Abstract

A quasi-periodic interface destroys the k invariance, introducing wavenumbers shifted by structure period harmonics, and allows transmission for supercritical incidence. An analytic method converting the interface boundary conditions into effective current source terms yields simple integral formulas for the transmitted fraction including quasi-periodic, subwavelength surface features with random variation of the feature size and period.

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  1. R. Windish, C. Rooman, B. Dutta, A. Knobloch, G. Borghs, G. H. Dohler, P. Heremans, “Light-extraction mechanisms in high-efficiency, surface-textured light-emitting diodes,” IEEE J. Sel. Top. Quantum Electron. 8, 248–255 (2002).
    [CrossRef]
  2. I. Schnitzer, E. Yablonovitch, C. Caneau, T. J. Gmitter, A. Scherer, “30% external quantum efficiency from surface textured, thin-film light-emitting diodes,” Appl. Phys. Lett. 63, 2174–2176 (1993).
    [CrossRef]
  3. A. G. Voronovich, Scattering from Rough Surfaces (Springer-Verlag, 1994).
    [CrossRef]
  4. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, 1991).
  5. J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, 1991).
  6. A. Sentenac, J.-J. Greffet, “Mean-field theory of light scattering by one-dimensional rough surfaces,” J. Opt. Soc. Am. A 15, 528–532 (1998).
    [CrossRef]
  7. See, for example, J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975), pp. 278–282.
  8. See, for example, J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975), pp. 432–438.
  9. See, for example, R. Kubo, M. Todda, W. Hasitsume, Statistical Physics II (Springer-Verlag, 1985), pp. 22–55.

2002 (1)

R. Windish, C. Rooman, B. Dutta, A. Knobloch, G. Borghs, G. H. Dohler, P. Heremans, “Light-extraction mechanisms in high-efficiency, surface-textured light-emitting diodes,” IEEE J. Sel. Top. Quantum Electron. 8, 248–255 (2002).
[CrossRef]

1998 (1)

1993 (1)

I. Schnitzer, E. Yablonovitch, C. Caneau, T. J. Gmitter, A. Scherer, “30% external quantum efficiency from surface textured, thin-film light-emitting diodes,” Appl. Phys. Lett. 63, 2174–2176 (1993).
[CrossRef]

Borghs, G.

R. Windish, C. Rooman, B. Dutta, A. Knobloch, G. Borghs, G. H. Dohler, P. Heremans, “Light-extraction mechanisms in high-efficiency, surface-textured light-emitting diodes,” IEEE J. Sel. Top. Quantum Electron. 8, 248–255 (2002).
[CrossRef]

Caneau, C.

I. Schnitzer, E. Yablonovitch, C. Caneau, T. J. Gmitter, A. Scherer, “30% external quantum efficiency from surface textured, thin-film light-emitting diodes,” Appl. Phys. Lett. 63, 2174–2176 (1993).
[CrossRef]

Dohler, G. H.

R. Windish, C. Rooman, B. Dutta, A. Knobloch, G. Borghs, G. H. Dohler, P. Heremans, “Light-extraction mechanisms in high-efficiency, surface-textured light-emitting diodes,” IEEE J. Sel. Top. Quantum Electron. 8, 248–255 (2002).
[CrossRef]

Dutta, B.

R. Windish, C. Rooman, B. Dutta, A. Knobloch, G. Borghs, G. H. Dohler, P. Heremans, “Light-extraction mechanisms in high-efficiency, surface-textured light-emitting diodes,” IEEE J. Sel. Top. Quantum Electron. 8, 248–255 (2002).
[CrossRef]

Gmitter, T. J.

I. Schnitzer, E. Yablonovitch, C. Caneau, T. J. Gmitter, A. Scherer, “30% external quantum efficiency from surface textured, thin-film light-emitting diodes,” Appl. Phys. Lett. 63, 2174–2176 (1993).
[CrossRef]

Greffet, J.-J.

Hasitsume, W.

See, for example, R. Kubo, M. Todda, W. Hasitsume, Statistical Physics II (Springer-Verlag, 1985), pp. 22–55.

Heremans, P.

R. Windish, C. Rooman, B. Dutta, A. Knobloch, G. Borghs, G. H. Dohler, P. Heremans, “Light-extraction mechanisms in high-efficiency, surface-textured light-emitting diodes,” IEEE J. Sel. Top. Quantum Electron. 8, 248–255 (2002).
[CrossRef]

Jackson, J. D.

See, for example, J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975), pp. 278–282.

See, for example, J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975), pp. 432–438.

Knobloch, A.

R. Windish, C. Rooman, B. Dutta, A. Knobloch, G. Borghs, G. H. Dohler, P. Heremans, “Light-extraction mechanisms in high-efficiency, surface-textured light-emitting diodes,” IEEE J. Sel. Top. Quantum Electron. 8, 248–255 (2002).
[CrossRef]

Kubo, R.

See, for example, R. Kubo, M. Todda, W. Hasitsume, Statistical Physics II (Springer-Verlag, 1985), pp. 22–55.

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, 1991).

Ogilvy, J. A.

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, 1991).

Rooman, C.

R. Windish, C. Rooman, B. Dutta, A. Knobloch, G. Borghs, G. H. Dohler, P. Heremans, “Light-extraction mechanisms in high-efficiency, surface-textured light-emitting diodes,” IEEE J. Sel. Top. Quantum Electron. 8, 248–255 (2002).
[CrossRef]

Scherer, A.

I. Schnitzer, E. Yablonovitch, C. Caneau, T. J. Gmitter, A. Scherer, “30% external quantum efficiency from surface textured, thin-film light-emitting diodes,” Appl. Phys. Lett. 63, 2174–2176 (1993).
[CrossRef]

Schnitzer, I.

I. Schnitzer, E. Yablonovitch, C. Caneau, T. J. Gmitter, A. Scherer, “30% external quantum efficiency from surface textured, thin-film light-emitting diodes,” Appl. Phys. Lett. 63, 2174–2176 (1993).
[CrossRef]

Sentenac, A.

Todda, M.

See, for example, R. Kubo, M. Todda, W. Hasitsume, Statistical Physics II (Springer-Verlag, 1985), pp. 22–55.

Voronovich, A. G.

A. G. Voronovich, Scattering from Rough Surfaces (Springer-Verlag, 1994).
[CrossRef]

Windish, R.

R. Windish, C. Rooman, B. Dutta, A. Knobloch, G. Borghs, G. H. Dohler, P. Heremans, “Light-extraction mechanisms in high-efficiency, surface-textured light-emitting diodes,” IEEE J. Sel. Top. Quantum Electron. 8, 248–255 (2002).
[CrossRef]

Yablonovitch, E.

I. Schnitzer, E. Yablonovitch, C. Caneau, T. J. Gmitter, A. Scherer, “30% external quantum efficiency from surface textured, thin-film light-emitting diodes,” Appl. Phys. Lett. 63, 2174–2176 (1993).
[CrossRef]

Appl. Phys. Lett. (1)

I. Schnitzer, E. Yablonovitch, C. Caneau, T. J. Gmitter, A. Scherer, “30% external quantum efficiency from surface textured, thin-film light-emitting diodes,” Appl. Phys. Lett. 63, 2174–2176 (1993).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

R. Windish, C. Rooman, B. Dutta, A. Knobloch, G. Borghs, G. H. Dohler, P. Heremans, “Light-extraction mechanisms in high-efficiency, surface-textured light-emitting diodes,” IEEE J. Sel. Top. Quantum Electron. 8, 248–255 (2002).
[CrossRef]

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

Other (6)

See, for example, J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975), pp. 278–282.

See, for example, J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975), pp. 432–438.

See, for example, R. Kubo, M. Todda, W. Hasitsume, Statistical Physics II (Springer-Verlag, 1985), pp. 22–55.

A. G. Voronovich, Scattering from Rough Surfaces (Springer-Verlag, 1994).
[CrossRef]

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, 1991).

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, 1991).

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Figures (12)

Fig. 1
Fig. 1

Induced current circulation due to across-the-surface discontinuity in the parallel electric displacement vector D = ϵ E .

Fig. 2
Fig. 2

(a) In flat interfaces, conservation of k = ϵ ( ω c ) sin θ = ϵ ( ω c ) sin θ yields Snell’s refraction law sin θ = ϵ ϵ sin θ . (b) When k exceeds the wavenumber k = ϵ ( ω c ) , for incidence angle sin θ > sin θ c ϵ ϵ , no transmission in the optically thin medium takes place. (c) For a periodically textured interface, the incident k is reduced by the period wavenumber q = 2 π D , k ( q ) = k n q , and thus transmission is allowed for supercritical incidence. Field components at a flat boundary, for TM incidence.

Fig. 3
Fig. 3

Surface geometry for TE incidence (E points out of the page). The network of effective current circulation I is decomposed in a flat-surface unperturbed I o plus periodic perturbation cells I 1 .

Fig. 4
Fig. 4

Plots of (a) form factor S N ( q D ) and (b) S N 2 ( q D ) for a finite periodic screen with increasing N = 3 (light dashed curve), 6, 9, 12, and 15 (solid curve). The integral of N S N 2 from q D = ( 0 , 2 π ) is unity, independent of N. (c) The Fourier component f 2 for N = 9 (dashed curve) and N = 19 .

Fig. 5
Fig. 5

Transmission coefficient versus incidence angle for (a) infinite flat surface. (b) Periodic interface of period k D = 4 π , width a D = 0.50 , and height b D = 0.48 . Solid curve corresponds to 50% polarization mix. N = 29 used for the computations.

Fig. 6
Fig. 6

Mixed polarization transmission coefficient versus θ for various incident wavelength values, for b D = 0.25 , and for the same other parameters as in Fig. 5b. Maximum transmission near λ D 2 = a .

Fig. 7
Fig. 7

Geometry illustration of a quasi-random textured interface showing the relation between incoming and scattered (transmitted) wavenumbers.

Fig. 8
Fig. 8

(a) Amplitude form factor, (b) intensity form factor, and (c) intensity Fourier component for a quasi-random screen of average width a D = 0.70 , height a D = 0.50 , and N = 9 . The rms deviation Δ 2 = Δ D 2 D 2 goes from zero (light dashed curve, periodic limit) to 1 (solid curve) in intervals of one fourth [zero to 0.8 by 1 5 in (c)]. Notice the transition from Bragg scattering to a near-uniform diffusion.

Fig. 9
Fig. 9

(a) Computed transmission coefficient versus incidence angle for the quasi-random interface; the experimental data (circles) cited in Ref. [1], corresponding to average width a D = 0.50 and height b D = 0.36 with incident λ D = 5 14 . The rms deviation Δ 2 = Δ D 2 D 2 = 0.50 . Solid curve corresponds to 50% polarization mix. (b) Same but not including the scattered wavenumber q effect on r and t.

Fig. 10
Fig. 10

Transmission coefficient of 50% mixed polarization versus incidence angle, for various incident wavelength values. Other parameters: tooth width a D = 0.70 and height b D = 0.48 , with rms deviation Δ 2 = Δ D 2 D 2 = 0.50 . The screen size value N = 29 used for all subsequent calculations. Maximum transmission near λ 2 D .

Fig. 11
Fig. 11

Transmission coefficient (50% polarization mix) versus incidence angle for (a) different tooth-width ratios a D and b D = 0.48 and (b) different tooth-height ratios b D and a D = 0.70 . The rms period deviation is Δ 2 = Δ D 2 D 2 = 0.25 , and height deviation σ 2 = Δ b 2 D 2 = 0.01 .

Fig. 12
Fig. 12

Transmission coefficient (50% polarization mix) versus incidence angle for (a) different tooth-width rms deviations (b) different tooth-height rms deviations as marked. The average values are a D = 0.70 and b D = 0.48 .

Equations (69)

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ϵ ω c sin θ q n < ϵ ω c .
ϵ ( E o + E o ) = ϵ E o , ( E o + E o ) = E o ,
( B o + B o ) = B o , ( B o + B o ) = B o ,
× × B = 1 c × [ D t + D t S ] = i ω c [ × D + × D S ] .
4 π c × J = 4 π c ( i ω 4 π ) × D S = i k o p ̂ o ( n ̂ × E ) δ ( z ) ,
[ 2 ω 2 c 2 ϵ ] B = i k o p ̂ o ( k ) ( n ̂ × E ) δ ( z ) I o ( k ) δ ( z ) .
I o TE TM = i k o p ̂ o ( ( E o + E o ) cos θ y ̂ ( E o + E o ) x ̂ ) = i k o p ̂ o E o ( ( 1 + r TM ) cos θ y ̂ ( 1 + r TE ) x ̂ ) .
( r t ) TM = ( cos θ ϵ ϵ cos θ cos θ + ϵ ϵ cos θ cos θ + cos θ cos θ + ϵ ϵ cos θ ) ,
( r t ) TE = ( ϵ ϵ cos θ cos θ ϵ ϵ cos θ + cos θ cos θ + cos θ ϵ ϵ cos θ + cos θ ) ,
p ̂ o TE TM = 2 ϵ t 1 + t ( cos θ cos θ cos 2 θ ) ,
p ̂ o TE TM = 2 ϵ r 1 + r ( 1 cos 2 θ ) .
p ̂ o ( q ) TE TM = 2 ϵ t ( q ) 1 + r ( 0 ) ( cos θ ( q ) cos θ cos 2 θ ( q ) ) ,
p ̂ o ( q ) TE TM = 2 ϵ r ( q ) 1 + r ( 0 ) ( cos θ ( q ) cos θ cos 2 θ ( q ) ) .
I 1 ( r ) = I o j = ( N 1 ) 2 ( N 1 ) 2 { H [ x ( R j a 2 ) ] H [ x ( R j + a 2 ) ] } [ δ ( z b ) δ ( z ) ] e i k x x + i k z z ,
I 1 ( k , q ) = I o ( k , q ) 1 2 π f ̂ ( q ) ,
f ̂ ( q ) = ( e i q z b 1 ) a D sin q x a 2 q x a 2 S N ( q x D ) ,
S N ( q n ) = j = ( N 1 ) 2 ( N 1 ) 2 e j ( i q n D ) = 1 N sin ( N q n D 2 ) sin ( q n D 2 )
B TE TM = E o ( y ̂ ϵ { < f ̂ ( q n ) e i κ n z i > f ̂ ( q n ) e κ n z } t ( q n ) cos θ ( q n ) ( ω c κ n ) e i ( k x q n ) x x ̂ ϵ { < f ̂ ( q n ) e i κ n z i > f ̂ ( q n ) e κ n z } t ( q n ) cos 2 θ ( q n ) ( ω c κ n ) e i ( k x q n ) x ) ,
E TE TM = E o ( x ̂ ϵ 1 ϵ { < f ̂ ( q n ) e i κ n z + > f ̂ ( q n ) e κ n z } t ( q n ) cos θ ( q n ) e i ( k x q n ) x y ̂ ϵ { < f ̂ ( q n ) e i κ n z + > f ̂ ( q n ) e κ n z } t ( q n ) cos 2 θ ( q n ) ( ω c κ n ) 2 e i ( k x q n ) x ) ,
B o ( q n ) TE TM = ϵ E o t ( q ) TE TM ( y ̂ x ̂ cos θ ( q n ) ) f ̂ ( q n ) e i ( k x q n ) x e i κ n z ,
E o ( q n ) TE TM = E o t ( q ) TE TM ( x ̂ cos θ ( q n ) y ̂ ) f ̂ ( q n ) e i ( k x q n ) x e i κ n z .
E tr = E o tr + E , E rf = E o rf + E .
P z TE TM = L c 8 π { < n + > n } ( E x ( q n ) B y * ( q n ) E y ( q n ) B x * ( q n ) ) + c.c. = L c 4 π < n ( E x ( q n ) B y * ( q n ) E y ( q n ) B x * ( q n ) ) .
T TE TM = { [ 1 + 2 R f ̂ ( 0 ) ] cos θ t ( 0 ) 2 TE TM + 1 2 π d q ¯ f ̂ ( q ¯ ) f ̂ * ( q ¯ ) cos θ ( q ¯ ) t ( q ¯ ) 2 TE TM } ϵ ϵ 1 cos θ ,
R TE TM = { [ 1 + 2 R f ̂ ( 0 ) ] cos θ r ( 0 ) 2 TE TM + 1 2 π d q ¯ f ̂ ( q ¯ ) f ̂ * ( q ¯ ) cos θ ( q ¯ ) r ( q ¯ ) 2 TE TM } 1 cos θ .
R + T r 2 ( 0 ) + t 2 ( 0 ) cos θ cos θ = 1 .
I 1 ( r ) = I o j = ( N 1 ) 2 ( N 1 ) 2 [ H ( x R j ) H ( x R j + ) ] [ δ ( z b j ) δ ( z ) ] e i k x x + i k z z ,
I 1 ( q ) = I o ( k , q ) j = ( N 1 ) 2 ( N 1 ) 2 1 2 π e i q z b j 1 1 N D 1 i q x ( e i q x R j e i q x R j + ) ,
e i q x R j e i q x R j + = e i q x j D q x 2 j Δ 2 2 ( e i q x a 2 e i q x a 2 ) e q x 2 δ 2 8 .
e i q z b j 1 = e i q z b e q z 2 σ 2 2 1 ,
I 1 ( k , q ) = I o ( k ) 1 2 π f ̂ ( q ) ,
f ̂ ( q ) = ( e i q z b e q z 2 σ 2 2 1 ) e q x 2 δ 2 8 sin ( q x a 2 ) q x a 2 a D 1 N j = ( N 1 ) 2 ( N 1 ) 2 e j ( i q x D q x 2 Δ 2 2 ) .
S ( q n ) 1 N j = ( N 1 ) 2 ( N 1 ) 2 e j ( i q n D q n 2 Δ 2 2 ) ,
= 1 N 1 e q n 2 Δ 2 + 2 e [ ( N + 1 ) 4 ] q n 2 Δ 2 ( e [ ( q n 2 Δ 2 ) 2 ] cos N 1 2 q n D cos N + 1 2 q n D ) 1 2 cos q n D e ( q n 2 Δ 2 2 ) + e q n 2 Δ 2 .
f ̂ x ( q ) f ̂ x * ( q ) = ( e i q z b j 1 ) ( e i q z b j 1 ) * 1 q x 2 1 N 2 D 2 j k ( e i q x R j e i q x R j + ) ( e i q x R k e i q x R k + ) * .
f ̂ ( q ) f ̂ * ( q ) = ( 1 + e q z 2 σ 2 2 cos ( q z b ) e ( q z 2 σ 2 2 ) ) 1 + e q x 2 δ 2 2 e ( q x 2 δ 2 2 ) cos ( q x a ) ( q x a ) 2 a 2 D 2 S 2 ( q x ) ,
S 2 ( q n ) = 1 N 2 { N ( 1 e q n 2 Δ 2 ) 1 2 cos q n D e ( q n 2 Δ 2 2 ) + e q n 2 Δ 2 + 4 e q n 2 Δ 2 ( 1 cos q n D cosh q n 2 Δ 2 2 ) ( 1 e ( N q n 2 Δ 2 2 ) cos N q n D ) sin q n D sinh q n 2 Δ 2 2 e ( N q n 2 Δ 2 2 ) sin N q n D ( 1 2 cos q n D e ( q n 2 Δ 2 2 ) + e q n 2 Δ 2 ) 2 } .
T TE TM = { [ 1 + 2 R f ̂ ( 0 ) ] cos θ t ( 0 ) 2 TE TM + 1 2 π d q ¯ f ̂ ( q ¯ ) f ̂ * ( q ¯ ) cos θ ( q ¯ ) t ( q ¯ ) 2 TE TM } ϵ ϵ 1 cos θ ,
R TE TM = { [ 1 + 2 R ̂ f ( 0 ) ] cos θ r ( 0 ) 2 TE TM + 1 2 π d q ¯ f ̂ ( q ¯ ) f ̂ * ( q ¯ ) cos θ ( q ¯ ) r ( q ¯ ) 2 TE TM } 1 cos θ ,
B ( q ) = i k o p ̂ o ( q x ) k z 2 + k x 2 ϵ ω 2 c 2 E o ( ( 1 + r TM ) cos θ y ̂ ( 1 + r TE ) x ̂ ) ,
B ( r ) = i n d k z k o p ̂ o ( q n ) e i k z z k z 2 κ 2 ( q n ) E o ( ( 1 + r TM ) cos θ y ̂ ( 1 + r TE ) x ̂ ) e i ( k x q n ) x ,
B TM = y ̂ p ̂ o 2 π 1 + r 2 E o cos θ 2 π { < ω c κ n f ̂ ( q n ) e i κ n z i > ω c κ n f ̂ ( q n ) e κ n z } e i ( k x q n ) x ,
B TE = x ̂ p ̂ o 2 π 1 + r 2 E o 2 π { < ω c κ n f ̂ ( q n ) e i κ n z i > ω c κ n f ̂ ( q n ) e κ n z } e i ( k x q n ) x ,
E TM = x ̂ p ̂ o 2 π 1 ϵ 1 + r 2 E o cos θ 2 π { < f ̂ ( q n ) e i κ n z + > f ̂ ( q n ) e κ n z } e i ( k x q n ) x ,
E TE = y ̂ p ̂ o 2 π 1 + r 2 E o 2 π { < ( ω c κ n ) 2 f ̂ ( q n ) e i κ n z + > ( ω c κ n ) 2 f ̂ ( q n ) e κ n z } e i ( k x q n ) x .
E TM = x ̂ p ̂ o 2 π 1 ϵ 1 + r 2 E o cos θ 2 π { < f ̂ ( q n ) e i κ n z + > f ̂ ( q n ) e + κ n z } e i ( k x q n ) x ,
E TE = y ̂ p ̂ o 2 π 1 + r 2 E o 2 π { < ( ω c κ n ) 2 f ̂ ( q n ) e i κ n z + > ( ω c κ n ) 2 f ̂ ( q n ) e + κ n z } e i ( k x q n ) x .
E ̂ o = E o q f ( q ) e i k x x
E o TM = E o p ̂ o 1 + r 2 ϵ f ̂ ( q n ) cos θ cos θ ( q n ) e i ( k x q n ) x ,
E o TE = E o p ̂ o 1 + r 2 ϵ f ̂ ( q n ) 1 cos 2 θ ( q n ) e i ( k x q n ) x ,
E o TM = E o p ̂ o 1 + r 2 ϵ f ̂ ( q n ) cos θ cos θ ( q n ) e i ( k x q n ) x ,
E o TE = E o p ̂ o 1 + r 2 ϵ f ̂ ( q n ) 1 cos 2 θ ( q n ) e i ( k x q n ) x .
E o tr TE TM = t E o ( x ̂ cos θ y ̂ ) , B o tr TE TM = t ϵ E o ( y ̂ x ̂ cos θ )
P z ( 0 ) = c L 8 π ( ( E x tr + E x ( 0 ) ) ( B y tr + B y ( 0 ) ) * ( E y tr + E y ( 0 ) ) ( B x tr + B x ( 0 ) ) * ) + c.c. = c L 4 π t ( 0 ) 2 ϵ E o 2 1 + f ̂ ( 0 ) 2 cos θ .
P z ( q 0 ) = c L 8 π q 0 ( E x ( q ) B y ( q ) * E y ( q ) B x ( q ) * ) + c.c. = c L 4 π ϵ E o 2 < q 0 t ( q ) 2 f ̂ 2 ( q ) cos θ ( q ) .
P z = c L 4 π ϵ E o 2 { t ( 0 ) 2 [ 1 + 2 R f ̂ ( 0 ) ] cos θ + < t ( q ) 2 f ̂ ( q ) 2 cos θ ( q ) } ,
P z = c L 4 π ϵ E o 2 { r ( 0 ) 2 [ 1 + 2 R f ̂ ( 0 ) ] cos θ + < r ( q ) 2 f ̂ ( q ) 2 cos θ ( q ) } ,
e i q R j = e i q R j e i q Δ R j .
e i q Δ R j = 1 + l = 1 1 l ! i l q l ( Δ R j ) l = 1 + k = 1 1 ( 2 k ) ! ( i ) 2 k q 2 k Δ R j 2 k = 1 + k = 1 1 k ! ( 1 ) k ( q 2 2 ) k Δ R j 2 k e q 2 Δ R j 2 2 .
e i q R j = e i q R j q 2 Δ R j 2 2 .
e i q R j = e i q j D j q 2 Δ 2 2 .
j = 1 N k = 1 N ( e i q x R j e i q x R j + ) ( e i q x R k e i q x R k + ) = j = 1 N k = 1 N { e i q x ( R j R k ) + e i q x ( R j + R k + ) e i q x ( R j R k + ) e i q x ( R j + R k ) } .
j = 1 N k = 1 N ( e i q x [ ( δ a j δ a k ) 2 ] + e i q x [ ( δ a j δ a k ) 2 ] e i q x a + i q x [ ( δ a j + δ a k ) 2 ] e i q x a i q x [ ( δ a j + δ a k ) 2 ] ) e i q x R j i q x R k .
j = 1 N k = 1 N e i q x [ ( δ a j δ a k ) 2 ] + e i q x [ ( δ a j δ a k ) 2 ] e i q x a + i q x [ ( δ a j + δ a k ) 2 ] e i q x a i q x [ ( δ a j + δ a k ) 2 ] e i q x R j i q x R k .
[ 1 + e 2 q x 2 δ a 2 2 e q x 2 δ a 2 2 ( e i q x a + e i q x a ) ] j = 1 N k = 1 N e i q x R j i q x R k .
e i q x R j i q x R k = e i q x [ ( j k ) D + l = 1 j δ D l l = 1 k δ D l ] = e i q x [ ( j k ) D + l = 1 l k δ D l ] = e i q x [ ( j k ) D j k Δ 2 2 ] .
N + ( N 1 ) ( e i q x D ( q x 2 Δ 2 2 ) + e i q x D ( q x 2 Δ 2 2 ) ) + + ( N m ) ( e i q x m D m ( q x 2 Δ 2 2 ) + e i q x m D m ( q x 2 Δ 2 2 ) ) + + ( N ( N 1 ) ) ( e i q x ( N 1 ) D ( N 1 ) ( q x 2 Δ 2 2 ) + e i q x ( N 1 ) D ( N 1 ) ( q x 2 Δ 2 2 ) ) = N + m = 1 N ( N m ) e m ( q x 2 Δ 2 2 ) ( e i q x m D e i q x m D ) = N ( m = 1 N e m ( q x 2 Δ 2 2 ) e i q x m D + c.c. ) ( m = 1 N m e m ( q x 2 Δ 2 2 ) e i q x m D + c.c. ) = N m = 1 N ( e i m β + + e i m β ) ( i β + ) m = 1 N e i m β + ( i β ) m = 1 N e i m β ,
j = 1 N k = 1 N e i q x R j i q x R k = N ( 1 e i N β + 1 e i β + + 1 e i N β 1 e i β 1 ) ( e i β + ( 1 e i N β + ) N e i N β + ( 1 e i β + ) ( 1 e i β + ) 2 + e i β ( 1 e i N β ) N e i N β ( 1 e i β ) ( 1 e i β ) 2 ) = N + N ( 1 e i β + ) e i β + ( 1 e i N β + ) ( 1 e i β + ) 2 + N ( 1 e i β ) e i β ( 1 e i N β ) ( 1 e i β ) 2 .
j = 1 N k = 1 N e i q x R j i q x R k = N ( 1 e q n 2 Δ 2 ) 1 2 cos q n D e ( q n 2 Δ 2 2 ) + e q n 2 Δ 2 + 4 e q n 2 Δ 2 ( 1 cos q n D cosh q n 2 Δ 2 2 ) ( 1 e ( N q n 2 Δ 2 2 ) cos N q n D ) sin q n D sinh q n 2 Δ 2 2 e ( N q n 2 Δ 2 2 ) sin N q n D ( 1 2 cos q n D e ( q n 2 Δ 2 2 ) + e q n 2 Δ 2 ) 2 .

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