Abstract

Calculated diffractive efficiencies in the visible spectral band from lossless planar holographic gratings are studied using the PSM and Kogelnik models of diffraction for the case of the σ-polarization. The results are numerically compared with rigorous coupled wave calculations over a wide parameter space covering both transmission and reflection geometries. For most reflection gratings, the PSM model is shown to consistently provide a marginally superior estimation of the diffractive efficiency. This is particularly evident in a clearly superior description of the diffractive sideband structure for most gratings, both in terms of angle and wavelength. For the transmission grating, the PSM model continues to provide a relatively good description of diffraction at low permittivity modulations and lower incidence angles with respect to the grating plane normal. However, overall Kogelnik’s theory is shown to provide a somewhat superior estimation of diffractive efficiency and a clearly superior description of the diffractive side-band structure in the transmission case.

© 2014 Optical Society of America

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References

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  1. D. Brotherton-Ratcliffe, “A treatment of the general volume holographic grating as an array of parallel stacked mirrors,” J. Mod. Opt. 59(13), 1113–1132 (2012).
    [Crossref]
  2. D. Brotherton-Ratcliffe, “Analytical treatment of the polychromatic spatially multiplexed volume holographic grating,” Appl. Opt. 51(30), 7188–7199 (2012).
    [Crossref] [PubMed]
  3. D. Brotherton-Ratcliffe, “A new type of coupled wave theory capable of analytically describing diffraction in polychromatic spatially multiplexed holographic gratings,” J. Phys. Conf. Ser. 415, 012034 (2013), doi:.
    [Crossref]
  4. H.Bjelkhagen and D.Brotherton-Ratcliffe, Ultra-Realistic Imaging – Advanced Techniques in Analogue and Digital Colour Holography (Taylor and Francis, 2012).
  5. M. P. Rouard, “Etudes des propriétés optiques des lames métalliques très minces,” Ann. Phys. (Paris) Ser. II 7, 291–384 (1837).
  6. F. Abeles, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés, Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640 (1950).
  7. M. G. Moharam and T. K. Gaylord, “Chain-matrix analysis of arbitrary-thickness dielectric reflection gratings,” J. Opt. Soc. Am. 72(2), 187–190 (1982).
    [Crossref]
  8. O. S. Heavens, “Optical Properties of thin films,” Rep. Prog. Phys. 23(301), 1 (1960).
    [Crossref]
  9. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48(9), 2909–2947 (1969).
    [Crossref]
  10. M. G. Moharam and T. K. Gaylord, “Rigorous coupled wave analysis of planar grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
  11. J. A. Kong, “Second order coupled mode equations for spatially periodic media,” J. Opt. Soc. Am. 67(6), 825–829 (1977).
    [Crossref]

2013 (1)

D. Brotherton-Ratcliffe, “A new type of coupled wave theory capable of analytically describing diffraction in polychromatic spatially multiplexed holographic gratings,” J. Phys. Conf. Ser. 415, 012034 (2013), doi:.
[Crossref]

2012 (2)

D. Brotherton-Ratcliffe, “A treatment of the general volume holographic grating as an array of parallel stacked mirrors,” J. Mod. Opt. 59(13), 1113–1132 (2012).
[Crossref]

D. Brotherton-Ratcliffe, “Analytical treatment of the polychromatic spatially multiplexed volume holographic grating,” Appl. Opt. 51(30), 7188–7199 (2012).
[Crossref] [PubMed]

1982 (1)

1981 (1)

1977 (1)

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48(9), 2909–2947 (1969).
[Crossref]

1960 (1)

O. S. Heavens, “Optical Properties of thin films,” Rep. Prog. Phys. 23(301), 1 (1960).
[Crossref]

1950 (1)

F. Abeles, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés, Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640 (1950).

1837 (1)

M. P. Rouard, “Etudes des propriétés optiques des lames métalliques très minces,” Ann. Phys. (Paris) Ser. II 7, 291–384 (1837).

Abeles, F.

F. Abeles, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés, Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640 (1950).

Brotherton-Ratcliffe, D.

D. Brotherton-Ratcliffe, “A new type of coupled wave theory capable of analytically describing diffraction in polychromatic spatially multiplexed holographic gratings,” J. Phys. Conf. Ser. 415, 012034 (2013), doi:.
[Crossref]

D. Brotherton-Ratcliffe, “A treatment of the general volume holographic grating as an array of parallel stacked mirrors,” J. Mod. Opt. 59(13), 1113–1132 (2012).
[Crossref]

D. Brotherton-Ratcliffe, “Analytical treatment of the polychromatic spatially multiplexed volume holographic grating,” Appl. Opt. 51(30), 7188–7199 (2012).
[Crossref] [PubMed]

Gaylord, T. K.

Heavens, O. S.

O. S. Heavens, “Optical Properties of thin films,” Rep. Prog. Phys. 23(301), 1 (1960).
[Crossref]

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48(9), 2909–2947 (1969).
[Crossref]

Kong, J. A.

Moharam, M. G.

Rouard, M. P.

M. P. Rouard, “Etudes des propriétés optiques des lames métalliques très minces,” Ann. Phys. (Paris) Ser. II 7, 291–384 (1837).

Ann. Phys. (Paris) (1)

F. Abeles, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés, Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640 (1950).

Ann. Phys. (Paris) Ser. II (1)

M. P. Rouard, “Etudes des propriétés optiques des lames métalliques très minces,” Ann. Phys. (Paris) Ser. II 7, 291–384 (1837).

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48(9), 2909–2947 (1969).
[Crossref]

J. Mod. Opt. (1)

D. Brotherton-Ratcliffe, “A treatment of the general volume holographic grating as an array of parallel stacked mirrors,” J. Mod. Opt. 59(13), 1113–1132 (2012).
[Crossref]

J. Opt. Soc. Am. (3)

J. Phys. Conf. Ser. (1)

D. Brotherton-Ratcliffe, “A new type of coupled wave theory capable of analytically describing diffraction in polychromatic spatially multiplexed holographic gratings,” J. Phys. Conf. Ser. 415, 012034 (2013), doi:.
[Crossref]

Rep. Prog. Phys. (1)

O. S. Heavens, “Optical Properties of thin films,” Rep. Prog. Phys. 23(301), 1 (1960).
[Crossref]

Other (1)

H.Bjelkhagen and D.Brotherton-Ratcliffe, Ultra-Realistic Imaging – Advanced Techniques in Analogue and Digital Colour Holography (Taylor and Francis, 2012).

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Figures (12)

Fig. 1
Fig. 1 Holographic recording (a) and replay (b) of a simple planar grating. Note that the effect of Snell’s law has not been explicitly displayed in the diagram. As such all incidence angles shown here are “internal” angles.
Fig. 2
Fig. 2 Comparison of the diffraction efficiencies (σ-polarization) predicted by (harmonic permittivity) RCW theory (red), PSM (blue) and Kogelnik (black) for a lossless reflection phase grating having a modulation of n1/n0 = 0.03. The 6 micron deep grating was recorded at λr = 500nm with reference beams at incidence angles to the grating substrate normal of Φr = 10° and Φo = −25° giving a grating slant of ψ = 7.5°. The grating was replayed at an incidence angle of Φc = −25°. n0 = 1.5.
Fig. 3
Fig. 3 Normalized cross-correlation functions (Eqs. (14) for a comparison of Kogelnik versus (harmonic permittivity) RCW theory and PSM versus (harmonic permittivity) RCW theory (σ-polarization). The graphs pertain to a grating of 8 microns thickness and having a modulation of n1/n0 = 0.03. Graphs (a)-(d) show different internal object beam recording incidence angles with respect to the grating substrate normal. (a) Φo = −40° (b) Φo = −27°, (c) Φo = −14°, (d) Φo = 0°. The x-axis represents the external incidence replay angle with respect to the grating substrate normal, Φc which is equal to the reference recording angle, Φr. Grating recorded at λr = 500nm. n0 = 1.5
Fig. 4
Fig. 4 Surface contour plot of the normalized cross-correlation measure defined in Eq. (15) for the case of an internal object beam recording angle of Φo = −30° (σ-polarization). The hashed area represents the region where the correlation measure is positive indicating that the PSM theory is to be preferred here. The large (blue) negative feature in the top right-hand corner is due to the rapidly rising difference between harmonic-index RCW theory and harmonic-permittivity RCW theory as discussed in the text.
Fig. 5
Fig. 5 Example of a rather extreme case (σ-polarization) in which there is a clearly saturated broadband response in a relatively thin reflection grating. Here Kogelnik’s theory appears closer to RCW theory than PSM only because the RCW calculation used here models a harmonic permittivity profile rather than a harmonic index profile. The grating modulation is n1/n0 = 0.15 and the grating thickness is 3 microns. Recording wavelength λr = 500nm. Grating recorded using internal object beam of Φo = −20° and internal reference beam Φr = −35°. Recording and replay geometries are the same. n0 = 1.5.
Fig. 6
Fig. 6 Typical graph showing diffraction efficiency (σ-polarization) as predicted by (harmonic-permittivity) RCW theory ( = + 1 mode), PSM and Kogelnik, against internal replay incidence angle with respect to the substrate normal for the fixed geometry case. The grating is 15 microns thick and has a modulation of n1/n0 = 0.02. It was recorded and replayed at λr = 500nm. Reference recording beam angles were Φo = 0 and Φr = −30° (internal). n0 = 1.5. The RCW calculation retained 7 modes.
Fig. 7
Fig. 7 Normalized cross-correlation functions (Eqs. (16) for a comparison of Kogelnik versus (harmonic permittivity) RCW theory and PSM versus (harmonic permittivity) RCW theory (σ-polarization). The graphs pertain to a grating of 8 microns thickness and having a modulation of n1/n0 = 0.03. Graphs (a)-(d) show different internal object beam recording incidence angles with respect to the grating normal. (a) Φo = −40°, (b) Φo = −27°, (c) Φo = −14° and (d) Φo = 0°. The x-axis represents the external incidence recording angle with respect to the grating normal. Grating recorded at λr = 500nm. n0 = 1.5.
Fig. 8
Fig. 8 Surface contour plot of the normalized cross-correlation measure defined in Eq. (17) for the case of an internal object beam recording angle of Φo = −30° (σ-polarization). The hashed area represents the region where the correlation measure is positive indicating that the PSM theory is to be preferred here. As in Fig. 4 the large (blue) negative feature in the top right-hand corner is due to the rapidly rising difference between harmonic-index RCW theory and harmonic-permittivity RCW theory as discussed in the text.
Fig. 9
Fig. 9 Diffractive efficiencies (σ-polarization) versus replay wavelength of nine unslanted transmission gratings of thickness 4 microns. The graphs cover three (internal) incidence angles with respect to the grating plane normal and three modulations. All gratings were recorded and replayed at the same angle. Recording wavelength was λr = 500nm. n0 = 1.5. (Harmonic permittivity) RCW theory: red, Kogelnik Model: black and PSM Model: blue.
Fig. 10
Fig. 10 Diffractive efficiencies (σ-polarization) versus replay wavelength of nine unslanted transmission gratings of thickness 8 microns. The graphs cover three (internal) incidence angles with respect to the grating plane normal and three modulations. All gratings were recorded and replayed at the same angle. Recording wavelength was λr = 500nm. n0 = 1.5. (Harmonic Permittivity) RCW theory: red, Kogelnik Model: black and PSM Model: blue.
Fig. 11
Fig. 11 Diffractive efficiencies (σ-polarization) versus replay wavelength of nine slanted transmission gratings of thickness 8 microns and slant angle 70°. The graphs cover three incidence (internal) angles with respect to the grating plane normal and three modulations. All gratings were recorded and replayed at the same angle. Recording wavelength was λr = 500nm. n0 = 1.5. (Harmonic Permittivity) RCW theory: red, Kogelnik Model: black and PSM Model: blue.
Fig. 12
Fig. 12 Diffractive efficiencies (σ-polarization) versus (internal) reconstruction angle of nine slanted transmission gratings having a slant angle of 70°. The graphs cover three grating thicknesses and three modulations. All gratings were recorded at 70° (internal angle) with respect to the grating plane normal and replayed at an angle between 60° and 80° (internal angles). Recording wavelength was λ r = 500nm. n0 = 1.5. (Harmonic Permittivity) RCW theory: red, Kogelnik Model: black and PSM Model: blue.

Equations (68)

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c R dR dy =iκS c S dS dy =iϑSiκR
c R(KOG) =cos( θ c ψ) c S(KOG) =cos( θ c ψ)2αcos θ r cosψ ϑ (KOG) =2αβcos θ r (cos θ c αcos θ r )
c R(PSM) = cos θ c cos( θ c ψ) αcos θ r c S(PSM) = cos θ c cos( θ c +ψ) αcos θ r ϑ (PSM) =2β(1 cos θ c αcos θ r ) cos 2 θ c
α= λ c / λ r
β= 2π n 0 λ c
κ= π n 1 λ c .
n= n 0 + n 1 2 { e iKr + e iKr }
ε= ε 0 + ε 1 2 { e iKr + e iKr }.
R(0)=1 S(d)=0
η σ = | c S | c R S(0) S (0)= κ 2 sin h 2 (dϒ) κ 2 sin h 2 (dϒ) c R c S ϒ 2
ϒ 2 = ϑ 2 4 c S 2 κ 2 c R c S
R(0)=1 S(0)=0
η σ = κ 2 2 ϒ 2 c S c R { cosh(2dϒ)1 }
Γ KOG = λ 1 λ 2 η KOG η RCW d λ c λ 1 λ 2 η KOG 2 d λ c λ 1 λ 2 η RCW 2 d λ c ; Γ PSM = λ 1 λ 2 η PSM η RCW d λ c λ 1 λ 2 η PSM 2 d λ c λ 1 λ 2 η RCW 2 d λ c
Θ(d ; n 1 / n 0 )= 1 Φ c2 Φ c1 Φ c = Φ c1 = 40 Φ c = Φ c2 = 40 ( Γ PSM ( Φ c ) Γ KOG ( Φ c ))d Φ c
Γ KOG = Φ 1 Φ 2 η KOG η RCW d Φ c Φ 1 Φ 2 η KOG 2 d Φ c Φ 1 Φ 2 η RCW 2 d Φ c ; Γ PSM = Φ 1 Φ 2 η PSM η RCW d Φ c Φ 1 Φ 2 η PSM 2 d Φ c Φ 1 Φ 2 η RCW 2 d Φ c
Θ(d ; n 1 / n 0 )= 1 Φ r2 Φ r1 Φ r = Φ r1 = 40 Φ r = Φ r2 = 40 ( Γ PSM ( Φ r ) Γ KOG ( Φ r ))d Φ r
2 u x 2 + 2 u y 2 γ 2 u=0
γ 2 = β 2 2βκ{ e iKr + e iKr }
u(y<0)= e i( k x x+ k y y)
k x =βsin( θ c ψ)=βsin( Φ c ) k y =βcos( θ c ψ)=βcos( Φ c )
u(x,y)= = + u (y) e i( k x + K x )x
{ ( k x + K x ) 2 β 2 } u (y) d 2 u d y 2 (y)=2βκ{ u 1 (y) e i K y y + u +1 (y) e i K y y }
u (y)= u ^ (y) e i( k y + K y )y
d 2 u ^ (y) d y 2 +2i( k y + K y ) d u ^ (y) dy = { ( k x + K x ) 2 + ( k y + K y ) 2 β 2 } u ^ (y)2βκ{ u ^ 1 (y)+ u ^ +1 (y) }
  γ 2 = β 2 (1+ n 1 2 n 0 [ e iKr + e iKr ]) 2 =( β 2 +2 κ 2 )2βκ[ e iKr + e iKr ] κ 2 [ e 2iKr + e 2iKr ]
u(x,y)= = + u (y) e i( k x + K x )x
{ ( k x + K x ) 2 ( β 2 +2 κ 2 ) } u (y) d 2 u d y 2 (y)= 2βκ{ u 1 (y) e i K y y + u +1 (y) e i K y y }+ κ 2 { u 2 (y) e 2i K y y + u +2 (y) e 2i K y y }
u (y)= u ^ (y) e i( k y + K y )y
d 2 u ^ (y) d y 2 +2i( k y + K y ) d u ^ (y) dy ={ ( k x + K x ) 2 + ( k y + K y ) 2 ( β 2 +2 κ 2 ) } u ^ (y) 2βκ{ u ^ 1 (y)+ u ^ +1 (y) } κ 2 { u ^ 2 (y)+ u ^ +2 (y) }
d 2 u ^ d y 2 +2i( k y + K y ) d u ^ dy { ( k x + K x ) 2 + ( k y + K y ) 2 β 2 } u ^ =0
u ^ = A e i( k y + K y )y+i { β 2 ( k x + K x ) 2 }y + B e i( k y + K y )yi { β 2 ( k x + K x ) 2 } y
u(x,y)= e i { β 2 k x 2 } y e i k x + = + u ^ e i( k y + K y )yi { β 2 ( k x + K x ) 2 } y e i( k x + K x )x e i( k y + K y )y = e i { β 2 k x 2 } y e i k x + = + u ^ e i { β 2 ( k x + K x ) 2 } y e i( k x + K x )x
u(x,y=0)= e i k x + = + u ^ e i( k x + K x )x
u(x,y=0 ) =i β 2 k x 2 e i k x i = + ( β 2 ( k x + K x ) 2 ) u ^ e i( k x + K x )x
u(x,y=0)= = + u ^ (y=0;) e i( k x + K x ) = e i k x + = + u ^ l (y=0;o) e i( k x + K x )x
u(x,y=0 ) = = + { i( k y + K y ) u ^ (y=0;)+ u ^ (y=0;) } e i( k x + K x ) =i β 2 k x 2 e i k x i = + ( β 2 ( k x + K x ) 2 ) u ^ (y=0;o) e i( k x + K x )x
u ^ 0 (y=0;)=1+ u ^ 0 (y=0;o)
i k y u ^ 0 (y=0;)+ u ^ 0 (y=0;)=i β 2 k x 2 i( β 2 k x 2 ) u ^ 0 (y=0;o)
u ^ 0 (y=0;)=2i k y { 1 u ^ 0 (y=0;) }
u ^ (y=0;)+i( k y + K y ) u ^ (y=0;)=i{ β 2 ( k x + K x ) 2 } u ^ (y=0;)
u(x,y)= = + u ^ e i( k y + K y )y+i { β 2 ( k x + K x ) 2 } y+i( k y + K y )y e i( k x + K x )x = = + u ^ e i { β 2 ( k x + K x ) 2 } y e i( k x + K x )x
u ^ (y=d;)+i( k y + K y ) u ^ (y=d;)=i{ β 2 ( k x + K x ) 2 } u ^ (y=d;)
d dy ( U 1, U 2, )=b( U 1, U 2, )
d dy ( U 1,2 U 1,1 U 1,0 U 1,1 U 1,2 U 2,2 U 2,1 U 2,0 U 2,1 U 2,2 )=( 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 b 2 a 0 0 0 c 2 0 0 0 0 a b 1 a 0 0 0 c 1 0 0 0 0 a b 0 a 0 0 0 c 0 0 0 0 0 a b 1 a 0 0 0 c 1 0 0 0 0 a b 2 0 0 0 0 c 2 )( U 1,2 U 1,1 U 1,0 U 1,1 U 1,2 U 2,2 U 2,1 U 2,0 U 2,1 U 2,2 )
a=2βκ b = ( k x + K x ) 2 + ( k y + K y ) 2 β 2 c =2i( k y + K y )
U α = m C m w αm e q m y
( U 1 U 2 U 3 U 4 U 5 U 6 U 7 U 8 U 9 U 10 )=( u ^ 2 u ^ 1 u ^ 0 u ^ 1 u ^ 2 u ^ 2 u ^ 1 u ^ 0 u ^ 1 u ^ 2 )=( w αm )( C 1 e q 1 y C 2 e q 2 y C 3 e q 3 y C 4 e q 4 y C e 5 q 5 y C 6 e q 6 y C 7 e q 7 y C 8 e q 8 y C 9 e q 9 y C 10 e q 10 y )
U 8 (0)= m=1 10 C m w 8m =2i k y (1 U 3 (0))=2i k y (1 m=1 10 C m w 3m )
U 6 (0)= m=1 10 C m w 6m ={ i( β 2 ( k x +2 K x ) 2 )i( k y +2 K y ) } m=1 10 C m w 1m
U 7 (0)= m=1 10 C m w 7m ={ i( β 2 ( k x + K x ) 2 )i( k y + K y ) } m=1 10 C m w 2m
U 9 (0)= m=1 10 C m w 9m ={ i( β 2 ( k x K x ) 2 )i( k y K y ) } m=1 10 C m w 4m
U 10 (0)= m=1 10 C m w 10m ={ i( β 2 ( k x 2 K x ) 2 )i( k y 2 K y ) } m=1 10 C m w 5m
U 6 (d)= m=1 10 C m w 6m e q m d ={ i( β 2 ( k x +2 K x ) 2 )i( k y +2 K y ) } m=1 10 C m w 1m e q m d
U 7 (d)= m=1 10 C m w 7m e q m d ={ i( β 2 ( k x + K x ) 2 )i( k y + K y ) } m=1 10 C m w 2m e q m d
U 8 (d)= m=1 10 C m w 8m e q m d =0
U 9 (d)= m=1 10 C m w 9m e q m d ={ i( β 2 ( k x K x ) 2 )i( k y K y ) } m=1 10 C m w 4m e q m d
U 10 (d)= m=1 10 C m w 10m e q m d ={ i( β 2 ( k x 2 K x ) 2 )i( k y 2 K y ) } m=1 10 C m w 5m e q m d
ZC=R+TC
Z=( w 61 w 62 w 63 w 64 w 65 w 66 w 67 w 68 w 69 w 610 w 71 w 72 w 73 w 74 w 75 w 76 w 77 w 78 w 79 w 719 w 81 w 91 w 101 w 61 e q 1 d w 62 e q 2 d w 63 e q 3 d w 64 e q 4 d w 71 e q 1 d w 72 e q 2 d w 73 e q 3 d w 74 e q 4 d w 81 w 91 w 101 )
T=( f 2 w 11 f 2 w 12 f 2 w 13 f 2 w 14 f 2 w 15 f 2 w 16 f 1 w 21 f 1 w 22 f 1 w 23 f 1 w 24 f 0 w 31 f 1 w 41 f 2 w 51 g 2 w 11 e q 1 d g 2 w 12 e q 2 d g 2 w 13 e q 3 d g 2 w 14 e q 4 d g 2 w 15 e q 5 d g 2 w 16 e q 6 d g 1 w 21 e q 1 d g 1 w 22 e q 2 d g 1 w 23 e q 3 d g 1 w 24 e q 4 d g 0 w 31 e q 1 d g 1 w 41 e q 1 d g 2 w 51 e q 1 d )
R=( 0 0 2i k y 0 0 0 0 0 0 0 )
f ={ i β 2 ( k x + K x ) 2 i( k y + K y ) } g ={ i β 2 ( k x + K x ) 2 i( k y + K y ) }
C= (ZT) 1 R
η (T)= β 2 ( k x + K x ) 2 k y u ^ (y=d) u ^ (y=d)
η (R)= β 2 ( k x + K x ) 2 k y u ^ (y=0) u ^ (y=0) 0 η 0 (R)=( u ^ (y=0)1)( u ^ (y=0)1) =0
d dy ( U 1,2 U 1,1 U 1,0 U 1,1 U 1,2 U 2,2 U 2,1 U 2,0 U 2,1 U 2,2 )=( 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 b 2 a κ 2 0 0 c 2 0 0 0 0 a b 1 a κ 2 0 0 c 1 0 0 0 κ 2 a b 0 a κ 2 0 0 c 0 0 0 0 κ 2 a b 1 a 0 0 0 c 1 0 0 0 κ 2 a b 2 0 0 0 0 c 2 )( U 1,2 U 1,1 U 1,0 U 1,1 U 1,2 U 2,2 U 2,1 U 2,0 U 2,1 U 2,2 )
a=2βκ b l = ( k x + K x ) 2 + ( k y + K y ) 2 ( β 2 +2 κ 2 ) c l =2i( k y + K y )

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