Abstract

The emission of light by sources, e.g., by luminescent centers, located in a thin nonabsorbing dielectric layer 0 between two half-spaces 1 and 2 is investigated theoretically. It is assumed that the light is emitted in electric or magnetic dipole transitions. But the theory is given in such a form that it can easily be extended to electric and magnetic quadrupole and higher-order multipole transitions. The electromagnetic boundary-value problem is solved rigorously for sources in layers 0 of arbitrary thickness. The radiation patterns, i.e., the angular distributions of light emitted into the half-spaces 1 and 2, are calculated. The theory takes into account the following effects that strongly influence the radiation patterns: (1) the wide-angle interferences that are a consequence of the coherence of the plane waves emitted into different directions, (2) the multiple-beam interferences that result from the multiple reflections of the plane waves between the interfaces 0/1 and 0/2, and (3) that evanescent waves present in the near field of the source radiate into media 1 and/or 2 if these media are denser than layer 0. This emission process is influenced by evanescent-wave effects analogous to the wide-angle interferences and the multiple-beam interferences of the plane waves. The limiting case of extremely thin layers 0 with optical thickness much smaller than the wavelength is also treated. Explicit analytical expressions are presented for the dipole radiation patterns in this case. Furthermore, the theory is generalized for sources in plane-stratified-layer systems. The dipole radiation patterns are derived for the case in which any numbers of loss-free or absorbing, dielectric or metallic thin films are present between the loss-free layer 0 of arbitrary thickness containing the source and the half-spaces 1 and 2.

© 1981 Optical Society of America

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References

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  1. W. Lukosz and R. E. Kunz, “Light emission by magnetic and electric dipoles close to a plane interface. I. Total radiated power,” J. Opt. Soc. Am. 67, 1607–1615 (1977); “II Radiation patterns of perpendicular oriented dipoles,” J. Opt. Soc. Am. 67, 1615–1619 (1977).
    [CrossRef]
  2. Reference 1 contains lists of references to related experimental and theoretical work.
  3. W. Lukosz, “Light emission by magnetic and electric dipoles close to a plane dielectric interface. III. Radiation patterns of dipoles with arbitrary orientation,” J. Opt. Soc. Am. 69, 1495–1503 (1979).
    [CrossRef]
  4. W. Lukosz and R. E. Kunz, “Changes in fluorescence lifetimes induced by variation of the radiating molecules’ optical environment,” Opt. Commun. 31, 42–46 (1979), R. E. Kunz and W. Lukosz, “Changes in fluorescence lifetimes induced by variable optical environments,” Phys. Rev. B 21, 4814–4828 (1980).
    [CrossRef]
  5. W. Lukosz, “Theory of optical-environment-dependent spontaneous-emission rates for emitters in thin layers,” Phys. Rev. B 22, 3030–3038 (1980).
    [CrossRef]
  6. W. Lukosz and R. E. Kunz, “New method for determining refractive index and thickness of fluorescent thin films,” Opt. Commun. 31, 251–256 (1979).
    [CrossRef]
  7. E. D. Palik, S. W. McKnight, R. T. Holm, W. Lukosz, and R. Thalmann, “Cathodo-luminescence multiple reflection effects in thin films,” J. Opt. Soc. Am. 70, 1626A (1980).
  8. When comparing Section 2.B of this paper with Refs. 1 and 3, please note that we now use the notation ϕ˜∞,+(H,E)(kx, ky) and ϕ˜∞,-(H,E)(kx, ky) to distinguish more clearly between the amplitudes of the waves in the half-spaces z≥ z0 and z< z0, respectively.
  9. The notation L∞(n0) is meant to indicate that the source is located in an infinite medium 0; Eqs (3.7) and (3.8) show that L∞(n0) depends not only on n0 but also on μ0.
  10. The total power L(z0) is a function of the position z0 of the source in layer 0 and depends on the optical properties of the media 1 and 2 (cf. Ref. 5). In the case in which the layer 0 is a dielectric waveguide, i.e., if n0 > n1, n2 and if the thickness d0 of the layer exceeds the cutoff thickness, the source not only radiates into media 1 and 2 but also excites guided modes of the waveguide. Then the power carried away by these modes is not included in L(z0) given by Eq (3.2) The total dipole power radiated by the dipole into media 1 and 2 and into the guided modes is given by the expressions for L(z0) derived in Section IV of Ref. 6.
  11. We use the subscripts 1, 0, 2 only in Section 5 of this paper to designate the radiation patterns of dipoles located in extremely thin layers 0 between the half-spaces 1 and 2. In Eqs (5.2) and (5.4) the subscript || denotes a single dipole whose dipole moment is parallel to the layer and not(as in Section 4.D) an ensemble of parallel dipoles with random orientations in the x–y plane.
  12. W. Lukosz and M. Meier, “Lifetimes and radiation patterns of luminescent centers close to a thin metal film,” Opt. Lett. 6, 251–253 (1981).
    [CrossRef] [PubMed]

1981 (1)

1980 (2)

W. Lukosz, “Theory of optical-environment-dependent spontaneous-emission rates for emitters in thin layers,” Phys. Rev. B 22, 3030–3038 (1980).
[CrossRef]

E. D. Palik, S. W. McKnight, R. T. Holm, W. Lukosz, and R. Thalmann, “Cathodo-luminescence multiple reflection effects in thin films,” J. Opt. Soc. Am. 70, 1626A (1980).

1979 (3)

W. Lukosz and R. E. Kunz, “Changes in fluorescence lifetimes induced by variation of the radiating molecules’ optical environment,” Opt. Commun. 31, 42–46 (1979), R. E. Kunz and W. Lukosz, “Changes in fluorescence lifetimes induced by variable optical environments,” Phys. Rev. B 21, 4814–4828 (1980).
[CrossRef]

W. Lukosz and R. E. Kunz, “New method for determining refractive index and thickness of fluorescent thin films,” Opt. Commun. 31, 251–256 (1979).
[CrossRef]

W. Lukosz, “Light emission by magnetic and electric dipoles close to a plane dielectric interface. III. Radiation patterns of dipoles with arbitrary orientation,” J. Opt. Soc. Am. 69, 1495–1503 (1979).
[CrossRef]

1977 (1)

Holm, R. T.

E. D. Palik, S. W. McKnight, R. T. Holm, W. Lukosz, and R. Thalmann, “Cathodo-luminescence multiple reflection effects in thin films,” J. Opt. Soc. Am. 70, 1626A (1980).

Kunz, R. E.

W. Lukosz and R. E. Kunz, “Changes in fluorescence lifetimes induced by variation of the radiating molecules’ optical environment,” Opt. Commun. 31, 42–46 (1979), R. E. Kunz and W. Lukosz, “Changes in fluorescence lifetimes induced by variable optical environments,” Phys. Rev. B 21, 4814–4828 (1980).
[CrossRef]

W. Lukosz and R. E. Kunz, “New method for determining refractive index and thickness of fluorescent thin films,” Opt. Commun. 31, 251–256 (1979).
[CrossRef]

W. Lukosz and R. E. Kunz, “Light emission by magnetic and electric dipoles close to a plane interface. I. Total radiated power,” J. Opt. Soc. Am. 67, 1607–1615 (1977); “II Radiation patterns of perpendicular oriented dipoles,” J. Opt. Soc. Am. 67, 1615–1619 (1977).
[CrossRef]

Lukosz, W.

W. Lukosz and M. Meier, “Lifetimes and radiation patterns of luminescent centers close to a thin metal film,” Opt. Lett. 6, 251–253 (1981).
[CrossRef] [PubMed]

W. Lukosz, “Theory of optical-environment-dependent spontaneous-emission rates for emitters in thin layers,” Phys. Rev. B 22, 3030–3038 (1980).
[CrossRef]

E. D. Palik, S. W. McKnight, R. T. Holm, W. Lukosz, and R. Thalmann, “Cathodo-luminescence multiple reflection effects in thin films,” J. Opt. Soc. Am. 70, 1626A (1980).

W. Lukosz and R. E. Kunz, “Changes in fluorescence lifetimes induced by variation of the radiating molecules’ optical environment,” Opt. Commun. 31, 42–46 (1979), R. E. Kunz and W. Lukosz, “Changes in fluorescence lifetimes induced by variable optical environments,” Phys. Rev. B 21, 4814–4828 (1980).
[CrossRef]

W. Lukosz and R. E. Kunz, “New method for determining refractive index and thickness of fluorescent thin films,” Opt. Commun. 31, 251–256 (1979).
[CrossRef]

W. Lukosz, “Light emission by magnetic and electric dipoles close to a plane dielectric interface. III. Radiation patterns of dipoles with arbitrary orientation,” J. Opt. Soc. Am. 69, 1495–1503 (1979).
[CrossRef]

W. Lukosz and R. E. Kunz, “Light emission by magnetic and electric dipoles close to a plane interface. I. Total radiated power,” J. Opt. Soc. Am. 67, 1607–1615 (1977); “II Radiation patterns of perpendicular oriented dipoles,” J. Opt. Soc. Am. 67, 1615–1619 (1977).
[CrossRef]

McKnight, S. W.

E. D. Palik, S. W. McKnight, R. T. Holm, W. Lukosz, and R. Thalmann, “Cathodo-luminescence multiple reflection effects in thin films,” J. Opt. Soc. Am. 70, 1626A (1980).

Meier, M.

Palik, E. D.

E. D. Palik, S. W. McKnight, R. T. Holm, W. Lukosz, and R. Thalmann, “Cathodo-luminescence multiple reflection effects in thin films,” J. Opt. Soc. Am. 70, 1626A (1980).

Thalmann, R.

E. D. Palik, S. W. McKnight, R. T. Holm, W. Lukosz, and R. Thalmann, “Cathodo-luminescence multiple reflection effects in thin films,” J. Opt. Soc. Am. 70, 1626A (1980).

J. Opt. Soc. Am. (3)

Opt. Commun. (2)

W. Lukosz and R. E. Kunz, “Changes in fluorescence lifetimes induced by variation of the radiating molecules’ optical environment,” Opt. Commun. 31, 42–46 (1979), R. E. Kunz and W. Lukosz, “Changes in fluorescence lifetimes induced by variable optical environments,” Phys. Rev. B 21, 4814–4828 (1980).
[CrossRef]

W. Lukosz and R. E. Kunz, “New method for determining refractive index and thickness of fluorescent thin films,” Opt. Commun. 31, 251–256 (1979).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. B (1)

W. Lukosz, “Theory of optical-environment-dependent spontaneous-emission rates for emitters in thin layers,” Phys. Rev. B 22, 3030–3038 (1980).
[CrossRef]

Other (5)

Reference 1 contains lists of references to related experimental and theoretical work.

When comparing Section 2.B of this paper with Refs. 1 and 3, please note that we now use the notation ϕ˜∞,+(H,E)(kx, ky) and ϕ˜∞,-(H,E)(kx, ky) to distinguish more clearly between the amplitudes of the waves in the half-spaces z≥ z0 and z< z0, respectively.

The notation L∞(n0) is meant to indicate that the source is located in an infinite medium 0; Eqs (3.7) and (3.8) show that L∞(n0) depends not only on n0 but also on μ0.

The total power L(z0) is a function of the position z0 of the source in layer 0 and depends on the optical properties of the media 1 and 2 (cf. Ref. 5). In the case in which the layer 0 is a dielectric waveguide, i.e., if n0 > n1, n2 and if the thickness d0 of the layer exceeds the cutoff thickness, the source not only radiates into media 1 and 2 but also excites guided modes of the waveguide. Then the power carried away by these modes is not included in L(z0) given by Eq (3.2) The total dipole power radiated by the dipole into media 1 and 2 and into the guided modes is given by the expressions for L(z0) derived in Section IV of Ref. 6.

We use the subscripts 1, 0, 2 only in Section 5 of this paper to designate the radiation patterns of dipoles located in extremely thin layers 0 between the half-spaces 1 and 2. In Eqs (5.2) and (5.4) the subscript || denotes a single dipole whose dipole moment is parallel to the layer and not(as in Section 4.D) an ensemble of parallel dipoles with random orientations in the x–y plane.

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Figures (4)

Fig. 1
Fig. 1

Source (radiating dipole) D located in a loss-free dielectric layer 0 of thickness d0 between media (half-spaces) 1 and 2. Θ, angle between dipole moment and z axis.

Fig. 2
Fig. 2

Definition of the angular coordinates α, φ and of the polarizations s and p appearing in the radiation patterns P(s,p)(α, φ); k ˆ ( k ˆ x = sin α cos φ , k ˆ y = sin α sin φ , k ˆ z = - cos α ) is the unit vector in the direction of observation. We call the xz plane defined by k ˆ and the z axis the plane of emission. For an s-polarized and a p-polarized wave emitted into direction k ˆ, the directions of the electric fields are indicated. The thin layer 0 and any intermediate layers between layer 0 and the half-spaces 1 and 2 (cf. Figs. 1 and 4) are not shown.

Fig. 3
Fig. 3

Interference effects influencing the emission of light by source D located in layer 0 into media 1 and 2: (a) wide-angle interferences, (b) multiple-beam interferences of plane waves emitted by D, and (c) and (d) the analogous effects for evanescent waves generated by D. Medium 1 is assumed to be optically denser than layer 0. Only two of the multiple reflections of the waves between the interfaces 0/1 and 0/2 are indicated in (b) and (d). See text for detailed explanations.

Fig. 4
Fig. 4

Source (radiating dipole) D in a system of thin layers. The source is located in a loss-free dielectric layer 0, as in the situation shown in Fig. 1. The theory allows for an arbitrary number of loss-free or absorbing layers ja, jb, … with thicknesses dja, djb, … between layer 0 and the half-spaces j (j = 1, 2).

Equations (97)

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Π ( E ) ( x ) = [ 0 , 0 , ϕ ( E ) ( x ) ] ,
Π ( H ) ( x ) = [ 0 , 0 , ϕ ( H ) ( x ) ] .
[ Δ + ( k j ) 2 ] ϕ ˜ ( E , H ) ( x ) = 0 ,
E ( x ) = i ω × Π ( H ) ( x ) + ( ˆ 0 j ) - 1 × Π ( E ) ( x ) ,
H ( x ) = ( μ ˆ 0 μ j ) - 1 × × Π ( H ) ( x ) - i ω × Π ( E ) ( x ) .
ϕ ( H , E ) ( x ) = - + ϕ ˜ , ± ( H , E ) ( k x , k y ) × exp { i [ k x x + k y y + k z , 0 ( z - z 0 ) ] } d k x d k y ,
k z , 0 = { + ( k 0 2 - k x 2 - k y 2 ) 1 / 2 if ( k x 2 + k y 2 ) 1 / 2 k 0 + i ( k x 2 + k y 2 - k 0 2 ) 1 / 2 if ( k x 2 + k y 2 ) 1 / 2 > k 0 ,
ϕ ˜ , ± ( H ) ( k x , k y ) = - i ( 8 π 2 ) - 1 μ ˆ 0 μ 0 p 0 ω k y × [ k z , 0 ( k x 2 + k y 2 ) ] - 1 sin Θ
ϕ ˜ , ± ( E ) ( k x , k y ) = i ( 8 π 2 ) - 1 p 0 × [ ( k z , 0 ) - 1 cos Θ k x ( k x 2 + k y 2 ) - 1 sin Θ ] ,
ϕ ˜ , ± ( H ) ( k x , k y ) = i ( 8 π 2 ) - 1 m 0 × [ ( k z , 0 ) - 1 cos Θ k x ( k x 2 + k y 2 ) - 1 sin Θ ]
ϕ ˜ , ± ( E ) ( k x , k y ) = i ( 8 π 2 ) - 1 ˆ 0 0 m 0 ω k y × [ k z , 0 ( k x 2 + k y 2 ) ] - 1 sin Θ .
ϕ 0 ( H , E ) ( x ) = ϕ ( H , E ) ( x ) + - + ϕ ˜ 0 , + ( H , E ) ( k x , k y ) × exp [ i ( k x x + k y y + k z , 0 z ) ] d k x d k y + - + ϕ ˜ 0 , - ( H , E ) ( k x , k y ) × exp [ i ( k x x + k y y - k z , 0 z ) ] d k x d k y ,
ϕ 1 ( H , E ) ( x ) = - + ϕ ˜ 1 ( H , E ) ( k x , k y ) × exp { i [ k x x + k y y + k z , 1 ( z - d 0 ) ] } d k x d k y
ϕ 2 ( H , E ) ( x ) = - + ϕ ˜ 2 ( H , E ) ( k x , k y ) × exp [ i ( k x x + k y y - k z , 2 z ) ] d k x d k y ,
k z , j = ( k j 2 - k x 2 - k y 2 ) 1 / 2 ,
ϕ ( E ) ( x ) , 1 ϕ ( E ) z ( x ) ϕ ( H ) ( x ) , 1 μ ϕ ( H ) z ( x ) } are continuous
ϕ ˜ 0 , + ( H , E ) ( k x , k y ) = m ( s , p ) r 0 , 2 ( s , p ) { ϕ ˜ , - ( H , E ) ( k x , k y ) exp ( i k z , 0 z 0 ) + r 0 , 1 ( s , p ) ϕ ˜ , + ( H , E ) ( k x , k y ) exp [ i k z , 0 ( 2 d 0 - z 0 ) ] } ,
ϕ ˜ 0 , - ( H , E ) ( k x , k y ) = m ( s , p ) r 0 , 1 ( s , p ) exp ( 2 i k z , 0 d 0 ) × [ ϕ ˜ , + ( H , E ) ( k x , k y ) exp ( - i k z , 0 z 0 ) + r 0 , 2 ( s , p ) ϕ ˜ , - ( H , E ) ( k x , k y ) exp ( i k z , o z 0 ) ] ,
m ( s , p ) = [ 1 - r 0 , 1 ( s , p ) r 0 , 2 ( s , p ) exp ( 2 i k z , 0 d 0 ) ] - 1
r 0 , j ( s , p ) = ( j / 0 ) ρ ( μ j / μ 0 ) 1 - ρ k z , 0 - k z , j ( j / 0 ) ρ ( μ j / μ 0 ) 1 - ρ k z , 0 + k z , j ,
ϕ ˜ 1 ( H , E ) ( k x , k y ) = t 0 , 1 ( s , p ) [ r 0 , 1 ( s , p ) ] - 1 × exp ( - i k z , 0 d 0 ) ϕ ˜ 0 , - ( H , E ) ( k x , k y ) = t 0 , 1 ( s , p ) m ( s , p ) exp [ + i k z , 0 ( d 0 - z 0 ) ] [ ϕ ˜ , + ( H , E ) ( k x , k y ) + r 0 , 2 ( s , p ) ϕ ˜ , - ( H , E ) ( k x , k y ) exp ( 2 i k z , 0 z 0 ) ] ,
ϕ ˜ 2 ( H , E ) ( k x , k y ) = t 0 , 2 ( s , p ) [ r 0 , 2 ( s , p ) ] - 1 ϕ ˜ 0 , + ( H , E ) ( k x , k y ) = t 0 , 2 ( s , p ) m ( s , p ) exp ( i k z , 0 z 0 ) { ϕ ˜ , - ( H , E ) ( k x , k y ) + r 0 , 1 ( s , p ) ϕ ˜ , + ( H , E ) ( k x , k y ) exp [ 2 i k z , 0 ( d 0 - z 0 ) ] } ,
t 0 , j ( s , p ) = 1 + r 0 , j ( s , p )
L ( n 0 ) P ( s , p ) ( α , φ ) d Ω = lim R ( A ) k ˆ · S ( s , p ) ( x ) d σ ,
Ω = 4 π [ P ( s ) ( α , φ ) + P ( p ) ( α , φ ) ] d Ω = L ( z 0 ) / L ( n 0 ) .
k x = k j sin α j cos φ , k y = k j sin α j sin φ ,
ϕ 1 ( H , E ) ( x ) = ( 2 π ) 2 ( i λ 1 R ) - 1 cos α 1 exp ( - i k 1 d 0 cos α 1 ) × ϕ ˜ 1 ( H , E ) ( k x = k 1 k ˆ x , k y = k 1 k ˆ z ) exp ( i k 1 k ˆ · x ) .
L ( n 0 ) P ( s ) ( α j , φ ) = ( 1 / 2 ) π 2 ˆ 0 j ( ω k j ) 3 × sin 2 2 α j ϕ ˜ j ( H ) ( k x , k y ) 2
L ( n 0 ) P ( p ) ( α j , φ ) = ( 1 / 2 ) π 2 μ ˆ 0 μ j ( ω k j ) 3 × sin 2 2 α j ϕ ˜ j ( E ) ( k x , k y ) 2 .
[ L ( n 0 ) ] e = n 0 μ 0 ( L vac ) e ,
[ L ( n 0 ) ] m = n 0 0 ( L vac ) m ,
( L vac ) e = p 0 2 ω 4 / 12 π ˆ 0 c 3 ,
( L vac ) m = m 0 2 ω 4 / 12 π μ ˆ 0 c 3
[ P ( s ) ( α , φ ) ] e = 3 8 π sin 2 Θ sin 2 φ ,
[ P ( p ) ( α , φ ) ] e = 3 8 π ( cos Θ sin α + sin Θ cos α cos φ ) 2 .
Ω = 4 π { [ P ( s ) ( α , φ ) ] e + [ P ( p ) ( α , φ ) ] e } d Ω = 1
e m , ( s ) ( p ) , ( p ) ( s ) .
P e ( s ) ( α 1 , φ ) = Q e ( s ) ( α 0 , φ ) M ( s ) ( α 0 ) T ˆ 0 , 1 ( s ) ( α 1 ) ,
Q e ( s ) ( α 0 , φ ) = 3 8 π sin 2 Θ sin 2 φ W + ( s ) ( α 0 )
P e ( p ) ( α 1 , φ ) = Q e ( p ) ( α 0 , φ ) M ( p ) ( α 0 ) T ˆ 0 , 1 ( p ) ( α 1 ) ,
Q e ( p ) ( α 0 , φ ) = 3 8 π [ cos 2 Θ sin 2 α 0 W + ( p ) ( α 0 ) + sin 2 Θ cos 2 φ cos 2 α 0 W - ( p ) ( α 0 ) - ( 1 / 2 ) sin 2 θ cos φ sin 2 α 0 T 0 , 2 ( p ) ( α 0 ) ] .
W ± ( s , p ) 1 ± r 0 , 2 ( s , p ) exp ( 2 i k z , 0 z 0 ) 2 = 1 + r 0 , 2 ( s , p ) 2 ± 2 r 0 , 2 ( s , p ) × cos [ 4 π ( n 0 z 0 / λ ) cos α 0 + δ 0 , 2 ( s , p ) ( α 0 ) ] ,
M ( s , p ) ( α 0 ) m ( s , p ) 2 = 1 - r 0 , 1 ( s , p ) r 0 , 2 ( s , p ) exp ( 2 i k z , 0 d 0 ) - 2 = 1 - r 0 , 1 ( s , p ) r 0 , 2 ( s . p ) exp [ 2 π i ( n 0 d 0 / λ ) cos α 0 ] - 2
T ˆ 0 , 1 ( s , p ) ( α 1 ) T 0 , 1 ( s , p ) ( α 0 ) d Ω 0 / d Ω 1 .
T 0 , 1 ( s , p ) ( α 0 ) t 0 , 1 ( s , p ) 2 ( μ ) 1 - ρ ( ) ρ k z , 1 / k z , 0 = 4 ( / μ ) ρ - 1 / 2 cos α 0 cos α 1 [ ( / μ ) ρ - 1 / 2 cos α 0 + cos α 1 ] 2 ,
d Ω 0 / d Ω 1 sin α 0 d α 0 / sin α 1 d α 1 = n 2 cos α 1 / cos α 0
T ˆ 0 , 1 ( s , p ) ( α 1 ) = t 01 ( s , p ) k 0 / k z , 0 2 ( μ / ) ρ - 1 / 2 n 2 cos 2 α 1 = 4 ( μ / ) ρ - 1 / 2 n 2 cos 2 α 1 [ cos α 0 + ( μ / ρ ) ρ - 1 / 2 cos α 1 ] 2 ,
T ˆ 0 , 1 ( s , p ) ( α 1 ) = 4 ( n ) 2 ρ + 1 cos 2 α 1 [ ( n ) 2 ρ - 1 cos α 0 + cos α 1 ] 2 .
T 0 , j ( s , p ) ( α 0 ) = 1 - r 0 , j ( s , p ) 2 ,
P e ( s ) ( α 1 , φ ) = Q e ( s ) ( α 1 , φ ) M ( s ) ( α 1 ) T ˆ 0 , 1 ( s ) ( α 1 ) × exp [ - 2 ( d 0 - z 0 ) / Δ z ( α 1 ) ] ,
Q e ( s ) ( α 1 , φ ) = 3 8 π sin 2 Θ sin 2 φ W + ( s ) ( α 1 )
P e ( p ) ( α 1 , φ ) = Q e ( p ) ( α 1 , φ ) M ( p ) ( α 1 ) T ˆ 0 , 1 ( p ) ( α 1 ) × exp [ - 2 ( d 0 - z 0 ) / Δ z ( α 1 ) ] ,
Q e ( p ) ( α 1 , φ ) = 3 8 π { n 2 cos 2 Θ sin 2 α 1 W + ( p ) ( α 1 ) + sin 2 Θ cos 2 φ ( n 2 sin 2 α 1 - 1 ) W - ( p ) ( α 1 ) - n sin 2 Θ cos φ sin α 1 ( n 2 sin 2 α 1 - 1 ) 1 / 2 × T 0 , 2 ( p ) exp [ - z 0 / Δ z ( α 1 ) ] } ,
Δ z ( α 1 ) 1 k z , 0 = ( λ / 2 π ) ( n 1 2 sin 2 α 1 - n 0 2 ) 1 / 2
W ± ( s , p ) ( α 1 ) 1 ± r 0 , 2 ( s , p ) exp [ - 2 z 0 / Δ z ( α 1 ) ] 2 ,
M ( s , p ) ( α 1 ) m ( s , p ) 2 = 1 - r 0 , 1 ( s , p ) r 0 , 2 ( s , p ) exp [ - 2 d 0 / Δ z ( α 1 ) ] - 2 .
T ˆ 0 , 1 ( s , p ) ( α 1 ) t 0 , 1 ( s , p ) k 0 / k z , 0 2 × ( μ / ) ρ - 1 / 2 n 2 cos 2 α 1 ,
T ˆ 0 , 1 ( s ) ( α 1 ) = 4 μ n 3 cos 2 α 1 ( μ 2 - 1 ) n 2 sin 2 α 1 + n 2 - μ 2
T ˆ 0 , 1 ( p ) ( α 1 ) = 4 n 3 cos 2 α 1 ( 2 - 1 ) n 2 sin 2 α 1 + n 2 - 2 .
T ˆ 0 , 1 ( s ) ( α 1 ) = 4 n 3 ( n 2 - 1 ) - 1 cos 2 α 1
T ˆ 0 , 1 ( p ) ( α 1 ) = T ˆ 0 , 1 ( s ) ( α 1 ) [ ( n 2 + 1 ) sin 2 α 1 - 1 ] - 1 ,
T 0 , 2 ( s , p ) = 2 Im [ r 0 , 2 ( s , p ) ] ,
T 0 , 2 ( s ) = 4 μ ¯ n ¯ cos α 2 ( n ¯ 2 sin 2 α 2 - 1 ) 1 / 2 ( μ ¯ 2 - 1 ) n ¯ 2 sin 2 α 2 + n ¯ 2 - μ ¯ 2 ,
T 0 , 2 ( p ) = 4 ¯ n ¯ cos α 2 ( n ¯ 2 sin 2 α 2 - 1 ) 1 / 2 ( ¯ 2 - 1 ) n ¯ 2 sin 2 α 2 + n ¯ 2 - ¯ 2 ,
1 2 ,             2 1 ; z 0 d 0 - z 0 ,             d 0 - z 0 z 0 , Θ 180 - Θ .
P ( s , p ) ( α , φ ) = P ( s , p ) ( α , - φ ) .
P e , ( p ) ( α , φ ) = P e , ( p ) ( α ) ,
P m , ( s ) ( α , φ ) = P m , ( s ) ( α ) .
P e ( s , p ) ( α = 90° , φ ) = P m ( s , p ) ( α = 90° , φ ) = 0.
P e , Θ ( s ) ( α ) = sin 2 Θ P e , ( s ) ( α ) ,
P e , Θ ( p ) ( α ) = cos 2 Θ P e , ( p ) ( α ) + sin 2 Θ P e , ( p ) ( α ) ,
P e , i ( s , p ) ( α ) = 1 / 2 Θ = 0 180° P e , Θ ( s , p ) ( α ) sin Θ d Θ .
P e , i ( s ) ( α ) = 2 / 3 P e , ( s ) ( α ) ,
P e , i ( p ) ( α ) = 1 / 3 P e , ( p ) ( α ) + 2 / 3 P e , ( p ) ( α ) .
[ P 1 , 0 , 2 ( s , p ) ( α , φ ) ] e , = ( n 1 / n 0 ) 3 ( 1 / 0 ) [ P 1 , 2 ( s , p ) ( α , φ ) ] e ,
[ P 1 , 0 , 2 ( s , p ) ( α , φ ) ] e , = ( n 1 μ 1 / n 0 μ 0 ) [ P 1 , 2 ( s , p ) ( α , φ ) ] e ,
[ P 1 , 0 , 2 ( s , p ) ( α , φ ) ] m , = ( n 1 / n 0 ) 3 ( μ 1 / μ 0 ) [ P 1 , 2 ( s , p ) ( α , φ ) ] m ,
[ P 1 , 0 , 2 ( s , p ) ( α , φ ) ] m , = ( n 1 1 / n 0 0 ) [ P 1 , 2 ( s , p ) ( α , φ ) ] m ,
[ P 1 , 0 , 2 ( s , p ) ( α , φ ) ] m = ( n 1 / n 0 ) 3 [ P 1 , 2 ( s , p ) ( α , φ ) ] m .
( L 1 , 0 , 2 ) e , = ( L 1 , 2 ) e , ,
( L 1 , 0 , 2 ) m , = ( L 1 , 2 ) m , .
( L 1 , 0 , 2 ) e , = ( 1 / 0 ) 2 ( L 1 , 2 ) e , ,
( L 1 , 0 , 2 ) m , = ( μ 1 / μ 0 ) 2 ( L 1 , 2 ) m , .
ϕ ˜ 1 ( H , E ) ( k x , k y ) = t 0 , 1 ( s , p ) [ 1 - r 0 , 1 ( s , p ) r 0 , 2 ( s , p ) ] - 1 × [ ϕ ˜ , + ( H , E ) ( k x , k y ) + r 0 , 2 ( s , p ) ϕ ˜ , - ( H , E ) ( k x , k y ) ] ,
ϕ ˜ 1 ( H , E ) ( k x , k y ) = [ ϕ ˜ , + ( H , E ) ( k x , k y ) ] + r 1 , 2 ( s , p ) [ ϕ ˜ , - ( H , E ) ( k x , k y ) ]
ϕ ˜ 2 ( H , E ) ( k x , k y ) = t 1 , 2 ( s , p ) [ ϕ ˜ , - ( H , E ) ( k x , k y ) ] ,
p 0 , x = p 0 , x ,             p 0 , z = ( 1 / 0 ) p 0 , z ,
m 0 , x = m 0 , x ,             m 0 , z = ( μ 1 / μ 0 ) m 0 , z .
r 0 , 1 a , 1 = r 0 , 1 a + r 1 a , 1 exp ( 2 i k z , 1 a d 1 a ) 1 + r 0 , 1 a r 1 a , 1 exp ( 2 i k z , 1 a d 1 a )
t 0 , 1 a , 1 = t 0 , 1 a t 1 a , 1 exp ( i k z , 1 a d 1 a ) 1 + r 0 , 1 a r 1 a , 1 exp ( 2 i k z , 1 a d 1 a ) ,
ϕ 1 ( H , E ) ( x ) = k x 2 + k y 2 k 1 2 ϕ ˜ 1 ( H , E ) ( k x , k y ) × exp ( - i k z , d 1 0 ) exp ( i k · x ) d k x d k y .
d k x d k y = k 1 2 cos α sin α d α d φ
k · x = k 1 R [ 1 - 1 2 ( k ˆ - k ˆ ) 2 ] ,
( k ˆ - k ˆ ) 2 = 4 { [ sin ( α - α ) / 2 ] 2 + sin α sin α [ sin ( φ - φ ) / 2 ] 2 } ( α - α ) 2 + ( φ - φ ) 2 sin 2 α .
ϕ ˜ 1 ( H E ) ( x ) = k 1 2 exp ( i k 1 R ) exp ( - i k 1 d 0 cos α 1 ) × α = 0 , π φ = 0 2 δ ϕ ˜ 1 ( H , E ) ( k x = k 1 k ˆ x , k y = k 1 k y ) × exp { - i π ( R / λ 1 ) [ ( α - α ) 2 + sin 2 α ( φ - φ ) 2 ] } cos α sin α d α d φ
- + exp ( i π x 2 ) d x = i ,
ϕ ˜ 1 ( H , E ) ( x ) = ( 2 π ) 2 ( i λ 1 R ) - 1 cos α 1 exp ( - i k 1 d 0 cos α 1 ) × ϕ ˜ 1 ( H , E ) ( k x = k 1 k ˆ x , k y = k 1 k ˆ y ) exp ( i k 1 R ) .