Abstract

When conventional Fresnel theory is applied to absorbing multilayer systems, a slight discrepancy with respect to the principle of conservation of energy arises. Here we solve this long-perceived problem and present rigorous expressions for the Fresnel equations generally applicable to interfaces between isotropic absorbing media. These equations satisfy the conservation law automatically and coincide naturally with the conventional ones in reflection by simple mirrors placed in a vacuum.

© 2009 Optical Society of America

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References

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  1. M. Dressel and G. Grüner, Electrodynamics of Solids: Optical Properties of Electrons in Matter (Cambridge U. Press, 2002), pp. 1-46.
    [CrossRef]
  2. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), pp. 453-471.
  3. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge U. Press, 1999), pp. 31-33.

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge U. Press, 1999), pp. 31-33.

Dressel, M.

M. Dressel and G. Grüner, Electrodynamics of Solids: Optical Properties of Electrons in Matter (Cambridge U. Press, 2002), pp. 1-46.
[CrossRef]

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), pp. 453-471.

Grüner, G.

M. Dressel and G. Grüner, Electrodynamics of Solids: Optical Properties of Electrons in Matter (Cambridge U. Press, 2002), pp. 1-46.
[CrossRef]

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), pp. 453-471.

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge U. Press, 1999), pp. 31-33.

Other (3)

M. Dressel and G. Grüner, Electrodynamics of Solids: Optical Properties of Electrons in Matter (Cambridge U. Press, 2002), pp. 1-46.
[CrossRef]

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), pp. 453-471.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge U. Press, 1999), pp. 31-33.

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Figures (2)

Fig. 1
Fig. 1

Right-handed Cartesian coordinate system depicting the plane interface formed by a pair of isotropic and homogeneous material media I and II. The plane of incidence and plane of reflection are defined in the figure with manifestation of the triple-wave scheme.

Fig. 2
Fig. 2

Relations between the four states (I) to (IV), the two kinds of operation (spacetime inversion and phase inversion), and the two kinds of symmetry of field (with respect to the helicity inversion and with respect to the spatial inversion).

Tables (1)

Tables Icon

Table 1 Change of Sign for the First and Fourth Stokes Parameters ( S 0 and S 3 ), Energy Flux Density ( S ̂ ) , Angular Momentum Density ( M ̂ ) , and Helicity ( h ̂ ) of the Uniform Field by Operations of (I) Identity, (II) Spacetime Inversion, (III) Phase Inversion and (IV) Composite Operation of the Spacetime Inversion and Phase Inversion on the Fourier Representation of Each Quantity a

Equations (66)

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S y I S y II = 0 ,
Δ S y = S y i + S y r S y t = 0 .
Δ S y = ( S y i + S y r S y I ) + ( S y I S y t ) = S y i + S y r S y I = δ S y ,
δ S y = E x i B z r + E x r B z i E z i B x r E z r B x i .
δ S y = S y i + S y r S y I = 0 .
f ( t , z ) = a 2 f ̂ ( ω , k ̃ y , k z ) exp [ i ( k z z ω t ) ] d ω d k z ,
{ A ̂ x , A ̂ y , A ̂ z } = [ + A ̂ s ( ω ) , + k z k ̃ A ̂ p ( ω ) , k ̃ y k ̃ A ̂ p ( ω ) ] .
k ̃ y = ± ( k ̃ 2 k z 2 ) 1 2 .
( S ̂ y ) s = 1 2 ( E ̂ x B ̂ z * + E ̂ x * B ̂ z ) ,
( S ̂ y ) p = 1 2 ( E ̂ z B ̂ x * + E ̂ z * B ̂ x ) .
( S ̂ y ) s = ω 2 ( k ̃ y + k ̃ y * ) A ̂ s ( ω ) A ̂ s * ( ω ) ,
( S ̂ y ) p = ω 2 k ̃ * 2 k ̃ y + k ̃ 2 k ̃ y * k ̃ k ̃ * A ̂ p ( ω ) A ̂ p * ( ω ) ,
( Δ S ̂ y ) s , p ( S ̂ y i ) s , p = 1 R s , p T s , p .
( δ S ̂ y ) s = 1 2 ( E ̂ x i B ̂ z r * + E ̂ x i * B ̂ z r + B ̂ z i E ̂ x r * + B ̂ z i * E ̂ x r ) ,
( δ S ̂ y ) p = 1 2 ( E ̂ z i B ̂ x r * + E ̂ z i * B ̂ x r + B ̂ x i E ̂ z r * + B ̂ x i * E ̂ z r ) .
{ E ̂ x i , E ̂ z i } = i ω i [ A ̂ s i ( ω i ) , k ̃ y i k ̃ I A ̂ p i ( ω i ) ] ,
{ B ̂ x i , B ̂ z i } = i [ k ̃ I A ̂ p i ( ω i ) , k ̃ y i A ̂ s i ( ω i ) ] ,
{ E ̂ x r , E ̂ z r } = i ω r [ A ̂ s r ( ω r ) , k ̃ y r k ̃ I A ̂ p r ( ω r ) ] ,
{ B ̂ x r , B ̂ z r } = i [ k ̃ I A ̂ p r ( ω r ) , k ̃ y r A ̂ s r ( ω r ) ] ,
{ E ̂ x t , E ̂ z t } = i ω t [ A ̂ s t ( ω t ) , k ̃ y t k ̃ II A ̂ p t ( ω t ) ] ,
{ B ̂ x t , B ̂ z t } = i [ k ̃ II A ̂ p t ( ω t ) , k ̃ y t A ̂ s t ( ω t ) ] .
E x i ( t , z ) + E x r ( t , z ) E x t ( t , z ) = 0 .
a 2 E ̂ x i ( ω i , k ̃ y i , k z i , ) exp [ i ( k z i z ω i t ) ] d ω i d k z i + a 2 E ̂ x r ( ω r , k ̃ y r , k z r ) exp [ i ( k z r z ω r t ) ] d ω r d k z r a 2 E ̂ x t ( ω t , k ̃ y t , k z t ) exp [ i ( k z t z ω t t ) ] d ω t d k z t = 0 .
E ̂ x i ( ω i , k ̃ y i , k z i , ) δ ( ω i ω , k z k z i ) d ω i d k z i + E ̂ x r ( ω r , k ̃ y r , k z r ) δ ( ω r ω , k z k z r ) d ω r d k z r E ̂ x t ( ω t , k ̃ y t , k z t ) δ ( ω t ω , k z k z t ) d ω t d k z t = 0 .
E ̂ x i ( ω , k ̃ y i , k z ) + E ̂ x r ( ω , k ̃ y r , k z ) E ̂ x t ( ω , k ̃ y t , k z ) = 0 .
ω [ A ̂ s i ( ω ) + A ̂ s r ( ω ) A ̂ s t ( ω ) ] = 0 .
k ̃ y i A ̂ s i ( ω ) + k ̃ y r A ̂ s r ( ω ) k ̃ y t A ̂ s t ( ω ) = 0 ,
k ̃ I A ̂ p i ( ω ) + k ̃ I A ̂ p r ( ω ) k ̃ II A ̂ p t ( ω ) = 0 ,
ω [ k ̃ y i k ̃ I A ̂ p i ( ω ) + k ̃ y r k ̃ I A ̂ p r ( ω ) k ̃ y t k ̃ II A ̂ p t ( ω ) ] = 0 .
1 + r ̂ s ex t ̂ s ex = 0 ,
k ̃ y i ( 1 r ̂ s ex ) k ̃ y t t ̂ s ex = 0 ,
k ̃ I ( 1 + r ̂ p ex ) k ̃ II t ̂ p ex = 0 ,
k ̃ II k ̃ y i ( 1 r ̂ p ex ) k ̃ I k ̃ y t t ̂ p ex = 0 ,
r ̂ s ex = k ̃ y i k ̃ y t k ̃ y i + k ̃ y t ,
t ̂ s ex = 2 k ̃ y i k ̃ y i + k ̃ y t .
r ̂ p ex = k ̃ II 2 k ̃ y i k ̃ I 2 k ̃ y t k ̃ II 2 k ̃ y i + k ̃ I 2 k ̃ y t ,
t ̂ p ex = 2 k ̃ I k ̃ II k ̃ y i k ̃ II 2 k ̃ y i + k ̃ I 2 k ̃ y t .
R s , p = r ̂ s , p ex r ̂ s , p ex * ,
T s = k ̃ y t + k ̃ y t * k ̃ y i + k ̃ y i * t ̂ s ex t ̂ s ex * ,
T p = k ̃ I k ̃ I * ( k ̃ II * 2 k ̃ y t + k ̃ II 2 k ̃ y t * ) k ̃ II k ̃ II * ( k ̃ I * 2 k ̃ y i + k ̃ I 2 k ̃ y i * ) t ̂ p ex t ̂ p ex * .
( Δ S ̂ y ) s ( S ̂ y i ) s = 2 ( k ̃ y i k ̃ y i * ) ( k ̃ y i * k ̃ y t k ̃ y i k ̃ y t * ) ( k ̃ y i + k ̃ y i * ) ( k ̃ y i + k ̃ y t ) ( k ̃ y i * + k ̃ y t * ) ,
( Δ S ̂ y ) p ( S ̂ y i ) p = 2 ( k ̃ I * 2 k ̃ y i k ̃ I 2 k ̃ y i * ) ( k ̃ I * 2 k ̃ II 2 k ̃ y i k ̃ y t * k ̃ I 2 k ̃ II * 2 k ̃ y i * k ̃ y t ) ( k ̃ I * 2 k ̃ y i + k ̃ I 2 k ̃ y i * ) ( k ̃ II 2 k ̃ y i + k ̃ I 2 k ̃ y t ) ( k ̃ II * 2 k ̃ y i * + k ̃ I * 2 k ̃ y t * ) .
( δ S ̂ y ) s , p = ω 2 ( k ̃ y i k ̃ y i * ) ( r ̂ s , p ex r ̂ s , p ex * ) A ̂ s , p i A ̂ s , p i * .
( I ) f ( t , z ) = a 2 f ̂ ( ω , k ̃ y , k z ) exp [ + i ( k z z ω t ) ] d ω d k z ,
( II ) f ( t , z ) = a 2 f ̂ ( ω , k ̃ y , k z ) exp [ i ( k z z ω t ) ] d ω d k z ,
( III ) f ( t , z ) = a 2 f ̂ * ( ω , k ̃ y , k z ) exp [ i ( k z z ω t ) ] d ω d k z ,
( IV ) f ( t , z ) = a 2 f ̂ * ( ω , k ̃ y , k z ) exp [ + i ( k z z ω t ) ] d ω d k z .
S ̂ = ω 2 ( k ̃ + k ̃ * ) S 0 ,
M ̂ = 1 4 ω ( k ̃ 2 + k ̃ * 2 ) S 3 ,
h ̂ = sgn [ S 3 S ̂ ] = sgn [ S 3 ω Re ( k ̃ ) S 0 ] .
f r ( t , z ) = a 2 f ̂ r ( ω r , k ̃ y r * , k z r ) exp [ i ( k z r z ω r t ) ] d ω r d k z r ,
{ E ̂ x r , E ̂ z r } = i ω r [ A ̂ s r ( ω r ) , k ̃ y r * k ̃ I * A ̂ p r ( ω r ) ] ,
{ B ̂ x r , B ̂ z r } = i { k ̃ I * A ̂ p r ( ω r ) , k ̃ y r * A ̂ s r ( ω r ) } ,
( δ S ̂ y ) s , p = ω 2 [ ( k ̃ y i + k ̃ y r ) A ̂ s , p i A ̂ s , p r * + ( k ̃ y i * + k ̃ y r * ) A ̂ s , p i * A ̂ s , p r ] .
ω ( A ̂ s i + A ̂ s r A ̂ s t ) = 0 ,
k ̃ y i A ̂ s i + k ̃ y r * A ̂ s r k ̃ y t A ̂ s t = 0 ,
k ̃ I A ̂ p i + k ̃ I * A ̂ p r k ̃ II A ̂ p t = 0 ,
ω [ k ̃ y i k ̃ I A ̂ p i + k ̃ y r * k ̃ I * A ̂ p r k ̃ y t k ̃ II A ̂ p t ] = 0 ,
1 + r ̂ s t ̂ s = 0 ,
k ̃ y i k ̃ y i * r ̂ s k ̃ y t t ̂ s = 0 ,
k ̃ I + k ̃ I * r ̂ p k ̃ II t ̂ p = 0 ,
k ̃ II k ̃ I * k ̃ y i k ̃ II k ̃ I k ̃ y i * r ̂ p k ̃ I k ̃ I * k ̃ y t t ̂ p = 0 .
r ̂ s = k ̃ y i k ̃ y t k ̃ y i * + k ̃ y t ,
t ̂ s = k ̃ y i + k ̃ y i * k ̃ y i * + k ̃ y t ,
r ̂ p = k ̃ I * ( k ̃ II 2 k ̃ y i k ̃ I 2 k ̃ y t ) k ̃ I ( k ̃ II 2 k ̃ y i * + k ̃ I * 2 k ̃ y t ) ,
t ̂ p = k ̃ II ( k ̃ I * 2 k ̃ y i + k ̃ I 2 k ̃ y i * ) k ̃ I ( k ̃ II 2 k ̃ y i * + k ̃ I * 2 k ̃ y t ) .

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