Abstract

The energy flux within an opaque medium near an interface is not the sum of an incident plus a reflected term, as there is a synergistic contribution to the time-averaged Poynting vector that involves simultaneously both the incident and reflected fields. Therefore, the well-known formula R+T=1, where R is the reflectance and T the transmittance, does not hold, and furthermore, R and T lose their accepted meanings. We illustrate the perils of assuming energy flux additivity by calculating the transmission and reflection spectrum of a film over a substrate normally illuminated by incoherent light at frequencies in the neighborhood of an optical resonance. We also show that the usual relation between the scattering, absorption, and extinction cross sections for particles immersed within a dissipative host have to be modified to account for the nonadditivity.

© 2005 Optical Society of America

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References

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  1. B. Rossi, Optics (Addison-Wesley, 1962), p. 414.
  2. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), p. 311.
  3. E. Hecht, Optics, 2nd ed. (Addison-Wesley, 1990), p. 250.
  4. L. E. R. Petersson, G. S. Smith, “Role of evanescent waves in power calculations for counterpropagating beams,” J. Opt. Soc. Am. A 20, 2378–2384 (2003).
    [CrossRef]
  5. M. Klein, Optics (Wiley, 1970), p. 573.
  6. J. Mathieu, Optics (Pergamon, 1975), p. 15.
  7. Ref. [3], p. 102.
  8. Ref. [2], p. 745.
  9. Ref. [1], p. 135.
  10. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975), Sect. 7.3.
  11. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983), Sect. 2.8.
  12. R. D. Mattuck, A Guide to Feynman Diagrams in the Many-Body Problem, 2nd ed. (Dover, 1992), Chap. 2.
  13. Ref. [3], Chap. 4.
  14. W. L. Mochán, Centro de Ciencias Físicas, Universidad Nacional Autónoma de México, Apartado Postal 48-3, 62251 Cuernavaca, Morelos, México, and G. P. Ortiz are preparing a manuscript to be called “Energy flow non-additivity for evanescent waves.”
  15. Ph. Balcou, L. Dutriaux, “Dual optical tunneling times in frustrated total internal reflection,” Phys. Rev. Lett. 78, 851–854 (1997).
    [CrossRef]
  16. Vera L. Brudny, Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Cd. Universitaria, Pabellón 1, C1428 EHA Buenos Aires, Argentina, and W. Luis Mochán are preparing a manuscript to be called “Frustrated total internal reflection and the illusion of superluminal propagation.”
  17. Vera L. Brudny, W. Luis Mochán, “On the apparent superluminality of evanescent waves,” Opt. Express 9, 561–566 (2001), (http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-11-561).
    [CrossRef] [PubMed]
  18. W. Luis Mochán, Vera L. Brudny, “Comment on ‘Noncausal time response in frustrated total internal reflection?’” Phys. Rev. Lett. 87, 119101 (2001).
    [CrossRef]
  19. V. Freilikher, M. Pustilnik, I. Yurkevich, “Statistical properties of the reflectance and transmittance of an amplifying random medium,” Phys. Rev. B 56, 5974–5977 (1997).
    [CrossRef]
  20. Wei Jiang, Changde Gong, “Two mechanisms, three stages of the localization of light in a disordered dielectric structure with photonic band gaps,” Phys. Rev. B 60, 12015–12022 (1999).
    [CrossRef]
  21. H. I. Pérez-Aguilar, Esparcimiento de Luz en un Medio Amplificador, Master’s thesis, (CICESE, Ensenada, B. C., México 2003).
  22. W. C. Mundy, J. A. Roux, A. M. Smith, “Mie scattering by spheres in an absorbing medium,” J. Opt. Soc. Am. 64, 1593–1597 (1974).
    [CrossRef]
  23. P. Chýlek, “Light scattering by small particles in an absorbing medium,” J. Opt. Soc. Am. 67, 561–563 (1977).
    [CrossRef]
  24. C. F. Bohren, D. P. Gilra, “Extinction by a spherical particle in an absorbing medium,” J. Colloid Interface Sci. 72, 215–221 (1979).
    [CrossRef]
  25. Q. Fu, W. Sun, “Mie theory for light scattering by a spherical particle in an absorbing medium,” Appl. Opt. 40, 1354–1361 (2001).
    [CrossRef]
  26. I. W. Sudiarta, P. Chýlek, “Mie-scattering formalism for spherical particles embedded in an absorbing medium,” J. Opt. Soc. Am. A 18, 1275–1278 (2001).
    [CrossRef]
  27. G. Barton, Elements of Green’s Functions and Propagation (Oxford U. Press, 1995), Chap. 2.
  28. Ref. [11], p. 73.

2003

2001

1999

Wei Jiang, Changde Gong, “Two mechanisms, three stages of the localization of light in a disordered dielectric structure with photonic band gaps,” Phys. Rev. B 60, 12015–12022 (1999).
[CrossRef]

1997

V. Freilikher, M. Pustilnik, I. Yurkevich, “Statistical properties of the reflectance and transmittance of an amplifying random medium,” Phys. Rev. B 56, 5974–5977 (1997).
[CrossRef]

Ph. Balcou, L. Dutriaux, “Dual optical tunneling times in frustrated total internal reflection,” Phys. Rev. Lett. 78, 851–854 (1997).
[CrossRef]

1979

C. F. Bohren, D. P. Gilra, “Extinction by a spherical particle in an absorbing medium,” J. Colloid Interface Sci. 72, 215–221 (1979).
[CrossRef]

1977

1974

Balcou, Ph.

Ph. Balcou, L. Dutriaux, “Dual optical tunneling times in frustrated total internal reflection,” Phys. Rev. Lett. 78, 851–854 (1997).
[CrossRef]

Barton, G.

G. Barton, Elements of Green’s Functions and Propagation (Oxford U. Press, 1995), Chap. 2.

Bohren, C. F.

C. F. Bohren, D. P. Gilra, “Extinction by a spherical particle in an absorbing medium,” J. Colloid Interface Sci. 72, 215–221 (1979).
[CrossRef]

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983), Sect. 2.8.

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), p. 311.

Brudny, Vera L.

Vera L. Brudny, W. Luis Mochán, “On the apparent superluminality of evanescent waves,” Opt. Express 9, 561–566 (2001), (http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-11-561).
[CrossRef] [PubMed]

W. Luis Mochán, Vera L. Brudny, “Comment on ‘Noncausal time response in frustrated total internal reflection?’” Phys. Rev. Lett. 87, 119101 (2001).
[CrossRef]

Vera L. Brudny, Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Cd. Universitaria, Pabellón 1, C1428 EHA Buenos Aires, Argentina, and W. Luis Mochán are preparing a manuscript to be called “Frustrated total internal reflection and the illusion of superluminal propagation.”

Chýlek, P.

Dutriaux, L.

Ph. Balcou, L. Dutriaux, “Dual optical tunneling times in frustrated total internal reflection,” Phys. Rev. Lett. 78, 851–854 (1997).
[CrossRef]

Freilikher, V.

V. Freilikher, M. Pustilnik, I. Yurkevich, “Statistical properties of the reflectance and transmittance of an amplifying random medium,” Phys. Rev. B 56, 5974–5977 (1997).
[CrossRef]

Fu, Q.

Gilra, D. P.

C. F. Bohren, D. P. Gilra, “Extinction by a spherical particle in an absorbing medium,” J. Colloid Interface Sci. 72, 215–221 (1979).
[CrossRef]

Gong, Changde

Wei Jiang, Changde Gong, “Two mechanisms, three stages of the localization of light in a disordered dielectric structure with photonic band gaps,” Phys. Rev. B 60, 12015–12022 (1999).
[CrossRef]

Hecht, E.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, 1990), p. 250.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983), Sect. 2.8.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975), Sect. 7.3.

Jiang, Wei

Wei Jiang, Changde Gong, “Two mechanisms, three stages of the localization of light in a disordered dielectric structure with photonic band gaps,” Phys. Rev. B 60, 12015–12022 (1999).
[CrossRef]

Klein, M.

M. Klein, Optics (Wiley, 1970), p. 573.

Mathieu, J.

J. Mathieu, Optics (Pergamon, 1975), p. 15.

Mattuck, R. D.

R. D. Mattuck, A Guide to Feynman Diagrams in the Many-Body Problem, 2nd ed. (Dover, 1992), Chap. 2.

Mochán, W. L.

W. L. Mochán, Centro de Ciencias Físicas, Universidad Nacional Autónoma de México, Apartado Postal 48-3, 62251 Cuernavaca, Morelos, México, and G. P. Ortiz are preparing a manuscript to be called “Energy flow non-additivity for evanescent waves.”

Mochán, W. Luis

Vera L. Brudny, W. Luis Mochán, “On the apparent superluminality of evanescent waves,” Opt. Express 9, 561–566 (2001), (http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-11-561).
[CrossRef] [PubMed]

W. Luis Mochán, Vera L. Brudny, “Comment on ‘Noncausal time response in frustrated total internal reflection?’” Phys. Rev. Lett. 87, 119101 (2001).
[CrossRef]

Mundy, W. C.

Pérez-Aguilar, H. I.

H. I. Pérez-Aguilar, Esparcimiento de Luz en un Medio Amplificador, Master’s thesis, (CICESE, Ensenada, B. C., México 2003).

Petersson, L. E. R.

Pustilnik, M.

V. Freilikher, M. Pustilnik, I. Yurkevich, “Statistical properties of the reflectance and transmittance of an amplifying random medium,” Phys. Rev. B 56, 5974–5977 (1997).
[CrossRef]

Rossi, B.

B. Rossi, Optics (Addison-Wesley, 1962), p. 414.

Roux, J. A.

Smith, A. M.

Smith, G. S.

Sudiarta, I. W.

Sun, W.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), p. 311.

Yurkevich, I.

V. Freilikher, M. Pustilnik, I. Yurkevich, “Statistical properties of the reflectance and transmittance of an amplifying random medium,” Phys. Rev. B 56, 5974–5977 (1997).
[CrossRef]

Appl. Opt.

J. Colloid Interface Sci.

C. F. Bohren, D. P. Gilra, “Extinction by a spherical particle in an absorbing medium,” J. Colloid Interface Sci. 72, 215–221 (1979).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Express

Phys. Rev. B

V. Freilikher, M. Pustilnik, I. Yurkevich, “Statistical properties of the reflectance and transmittance of an amplifying random medium,” Phys. Rev. B 56, 5974–5977 (1997).
[CrossRef]

Wei Jiang, Changde Gong, “Two mechanisms, three stages of the localization of light in a disordered dielectric structure with photonic band gaps,” Phys. Rev. B 60, 12015–12022 (1999).
[CrossRef]

Phys. Rev. Lett.

W. Luis Mochán, Vera L. Brudny, “Comment on ‘Noncausal time response in frustrated total internal reflection?’” Phys. Rev. Lett. 87, 119101 (2001).
[CrossRef]

Ph. Balcou, L. Dutriaux, “Dual optical tunneling times in frustrated total internal reflection,” Phys. Rev. Lett. 78, 851–854 (1997).
[CrossRef]

Other

Vera L. Brudny, Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Cd. Universitaria, Pabellón 1, C1428 EHA Buenos Aires, Argentina, and W. Luis Mochán are preparing a manuscript to be called “Frustrated total internal reflection and the illusion of superluminal propagation.”

B. Rossi, Optics (Addison-Wesley, 1962), p. 414.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), p. 311.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, 1990), p. 250.

H. I. Pérez-Aguilar, Esparcimiento de Luz en un Medio Amplificador, Master’s thesis, (CICESE, Ensenada, B. C., México 2003).

M. Klein, Optics (Wiley, 1970), p. 573.

J. Mathieu, Optics (Pergamon, 1975), p. 15.

Ref. [3], p. 102.

Ref. [2], p. 745.

Ref. [1], p. 135.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975), Sect. 7.3.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983), Sect. 2.8.

R. D. Mattuck, A Guide to Feynman Diagrams in the Many-Body Problem, 2nd ed. (Dover, 1992), Chap. 2.

Ref. [3], Chap. 4.

W. L. Mochán, Centro de Ciencias Físicas, Universidad Nacional Autónoma de México, Apartado Postal 48-3, 62251 Cuernavaca, Morelos, México, and G. P. Ortiz are preparing a manuscript to be called “Energy flow non-additivity for evanescent waves.”

G. Barton, Elements of Green’s Functions and Propagation (Oxford U. Press, 1995), Chap. 2.

Ref. [11], p. 73.

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Figures (6)

Fig. 1
Fig. 1

Film of thickness d on which monochromatic light is normally incident from the left. There are two interfaces between the three media [vacuum (1), film (2), and substrate (3)], with refractive indices ( n i ) . The wave vectors ( ± k i , i = 1 3 ) for waves traveling toward the right and left within each medium are shown. The coordinate system is also indicated.

Fig. 2
Fig. 2

Reflectance R M (solid curve) calculated for a film normally illuminated with monochromatic light of frequency ω and its incoherent counterpart R I (dashed curve) as a function of frequency ω. Light is incident from vacuum ( n 1 = 1 ) and the film lies on a substrate with n 3 = 2 . The thickness of the film is d = λ T 2 , and it has a dissipation factor γ = 0.01 ω T and effective weight ω p 2 = ω T 2 . The longitudinal and transverse frequencies ω T and ω L are indicated.

Fig. 3
Fig. 3

Reflectance R M (solid curve) calculated for a film as in Fig. 2 but five times thinner, of width d = λ T 10 , normally illuminated by monochromatic light of frequency ω, and its incoherent counterpart R I (dashed curve).

Fig. 4
Fig. 4

Incoherent reflectance R I of free-standing films ( n 1 = n 3 = 1 ) of widths d = 10 λ p , 11 λ p ; dissipation factors γ = 10 8 , 2 × 10 8 ; and transition weight ω p 2 = ω T 2 .

Fig. 5
Fig. 5

Transmittance T 21 of a single interface 21 for a wave incident from a medium with dielectric response given by Eq. (1) on the interface that separates it from vacuum, as a function of frequency. We took ω p 2 = ω T 2 and γ = 0.01 ω T , as in Figs. 2, 3.

Fig. 6
Fig. 6

Pulse of length 2 L = v g τ 0 incident on (solid curve) and reflected from (dashed curve) an interface (vertical solid line) at different times increasing from the bottom toward the top. Beyond the pseudointerface of width w L (gray section) additivity holds, as there is no overlap and therefore no interference between the incident and the reflected pulses. The directions of motion are indicated (arrows).

Equations (69)

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ϵ 2 ( ω ) = 1 + ω p 2 ω T 2 ω 2 i ω γ ,
r = r 12 + t 12 r 23 t 21 exp ( 2 i k 2 d ) l = 1 [ r 21 r 23 exp ( 2 i k 2 d ) ] l 1 ,
r i j = n i n j n i + n j ,
t i j = 2 n i n i + n j ,
k 2 = n 2 ω c .
r = r 12 + t 12 t 21 r 23 exp ( 2 i k 2 d ) 1 r 21 r 23 exp ( 2 i k 2 d ) .
t = t 12 t 23 exp ( i k 2 d ) 1 r 21 r 23 exp ( i 2 k 2 d ) .
R I = R 12 + T 12 R 23 T 21 exp ( 4 k 2 d ) l = 1 [ R 21 R 23 exp ( 4 k 2 d ) ] l 1 ,
R i j = r i j 2 = n i n j 2 n i + n j 2 ,
T i j = ( n j n i ) t i j 2 = 4 n j n i ( n i ) 2 + ( n i ) 2 n i + n j 2 ,
R I = R 12 + T 12 T 21 R 23 exp ( 4 k 2 d ) 1 R 12 R 23 exp ( 4 k 2 d ) .
T I = T 12 T 23 exp ( 2 k 2 d ) 1 R 12 R 23 exp ( 4 k 2 d ) .
0 R I 1 , 0 T I 1 , 0 R I + T I 1 .
E ( z ) = E i exp ( i k 2 z ) + E r exp ( i k 2 z ) ,
S z ( z ) = I i ( z ) + I r ( z ) + I int ( z ) ,
I int ( z ) = 2 n 2 n 2 Im ( r 21 exp ( 2 i k 2 z ) ) I i = 2 n 2 n 2 [ r 21 sin ( 2 k 2 z ) + r 21 cos ( 2 k 2 z ) ] I i ,
P ( z ) = ω ϵ 2 8 π E ( z ) 2 ,
P int ( z ) = 2 P i Re [ r 21 exp ( 2 i k 2 z ) ]
S z ( z ) = I t = c 8 π n 1 E t 2 = T 21 I i .
T 21 = 1 R 21 + 2 n 2 n 2 r 21 ,
2 L = v g τ 0 ,
E ( z , t ) = 0 d ω ( 2 π ) A ( ω ) [ exp ( i k 2 z ) + r 21 exp ( i k 2 z ) ] exp ( i ω t ) + c.c.
G 2 ( ω ) = 1 2 π Δ ω exp { [ ( ω ω 0 ) 2 2 Δ ω 2 ] }
k 2 ( ω ) k 2 ( ω 0 ) + ( ω ω 0 ) d k 2 d ω ω 0 ,
E ( z ) = E i ( z ) + E r ( z ) + E int ( z ) ,
E int = 2 n 2 n 2 exp ( z 2 2 L 2 ) [ r 21 sin ( 2 k 2 z ) + r 21 cos ( 2 k 2 z ) ] E i
R ̃ 21 S r ( w ) S i ( w ) = R 21 exp ( 4 k 2 w ) ,
T ̃ 21 S t ( 0 ) S i ( w ) = T 21 exp ( 2 k 2 w ) .
R I = R 12 + T ̃ 12 T ̃ 21 R ̃ 23 exp ( 4 k 2 d ̃ ) 1 R ̃ 12 R ̃ 23 exp ( 4 k 2 d ̃ ) ,
T I = T ̃ 12 T ̃ 23 exp ( 2 k 2 d ̃ ) 1 R ̃ 12 R ̃ 23 exp ( 4 k 2 d ̃ ) .
d 2 L ,
Δ j = i ω ϵ 2 ϵ 4 π E i
G ( z , z ) = i exp ( i k z z ) 2 k ,
E ± ( z ) = i ω c A ± ( z ) = s ± E i exp ( ± i k z ) ,
s + = s = i ω d 2 n c ( ϵ 2 ϵ )
I a ω d ϵ 2 ϵ 8 π E i 2 = ω d n c ( ϵ 2 ϵ ) I i ,
I t ( z ) = c 8 π n E i ( z ) + E + ( z ) 2 ( 1 + 2 s + ) I i exp ( 2 k z ) = I i ( z ) I e ( z ) , ( z > 0 ) ,
I e ( z ) = 2 s + I i exp ( 2 k z ) ,
σ s A = 0 ,
σ a A = I a I i = ω d n c ( ϵ 2 ϵ ) ,
σ e A = I e ( 0 ) I i = 2 s +
σ e = σ s + σ a , ( wrong )
( σ e σ s σ a ) A = ω c d [ Im ( ϵ 2 ϵ n ) ϵ 2 ϵ n ] 0
I t ( z ) = c 8 π n E i ( z ) + E ( z ) 2 { exp ( 2 k z ) + 2 n n Im [ s exp ( 2 i k z ) ] } I i = I i ( z ) I int ( z ) , ( z < 0 )
I int ( z ) = 2 n n Im [ s exp ( 2 i k z ) ] I i
σ int A = I int ( 0 ) I i = 2 n n s ,
σ e = σ a + σ s + σ int .
s + = t exp ( i k d ) 1 ,
s = r ,
I s ( z ) = { I s + ( z ) = s + 2 exp ( 2 k z ) I i , ( z d ) I s ( z ) = s 2 exp ( 2 k z ) I i ( z 0 ) } .
I a ( z , z + ) = { s 2 exp ( 2 k z ) 2 n n Im [ s exp ( 2 i k z ) ] + ( s + 2 + 2 s + ) exp ( 2 k z + ) } I i ,
I t ( z ) = I i ( z ) + I s + ( z ) I e ( z ) , ( z > d ) ,
I e ( z ) = 2 s + exp ( 2 k z ) I i , ( z > d ) .
σ s A = [ I s ( 0 ) + I s + ( 0 ) ] I i = s + 2 + s 2 ,
σ a A = I a ( 0 , 0 ) I i = ( s 2 + s + 2 ) + 2 n n s 2 s + ,
σ e A = I e ( 0 ) I i = 2 s + .
E i ( r ) = E i x ̂ exp ( i k z ) .
E s ( r ) = i s ( Ω ) E i exp ( i k r ) k r ,
I ( r ) = S z ( r ) = { 1 2 Im [ s x ( 0 ) exp ( i k * r θ 2 2 ) k r ] + s ( 0 ) 2 k 2 r 2 } I i exp ( 2 k r ) ,
P d ( r ) = [ A d 4 π k 2 s x ( 0 ) ] I i exp ( 2 k r ) ,
σ e = 4 π k 2 s x ( 0 ) ,
P e ( r ) = 4 π k 2 s x ( 0 ) I i exp ( 2 k r ) .
σ e ( z ) = 4 π Re [ s x ( 0 ) k 2 ] ( flat detector ) ,
I ( r ) = S z ( r ) = { exp ( 2 k r ) + 2 n n Re [ s x ( π ) exp ( 2 i k r ) exp ( i k * r ξ 2 2 ) k r ] s ( π ) 2 k 2 r 2 exp ( 2 k r ) } I i ,
P int ( r ) = 4 π k 2 n n Im [ s x ( π ) exp ( 2 i k r ) ] I i .
σ int P int ( r 0 ) I i = 4 π k 2 n n s x ( π ) .
P s ( r ) = d Ω s ( Ω ) 2 k 2 exp ( 2 k r )
σ s P s ( r 0 ) I i = d Ω s ( Ω ) 2 k 2 ,
σ a P a ( r 0 ) I i .

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