Abstract

For plane-wave light incident upon multilayer optical components, we use first-order perturbation theory to calculate two effects of interface roughness: (1) angle-resolved scattering and (2) guided-mode coupling. The interface roughness, which is assumed to be a random variable having root-mean-square roughness much less than the incident wavelength, is the perturbation parameter. When guided-wave mode resonances are inherent in optical component design, we show that fractions of incident energy scattered directly and coupled into guided-wave modes are comparable. It follows that, in consideration of multilayer optics, energy coupled into guided-wave modes can be an equally important consideration, as this energy may then end up as an additional contribution to scattering and absorption.

© 1995 Optical Society of America

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References

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  1. J. M. Elson, J. P. Rahn, J. M. Bennett, “Relationship of the total integrated scattering from multilayer-coated optics to angle of incidence, polarization, correlation length, and roughness cross-correlation properties,” Appl. Opt. 22, 3207–3219 (1983).
    [CrossRef] [PubMed]
  2. J. M. Elson, L. F. DeSandre, J. L. Stanford, “Analysis of anomalous resonance effects in multilayer-overcoated, low-efficiency gratings,” J. Opt. Soc. Am. A 5, 74–88 (1988).
    [CrossRef]
  3. L. F. DeSandre, J. M. Elson, “Extinction-theorem analysis of diffraction anomalies in overcoated gratings,” J. Opt. Soc. Am. A 8, 763–777 (1991).
    [CrossRef]
  4. J. M. Elson, L. F. DeSandre, P. C. Archibald, J. L. Stanford, “Comparison of theory and experiment of anomalous resonance effects associated with a nine-layer overcoated, low-efficiency grating,” NWCTP 6783 (Naval Air Warfare Center, China Lake, Calif., December1986).
  5. J. M. Elson, “Low efficiency diffraction grating theory,” Tech. Rep. AFWL-TR-75-210 (Air Force Weapons Laboratory, Kirtland Air Force Base, N.M., 1976);“Light scattering from surfaces with a single dielectric overlayer,” J. Opt. Soc. Am. 66, 682–694 (1976);“Infrared light scattering from surfaces covered with multiple dielectric overlayers,” Appl. Opt. 16, 2872–2881 (1977);“Angle resolved light scattering from composite optical surfaces,” in Periodic Structures, Moire Patterns, and Diffraction Phenomena I, C. H. Chi, E. G. Loewen, C. L. O’Bryan, eds., Proc. Soc. Photo-Opt. Instrum. Eng.240, 296–305 (1980).
    [PubMed]
  6. C. Amra, “Light scattering from multilayer optics. I. Tools of investigation,” J. Opt. Soc. Am. 11, 197–210 (1994);“Light scattering from multilayer optics. II. Application to experiment,” J. Opt. Soc. Am. 11, 211–226 (1994).
    [CrossRef]
  7. J. M. Elson, J. P. Rahn, J. M. Bennett, “Light scattering from multilayer optics: comparison of theory and experiment,” Appl. Opt. 19, 669–679 (1980).
    [CrossRef] [PubMed]
  8. E. Kroger, E. Kretschmann, “Scattering of light by slightly rough surfaces of thin films including plasma resonance emission,” Z. Phys. 237, 1–15 (1970).
    [CrossRef]
  9. J. M. Elson, R. H. Ritchie, “Diffuse scattering and surface-plasmon generation by photons at a rough dielectric surface,” Phys. Status Solidi (b) 62, 461–468 (1974).
    [CrossRef]
  10. J. M. Elson, “Diffraction and diffuse scattering from dielectric multilayers,” J. Opt. Soc. Am. 69, 48–54 (1979).
    [CrossRef]

1994 (1)

C. Amra, “Light scattering from multilayer optics. I. Tools of investigation,” J. Opt. Soc. Am. 11, 197–210 (1994);“Light scattering from multilayer optics. II. Application to experiment,” J. Opt. Soc. Am. 11, 211–226 (1994).
[CrossRef]

1991 (1)

1988 (1)

1983 (1)

1980 (1)

1979 (1)

1974 (1)

J. M. Elson, R. H. Ritchie, “Diffuse scattering and surface-plasmon generation by photons at a rough dielectric surface,” Phys. Status Solidi (b) 62, 461–468 (1974).
[CrossRef]

1970 (1)

E. Kroger, E. Kretschmann, “Scattering of light by slightly rough surfaces of thin films including plasma resonance emission,” Z. Phys. 237, 1–15 (1970).
[CrossRef]

Amra, C.

C. Amra, “Light scattering from multilayer optics. I. Tools of investigation,” J. Opt. Soc. Am. 11, 197–210 (1994);“Light scattering from multilayer optics. II. Application to experiment,” J. Opt. Soc. Am. 11, 211–226 (1994).
[CrossRef]

Archibald, P. C.

J. M. Elson, L. F. DeSandre, P. C. Archibald, J. L. Stanford, “Comparison of theory and experiment of anomalous resonance effects associated with a nine-layer overcoated, low-efficiency grating,” NWCTP 6783 (Naval Air Warfare Center, China Lake, Calif., December1986).

Bennett, J. M.

DeSandre, L. F.

L. F. DeSandre, J. M. Elson, “Extinction-theorem analysis of diffraction anomalies in overcoated gratings,” J. Opt. Soc. Am. A 8, 763–777 (1991).
[CrossRef]

J. M. Elson, L. F. DeSandre, J. L. Stanford, “Analysis of anomalous resonance effects in multilayer-overcoated, low-efficiency gratings,” J. Opt. Soc. Am. A 5, 74–88 (1988).
[CrossRef]

J. M. Elson, L. F. DeSandre, P. C. Archibald, J. L. Stanford, “Comparison of theory and experiment of anomalous resonance effects associated with a nine-layer overcoated, low-efficiency grating,” NWCTP 6783 (Naval Air Warfare Center, China Lake, Calif., December1986).

Elson, J. M.

L. F. DeSandre, J. M. Elson, “Extinction-theorem analysis of diffraction anomalies in overcoated gratings,” J. Opt. Soc. Am. A 8, 763–777 (1991).
[CrossRef]

J. M. Elson, L. F. DeSandre, J. L. Stanford, “Analysis of anomalous resonance effects in multilayer-overcoated, low-efficiency gratings,” J. Opt. Soc. Am. A 5, 74–88 (1988).
[CrossRef]

J. M. Elson, J. P. Rahn, J. M. Bennett, “Relationship of the total integrated scattering from multilayer-coated optics to angle of incidence, polarization, correlation length, and roughness cross-correlation properties,” Appl. Opt. 22, 3207–3219 (1983).
[CrossRef] [PubMed]

J. M. Elson, J. P. Rahn, J. M. Bennett, “Light scattering from multilayer optics: comparison of theory and experiment,” Appl. Opt. 19, 669–679 (1980).
[CrossRef] [PubMed]

J. M. Elson, “Diffraction and diffuse scattering from dielectric multilayers,” J. Opt. Soc. Am. 69, 48–54 (1979).
[CrossRef]

J. M. Elson, R. H. Ritchie, “Diffuse scattering and surface-plasmon generation by photons at a rough dielectric surface,” Phys. Status Solidi (b) 62, 461–468 (1974).
[CrossRef]

J. M. Elson, L. F. DeSandre, P. C. Archibald, J. L. Stanford, “Comparison of theory and experiment of anomalous resonance effects associated with a nine-layer overcoated, low-efficiency grating,” NWCTP 6783 (Naval Air Warfare Center, China Lake, Calif., December1986).

J. M. Elson, “Low efficiency diffraction grating theory,” Tech. Rep. AFWL-TR-75-210 (Air Force Weapons Laboratory, Kirtland Air Force Base, N.M., 1976);“Light scattering from surfaces with a single dielectric overlayer,” J. Opt. Soc. Am. 66, 682–694 (1976);“Infrared light scattering from surfaces covered with multiple dielectric overlayers,” Appl. Opt. 16, 2872–2881 (1977);“Angle resolved light scattering from composite optical surfaces,” in Periodic Structures, Moire Patterns, and Diffraction Phenomena I, C. H. Chi, E. G. Loewen, C. L. O’Bryan, eds., Proc. Soc. Photo-Opt. Instrum. Eng.240, 296–305 (1980).
[PubMed]

Kretschmann, E.

E. Kroger, E. Kretschmann, “Scattering of light by slightly rough surfaces of thin films including plasma resonance emission,” Z. Phys. 237, 1–15 (1970).
[CrossRef]

Kroger, E.

E. Kroger, E. Kretschmann, “Scattering of light by slightly rough surfaces of thin films including plasma resonance emission,” Z. Phys. 237, 1–15 (1970).
[CrossRef]

Rahn, J. P.

Ritchie, R. H.

J. M. Elson, R. H. Ritchie, “Diffuse scattering and surface-plasmon generation by photons at a rough dielectric surface,” Phys. Status Solidi (b) 62, 461–468 (1974).
[CrossRef]

Stanford, J. L.

J. M. Elson, L. F. DeSandre, J. L. Stanford, “Analysis of anomalous resonance effects in multilayer-overcoated, low-efficiency gratings,” J. Opt. Soc. Am. A 5, 74–88 (1988).
[CrossRef]

J. M. Elson, L. F. DeSandre, P. C. Archibald, J. L. Stanford, “Comparison of theory and experiment of anomalous resonance effects associated with a nine-layer overcoated, low-efficiency grating,” NWCTP 6783 (Naval Air Warfare Center, China Lake, Calif., December1986).

Appl. Opt. (2)

J. Opt. Soc. Am. (2)

C. Amra, “Light scattering from multilayer optics. I. Tools of investigation,” J. Opt. Soc. Am. 11, 197–210 (1994);“Light scattering from multilayer optics. II. Application to experiment,” J. Opt. Soc. Am. 11, 211–226 (1994).
[CrossRef]

J. M. Elson, “Diffraction and diffuse scattering from dielectric multilayers,” J. Opt. Soc. Am. 69, 48–54 (1979).
[CrossRef]

J. Opt. Soc. Am. A (2)

Phys. Status Solidi (b) (1)

J. M. Elson, R. H. Ritchie, “Diffuse scattering and surface-plasmon generation by photons at a rough dielectric surface,” Phys. Status Solidi (b) 62, 461–468 (1974).
[CrossRef]

Z. Phys. (1)

E. Kroger, E. Kretschmann, “Scattering of light by slightly rough surfaces of thin films including plasma resonance emission,” Z. Phys. 237, 1–15 (1970).
[CrossRef]

Other (2)

J. M. Elson, L. F. DeSandre, P. C. Archibald, J. L. Stanford, “Comparison of theory and experiment of anomalous resonance effects associated with a nine-layer overcoated, low-efficiency grating,” NWCTP 6783 (Naval Air Warfare Center, China Lake, Calif., December1986).

J. M. Elson, “Low efficiency diffraction grating theory,” Tech. Rep. AFWL-TR-75-210 (Air Force Weapons Laboratory, Kirtland Air Force Base, N.M., 1976);“Light scattering from surfaces with a single dielectric overlayer,” J. Opt. Soc. Am. 66, 682–694 (1976);“Infrared light scattering from surfaces covered with multiple dielectric overlayers,” Appl. Opt. 16, 2872–2881 (1977);“Angle resolved light scattering from composite optical surfaces,” in Periodic Structures, Moire Patterns, and Diffraction Phenomena I, C. H. Chi, E. G. Loewen, C. L. O’Bryan, eds., Proc. Soc. Photo-Opt. Instrum. Eng.240, 296–305 (1980).
[PubMed]

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Figures (4)

Fig. 1
Fig. 1

Schematic showing multilayer system nomenclature consisting of a substrate, L layers, and a superstrate The superstrate contains the incident beam and the reflected scattered field. The substrate contains the transmitted scattered field, if any, and there may also be a transmitted specular beam. Although this is not illustrated here, the multilayer system may support guided-wave modes. The angles θ0 and θ are the polar angles of incidence and scattering, respectively. The incident beam is assumed to be in the (x, y) plane. The scattered field has direction (θ, ϕ); the azimuth angle ϕ is not shown here. The layers of the multilayer system have complex dielectric constants j, where j = 1 is the substrate, j = 2 → L + 1 are the layers of the stack with mean physical thicknesses τj, and j = L + 2 is the superstrate. The mean surface level of interface j is z = dj, and Δzj(x, y) is a random function that describes the roughness.

Fig. 2
Fig. 2

(a) Log(ARS) versus polar scattering angle θ for case 1 (glass–thin metal film–vacuum) as in Eqs. (5.9). The incident angle is 35 deg (which is less than the glass–air total internal reflection angle), and the scattering is in the plane of incidence. Solid curve, reflection ARS, correlated roughness, and pp; dotted curve, reflection ARS, uncorrelated roughness, and pp; dashed curve, transmission ARS, correlated roughness, and ss; dashed–dotted curve, transmission ARS, uncorrelated roughness, and ss. (b) Log(ARS) versus polar scattering angle θ for case 1 (glass–thin metal film–vacuum). The only difference between the calculation of these results and those in (a) is that the angle of incidence is now 45 deg, which is greater than the glass–air total internal reflection angle. (c) GWC versus azimuthal angle ϕ for case 1. This plot is in polar coordinates, with the right-hand horizontal axis being ϕ = 0 deg. The dashed curve is for uncorrelated roughness, λ = 0.633 μm, and angle of incidence θ0 = 35 deg. (d) GWC versus azimuthal angle ϕ for case 1. This plot is identical to (c) that in except that θ0 = 45 deg.

Fig. 3
Fig. 3

(a) Log(ARS) versus polar scattering angle θ for the 30-layer stack as described for case 2 [air–(HL)15–metal]. The incident angle is 65 deg, and the scattering is in the plane of incidence. Solid curve, reflection ARS, correlated roughness, and pp; dotted curve, reflection ARS, uncorrelated roughness, and pp; dashed curve, reflection ARS, correlated roughness, and ss; dashed–dotted curve, reflection ARS, uncorrelated roughness, and ss. (b) GWC versus azimuthal angle ϕ for case 2. This plot is in polar coordinates, with the right-hand horizontal axis being ϕ = 0 deg. The solid and dashed curves are for correlated and uncorrelated roughness, respectively. λ = 0.633 μm, and θ0 = 65 deg.

Fig. 4
Fig. 4

(a) Log(ARS) versus polar scattering angle θ for the 30-layer stack as described for case 3 [air–(HL)15–dielectric]. The incident angle is 65 deg, and the scattering is reflective ARS in the plane of incidence. Solid curve, reflection ARS, correlated roughness, and pp; dotted curve, reflection ARS, uncorrelated roughness, and pp; dashed curve, reflection ARS, correlated roughness, and ss; dashed–dotted curve, reflection ARS, uncorrelated roughness, and ss. (b) Calculation of log(ARS) versus polar scattering angle θ for the 30-layer stack as described for case 3 [air–(HL)15–dielectric]. These data are analogous to those in (a), except for transmission ARS. Solid curve, transmission ARS, correlated roughness, and pp; dotted curve, transmission ARS, uncorrelated roughness, and pp; dashed curve, transmission ARS, correlated roughness, and ss; dashed–dotted curve, transmission ARS, uncorrelated roughness, and ss. (c) GWC versus azimuthal angle ϕ for case 3. This plot is in polar coordinates, with the right-hand horizontal axis being ϕ = 0 deg. The dashed curve is for uncorrelated roughness, λ = 0.633 μm, and θ0 = 65 deg.

Equations (159)

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n ̂ j = x ̂ [ Δ z j ( ρ ) x ] ŷ [ Δ z j ( ρ ) y ] .
E j [ ρ , d j + Δ z ( ρ ) ] = E j ( 0 ) ( ρ , d j ) + Δ z j ( ρ ) × z E j ( 0 ) ( ρ , z ) z = d j + E j ( 1 ) ( ρ , d j ) .
n j × Δ E j + 1 , j [ ρ , d j + Δ z j ( ρ ) ] = 0 ,
Δ E j + 1 , j [ ρ , d j + Δ z j ( ρ ) ] = E j + 1 , j [ ρ , d j + Δ z j ( ρ ) ] E j [ ρ , d j + Δ z j ( ρ ) ]
Δ E j + 1 , j , x ( 1 ) [ ρ , Δ z j ( ρ ) ] = [ x Δ z j ( ρ ) ] Δ E j + 1 , j , z ( 0 ) ( ρ , d j ) Δ z j ( ρ ) Δ [ z E j + 1 , j , x ( 0 ) ( ρ , z ) ] z = d j ,
Δ E j + 1 , j , y ( 1 ) [ ρ , Δ z j ( ρ ) ] = [ y Δ z j ( ρ ) ] Δ E j + 1 , j , z ( 0 ) ( ρ , d j ) Δ z j ( ρ ) Δ [ z E j + 1 , j , y ( 0 ) ( ρ , z ) ] z = d j .
E j ( 0 ) ( ρ , z ) = e j ( 0 ) ( k 0 , z ) exp ( i k 0 x ) ,
E j ( 1 ) ( ρ , z ) = 1 4 π 2 d 2 k e j ( 1 ) ( k , z ) exp ( i k ρ ) ,
( 2 π ) 2 Δ e j + 1 , j , x ( 1 ) ( k , d j ) = i k x Δ e j + 1 , j , z ( 0 ) ( k 0 , d j ) Δ Z j ( k 0 k ) ,
( 2 π ) 2 Δ e j + 1 , j , y ( 1 ) ( k , d j ) = i k y Δ e j + 1 , j , z ( 0 ) ( k 0 , d j ) Δ Z j ( k 0 k ) ,
Δ Z j ( k 0 k ) = d 2 ρ Δ z j ( ρ ) exp [ i ( k 0 k ) ρ ]
( 2 π ) 2 Δ h j + 1 , j , x ( 1 ) ( k , d j ) = i ( ω / c ) Δ d j + 1 , j , y ( 0 ) ( k 0 , d j ) × Δ Z j ( k 0 k ) ,
( 2 π ) 2 Δ h j + 1 , j , y ( 1 ) ( k , d j ) = i ( ω / c ) Δ d j + 1 , j , x ( 0 ) ( k 0 , d j ) × Δ Z j ( k 0 k ) ,
Δ d j + 1 , j ( 0 ) ( k 0 , d j ) = j + 1 e j + 1 ( 0 ) ( k 0 , d j ) j e j ( 0 ) ( k 0 , d j ) .
[ 2 + ( ω / c ) 2 j ] E j ( 0 ) ( ρ , z ) = 0 ,
E j ( 0 ) ( ρ , z ) = E j , p ( 0 ) ( ρ , z ) + E j , s ( 0 ) ( ρ , z ) .
E j , p ( 0 ) ( ρ , z ) = [ x ̂ e j , x ( 0 ) ( z ) + e j , z ( 0 ) ( z ) ] exp ( i k 0 x ) ,
E j , s ( 0 ) ( ρ , z ) = ŷ e j , y ( 0 ) ( z ) exp ( i k 0 x ) exp ( i k 0 x ) ,
e j , x ( 0 ) ( z ) = q j ( 0 ) cos ψ ( ω / c ) j [ a j ( 0 ) exp ( i q j ( 0 ) z ) + b j ( 0 ) exp ( i q j ( 0 ) z ) ] ,
e j , z ( 0 ) ( z ) = k 0 cos ψ ( ω / c ) j [ a j ( 0 ) exp ( i q j ( 0 ) z ) b j ( 0 ) exp ( i q j ( 0 ) z ) ] ,
e j , y ( 0 ) ( z ) = ŷ sin ψ [ g j ( 0 ) exp ( i q j ( 0 ) z ) + f j ( 0 ) exp ( i q j ( 0 ) z ) ] ,
q j ( 0 ) = [ ( ω / c ) 2 j k 0 2 ] 1 / 2 .
a j ( 0 ) = q L + 2 ( 0 ) P 0 , 12 ( 1 , j 1 ) q j ( 0 ) P 0 , 11 ( 1 , L + 1 ) ,
b j ( 0 ) = q L + 2 ( 0 ) P 0 , 11 ( 1 , j 1 ) q j ( 0 ) P 0 , 11 ( 1 , L + 1 ) ,
g j ( 0 ) = q L + 2 ( 0 ) S 0 , 12 ( 1 , j 1 ) q j ( 0 ) S 0 , 11 ( 1 , L + 1 ) ,
f j ( 0 ) = q L + 2 ( 0 ) S 0 , 11 ( 1 , j 1 ) q j ( 0 ) S 0 , 11 ( 1 , L + 1 ) .
P 0 ( 1 , j 1 ) = [ P 0 , 11 ( 1 , j 1 ) P 0 , 12 ( 1 , j 1 ) P 0 , 21 ( 1 , j 1 ) P 0 , 22 ( 1 , j 1 ) ] = p 0 ( 1 ) p 0 ( 2 ) p 0 ( j 1 ) ,
P 0 ( 1 , 0 ) = [ P 0 , 11 ( 1 , 0 ) P 0 , 12 ( 1 , 0 ) P 0 , 21 ( 1 , 0 ) P 0 , 22 ( 1 , 0 ) ] = [ 1 0 0 1 ] .
p 0 ( m ) = [ p 0 , 11 ( m ) p 0 , 12 ( m ) p 0 , 21 ( m ) p 0 , 22 ( m ) ] ,
p 0 , 11 ( m ) = [ m q m + 1 ( 0 ) + m + 1 q m ( 0 ) 2 q m ( 0 ) m m + 1 ] exp [ i ( q m + 1 ( 0 ) q m ( 0 ) ) d m ] ,
p 0 , 12 ( m ) = [ m q m + 1 ( 0 ) m + 1 q m ( 0 ) 2 q m ( 0 ) m m + 1 ] exp [ i ( q m + 1 ( 0 ) + q m ( 0 ) ) d m ] ,
p 0 , 21 ( m ) = [ m q m + 1 ( 0 ) m + 1 q m ( 0 ) 2 q m ( 0 ) m m + 1 ] exp [ i ( q m + 1 ( 0 ) + q m ( 0 ) ) d m ] ,
p 0 , 22 ( m ) = [ m q m + 1 ( 0 ) + m + 1 q m ( 0 ) 2 q m ( 0 ) m m + 1 ] exp [ i ( q m + 1 ( 0 ) q m ( 0 ) ) d m ] .
S 0 ( 1 , j 1 ) = [ S 0 , 11 ( 1 , j 1 ) S 0 , 21 ( 1 , j 1 ) S 0 , 21 ( 1 , j 1 ) S 0 , 22 ( 1 , j 1 ) ] = s 0 ( 1 ) s 0 ( 2 ) s 0 ( j 1 ) ,
S 0 ( 1 , 0 ) = [ S 0 , 11 ( 1 , 0 ) S 0 , 12 ( 1 , 0 ) S 0 , 21 ( 1 , 0 ) S 0 , 22 ( 1 , 0 ) ] = [ 1 0 0 1 ] .
s 0 ( m ) = [ s 0 , 11 ( m ) s 0 , 12 ( m ) s 0 , 21 ( m ) s 0 , 22 ( m ) ] ,
s 0 , 11 ( m ) = ( q m ( 0 ) + q m + 1 ( 0 ) 2 q m ( 0 ) ) exp [ i ( q m + 1 ( 0 ) q m ( 0 ) ) d m ] ,
s 0 , 12 ( m ) = ( q m ( 0 ) q m + 1 ( 0 ) 2 q m ( 0 ) ) exp [ i ( q m + 1 ( 0 ) + q m ( 0 ) ) d m ] ,
s 0 , 21 ( m ) = ( q m ( 0 ) q m + 1 ( 0 ) 2 q m ( 0 ) ) exp [ i ( q m + 1 ( 0 ) + q m ( 0 ) ) d m ] ,
s 0 , 22 ( m ) = ( q m ( 0 ) + q m + 1 ( 0 ) 2 q m ( 0 ) ) exp [ i ( q m + 1 ( 0 ) q m ( 0 ) ) d m ] .
P 0 = c 8 π A d 2 ρ ( E L + 2 ( 0 ) × H L + 2 ( 0 ) ) = A c 2 q L + 2 ( 0 ) 8 π ω ,
E j ( 1 ) ( ρ , z ) = 1 4 π 2 0 k d k 0 2 π d ϕ × [ e j , p ( 1 ) ( z ) + e j , s ( 1 ) ( z ) ] exp ( i k ρ ) ,
H j ( 1 ) ( ρ , z ) = × E j ( 1 ) ( ρ , z ) i ( ω / c ) ,
e j , p ( 1 ) ( z ) = a j ( 1 ) ( k q j ( 1 ) k ) ( ω / c ) j exp ( i q j ( 1 ) z ) + b j ( 1 ) ( k ̂ q j ( 1 ) k ) ( ω / c ) j × exp ( i q j ( 1 ) z ) ,
e j , s ( 1 ) ( z ) = ( k ̂ × ) [ g j ( 1 ) exp ( i q j ( 1 ) z ) + f j ( 1 ) exp ( i q j ( 1 ) z ) ] ,
a j ( 1 ) = P 1 , 11 ( j , L + 1 ) a L + 2 ( 1 ) i ( ω / c ) 2 n = 1 L + 2 j Δ Z j + n 1 ( k 0 k ) q j + n 1 ( 1 ) j + n 1 × ( k j + n 1 Y 1 ( j + n 1 ) η 1 ( j , j + n 2 ) q n + j 1 ( 1 ) Y 2 ( j + n 1 ) η 2 ( j , j + n 2 ) ) ,
b j ( 1 ) = P 1 , 21 ( j , L + 1 ) a L + 2 ( 1 ) i ( ω / c ) 2 n = 1 L + 2 j Δ Z j + n 1 ( k 0 k ) q j + n 1 ( 1 ) j + n 1 × ( k j + n 1 Y 1 ( j + n 1 ) η 3 ( j , j + n 2 ) q n + j 1 ( 1 ) Y 2 ( j + n 1 ) η 4 ( j , j + n 2 ) ) ,
g j ( 1 ) = S 1 , 11 ( j , L + 1 ) g L + 2 ( 1 ) i ( ω / c ) 2 2 n = 1 L + 2 j Δ Z j + n 1 ( k 0 k ) q j + n 1 ( 1 ) × X ( j + n 1 ) ξ 1 ( j , j + n 2 ) ,
f j ( 1 ) = S 1 , 21 ( j , L + 1 ) g L + 2 ( 1 ) + i ( ω / c ) 2 2 n = 1 L + 2 j Δ Z j + n 1 ( k 0 k ) q j + n 1 ( 1 ) × X ( j + n 1 ) ξ 2 ( j , j + n 2 ) .
a L + 2 ( 1 ) = i ( ω / c ) 2 P 1 , 11 ( 1 , L + 1 ) j = 1 L + 1 μ p ( j ) Δ Z j ( k 0 k ) ,
g L + 2 ( 1 ) = i ( ω / c ) 2 S 1 , 11 ( 1 , L + 1 ) j = 1 L + 1 μ s ( j ) Δ Z j ( k 0 k ) ,
μ p ( j ) = k j Y 1 ( j ) η 1 ( 1 , j 1 ) q j ( 1 ) Y 2 ( j ) η 2 ( 1 , j 1 ) q j ( 1 ) j ,
μ s ( j ) = ( ω / c ) X ( j ) ξ 1 ( 1 , j 1 ) q j ( 1 ) .
b 1 ( 1 ) = P 1 , 21 ( 1 , L + 1 ) α L + 2 ( 1 ) i ( ω / c ) 2 j = 1 L + 1 Δ Z j ( k 0 k ) q j ( 1 ) j × ( k j Y 1 ( j ) η 4 ( 1 , j 1 ) q j ( 1 ) Y 2 ( j ) η 3 ( 1 , j 1 ) ) = i ( ω / c ) 2 P 1 , 11 ( 1 , L + 1 ) j = 1 L + 1 α p ( j ) Δ Z j ( k 0 k ) ,
f 1 ( 1 ) = S 1 , 21 ( 1 , L + 1 ) g L + 2 ( 1 ) + i ( ω / c ) 2 j = 1 L + 1 Δ Z j ( k 0 k ) q j ( 1 ) × X ( j ) ξ 2 ( 1 , j 1 ) = i ( ω / c ) 2 S 1 , 11 ( 1 ) j = 1 L + 1 α s ( j ) Δ Z j ( k 0 k ) ,
α p ( j ) = k j Y 1 ( j ) q j ( 1 ) ( η 1 ( 1 , j 1 ) P 1 , 21 ( 1 , L + 1 ) η 4 ( 1 , j 1 ) P 1 , 11 ( 1 , L + 1 ) ) Y 2 ( j ) j ( η 2 ( 1 , j 1 ) P 1 , 21 ( 1 , L + 1 ) η 3 ( 1 , j 1 ) P 1 , 11 ( 1 , L + 1 ) ) ,
α s ( j ) = ( ω / c ) X ( j ) q j ( 1 ) ( ξ 2 ( 1 , j 1 ) S 1 , 11 ( 1 , L + 1 ) ξ 1 ( 1 , j 1 ) S 1 , 21 ( 1 , L + 1 ) ) .
X ( j ) = ( j + 1 j ) [ e j , x ( 0 ) ( d j ) sin ϕ e j , y ( 0 ) ( d j ) cos ϕ ] ,
Y 1 ( j ) = ( j + 1 j j + 1 ) e j , z ( 0 ) ( d j ) ,
Y 2 ( j ) = ( j + 1 j ) [ e j , x ( 0 ) ( d j ) cos ϕ + e j , y ( 0 ) ( d j ) sin ϕ ] .
ξ 1 ( 1 , j 1 ) = S 1 , 11 ( 1 , j 1 ) exp ( i q j ( 1 ) d j ) S 1 , 12 ( 1 , j 1 ) exp ( i q j ( 1 ) d j ) ,
ξ 2 ( 1 , j 1 ) = S 1 , 21 ( 1 , j 1 ) exp ( i q j ( 1 ) d j ) S 1 , 22 ( 1 , j 1 ) exp ( i q j ( 1 ) d j ) ,
η 1 ( 1 , j 1 ) = P 1 , 11 ( 1 , j 1 ) exp ( i q j ( 1 ) d j ) + P 1 , 12 ( 1 , j 1 ) exp ( i q j ( 1 ) d j ) ,
η 2 ( 1 , j 1 ) = P 1 , 11 ( 1 , j 1 ) exp ( i q j ( 1 ) d j ) P 1 , 12 ( 1 , j 1 ) exp ( i q j ( 1 ) d j ) ,
η 3 ( 1 , j 1 ) = P 1 , 21 ( 1 , j 1 ) exp ( i q j ( 1 ) d j ) + P 1 , 22 ( 1 , j 1 ) exp ( i q j ( 1 ) d j ) ,
η 4 ( 1 , j 1 ) = P 1 , 21 ( 1 , j 1 ) exp ( i q j ( 1 ) d j ) P 1 , 22 ( 1 , j 1 ) exp ( i q j ( 1 ) d j ) .
P 1 ( m , n ) = [ P 1 , 11 ( m , n ) P 1 , 12 ( m , n ) P 1 , 21 ( m , n ) P 1 , 22 ( m , n ) ] = p 1 ( m ) p 1 ( m + 1 ) p 1 ( n 1 ) p 1 ( n ) ,
P 1 ( 1 , 0 ) = [ P 1 , 11 ( 1 , 0 ) P 1 , 12 ( 1 , 0 ) P 1 , 21 ( 1 , 0 ) P 1 , 22 ( 1 , 0 ) ] = [ 1 0 0 1 ] , P 1 ( L + 2 , L + 1 ) = [ P 1 , 11 ( L + 2 , L + 1 ) P 1 , 12 ( L + 2 , L + 1 ) P 1 , 21 ( L + 2 , L + 1 ) P 1 , 22 ( L + 2 , L + 1 ) ] = [ 1 0 0 1 ] .
p 1 , 11 ( j ) = ( j q j + 1 ( 1 ) + j + 1 q j ( 1 ) 2 q j ( 1 ) j j + 1 ) exp [ i ( q j + 1 ( 1 ) q j ( 1 ) ) d j ] ,
p 1 , 12 ( j ) = ( j q j + 1 ( 1 ) j + 1 q j ( 1 ) 2 q j ( 1 ) j j + 1 ) exp [ i ( q j + 1 ( 1 ) + q j ( 1 ) ) d j ] ,
p 1 , 21 ( j ) = ( j q j + 1 ( 1 ) j + 1 q j ( 1 ) 2 q j ( 1 ) j j + 1 ) exp [ i ( q j + 1 ( 1 ) + q j ( 1 ) ) d j ] ,
p 1 , 22 ( j ) = ( j q j + 1 ( 1 ) + j + 1 q j ( 1 ) 2 q j ( 1 ) j j + 1 ) exp [ i ( q j + 1 ( 1 ) q j ( 1 ) ) d j ] ,
S 1 ( m , n ) = [ S 1 , 11 ( m , n ) S 1 , 12 ( m , n ) S 1 , 21 ( m , n ) S 1 , 22 ( m , n ) ] = s 1 ( m ) s 1 ( m + 1 ) s 1 ( n 1 ) s 1 ( n ) ,
S 1 ( 1 , 0 ) = [ S 1 , 11 ( 1 , 0 ) S 1 , 12 ( 1 , 0 ) S 1 , 21 ( 1 , 0 ) S 1 , 22 ( 1 , 0 ) ] = [ 1 0 0 1 ] ,
S 1 ( L + 2 , L + 1 ) = [ S 1 , 11 ( L + 2 , L + 1 ) S 1 , 12 ( L + 2 , L + 1 ) S 1 , 21 ( L + 2 , L + 1 ) S 1 , 22 ( L + 2 , L + 1 ) ] = [ 1 0 0 1 ] .
s 1 , 11 ( j ) = ( q j ( 1 ) + q j + 1 ( 1 ) 2 q j ( 1 ) ) exp [ i ( q j + 1 ( 1 ) q j ( 1 ) ) d j ] ,
s 1 , 12 ( j ) = ( q j ( 1 ) q j + 1 ( 1 ) 2 q j ( 1 ) ) exp [ i ( q j + 1 ( 1 ) + q j ( 1 ) ) d j ] ,
s 1 , 21 ( j ) = ( q j ( 1 ) q j + 1 ( 1 ) 2 q j ( 1 ) ) exp [ i ( q j + 1 ( 1 ) + q j ( 1 ) ) d j ] ,
s 1 , 22 ( j ) = ( q j ( 1 ) + q j + 1 ( 1 ) 2 q j ( 1 ) ) exp [ i ( q j + 1 ( 1 ) q j ( 1 ) ) d j ] .
ARS = BRDF cos θ = 1 P 0 d P d Ω .
P j = c 8 π A d 2 ρ ( E j ( 1 ) × H j ( 1 ) * ) .
P L + 2 = c 2 32 π 3 ω d 2 k q L + 2 ( 1 ) ( | a L + 2 ( 1 ) | 2 + | g L + 2 ( 1 ) | 2 ) × exp [ 2 z S ( q L + 2 ( 1 ) ) ] ,
P 1 = c 2 32 π 3 ω d 2 k q 1 ( 1 ) ( | b 1 ( 1 ) | 2 + | f 1 ( 1 ) | 2 ) × exp [ 2 z S ( q 1 ( 1 ) ) ] .
k = ( k x , k y ) = ( ω / c ) L + 2 sin θ ( cos ϕ , sin ϕ ) ,
d 2 k = ( ω / c ) 2 L + 1 cos θ sin θ d θ d ϕ = ( ω / c ) 2 L + 2 cos θ d Ω ,
( k x , k y ) = ( ω / c ) 1 sin θ ( cos ϕ , sin ϕ ) ,
d 2 k = ( ω / c ) 2 1 cos θ sin θ d θ d ϕ = ( ω / c ) 2 1 cos θ d Ω .
1 p 0 d P L + 2 d Ω = ( ω / c ) 2 q L + 2 ( 1 ) L + 2 cos θ ( 2 π ) 2 q L + 2 ( 0 ) × { | a L + 2 ( 1 ) | 2 + | g L + 2 ( 1 ) | 2 } A ,
1 p 0 d P 1 d Ω = ( ω / c ) 2 q 1 ( 1 ) 1 cos θ ( 2 π ) 2 q L + 2 ( 0 ) { | b 1 ( 1 ) | 2 + | f 1 ( 1 ) | 2 } A .
| a L + 2 ( 1 ) | 2 = ( ω / c ) 2 4 | P 1 , 11 ( 1 , L + 1 ) | 2 | A p ( k 0 k ) | 2 ,
| g L + 2 ( 1 ) | 2 = ( ω / c ) 2 4 | S 1 , 11 ( 1 , L + 1 ) | 2 | A s ( k 0 k ) | 2 ,
| b 1 ( 1 ) | 2 = ( ω / c ) 2 4 | P 1 , 11 ( 1 , L + 1 ) | 2 | B p ( k 0 k ) | 2 ,
| f 1 ( 1 ) | 2 = ( ω / c ) 2 4 | S 1 , 11 ( 1 , L + 1 ) | 2 | B s ( k 0 k ) | 2 ,
A p ( k 0 k ) = n = 1 L + 1 μ p ( n ) Δ Z n ( k 0 k ) ,
A s ( k 0 k ) = n = 1 L + 1 μ s ( n ) Δ Z n ( k 0 k ) ,
B p ( k 0 k ) = n = 1 L + 1 α p ( n ) Δ Z n ( k 0 k ) ,
B s ( k 0 k ) = n = 1 L + 1 α s ( n ) Δ Z n ( k 0 k ) .
1 P 0 d P L + 2 d Ω = ( ω / c ) 4 q L + 2 ( 1 ) L + 2 cos θ 16 π 2 q L + 2 ( 0 ) A × { | A p ( k 0 k ) | 2 | P 1 , 11 ( 1 , L + 1 ) | 2 + | A s ( k 0 k ) | 2 | S 1 , 11 ( 1 , L + 1 ) | 2 }
1 P 0 d P 1 d Ω = ( ω / c ) 4 q 1 ( 1 ) 1 cos θ 16 π 2 q L + 2 ( 0 ) A × { | B p ( k 0 k ) | 2 | P 1 , 11 ( 1 , L + 1 ) | 2 + | B s ( k 0 k ) | 2 | S 1 , 11 ( 1 , L + 1 ) | 2 }
W = ½ V E ( 1 ) J ( 1 ) * d 2 ρ d z ,
J ( 1 ) ( ρ , z ) = ( 1 ) 4 π E ( 1 ) ( ρ , z ) t = i ω ( 1 ) 4 π E ( 1 ) ( ρ , z ) ,
W = ω 8 π lim δ [ δ V | E ( 1 ) ( ρ , z ) | 2 d 2 ρ d z ] .
W P 0 = ( ω / c ) 2 A q ( L + 2 ) ( 0 ) lim δ 0 [ δ j = 1 L + 2 d 2 ρ d j 1 d j d z | E j ( 1 ) ( ρ , z ) | 2 ] ,
ã j ( 1 ) = P 1 , 11 ( j , L + 1 ) a L + 2 = i ω 2 c P 1 , 11 ( j , L + 1 ) P 1 , 11 ( 1 , L + 1 ) A p ( k 0 k ) ,
b j ( 1 ) = P 1 , 21 ( j , L + 1 ) a L + 2 = i ω 2 c P 1 , 21 ( j , L + 1 ) P 1 , 11 ( 1 , L + 1 ) A p ( k 0 k ) ,
g j ( 1 ) = S 1 , 11 ( j , L + 1 ) g L + 2 = i ω 2 c S 1 , 11 ( j , L + 1 ) S 1 , 11 ( 1 , L + 1 ) A s ( k 0 k ) ,
f j ( 1 ) = S 1 , 21 ( j , L + 1 ) g L + 2 = i ω 2 c S 1 , 21 ( j , L + 1 ) S 1 , 11 ( 1 , L + 1 ) A s ( k 0 k ) .
W p P 0 = 1 16 π 2 ( ω / c ) 4 q L + 2 ( 0 ) lim δ 0 δ j = 1 L + 2 d 2 k × F p ( j ) ( k ) | P 1 , 11 ( 1 , L + 1 ) | 2 | A p ( k 0 k ) | 2 A ,
W s P 0 = 1 16 π 2 ( ω / c ) 4 q L + 2 ( 0 ) lim δ 0 δ j = 1 L + 2 d 2 k × F s ( j ) ( k ) | S 1 , 11 ( 1 , L + 1 ) | 2 | A s ( k 0 k ) | 2 A ,
I = lim δ 0 δ | P 1 , 11 ( 1 , L + 1 ) | 2 f ( k ) d k ,
P 1 , 11 ( 1 , L + 1 ) ( k m ( p ) ) = 0 .
P 1 ( 1 , L + 1 ) ( k ) K p ( m ) ( k k m ( p ) ) + i E p ( m ) δ ,
K p ( m ) = ( P 1 ( 1 , L + 1 ) k ) , E p ( m ) = ( P 1 ( 1 , L + 1 ) ) .
P 1 ( 1 , L + 1 ) k = j = 1 L + 2 P 1 ( 1 , j 1 ) ( P 1 ( j ) k ) P 1 ( j + 1 , L + 1 ) ,
P 1 ( 1 , L + 1 ) = j = 1 L + 2 P 1 ( 1 , j 1 ) ( p 1 ( j ) j + p 1 ( j ) j + 1 ) P 1 ( j + 1 , L + 1 ) ,
I = lim δ 0 δ | K p ( m ) ( k k m ( p ) ) + i E p ( m ) δ | 2 f ( k ) d k = π f ( k m ( p ) ) | ( K p ( m ) E p * ( m ) ) | ,
lim δ 0 δ | K p ( m ) ( k k m ( p ) ) + i E p ( m ) δ | 2 = π δ ( k k m ( p ) ) | ( K p ( m ) E p * ( m ) ) | .
1 P 0 d W p d ϕ = 1 16 π ( ω / c ) 4 q L + 2 ( 0 ) j = 1 L + 2 m = 1 M p k m ( p ) F p ( j ) ( k m ( p ) ) | ( K p ( m ) E p * ( m ) ) | × | A P ( k 0 k m ( p ) ) | 2 A ,
1 P 0 d W s d ϕ = 1 16 π ( ω / c ) 4 q L + 2 ( 0 ) j = 1 L + 2 m = 1 M s k m ( s ) F s ( j ) ( k m ( s ) ) | ( K s ( m ) E s * ( m ) ) | × | A s ( k 0 k m ( s ) ) | 2 A ,
1 P 0 d W d ϕ = 1 P 0 d W p d ϕ + 1 P 0 d W s d ϕ .
T ( k 0 k ) = j = 1 L + 1 t j Δ Z j ( k 0 k ) .
| T ( k 0 k ) | 2 A = 1 A | j = 1 L + 1 t j Δ Z j ( k 0 k ) | 2 = 1 A j = 1 L + 1 j = 1 L + 1 t j t j * × Δ Z j ( k 0 k ) Δ Z j * ( k 0 k ) .
g j j ( k 0 k ) = Δ Z j ( k 0 k ) Δ Z j * ( k 0 k ) A ,
g j j ( k 0 k ) = d 2 τ { lim A 1 A d 2 ρ Δ z j ( ρ + τ ) Δ z j ( ρ ) } × exp [ i ( k 0 k ) τ ] = d 2 τ G j j ( τ ) exp [ i ( k 0 k ) τ ] ,
G j j ( τ ) = Δ z j ( ρ + τ ) Δ z j ( ρ ) = lim A [ 1 A d 2 ρ Δ z j ( ρ + τ ) Δ z j ( ρ ) ]
G j j ( τ ) = G ( τ ) correlated ,
G j j ( τ ) = δ j j G ( τ ) uncorrelated ,
G ( τ ) = δ L 2 exp ( | τ | / σ L ) + δ S 2 exp [ ( τ / σ S ) 2 ] ,
g ( k 0 k ) = 2 π δ L 2 σ L 2 ( 1 + | k 0 k | 2 σ L 2 ) 3 / 2 + π δ S 2 σ S 2 exp ( | k 0 k | 2 σ S 2 / 4 ) ,
| T ( k 0 k ) | 2 = g ( k 0 k ) | j = 1 L + 1 t j | 2 correlated model
| T ( k 0 k ) | 2 = g ( k 0 k ) j = 1 L + 1 | t j | 2 uncorrelated model .
| A p ( k 0 k ) | = g ( k 0 k ) { | n = 1 L + 1 μ p ( n ) | 2 correlated n = 1 L + 1 | μ p ( n ) | 2 uncorrelated ,
A s ( k 0 k ) = g ( k 0 k ) { | n = 1 L + 1 μ s ( n ) | 2 correlated n = 1 L + 1 | μ s ( n ) | 2 uncorrelated ,
B p ( k 0 k ) = g ( k 0 k ) { | n = 1 L + 1 α p ( n ) | 2 correlated n = 1 L + 1 | α p ( n ) | 2 uncorrelated ,
B s ( k 0 k ) = g ( k 0 k ) { | n = 1 L + 1 α s ( n ) | 2 correlated n = 1 L + 1 | α s ( n ) | 2 uncorrelated .
p j , 11 ( 1 ) k = k ( j + 1 j ) 2 q j 3 q j + 1 j j + 1 [ ( ω / c ) 2 j + i q j d j ( k 2 + q j q j + 1 ) ] × exp [ i ( q j + 1 q j ) d j ] ,
p j , 12 ( 1 ) k = k ( j + 1 j ) 2 q j 3 q j + 1 j j + 1 [ ( ω / c ) 2 j + i q j d j ( k 2 q j q j + 1 ) ] × exp [ i ( q j + 1 + q j ) d j ] ,
p j , 21 ( 1 ) k = k ( j + 1 j ) 2 q j 3 q j + 1 j j + 1 [ ( ω / c ) 2 j i q j d j ( k 2 q j q j + 1 ) ] × exp [ i ( q j + 1 + q j ) d j ] ,
p j , 22 ( 1 ) k = k ( j + 1 j ) 2 q j 3 q j + 1 j j + 1 [ ( ω / c ) 2 j i q j d j ( k 2 + q j q j + 1 ) ] × exp [ i ( q j + 1 q j ) d j ] ;
p j , 11 ( 1 ) j + p j , 11 ( 1 ) j + 1 = ( j + 1 j ) 4 q j 3 q j + 1 j j + 1 × [ ( ω / c ) 2 ( k 2 + q j q j + 1 ) ( 1 + i q j d j ) k 2 q j ( q j + 1 j q j j + 1 ) ] exp [ i ( q j + 1 q j ) d j ] ,
p j , 12 ( 1 ) j + p j , 12 ( 1 ) j + 1 = ( j + 1 j ) 4 q j 3 q j + 1 j j + 1 × [ ( ω / c ) 2 ( k 2 q j q j + 1 ) ( 1 + i q j d j ) + k 2 q j ( q j + 1 j + q j j + 1 ) ] exp [ i ( q j + 1 + q j ) d j ] ,
p j , 21 ( 1 ) j + p j , 21 ( 1 ) j + 1 = ( j + 1 j ) 4 q j 3 q j + 1 j j + 1 × [ ( ω / c ) 2 ( k 2 q j q j + 1 ) ( 1 i q j d j ) + k 2 q j ( q j + 1 j + q j j + 1 ) ] exp [ i ( q j + 1 + q j ) d j ] ,
p j , 22 ( 1 ) j + p j , 22 ( 1 ) j + 1 = ( j + 1 j ) 4 q j 3 q j + 1 j j + 1 × [ ( ω / c ) 2 ( k 2 + q j q j + 1 ) ( 1 i q j d j ) k 2 q j ( q j + 1 j q j j + 1 ) ] exp [ i ( q j + 1 q j ) d j ] .
K p ( m ) = P 1 ( 1 , L + 1 ) k = j = 1 L + 2 P 1 ( 1 , j 1 ) ( p 1 ( j ) k ) P 1 ( j + 1 , L + 1 ) ,
E p ( m ) = P 1 ( 1 , L + 1 ) = j = 1 L + 2 P 1 ( 1 , j 1 ) ( p 1 ( j ) j + p 1 ( j ) j + 1 ) P 1 ( j + 1 , L + 1 ) ,
S j , 11 ( 1 ) k = k ( ω / c ) 2 ( j + 1 j ) ( 1 + i q j d j ) 2 q j 3 q j + 1 × exp [ i ( q j + 1 q j ) d j ] ,
s j , 12 ( 1 ) k = k ( ω / c ) 2 ( j + 1 j ) ( 1 + i q j d j ) 2 q j 3 q j + 1 × exp [ i ( q j + 1 + q j ) d j ] ,
S j , 21 ( 1 ) k = k ( ω / c ) 2 ( j + 1 j ) ( 1 i q j d j ) 2 q j 3 q j + 1 × exp [ i ( q j + 1 + q j ) d j ] ,
S j , 22 ( 1 ) k = k ( ω / c ) 2 ( j + 1 j ) ( 1 i q j d j ) 2 q j 3 q j + 1 × exp [ i ( q j + 1 q j ) d j ] ;
S j , 11 ( 1 ) j + S j , 11 ( 1 ) j + 1 = ( ω / c ) 4 ( j + 1 j ) ( 1 + i q j d j ) 4 q j 3 q j + 1 × exp [ i ( q j + 1 q j ) d j ] ,
s j , 12 ( 1 ) j + s j , 12 ( 1 ) j + 1 = ( ω / c ) 4 ( j + 1 j ) ( 1 + i q j d j ) 4 q j 3 q j + 1 × exp [ i ( q j + 1 + q j ) d j ] ,
s j , 21 ( 1 ) j + s j , 21 ( 1 ) j + 1 = ( ω / c ) 4 ( j + 1 j ) ( 1 i q j d j ) 4 q j 3 q j + 1 × exp [ i ( q j + 1 + q j ) d j ] ,
s j , 22 ( 1 ) j + s j , 22 ( 1 ) j + 1 = ( ω / c ) 4 ( j + 1 j ) ( 1 i q j d j ) 4 q j 3 q j + 1 × exp [ i ( q j + 1 q j ) d j ] ,
K s ( m ) = S 1 ( 1 , L + 1 ) k = j = 1 L + 2 S 1 ( 1 , j 1 ) ( s 1 ( j ) k ) S 1 ( j + 1 , L + 1 ) ,
E s ( m ) = S 1 ( 1 , L + 1 ) = j = 1 L + 2 S 1 ( 1 , j 1 ) ( s 1 ( j ) j + s 1 ( j ) j + 1 ) S 1 ( j + 1 , L + 1 ) .
F p ( j ) ( k ) = ( k 2 + | q 1 ( j ) | 2 | j | ( ω / c ) 2 ) ( | P 1,11 ( j , L + 1 ) | 2 β j ( ) + | P 1 , 21 ( j , L + 1 ) | 2 β j ( + ) ) 2 ( k 2 | q 1 ( j ) | 2 | j | ( ω / c ) 2 ) [ P 1 , 11 ( j , L + 1 ) ( P 1 , 21 ( j , L + 1 ) ) * γ j ] ,
F s ( j ) ( k ) = | S 1 , 11 ( j , L + 1 ) | 2 β j ( ) + | S 1 , 21 ( j , L + 1 ) | 2 β j ( + ) + 2 [ S 1 , 11 ( j , L + 1 ) ( S 1 , 21 ( j , L + 1 ) ) * γ j ] ,
β j ( ± ) = ± exp [ ± 2 S ( q 1 ( j ) ) d j ] exp [ ± 2 S ( q 1 ( j ) ) d j 1 ] 2 S ( q 1 ( j ) ) ,
γ j = exp [ 2 i ( q 1 ( j ) ) d j ] exp [ 2 i ( q 1 ( j ) ) d j 1 ] 2 i ( q 1 ( j ) ) .

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