Abstract

The S matrix is first introduced within the framework of the angular spectrum representation of wave fields interacting with linear dielectric bodies of arbitrary shape. By using some universal properties of the S matrix, a number of relations involving certain generalized reflection and transmission coefficients are derived. These relations may be regarded as generalizations of two well-known classic reciprocity relations due to G. G. Stokes. Two reciprocity relations involving the reflection and the transmission coefficients for interaction of a plane electromagnetic wave with a stratified dielectric medium are obtained as special cases.

© 1986 Optical Society of America

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References

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  1. G. G. Stokes, Cambridge Dublin Math. J. 4, 1 (1849). Reprinted in Mathematical and Physical Papers of G. G. Stokes (Cambridge U. Press, Cambridge, 1883), Vol. II, pp. 89–103. For modern treatments of the Stokes relations see, for example, F. A. Jenkins, H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, New York, 1976), pp. 286–288, or E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass, 1974), pp. 91–93.
  2. A. Vasíček, Optics of Thin Films (North-Holland, Amsterdam, 1960), p. 173.
  3. M. Nazarathy, “A Fabry–Perot interferometer with one phase-conjugate mirror,” Opt. Commun. 45, 117-121 (1983).
    [CrossRef]
  4. A. T. Friberg, P. D. Drummond, “Reflection of a linearly polarized plane wave from a lossless stratified mirror in the presence of a phase-conjugate mirror,” J. Opt. Soc. Am. 73, 1216–1219 (1983); P. D. Drummond, A. T. Friberg, “Specular reflection cancellation in an interferometer with a phase-conjugate mirror,” J. Appl. Phys. 54, 5618–5625 (1983).
    [CrossRef]
  5. D. M. Kerns, Plane-Wave Scattering-Matrix Theory of Antennas and Antenna-Antenna Interactions (U.S. Department of Commerce, National Bureau of Standards, Washington, 1981), Monograph 162.
  6. R. Mittra, T. M. Habashy, “Theory of wave-front-distortion correction by phase conjugation,” J. Opt. Soc. Am. A 1, 1103–1109 (1984).
    [CrossRef]
  7. E. Wolf, “A scalar representation of electromagnetic fields: II,” Proc. Phys. Soc. London 74, 269–280 (1959), App.
    [CrossRef]
  8. K. Miyamoto, E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave—Part I,” J. Opt. Soc. Am. 52, 615–625 (1962), App.
    [CrossRef]
  9. In the analogous problem in the theory of quantum-mechanical potential scattering, C(±)are associated with the incoming state and D(±)with the outgoing state [cf. P. Roman, Advanced Quantum Theory (Addison-Wesley, Reading, Mass., 1965), pp. 282 and 294]. A word of caution regarding this terminology is in order here. A plane homogeneous waveU(i)(r)=eikn0·r formally has the asymptotic behavior (see Appendix A)eikn0·r∼2πik[Δ(n−n0)eikrr−Δ(n+n0)e−ikrr] as kr→ ∞, with the unit vector n fixed, and Δ is the “spherical” delta function, defined by Eq. (2.12). Hence a plane wave provides both incoming and outgoing contributions at infinity.
  10. The minus sign is included on the right-hand side of Eq. (2.4) in order that S reduce to the unit matrix in the absence of scattering.
  11. E. Gerjuoy, D. S. Saxon, “Variational principles for the acoustic field,” Phys. Rev. 94, 1445–1458 (1954).
    [CrossRef]
  12. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 13.5, Eq. (107).
  13. E. Wolf, “New theory of radiative energy transfer in free electromagnetic fields,” Phys. Rev. D 13, 869–886 (1976), App. B.
    [CrossRef]
  14. D. S. Jones, “Removal of an inconsistency in the theory of diffraction,” Proc. Cambr. Phil. Soc. 48, 733–741 (1952), lemma on p. 736. See also Ref. 12.
    [CrossRef]
  15. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 111.

1984

1983

1976

E. Wolf, “New theory of radiative energy transfer in free electromagnetic fields,” Phys. Rev. D 13, 869–886 (1976), App. B.
[CrossRef]

1962

1959

E. Wolf, “A scalar representation of electromagnetic fields: II,” Proc. Phys. Soc. London 74, 269–280 (1959), App.
[CrossRef]

1954

E. Gerjuoy, D. S. Saxon, “Variational principles for the acoustic field,” Phys. Rev. 94, 1445–1458 (1954).
[CrossRef]

1952

D. S. Jones, “Removal of an inconsistency in the theory of diffraction,” Proc. Cambr. Phil. Soc. 48, 733–741 (1952), lemma on p. 736. See also Ref. 12.
[CrossRef]

1849

G. G. Stokes, Cambridge Dublin Math. J. 4, 1 (1849). Reprinted in Mathematical and Physical Papers of G. G. Stokes (Cambridge U. Press, Cambridge, 1883), Vol. II, pp. 89–103. For modern treatments of the Stokes relations see, for example, F. A. Jenkins, H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, New York, 1976), pp. 286–288, or E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass, 1974), pp. 91–93.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 13.5, Eq. (107).

Drummond, P. D.

Friberg, A. T.

Gerjuoy, E.

E. Gerjuoy, D. S. Saxon, “Variational principles for the acoustic field,” Phys. Rev. 94, 1445–1458 (1954).
[CrossRef]

Habashy, T. M.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 111.

Jones, D. S.

D. S. Jones, “Removal of an inconsistency in the theory of diffraction,” Proc. Cambr. Phil. Soc. 48, 733–741 (1952), lemma on p. 736. See also Ref. 12.
[CrossRef]

Kerns, D. M.

D. M. Kerns, Plane-Wave Scattering-Matrix Theory of Antennas and Antenna-Antenna Interactions (U.S. Department of Commerce, National Bureau of Standards, Washington, 1981), Monograph 162.

Mittra, R.

Miyamoto, K.

Nazarathy, M.

M. Nazarathy, “A Fabry–Perot interferometer with one phase-conjugate mirror,” Opt. Commun. 45, 117-121 (1983).
[CrossRef]

Roman, P.

In the analogous problem in the theory of quantum-mechanical potential scattering, C(±)are associated with the incoming state and D(±)with the outgoing state [cf. P. Roman, Advanced Quantum Theory (Addison-Wesley, Reading, Mass., 1965), pp. 282 and 294]. A word of caution regarding this terminology is in order here. A plane homogeneous waveU(i)(r)=eikn0·r formally has the asymptotic behavior (see Appendix A)eikn0·r∼2πik[Δ(n−n0)eikrr−Δ(n+n0)e−ikrr] as kr→ ∞, with the unit vector n fixed, and Δ is the “spherical” delta function, defined by Eq. (2.12). Hence a plane wave provides both incoming and outgoing contributions at infinity.

Saxon, D. S.

E. Gerjuoy, D. S. Saxon, “Variational principles for the acoustic field,” Phys. Rev. 94, 1445–1458 (1954).
[CrossRef]

Stokes, G. G.

G. G. Stokes, Cambridge Dublin Math. J. 4, 1 (1849). Reprinted in Mathematical and Physical Papers of G. G. Stokes (Cambridge U. Press, Cambridge, 1883), Vol. II, pp. 89–103. For modern treatments of the Stokes relations see, for example, F. A. Jenkins, H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, New York, 1976), pp. 286–288, or E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass, 1974), pp. 91–93.

Vasícek, A.

A. Vasíček, Optics of Thin Films (North-Holland, Amsterdam, 1960), p. 173.

Wolf, E.

E. Wolf, “New theory of radiative energy transfer in free electromagnetic fields,” Phys. Rev. D 13, 869–886 (1976), App. B.
[CrossRef]

K. Miyamoto, E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave—Part I,” J. Opt. Soc. Am. 52, 615–625 (1962), App.
[CrossRef]

E. Wolf, “A scalar representation of electromagnetic fields: II,” Proc. Phys. Soc. London 74, 269–280 (1959), App.
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 13.5, Eq. (107).

Cambridge Dublin Math. J.

G. G. Stokes, Cambridge Dublin Math. J. 4, 1 (1849). Reprinted in Mathematical and Physical Papers of G. G. Stokes (Cambridge U. Press, Cambridge, 1883), Vol. II, pp. 89–103. For modern treatments of the Stokes relations see, for example, F. A. Jenkins, H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, New York, 1976), pp. 286–288, or E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass, 1974), pp. 91–93.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

M. Nazarathy, “A Fabry–Perot interferometer with one phase-conjugate mirror,” Opt. Commun. 45, 117-121 (1983).
[CrossRef]

Phys. Rev.

E. Gerjuoy, D. S. Saxon, “Variational principles for the acoustic field,” Phys. Rev. 94, 1445–1458 (1954).
[CrossRef]

Phys. Rev. D

E. Wolf, “New theory of radiative energy transfer in free electromagnetic fields,” Phys. Rev. D 13, 869–886 (1976), App. B.
[CrossRef]

Proc. Cambr. Phil. Soc.

D. S. Jones, “Removal of an inconsistency in the theory of diffraction,” Proc. Cambr. Phil. Soc. 48, 733–741 (1952), lemma on p. 736. See also Ref. 12.
[CrossRef]

Proc. Phys. Soc. London

E. Wolf, “A scalar representation of electromagnetic fields: II,” Proc. Phys. Soc. London 74, 269–280 (1959), App.
[CrossRef]

Other

In the analogous problem in the theory of quantum-mechanical potential scattering, C(±)are associated with the incoming state and D(±)with the outgoing state [cf. P. Roman, Advanced Quantum Theory (Addison-Wesley, Reading, Mass., 1965), pp. 282 and 294]. A word of caution regarding this terminology is in order here. A plane homogeneous waveU(i)(r)=eikn0·r formally has the asymptotic behavior (see Appendix A)eikn0·r∼2πik[Δ(n−n0)eikrr−Δ(n+n0)e−ikrr] as kr→ ∞, with the unit vector n fixed, and Δ is the “spherical” delta function, defined by Eq. (2.12). Hence a plane wave provides both incoming and outgoing contributions at infinity.

The minus sign is included on the right-hand side of Eq. (2.4) in order that S reduce to the unit matrix in the absence of scattering.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 13.5, Eq. (107).

D. M. Kerns, Plane-Wave Scattering-Matrix Theory of Antennas and Antenna-Antenna Interactions (U.S. Department of Commerce, National Bureau of Standards, Washington, 1981), Monograph 162.

A. Vasíček, Optics of Thin Films (North-Holland, Amsterdam, 1960), p. 173.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 111.

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Figures (4)

Fig. 1
Fig. 1

Illustrating the notation.

Fig. 2
Fig. 2

The far fields in the half-spaces R + and R on either side of the scatterer.

Fig. 3
Fig. 3

Illustrating the significance of the elements of the partitioned S matrix as generalized transmission and reflection coefficients.

Fig. 4
Fig. 4

Illustrating the notation relating to the derivation of the Stokes relations for stratified dielectric media.

Equations (106)

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τ t + r 2 = 1 ,
ρ + r = 0.
U ( r ) = i k 2 π σ ( + ) C ( ) ( n ) e i k n · r d Ω + i k 2 π σ ( ) D ( ) ( n ) e i k n · r d Ω ,
U ( r ) = i k 2 π σ ( ) C ( + ) ( n ) e i k n · r d Ω + i k 2 π σ ( + ) D ( + ) ( n ) e i k n · r d Ω .
σ ( + ) : n 2 = 1 , n z 0 ,
σ ( ) : n 2 = 1 , n z 0 .
U ( r u ) C ( ± ) ( u ) e i k r r + D ( ± ) ( u ) e i k r r as k r ,
D ( n ) = S ( n , n ) C ( n ) d Ω ,
C ( n ) = [ C ( ) ( n ) C ( + ) ( n ) ] , D ( n ) = [ D ( + ) ( n ) D ( ) ( n ) ]
D ( + ) ( n ) = σ ( + ) S ( + ) ( n , n ) C ( ) ( n ) d Ω σ ( ) S ( + + ) ( n , n ) C ( + ) ( n ) d Ω ,
D ( ) ( n ) = σ ( + ) S ( ) ( n , n ) C ( ) ( n ) d Ω σ ( ) S ( + ) ( n , n ) C ( + ) ( n ) d Ω ,
S ( n , n ) = [ S ( + ) ( n , n ) S ( + + ) ( n , n ) S ( ) ( n , n ) S ( + ) ( n , n ) ] .
S ( + ) ( n , n ) : n z > 0 , n z > 0 ,
S ( + + ) ( n , n ) : n z > 0 , n z > 0 ,
S ( ) ( n , n ) : n z > 0 , n z > 0 ,
S ( + ) ( n , n ) : n z > 0 , n z > 0 .
U ( r u ) F 1 ( u ) e i k r r + F 2 ( u ) e i k r r , as k r ,
F 1 ( n ) = σ S ( n , n ) F 2 ( n ) d Ω ,
σ S * ( n , n ) S ( n , n ) d Ω = Δ ( n , n ) ,
σ S ( n , n ) S * ( n , n ) d Ω = Δ ( n n ) ,
Δ ( n n ) = δ ( θ θ ) δ ( φ φ ) | sin θ | ,
S ( n , n ) = S ( n , n ) .
U ( i ) ( r ) = e i k n 0 · r ( n 0 2 = 1 ) ,
F 1 ( n ) = 2 π i k S ( n , n 0 ) ,
F 2 ( n ) = 2 π i k Δ ( n + n 0 ) .
U ( i ) ( r ) = e i k n 0 · r
C ( ± ) ( n ) = 2 π i k Δ ( n n 0 ) ,
D ( ± ) ( n ) = 2 π i k S ( n , n 0 ) ,
U out ( r ; n 0 = σ ( ) S ( n , n 0 ) e i k n · r d Ω when r R ,
= σ ( ) S ( n , n 0 ) e i k n · r d Ω when r R + ,
t ( n , n 0 ) S ( + ) ( n , n 0 ) n z > 0 , n 0 z > 0 ,
r ( n , n 0 ) S ( ) ( n , n 0 ) n z > 0 , n 0 z > 0 .
τ ( n , n 0 ) S ( + ) ( n , n 0 ) n z > 0 , n 0 z > 0 ,
ρ ( n , n 0 ) S ( + + ) ( n , n 0 ) n z > 0 , n 0 z > 0
S ( n , n ) = [ t ( n , n ) ρ ( n , n ) r ( n , n ) τ ( n , n ) ] .
t ( n , n ) = τ ( n , n ) ,
τ ( n , n ) = t ( n , n ) ,
ρ ( n , n ) = ρ ( n , n ) ,
r ( n , n ) = r ( n , n ) .
σ ( ) S ( n , n ) S ( n , n ) d Ω + σ ( + ) S ( n , n ) S ( n , n ) d Ω = Δ ( n n ) .
σ ( ) r ( n , n ) r ( n , n ) d Ω + σ ( + ) t ( n , n ) t ( n , n ) d Ω = Δ ( n n ) .
σ ( ) τ * ( n , n ) r ( n , n ) d Ω + σ ( + ) ρ ( n , n ) t ( n , n ) d Ω = 0.
σ ( ) r * ( n , n ) τ ( n , n ) d Ω + σ ( + ) t ( n , n ) ρ ( n , n ) d Ω = 0.
σ ( ) τ ( n , n ) τ ( n , n ) d Ω + σ ( + ) ρ ( n , n ) ρ ( n , n ) d Ω = Δ ( n n ) .
σ S ( n , n ) S ( n , n ) d Ω = I Δ ( n n ) .
σ ( ) ρ ( n , n ) ρ * ( n , n ) d Ω + σ ( + ) t ( n , n ) t * ( n , n ) d Ω = Δ ( n n ) ,
σ ( ) τ ( n , n ) ρ * ( n , n ) d Ω + σ ( + ) r ( n , n ) t * ( n , n ) d Ω = 0.
σ ( ) ρ ( n , n ) τ * ( n , n ) d Ω + σ ( + ) t ( n , n ) r * ( n , n ) d Ω = 0.
σ ( ) τ ( n , n ) τ * ( n , n ) d Ω + σ ( + ) r ( n , n ) r * ( n , n ) d Ω = Δ ( n n ) .
σ S ( n , n ) S ( n , n ) d Ω = I Δ ( n , n ) .
σ ( ) ρ * ( n , n ) ρ ( n , n ) d Ω + σ ( + ) τ * ( n , n ) τ ( n , n ) d Ω = Δ ( n n ) .
σ ( + ) ρ * ( n , n ) ρ ( n , n ) d Ω + σ ( ) τ * ( n , n ) τ ( n , n ) d Ω = Δ ( n n ) ,
σ ( + ) σ ( ) ,
t τ ,
ρ r .
r ( n , n ) = r ¯ ( n ) Δ [ n n r ( n ) ]
t ( n , n ) = t ¯ ( n ) Δ [ n n t ( n ) ] ,
σ ( ) r * ( n , n ) r ( n , n ) d Ω = σ ( ) r * ( n ) r ( n ) Δ [ n n r ( n ) ] Δ [ n n r ( n ) ] d Ω .
n r ( n ) ( π θ , φ ) , n r ( n ) ( π θ , φ ) ,
σ ( ) r * ( n , n ) r ( n , n ) d Ω = r * ( n ) r ( n ) σ ( ) Δ [ n n r ( n ) ] Δ [ n n r ( n ) ] d Ω .
σ ( ) Δ [ n n r ( n ) ] Δ [ n n r ( n ) ] d Ω = Δ [ n r ( n ) n r ( n ) ] .
σ ( ) Δ [ n n r ( n ) ] Δ [ n n r ( n ) ] d Ω = δ [ π θ ( π θ ) ] δ ( φ φ ) | sin θ | = Δ ( n n ) .
σ ( ) r * ( n , n ) r ( n , n ) d Ω = r ¯ * ( n ) r ¯ ( n ) Δ ( n n ) .
σ ( + ) r * ( n , n ) t ( n , n ) d Ω = t ¯ * ( n ) t ¯ ( n ) Δ ( n n ) .
r ¯ * ( n ) r ¯ ( n ) + t ¯ * ( n ) t ¯ ( n ) = 1.
ρ ( n , n ) = ρ ¯ ( n ) Δ [ n n ρ ( n ) ] ,
τ ( n , n ) = τ ¯ ( n ) Δ [ n n τ ( n ) ] .
τ ( n , n ) = τ ¯ ( n ) Δ [ n n τ ( n ) ] .
τ ( n , n ) = τ ¯ ( n ) Δ ( n + n ) .
t ( n , n ) = t ¯ ( n ) Δ ( n n ) .
τ ¯ ( n ) = t ¯ ( n ) .
r ¯ * ( n ) r ¯ ( n ) + t ¯ * ( n ) τ ¯ ( n ) = 1.
σ ( ) τ * ( n , n ) r ( n , n ) d Ω = τ ¯ * ( n ) r ¯ ( n ) σ ( ) Δ [ n n τ ( n ) ] Δ [ n n r ( n ) ] d Ω .
σ ( ) τ * ( n , n ) r ( n , n ) d Ω = τ ¯ * ( n ) r ¯ ( n ) δ [ θ ( π 0 ) ] δ ( φ φ ) | sin θ | .
σ ( ) ρ * ( n , n ) t ( n , n ) d Ω = ρ ¯ * ( n ) t ¯ ( n ) δ [ θ + θ π ] δ ( φ φ ) | sin θ | .
τ ¯ * ( n r ) r ¯ ( n ) + ρ ¯ * ( n r ) t ¯ ( n ) = 0 ,
t ¯ * ( n r ) r ¯ ( n ) + ρ ¯ * ( n r ) t ¯ ( n ) = 0.
t ¯ * ( n ) r ¯ ( n ) + t ¯ ( n ) ρ ¯ * ( n ) = 0.
2 V ( r ) + k 2 V ( r ) = 0
V ( r ) = σ a ( n ) e i k n · r d Ω .
a ( n ) = k 2 ( 2 π ) 3 lim + 0 k k + d K V ( r ) e i K n · r d 3 r .
V ( r u ) 2 π i k [ a ( u ) e i k r r a ( u ) e i k r r ] as k r .
V ( r ) = e i k n 0 · r .
V ( r ) e i K n · r d 3 r = exp [ i ( k n 0 K n ) · r ] d 3 r = ( 2 π ) 3 δ ( 3 ) ( k n 0 K n ) ,
a ( n ) = k 2 lim + 0 k k + δ ( 3 ) ( k n 0 K n ) d K .
δ ( 3 ) ( k n 0 K n ) = ( 1 / k 2 ) Δ ( n n 0 ) δ ( k K ) ,
a ( n ) = Δ ( n n 0 ) .
e i k n 0 · r 2 π i k [ Δ ( n n 0 ) e i k r r Δ ( n + n 0 ) e i k r r ] as k r .
C ( ± ) ( n ) = 2 π i k Δ ( n n 0 ) ,
D ( ± ) ( n ) = 2 π i k Δ ( n n 0 ) .
e i k n 0 · r = σ ( + ) Δ ( n n 0 ) e i k n · r d Ω + σ ( ) Δ ( n n 0 ) e i k n · r d Ω ,
e i k n 0 · r = σ ( ) Δ ( n n 0 ) e i k n · r d Ω + σ ( + ) Δ ( n n 0 ) e i k n · r d Ω .
U ( i ) ( r ) = e i k n 0 · r .
U ( r n ) e i k n 0 · r + A ( n , n 0 ) e i k r r , k r ,
U ( r n ) F 1 ( n ) e i k r r + F 2 ( n ) e i k r r , as k r ,
F 1 ( n ) = 2 π i k Δ ( n n 0 ) + A ( n , n 0 ) ,
F 2 ( n ) = 2 π i k Δ ( n + n 0 ) .
A ( n , n 0 ) = 2 π i k [ S ( n , n 0 ) Δ ( n n 0 ) ] .
F 1 ( n ) = 2 π i k S ( n , n 0 ) .
Δ ( n n ) = δ ( θ θ ) δ ( φ φ ) | sin θ | ,
( 4 π ) Δ ( n n ) Δ ( n n ) d Ω = 1 | sin θ ' | 0 π 0 2 π δ ( θ θ ) δ ( φ φ ) δ ( θ θ ) δ ( φ φ ) | sin θ | sin θ d θ d φ .
( 4 π ) Δ ( n n ) Δ ( n n ) d Ω = 0 π δ ( θ θ ) δ ( θ θ ) d θ × 0 2 π δ ( φ φ ) δ ( φ φ ) d φ .
( 4 π ) Δ ( n n ) Δ ( n n ) d Ω = δ ( θ θ ) δ ( φ φ ) | sin θ |
( 4 π ) Δ ( n n ) Δ ( n n ) d Ω = Δ ( n n ) .
U(i)(r)=eikn0·r
eikn0·r2πik[Δ(nn0)eikrrΔ(n+n0)eikrr]

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