Abstract

Energy balance and reciprocity relations are studied for harmonic inhomogeneous plane waves that are incident upon a stack of continuous absorbing dielectric media that are macroscopically characterized by their electric and magnetic permittivities and their conductivities. New cross terms between parallel electric and parallel magnetic modes are identified in the fully generalized Poynting vector. The symmetry and the relations between the general Fresnel coefficients are investigated in the context of energy balance at the interface. The contributions of the so-called mixed Poynting vector are discussed in detail. In particular a new transfer matrix is introduced for energy fluxes in thin-film optics based on the Poynting and mixed Poynting vectors. Finally, the study of reciprocity relations leads to a generalization of a theorem of reversibility for conducting and dielectric media.

© 1994 Optical Society of America

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References

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  1. M. A. Dupertuis, M. Proctor, B. Acklin, “Generalization of complex Snell–Descartes and Fresnel laws,” J. Opt. Soc. Am. A 11, 1159–1166 (1994).
    [CrossRef]
  2. Z. Knittl, “The principle of reversibility and thin film optics,” Opt. Acta 17, 33–45 (1962).
    [CrossRef]
  3. B. Chen, “Wavevector space method for wave propagation in bounded media,” Ph.D. dissertation (Worcester Polytechnic Institute, Worcester, Mass., 1992).
  4. B. Chen, D. F. Nelson, “Wavevector space method and its application to the optics near an exciton resonance,” Solid State Commun. 86, 769–773 (1993).
    [CrossRef]
  5. Z. Y. Ou, L. Mandel, “Derivation of reciprocity relations for a beam splitter from energy balance,” Am. J. Phys. 57, 66–67 (1989).
    [CrossRef]
  6. I. Santavý, “On the reversibility of light beams in conducting media,” Opt. Acta 8, 301–307 (1961).
    [CrossRef]
  7. Z. Knittl, “On the energy balance in the optics of metals,” Czech. J. Phys. 9, 133–145 (1959).
    [CrossRef]
  8. M. Born, R. Ladenburg, “Über das Verhältnis von Emissions- und Absorptionsvermögen bei Stark absorbieren- den Körpern,” Phys. Z. 12, 198–202 (1911).
  9. C. von Fragstein, “Energieübergang an der Grenze zweier absorbierender Medien mit einer Anwendung auf die Wärmestrahlung in absorbierenden Körpern,” Ann. Phys. Leipz. 7(6), 63–72 (1950).
    [CrossRef]
  10. C. von Fragstein, “The history of the mixed Poynting vector,” Abh. Brauns. Wiss. Ges. 34, 25–29 (1987).
  11. H. C. Chen, “Theory of electromagnetic waves,” (McGraw-Hill, New York, 1983).
  12. K. H. Beckman, B. Caspar, “On the energy balance for the passage of light through a thin absorbing film,” Philips Res. Rep. 20, 190–205 (1965).
  13. A. Vasicek, Optics of Thin Films (North-Holland, Amsterdam, 1960), p. 326;“The general definition of reflected and transmitted light intensity for a boundary surface between two arbitrary media,” Thin Films 1, 7–23 (1968), and references therein.
  14. C. von Fragstein, F. R. Kessler, “Comment on the article by A. K. S. Thakur: transmission of electromagnetic waves at the boundary of two lossy media,” Optik 66, 9–17 (1983).
  15. H. D. Geiler, K. Hehl, D. Stock, “Deposition of laser energy into inhomogeneous layer systems,” Phys. Status Solidi A 78, 193–199 (1983).
    [CrossRef]
  16. A. Yariv, P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, New York, 1984).

1994 (1)

1993 (1)

B. Chen, D. F. Nelson, “Wavevector space method and its application to the optics near an exciton resonance,” Solid State Commun. 86, 769–773 (1993).
[CrossRef]

1989 (1)

Z. Y. Ou, L. Mandel, “Derivation of reciprocity relations for a beam splitter from energy balance,” Am. J. Phys. 57, 66–67 (1989).
[CrossRef]

1987 (1)

C. von Fragstein, “The history of the mixed Poynting vector,” Abh. Brauns. Wiss. Ges. 34, 25–29 (1987).

1983 (2)

C. von Fragstein, F. R. Kessler, “Comment on the article by A. K. S. Thakur: transmission of electromagnetic waves at the boundary of two lossy media,” Optik 66, 9–17 (1983).

H. D. Geiler, K. Hehl, D. Stock, “Deposition of laser energy into inhomogeneous layer systems,” Phys. Status Solidi A 78, 193–199 (1983).
[CrossRef]

1965 (1)

K. H. Beckman, B. Caspar, “On the energy balance for the passage of light through a thin absorbing film,” Philips Res. Rep. 20, 190–205 (1965).

1962 (1)

Z. Knittl, “The principle of reversibility and thin film optics,” Opt. Acta 17, 33–45 (1962).
[CrossRef]

1961 (1)

I. Santavý, “On the reversibility of light beams in conducting media,” Opt. Acta 8, 301–307 (1961).
[CrossRef]

1959 (1)

Z. Knittl, “On the energy balance in the optics of metals,” Czech. J. Phys. 9, 133–145 (1959).
[CrossRef]

1950 (1)

C. von Fragstein, “Energieübergang an der Grenze zweier absorbierender Medien mit einer Anwendung auf die Wärmestrahlung in absorbierenden Körpern,” Ann. Phys. Leipz. 7(6), 63–72 (1950).
[CrossRef]

1911 (1)

M. Born, R. Ladenburg, “Über das Verhältnis von Emissions- und Absorptionsvermögen bei Stark absorbieren- den Körpern,” Phys. Z. 12, 198–202 (1911).

Acklin, B.

Beckman, K. H.

K. H. Beckman, B. Caspar, “On the energy balance for the passage of light through a thin absorbing film,” Philips Res. Rep. 20, 190–205 (1965).

Born, M.

M. Born, R. Ladenburg, “Über das Verhältnis von Emissions- und Absorptionsvermögen bei Stark absorbieren- den Körpern,” Phys. Z. 12, 198–202 (1911).

Caspar, B.

K. H. Beckman, B. Caspar, “On the energy balance for the passage of light through a thin absorbing film,” Philips Res. Rep. 20, 190–205 (1965).

Chen, B.

B. Chen, D. F. Nelson, “Wavevector space method and its application to the optics near an exciton resonance,” Solid State Commun. 86, 769–773 (1993).
[CrossRef]

B. Chen, “Wavevector space method for wave propagation in bounded media,” Ph.D. dissertation (Worcester Polytechnic Institute, Worcester, Mass., 1992).

Chen, H. C.

H. C. Chen, “Theory of electromagnetic waves,” (McGraw-Hill, New York, 1983).

Dupertuis, M. A.

Geiler, H. D.

H. D. Geiler, K. Hehl, D. Stock, “Deposition of laser energy into inhomogeneous layer systems,” Phys. Status Solidi A 78, 193–199 (1983).
[CrossRef]

Hehl, K.

H. D. Geiler, K. Hehl, D. Stock, “Deposition of laser energy into inhomogeneous layer systems,” Phys. Status Solidi A 78, 193–199 (1983).
[CrossRef]

Kessler, F. R.

C. von Fragstein, F. R. Kessler, “Comment on the article by A. K. S. Thakur: transmission of electromagnetic waves at the boundary of two lossy media,” Optik 66, 9–17 (1983).

Knittl, Z.

Z. Knittl, “The principle of reversibility and thin film optics,” Opt. Acta 17, 33–45 (1962).
[CrossRef]

Z. Knittl, “On the energy balance in the optics of metals,” Czech. J. Phys. 9, 133–145 (1959).
[CrossRef]

Ladenburg, R.

M. Born, R. Ladenburg, “Über das Verhältnis von Emissions- und Absorptionsvermögen bei Stark absorbieren- den Körpern,” Phys. Z. 12, 198–202 (1911).

Mandel, L.

Z. Y. Ou, L. Mandel, “Derivation of reciprocity relations for a beam splitter from energy balance,” Am. J. Phys. 57, 66–67 (1989).
[CrossRef]

Nelson, D. F.

B. Chen, D. F. Nelson, “Wavevector space method and its application to the optics near an exciton resonance,” Solid State Commun. 86, 769–773 (1993).
[CrossRef]

Ou, Z. Y.

Z. Y. Ou, L. Mandel, “Derivation of reciprocity relations for a beam splitter from energy balance,” Am. J. Phys. 57, 66–67 (1989).
[CrossRef]

Proctor, M.

Santavý, I.

I. Santavý, “On the reversibility of light beams in conducting media,” Opt. Acta 8, 301–307 (1961).
[CrossRef]

Stock, D.

H. D. Geiler, K. Hehl, D. Stock, “Deposition of laser energy into inhomogeneous layer systems,” Phys. Status Solidi A 78, 193–199 (1983).
[CrossRef]

Vasicek, A.

A. Vasicek, Optics of Thin Films (North-Holland, Amsterdam, 1960), p. 326;“The general definition of reflected and transmitted light intensity for a boundary surface between two arbitrary media,” Thin Films 1, 7–23 (1968), and references therein.

von Fragstein, C.

C. von Fragstein, “The history of the mixed Poynting vector,” Abh. Brauns. Wiss. Ges. 34, 25–29 (1987).

C. von Fragstein, F. R. Kessler, “Comment on the article by A. K. S. Thakur: transmission of electromagnetic waves at the boundary of two lossy media,” Optik 66, 9–17 (1983).

C. von Fragstein, “Energieübergang an der Grenze zweier absorbierender Medien mit einer Anwendung auf die Wärmestrahlung in absorbierenden Körpern,” Ann. Phys. Leipz. 7(6), 63–72 (1950).
[CrossRef]

Yariv, A.

A. Yariv, P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, New York, 1984).

Yeh, P.

A. Yariv, P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, New York, 1984).

Abh. Brauns. Wiss. Ges. (1)

C. von Fragstein, “The history of the mixed Poynting vector,” Abh. Brauns. Wiss. Ges. 34, 25–29 (1987).

Am. J. Phys. (1)

Z. Y. Ou, L. Mandel, “Derivation of reciprocity relations for a beam splitter from energy balance,” Am. J. Phys. 57, 66–67 (1989).
[CrossRef]

Ann. Phys. Leipz. (1)

C. von Fragstein, “Energieübergang an der Grenze zweier absorbierender Medien mit einer Anwendung auf die Wärmestrahlung in absorbierenden Körpern,” Ann. Phys. Leipz. 7(6), 63–72 (1950).
[CrossRef]

Czech. J. Phys. (1)

Z. Knittl, “On the energy balance in the optics of metals,” Czech. J. Phys. 9, 133–145 (1959).
[CrossRef]

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

Opt. Acta (2)

Z. Knittl, “The principle of reversibility and thin film optics,” Opt. Acta 17, 33–45 (1962).
[CrossRef]

I. Santavý, “On the reversibility of light beams in conducting media,” Opt. Acta 8, 301–307 (1961).
[CrossRef]

Optik (1)

C. von Fragstein, F. R. Kessler, “Comment on the article by A. K. S. Thakur: transmission of electromagnetic waves at the boundary of two lossy media,” Optik 66, 9–17 (1983).

Philips Res. Rep. (1)

K. H. Beckman, B. Caspar, “On the energy balance for the passage of light through a thin absorbing film,” Philips Res. Rep. 20, 190–205 (1965).

Phys. Status Solidi A (1)

H. D. Geiler, K. Hehl, D. Stock, “Deposition of laser energy into inhomogeneous layer systems,” Phys. Status Solidi A 78, 193–199 (1983).
[CrossRef]

Phys. Z. (1)

M. Born, R. Ladenburg, “Über das Verhältnis von Emissions- und Absorptionsvermögen bei Stark absorbieren- den Körpern,” Phys. Z. 12, 198–202 (1911).

Solid State Commun. (1)

B. Chen, D. F. Nelson, “Wavevector space method and its application to the optics near an exciton resonance,” Solid State Commun. 86, 769–773 (1993).
[CrossRef]

Other (4)

B. Chen, “Wavevector space method for wave propagation in bounded media,” Ph.D. dissertation (Worcester Polytechnic Institute, Worcester, Mass., 1992).

A. Yariv, P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, New York, 1984).

A. Vasicek, Optics of Thin Films (North-Holland, Amsterdam, 1960), p. 326;“The general definition of reflected and transmitted light intensity for a boundary surface between two arbitrary media,” Thin Films 1, 7–23 (1968), and references therein.

H. C. Chen, “Theory of electromagnetic waves,” (McGraw-Hill, New York, 1983).

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Equations (99)

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r + r + + t + t = 1 ,
r + t + + t + r = 0 ,
r r + t t + = 1 ,
r t + t r + = 0 .
r + r + * + t + t * = 1 ,
r + t + * + t + r * = 0 ,
r r * + t t + * = 1 ,
r t * + t t + * = 0 ,
r + r + * = 1
E ( r , t ) = ½ [ E ( r ) exp ( i ω t ) + c.c . ] ,
H ( r , t ) = ½ [ H ( r ) exp ( i ω t ) + c.c . ] ,
S ( r , t ) = E ( r , t ) H ( r , t ) = ½ Re [ E ( r ) H * ( r ) ] = Re [ S ( r ) ] .
S ( r ) + ½ J * ( r ) E ( r ) 2 i ω [ w e ( r ) w m ( r ) ] = 0 ,
w e ( r ) = ¼ E ( r ) D * ( r ) ,
w m ( r ) = ¼ B ( r ) H * ( r ) ,
E ( r ) = E exp ( i k r ) ,
D ( r ) = 0 E ( r ) ,
H ( r ) = H exp ( i k r ) ,
B ( r ) = μ 0 μ H ( r ) ,
k = ñ k 0 n ,
S ( r ) = S exp [ 2 Im ( k ) r ] = 1 2 ω μ 0 μ [ | E | 2 k * ( E k * ) E * ] exp [ 2 Im ( k ) r ] ,
w e = ¼ 0 | E | 2 exp [ 2 Im ( k ) r ] ,
w m = ¼ μ 0 μ | H | 2 exp [ 2 Im ( k ) r ] .
| H | 2 = 2 k S ω μ 0 μ = 1 ω μ 0 μ ( | k | 2 | E | 2 | k E * | 2 ) .
S ( r ) = i ω 2 ( 0 * | E | 2 μ 0 μ | H | 2 ) exp [ 2 Im ( k ) r ] ,
k S = 1 2 ω μ 0 μ [ | E | 2 ( k k * ) ( E k * ) ( k E * ) ] ,
S L ( r ) = ŝ S L ( r ) = ŝ S R ( r ) = S R ( r ) .
E PE = E PE q 1 , H PE = H PE q 2 ,
E PM = E PM q 2 , H PM = H PM q 1 ,
q 1 = ŝ n [ ( ŝ n ) 2 ] 1 / 2 , q 2 = n q 1 .
E PE = q 1 E , H PE = q 2 H ,
E PM = q 2 E , H PM = q 1 H ,
E PE = ñ μ 0 0 H PM ,
H PE = ñ μ 0 μ 0 E PE .
S = ½ ( E H * ) = S PE + S PM + S X ,
S PE = ½ E PE H PE * ( q 1 q 2 * ) ,
S PM = ½ E PM H PM * ( q 2 q 1 * ) ,
S X = ½ [ E PE H PM * ( q 1 q 1 * ) + E PM H PE * ( q 2 q 2 * ) ] .
q 1 q 1 * = ( n n * ) ŝ | ( ŝ n ) 2 | ŝ ,
q 2 q 2 * = [ ( n n * ) q 1 * ] q 1 + [ ( q 1 q 1 * ) n * ] n .
ŝ ( n n * ) = 0 .
q 1 q 1 * = 0 , q 2 q 2 * = 0 .
S i + S r + S m = S t = S t ,
S j = ½ ( E j H j * ) , j = i , r , t ,
S i = ½ ŝ ( E i H r * + E r H i * ) .
S i = S i ,
S i = T S i .
ϕ i = Re ( S i ) = 1 2 ω μ 0 μ i Re [ | E i | 2 k i * ( E i k i * ) E i * ] ,
ϕ r = Re ( S i ) = ϕ i ,
ϕ t = Re ( S i ) = Re ( T S i ) = T ϕ i .
S i = S i ,
ϕ m = Re ( S m ) = Re ( S i ) = ϕ i ,
ϕ i + ϕ r + ϕ m = ϕ i .
PE = PE = | r PE | 2 ,
T PE = | t PE | 2 ŝ k t * / μ t ŝ k i * / μ i ,
PE = r PE r PE * = 2 i Im ( r PE ) .
X = r PE * r PM ,
T X = t PE * t PM | s k t | 2 | ŝ k i | 2 μ i i μ t t ,
X = ( r PE * + r PM ) .
J + T J = 1 + J , J = PE , PM , X .
( ŝ k i ) ( ŝ k t * ) = ( ŝ k i * ) ( ŝ k t ) ,
ŝ k i i ŝ k t * t * = ŝ k i * i * ŝ k t t
n i κ t = n t κ i ,
= | ñ i ñ t ñ i + ñ t | 2 = ( n i n t ) 2 + ( κ i κ t ) 2 ( n i + n t ) 2 + ( κ i + κ t ) 2 ,
T = 1 n i Re ( T ñ i * ) = 1 n i Re ( 4 ñ i ñ t * | ñ i + ñ t | 2 ñ i * ) = 4 n i n t ( 1 + κ i 2 n i 2 ) ( n i + n t ) 2 + ( κ i + κ t ) 2 ,
= 4 κ i ( n i κ t n t κ i ) n i [ ( n i + n t ) 2 + ( κ i + κ t ) 2 ] .
S m ( r ) = ½ ( E i H r * ) exp [ i ( k i k r * ) r ] + ½ ( E i H i * ) exp [ i ( k r k i * ) r ] = ½ ( E i H r * ) exp { 2 i Re ( ŝ k i ) ( ŝ r ) 2 Im [ ŝ ( ŝ k i ) ] r } + ½ ( E r H i * ) exp { + 2 i Re ( ŝ k i ) ( ŝ r )
2 Im [ ŝ ( ŝ k i ) ] r } ,
S i ( r ) = ½ { r PE exp [ 2 i n i k 0 ( ŝ r ) ] c.c . }
κ i n i > κ t n t > 0.1 .
E ( n ) ( r ) = [ E PE + ( n ) + E PM + ( n ) ] exp [ i k + ( n ) ( r r n 1 ) ] + [ E PE ( n ) + E PM ( n ) ] exp [ i k ( n ) ( r r n 1 ) ] ,
H ( n ) ( r ) = [ H PE + ( n ) + H PM + ( n ) ] exp [ i k + ( n ) ( r r n 1 ) ] + [ H PE ( n ) + H PM ( n ) ] exp [ i k ( n ) ( r r n 1 ) ] ,
ŝ k + ( n ) = ŝ k ( n ) = ŝ k + ( m ) .
s k + ( n ) = s k ( n ) = { [ k + ( n ) ] 2 [ ŝ ( ŝ k + ( m ) ) ] 2 } 1 / 2 .
E PE + ( n ) exp [ i ŝ k + n d n ] + E PE n exp [ i ŝ k ( n ) d n ] = E PE + ( n + 1 ) + E PE ( n + 1 ) ,
1 μ ( n ) { [ ŝ k + ( n ) ] E PE + ( n ) exp [ i ŝ k ( n ) d n ] + [ ŝ k ( n ) ] E PE ( n ) exp [ i ŝ k ( n ) d n ] } = 1 μ ( n + 1 ) { [ ŝ k + ( n + 1 ) ] E PE + ( n + 1 ) + [ ŝ k ( n + 1 ) ] E PE ( n + 1 ) } .
( E PE + ( n ) E PE ( n ) ) = [ A PE ( n ) PE ( n ) C PE ( n ) D PE ( n ) ] ( E PE + ( n + 1 ) E PE ( n + 1 ) ) ,
A PE ( n ) = ½ exp ( i ŝ k + ( n ) d n ) [ 1 + μ ( n ) μ ( n + 1 ) ŝ k + ( n + 1 ) ŝ k + ( n ) ] ,
PE ( n ) = ½ exp ( i ŝ k + ( n ) d n ) [ 1 μ ( n ) μ ( n + 1 ) ŝ k + ( n + 1 ) ŝ k + ( n ) ] ,
C PE ( n ) = ½ exp ( i ŝ k + ( n ) d n ) [ 1 μ ( n ) μ ( n + 1 ) ŝ k + ( n + 1 ) ŝ k + ( n ) ] ,
D PE ( n ) = ½ exp ( i ŝ k + ( n ) d n ) [ 1 + μ ( n ) μ ( n + 1 ) ŝ k + ( n + 1 ) ŝ k + ( n ) ] ,
S PE + ( n + 1 ) = T PE + ( n + 1 ) S PE + ( n ) PE ( n + 1 ) S PE ( n + 1 ) + PE m + ( n + 1 ) S PE m + ( n + 1 ) + PE m ( n + 1 ) S PE m ( n + 1 ) ,
S PE ( n ) = T PE ( n ) S PE ( n + 1 ) PE + ( n ) S PE + ( n ) PE m + ( n ) S PE m + ( n + 1 ) + PE m ( n ) S PE m ( n + 1 ) ,
S PE m + ( n ) = PE + ( n ) S PE + ( n ) + PE ( n ) S PE ( n ) PE m + ( n ) S PE m + ( n + 1 ) ,
S PE m ( n ) = ( S PE m + ( n ) ) * ,
( S PE + ( n ) S PE m + ( n ) S PE m ( n ) S PE ( n ) ) = [ PE ( n ) PE ( n ) G PE ( n ) PE ( n ) PE ( n ) J PE ( n ) K PE ( n ) PE ( n ) N PE ( n ) O PE ( n ) P PE ( n ) Q PE ( n ) T PE ( n ) u PE ( n ) V PE ( n ) W PE ( n ) ] ( S PE + ( n + 1 ) S PE m + ( n + 1 ) S PE m ( n + 1 ) S PE ( n + 1 ) ) ,
E = μ 0 μ t H , ( 0 E ) = ρ , H = σ E + 0 t E , ( μ H ) = 0 .
E = E , H = H σ = σ .
k = ñ k 0 n = ñ * k 0 n * = k * .
( E PE ( 0 ) E PE + ( N + 1 ) ) = [ PE + Θ PE Θ PE + PE ] ( E PE + ( 0 ) E PE ( N + 1 ) ) = PE ( E PE + ( 0 ) E PE ( N + 1 ) ) .
( E PE ( 0 ) * E PE + ( N + 1 ) * ) = PE * ( E PE ( N + 1 ) * E PE + ( 0 ) * ) .
{ exp [ i ( k r ω t ) ] } * = exp [ + i ( k r + ω t ) ] .
PE PE * = 1 .
PE + PE + * + Θ PE + Θ PE * = 1 ,
PE + PE + * + Θ PE + PE * = 0 ,
PE PE * + Θ PE Θ PE + * = 1 ,
PE Θ PE * + Θ PE PE * = 0 ,
Θ PE + Θ PE * = Θ PE Θ PE + * ,
PE + PE + * = PE PE * ,

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