z = 7.8 ) [ k = 0 4 ( p a 1 , 4 k , 13 , p λ ( 0.12 ) p p a 2 , 4 k , 13 , p λ ( 0.12 ) p p p a 1 , 4 k , 13 , p λ ( 0.12 ) p 1 p p a 2 , 4 k , 13 , p λ ( 0.12 ) p 1 ) ] 13 ( 1 i 2 π 1.6 λ ) .
E r ( 0 , ω ) = i ω ε 1 E p , 2 π c ω i ( L ) + c E p , 2 π c ω i z z = L i ω ε 1 E p , 2 π c ω i ( L ) c E p , 2 π c ω i z z = L E i ( 0 , ω ) ,
E t ( L , ω ) = 2 i ω ε 1 i ω ε 1 E p , 2 π c ω i ( L ) c E p , 2 π c ω i z z = L E i ( 0 , ω ) .
E r ( 0 , t ) = 1 2 π + E r ( 0 , ω ) e i ω t d ω = 1 π Re [ 0 + E r ( 0 , ω ) e i ω t d ω ] ,
E t ( L , t ) = 1 2 π + E t ( L , ω ) e i ω t d ω = 1 π Re [ 0 + E t ( L , ω ) e i ω t d ω ] .
E i ( 0 , t ) = A e [ ( c t ) 2 2 ( W 0 ε 1 ) 2 ] cos ( ω 0 t ) ,
E i ( 0 , ω ) = A W 0 2 π ε 1 c [ e 2 ε 1 T 0 2 ( ω + ω 0 ) 2 + e 2 ε 1 T 0 2 ( ω ω 0 ) 2 ] .
E r ( 0 , ω ) = i ( ε 1 ε 3 ) cos ( ω ε 2 c L ) + ( ε 1 ε 3 ε 2 ε 2 ) sin ( ω ε 2 c L ) i ( ε 1 + ε 3 ) cos ( ω ε 2 c L ) + ( ε 1 ε 3 ε 2 + ε 2 ) sin ( ω ε 2 c L ) E i ( 0 , ω ) ,
E t ( L , ω ) = 2 i ε 1 i ( ε 1 + ε 3 ) cos ( ω ε 2 c L ) + ( ε 1 ε 3 ε 2 + ε 2 ) sin ( ω ε 2 c L ) E i ( 0 , ω ) .
ε 1 = ε 3 = 1 , ε 2 ( z ) = ε 2 = 10.89 ,
for each z [ 0 , L ] , L = 4 μ m .
A = 1 V m , ω 0 = 2.9788 PHz ( λ 0 = 632.8 nm ) ,
W 0 = 1.2 μ m .
A = 1 V m , ω 0 = 2.9788 PHz ( λ 0 = 632.8 nm ) ,
W 0 = 1.5 μ m .
( E p , λ i ( 10.2 ) E p , λ i z z = 10.2 ) [ k = 0 4 ( p a 1 , 4 k , 13 , p λ ( 0.12 ) p p a 2 , 4 k , 13 , p λ ( 0.12 ) p p p a 1 , 4 k , 13 , p λ ( 0.12 ) p 1 p p a 2 , 4 k , 13 , p λ ( 0.12 ) p 1 ) ] 17 ( 1 i 2 π 1.6 λ ) .
( f λ , 1 ( 0 ) f λ , 1 ( 0 ) ) = ( f λ , 0 ( L d ) f λ , 0 ( L d ) ) = ( f λ , 1 , 0 ( L d ) f λ , 2 , 0 ( L d ) f λ , 1 , 0 ( L d ) f λ , 2 , 0 ( L d ) ) ( e 1 λ e 2 λ ) .
( f λ , k ( L d ) f λ , k ( L d ) ) = ( f λ , 1 , k 1 ( L d ) f λ , 2 , k 1 ( L d ) f λ , 1 , k 1 ( L d ) f λ , 2 , k 1 ( L d ) ) ( f λ , k 1 ( L d ) f λ , k 1 ( L d ) ) ;
k > 0 .
( f λ , K 1 ( L d ) f λ , K 1 ( L d ) ) = [ k = 0 K 1 ( f λ , 1 , K 1 k ( L d ) f λ , 2 , K 1 k ( L d ) f λ , 1 , K 1 k ( L d ) f λ , 2 , K 1 k ( L d ) ) ] ( e 1 λ e 2 λ ) .
f λ , k ( z ) y λ , k ( z ) , for z [ 0 , L d ] , k S K , λ S .
( f λ , K 1 ( L d ) f λ , K 1 ( L d ) ) ( y λ , K 1 ( L d ) y λ , K 1 ( L d ) ) = ( y λ ( L ) y λ ( L ) ) .
φ λ , i , k ( f ) ( z ) φ λ , i , k ( g ) ( z ) 0 z ( 0 x M λ max z [ 0 , L d ] f ( z ) g ( z ) d τ )
d x z 2 M λ 2 max z [ 0 , L d ] f ( z ) g ( z ) .
φ λ , i , k ( f ) ( z ) φ λ , i , k ( g ) ( z ) z 2 M λ Λ 2 ! ; z [ 0 , L d ] .
φ λ , i , k n + 1 ( f ) ( z ) φ λ , i , k n + 1 ( g ) ( z ) 0 z ( 0 x τ 2 n M λ n + 1 Λ ( 2 n ) ! d τ ) d x = z 2 ( n + 1 ) M λ n + 1 Λ ( 2 ( n + 1 ) ) ! ,
φ λ , i , k n ( f ) ( z ) φ λ , i , k n ( g ) ( z ) ( z M λ ) 2 n ( 2 n ) ! max z [ 0 , L d ] f ( z ) g ( z ) ;
z [ 0 , L d ] .
max z [ 0 , L d ] f λ , i , k , n + 1 ( z ) f λ , i , k , n ( z ) ( L d M λ ) 2 n ( 2 n ) ! L d i 1 .
lim n + f λ , i , k , n ( z ) = f λ , i , k ( z ) , z [ 0 , L d ] .
f λ , i , k ( z ) f λ , i , k , n ( z ) = r = n f λ , i , k , r + 1 ( z ) f λ , i , k , r ( z ) r = n f λ , i , k , r + 1 ( z ) f λ , i , k , r ( z ) ( L d M λ ) 2 n ( 2 n ) ! L d i 1 r = 0 ( L d M λ ) 2 r ( 2 r ) ! = ( L d M λ ) 2 n ( 2 n ) ! L d i 1 cosh ( L d M λ ) , for each z [ 0 , L d ] .

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