Abstract

Total internal reflection occurs for large angles of incidence, when light is incident from a high-refractive-index medium onto a low-index medium. We consider the situation where the low-index medium is active. By invoking causality in its most fundamental form, we argue that evanescent gain may or may not appear, depending on the analytic and global properties of the permittivity function. For conventional, weak gain media, we show that there is an absolute instability associated with infinite transverse dimensions. This instability can be ignored or eliminated in certain cases, for which evanescent gain prevails.

© 2011 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. N. Romanov and S. S. Shakhidzhanov, “Amplification of electromagnetic field in total internal reflection from a region of inverted population,” JETP Lett. 16, 209–211 (1972).
  2. A. A. Kolokolov, “Reflection of plane-waves from an amplifying medium,” JETP Lett. 21, 312–313 (1975).
  3. P. R. Callary and C. K. Carniglia, “Internal-reflection from an amplifying layer,” J. Opt. Soc. Am. 66, 775–779 (1976).
    [CrossRef]
  4. W. Lukosz and P. P. Herrmann, “Amplification by reflection from an active medium,” Opt. Commun. 17, 192–195 (1976).
    [CrossRef]
  5. R. F. Cybulski and C. K. Carniglia, “Internal-reflection from an exponential amplifying region,” J. Opt. Soc. Am. 67, 1620–1627 (1977).
    [CrossRef]
  6. R. F. Cybulski and M. P. Silverman, “Investigation of light amplification by enhanced internal-reflection .1. theoretical reflectance and transmittance of an exponentially nonuniform gain region,” J. Opt. Soc. Am. 73, 1732–1738 (1983).
    [CrossRef]
  7. A. A. Kolokolov, “Determination of the reflection coefficient of a plane monochromatic wave,” J. Commun. Technol. Electron. 43, 837–845 (1998).
  8. A. A. Kolokolov, “Fresnel formulas and the principle of causality,” Phys. Usp. 42, 931–940 (1999).
    [CrossRef]
  9. J. Fan, A. Dogariu, and L. J. Wang, “Amplified total internal reflection,” Opt. Express 11, 299–308 (2003).
    [CrossRef] [PubMed]
  10. K. J. Willis, J. B. Schneider, and S. C. Hagness, “Amplified total internal reflection: theory, analysis, and demonstration of existence via FDTD,” Opt. Express 16, 1903–1913 (2008).
    [CrossRef] [PubMed]
  11. A. Siegman, “Fresnel reflection, Lenserf reflection, and evanescent gain,” Opt. Photon. News 21, 38–45 (2010).
    [CrossRef]
  12. C. J. Koester, “Laser action by enhanced total internal reflection,” IEEE J. Quantum Electron. 2, 580–584 (1966).
    [CrossRef]
  13. B. Y. Kogan, V. M. Volkov, and S. A. Lebedev, “Superluminescence and generation of stimulated radiation under internal-reflection conditions,” JETP Lett. 16, 100–101 (1972).
  14. S. A. Lebedev, V. M. Volkov, and B. Y. Kogan, “Value of the gain for light internally reflected from a medium with inverted population,” Opt. Spectrosc. 35, 565–566 (1973).
  15. M. P. Silverman and J. R. F. Cybulski, “Investigation of light amplification by enhanced internal reflection. Part II. Experimental determination of the single-pass reflectance of an optically pumped gain region,” J. Opt. Soc. Am. 73, 1739–1743 (1983).
    [CrossRef]
  16. B. Nistad and J. Skaar, “Causality and electromagnetic properties of active media,” Phys. Rev. E 78, 036603 (2008).
    [CrossRef]
  17. J. Skaar, “Fresnel equations and the refractive index of active media,” Phys. Rev. E 73, 026605 (2006).
    [CrossRef]
  18. A. Lakhtakia, J. B. Geddes, and T. G. Mackay, “When does the choice of the refractive index of a linear, homogeneous, isotropic, active, dielectric medium matter?” Opt. Express 15, 17709–17714 (2007).
    [CrossRef] [PubMed]
  19. L. V. Ahlfors, Complex Analysis (McGraw-Hill International Editions, 1979).
  20. P. A. Sturrock, “Kinematics of growing waves,” Phys. Rev. 112, 1488–1503 (1958).
    [CrossRef]
  21. R. J. Briggs, Electron-Stream Interactions with Plasmas (MIT Press, 1964).

2010 (1)

A. Siegman, “Fresnel reflection, Lenserf reflection, and evanescent gain,” Opt. Photon. News 21, 38–45 (2010).
[CrossRef]

2008 (2)

2007 (1)

2006 (1)

J. Skaar, “Fresnel equations and the refractive index of active media,” Phys. Rev. E 73, 026605 (2006).
[CrossRef]

2003 (1)

1999 (1)

A. A. Kolokolov, “Fresnel formulas and the principle of causality,” Phys. Usp. 42, 931–940 (1999).
[CrossRef]

1998 (1)

A. A. Kolokolov, “Determination of the reflection coefficient of a plane monochromatic wave,” J. Commun. Technol. Electron. 43, 837–845 (1998).

1983 (2)

1977 (1)

1976 (2)

P. R. Callary and C. K. Carniglia, “Internal-reflection from an amplifying layer,” J. Opt. Soc. Am. 66, 775–779 (1976).
[CrossRef]

W. Lukosz and P. P. Herrmann, “Amplification by reflection from an active medium,” Opt. Commun. 17, 192–195 (1976).
[CrossRef]

1975 (1)

A. A. Kolokolov, “Reflection of plane-waves from an amplifying medium,” JETP Lett. 21, 312–313 (1975).

1973 (1)

S. A. Lebedev, V. M. Volkov, and B. Y. Kogan, “Value of the gain for light internally reflected from a medium with inverted population,” Opt. Spectrosc. 35, 565–566 (1973).

1972 (2)

B. Y. Kogan, V. M. Volkov, and S. A. Lebedev, “Superluminescence and generation of stimulated radiation under internal-reflection conditions,” JETP Lett. 16, 100–101 (1972).

G. N. Romanov and S. S. Shakhidzhanov, “Amplification of electromagnetic field in total internal reflection from a region of inverted population,” JETP Lett. 16, 209–211 (1972).

1966 (1)

C. J. Koester, “Laser action by enhanced total internal reflection,” IEEE J. Quantum Electron. 2, 580–584 (1966).
[CrossRef]

1958 (1)

P. A. Sturrock, “Kinematics of growing waves,” Phys. Rev. 112, 1488–1503 (1958).
[CrossRef]

Ahlfors, L. V.

L. V. Ahlfors, Complex Analysis (McGraw-Hill International Editions, 1979).

Briggs, R. J.

R. J. Briggs, Electron-Stream Interactions with Plasmas (MIT Press, 1964).

Callary, P. R.

Carniglia, C. K.

Cybulski, J. R. F.

Cybulski, R. F.

Dogariu, A.

Fan, J.

Geddes, J. B.

Hagness, S. C.

Herrmann, P. P.

W. Lukosz and P. P. Herrmann, “Amplification by reflection from an active medium,” Opt. Commun. 17, 192–195 (1976).
[CrossRef]

Koester, C. J.

C. J. Koester, “Laser action by enhanced total internal reflection,” IEEE J. Quantum Electron. 2, 580–584 (1966).
[CrossRef]

Kogan, B. Y.

S. A. Lebedev, V. M. Volkov, and B. Y. Kogan, “Value of the gain for light internally reflected from a medium with inverted population,” Opt. Spectrosc. 35, 565–566 (1973).

B. Y. Kogan, V. M. Volkov, and S. A. Lebedev, “Superluminescence and generation of stimulated radiation under internal-reflection conditions,” JETP Lett. 16, 100–101 (1972).

Kolokolov, A. A.

A. A. Kolokolov, “Fresnel formulas and the principle of causality,” Phys. Usp. 42, 931–940 (1999).
[CrossRef]

A. A. Kolokolov, “Determination of the reflection coefficient of a plane monochromatic wave,” J. Commun. Technol. Electron. 43, 837–845 (1998).

A. A. Kolokolov, “Reflection of plane-waves from an amplifying medium,” JETP Lett. 21, 312–313 (1975).

Lakhtakia, A.

Lebedev, S. A.

S. A. Lebedev, V. M. Volkov, and B. Y. Kogan, “Value of the gain for light internally reflected from a medium with inverted population,” Opt. Spectrosc. 35, 565–566 (1973).

B. Y. Kogan, V. M. Volkov, and S. A. Lebedev, “Superluminescence and generation of stimulated radiation under internal-reflection conditions,” JETP Lett. 16, 100–101 (1972).

Lukosz, W.

W. Lukosz and P. P. Herrmann, “Amplification by reflection from an active medium,” Opt. Commun. 17, 192–195 (1976).
[CrossRef]

Mackay, T. G.

Nistad, B.

B. Nistad and J. Skaar, “Causality and electromagnetic properties of active media,” Phys. Rev. E 78, 036603 (2008).
[CrossRef]

Romanov, G. N.

G. N. Romanov and S. S. Shakhidzhanov, “Amplification of electromagnetic field in total internal reflection from a region of inverted population,” JETP Lett. 16, 209–211 (1972).

Schneider, J. B.

Shakhidzhanov, S. S.

G. N. Romanov and S. S. Shakhidzhanov, “Amplification of electromagnetic field in total internal reflection from a region of inverted population,” JETP Lett. 16, 209–211 (1972).

Siegman, A.

A. Siegman, “Fresnel reflection, Lenserf reflection, and evanescent gain,” Opt. Photon. News 21, 38–45 (2010).
[CrossRef]

Silverman, M. P.

Skaar, J.

B. Nistad and J. Skaar, “Causality and electromagnetic properties of active media,” Phys. Rev. E 78, 036603 (2008).
[CrossRef]

J. Skaar, “Fresnel equations and the refractive index of active media,” Phys. Rev. E 73, 026605 (2006).
[CrossRef]

Sturrock, P. A.

P. A. Sturrock, “Kinematics of growing waves,” Phys. Rev. 112, 1488–1503 (1958).
[CrossRef]

Volkov, V. M.

S. A. Lebedev, V. M. Volkov, and B. Y. Kogan, “Value of the gain for light internally reflected from a medium with inverted population,” Opt. Spectrosc. 35, 565–566 (1973).

B. Y. Kogan, V. M. Volkov, and S. A. Lebedev, “Superluminescence and generation of stimulated radiation under internal-reflection conditions,” JETP Lett. 16, 100–101 (1972).

Wang, L. J.

Willis, K. J.

IEEE J. Quantum Electron. (1)

C. J. Koester, “Laser action by enhanced total internal reflection,” IEEE J. Quantum Electron. 2, 580–584 (1966).
[CrossRef]

J. Commun. Technol. Electron. (1)

A. A. Kolokolov, “Determination of the reflection coefficient of a plane monochromatic wave,” J. Commun. Technol. Electron. 43, 837–845 (1998).

J. Opt. Soc. Am. (4)

JETP Lett. (3)

B. Y. Kogan, V. M. Volkov, and S. A. Lebedev, “Superluminescence and generation of stimulated radiation under internal-reflection conditions,” JETP Lett. 16, 100–101 (1972).

G. N. Romanov and S. S. Shakhidzhanov, “Amplification of electromagnetic field in total internal reflection from a region of inverted population,” JETP Lett. 16, 209–211 (1972).

A. A. Kolokolov, “Reflection of plane-waves from an amplifying medium,” JETP Lett. 21, 312–313 (1975).

Opt. Commun. (1)

W. Lukosz and P. P. Herrmann, “Amplification by reflection from an active medium,” Opt. Commun. 17, 192–195 (1976).
[CrossRef]

Opt. Express (3)

Opt. Photon. News (1)

A. Siegman, “Fresnel reflection, Lenserf reflection, and evanescent gain,” Opt. Photon. News 21, 38–45 (2010).
[CrossRef]

Opt. Spectrosc. (1)

S. A. Lebedev, V. M. Volkov, and B. Y. Kogan, “Value of the gain for light internally reflected from a medium with inverted population,” Opt. Spectrosc. 35, 565–566 (1973).

Phys. Rev. (1)

P. A. Sturrock, “Kinematics of growing waves,” Phys. Rev. 112, 1488–1503 (1958).
[CrossRef]

Phys. Rev. E (2)

B. Nistad and J. Skaar, “Causality and electromagnetic properties of active media,” Phys. Rev. E 78, 036603 (2008).
[CrossRef]

J. Skaar, “Fresnel equations and the refractive index of active media,” Phys. Rev. E 73, 026605 (2006).
[CrossRef]

Phys. Usp. (1)

A. A. Kolokolov, “Fresnel formulas and the principle of causality,” Phys. Usp. 42, 931–940 (1999).
[CrossRef]

Other (2)

L. V. Ahlfors, Complex Analysis (McGraw-Hill International Editions, 1979).

R. J. Briggs, Electron-Stream Interactions with Plasmas (MIT Press, 1964).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

A wave is incident from a high-index medium to a low index medium with gain. The source produces a single, spatial frequency kx. The electromagnetic boundary conditions require preservation of the wavenumber kx parallel to the interface. The longitudinal wavenumbers are denoted k1z and k2z. Note that since the excitation is assumed to be causal, it contains a band of frequencies, and therefore also a band of k1z’s and k2z’s.

Fig. 2
Fig. 2

The two possible solutions for the wavenumber k2z for monochromatic analysis and a gainy medium. The arrows indicate the two possible wavenumbers in the complex plane, as the gain tends to zero. For a lossy medium, we always have a solution that tends to the upper alternative + i k x 2 ω 1 2 / c 2 in the limit of zero loss. For simplicity we have taken Reɛ2 = 1 here.

Fig. 3
Fig. 3

The complex ω-plane. For conventional, weak gain media, there are branch points right above ω = ±(kxc +δ′) ≈ ±kxc. The branch cuts can be chosen arbitrarily; however, the shown, vertical cuts minimize the integral around the part of the branch cuts in the upper half-plane.

Fig. 4
Fig. 4

The function k 2 z 2 ( ω ) = ɛ 2 ( ω ) ω 2 / c 2 k x 2 for a typical gain medium, plotted in the complex k 2 z 2-plane. To identify k2z, we require it to be +ω/c at ω = ∞, continuous as ω decreases towards zero, except at the branch cut at ω = kxc where it changes sign.

Fig. 5
Fig. 5

The reflected electric field (solid line, z < 0) and transmitted electric field (dashed line, z > 0) for a plane wave incident to a weak Lorentzian gain medium (20). The parameters used: F = 0.01, Γ = 0.1ω0, kxc = 2ω0, ω1 = ω0, γ = 0.001, and ɛ1 = 4.7. The field is plotted for x = 0 and ω0t = 103, and normalized to the incident field. The amplitude of the reflected field is 1.01. Note that the field is discontinuous at z = 0 because the incident wave is not included.

Fig. 6
Fig. 6

The real and imaginary parts of k 2 z ( ω ) = ɛ 2 ω 2 / c 2 k x 2, with ɛ2 given by Eq. (15). We have set ω2 = kxc, N = kxc(6/10 – i/1000) and P = kxc(7/10 – i/1000).

Fig. 7
Fig. 7

The reflected electric field (solid line, z < 0) and transmitted electric field (dashed line, z > 0) for a plane wave incident to the gain medium described by Eq. (15) and Fig. 6. The field is plotted for x = 0 and kxct = 105, and normalized to the incident field. The amplitude of the reflected field is 0.98. The parameters used: ω1 = 0.65kxc, γ = 10−6, and ɛ1 = 4.

Fig. 8
Fig. 8

The semi-infinite gain medium can be replaced by a finite size gain medium, provided we only consider times t < d/c.

Fig. 9
Fig. 9

Circles and stars mark zeros and poles of k2z respectively. The cross marks the pole at ω = ω1. Branch cuts are arbitrarily chosen to lie parallel to the imaginary axis, extending into the lower half-plane Im(ω) < 0. The integration path C is shown with the dashed line. Contributing branch cuts and poles are enclosed by paths C1, bc− and bc+. For t ≳ 2/Γ, we have ∮C f(ω)dω ≈ ∮C1 f(ω)dω + ∫bc− f(ω)dω + ∫bc+ f(ω)dω.

Equations (29)

Equations on this page are rendered with MathJax. Learn more.

k 1 z = ± ɛ 1 ω 2 / c 2 k x 2 ,
k 2 z = ± ɛ 2 ω 2 / c 2 k x 2 .
E ( ω ) = 0 ( t ) exp ( i ω t ) d t .
( t ) = 1 2 π + i γ + + i γ E ( ω ) exp ( i ω t ) d ω .
ρ = k 1 z k 2 z k 1 z + k 2 z ,
τ = 2 k 1 z k 1 z + k 2 z exp ( i k 2 z z ) ,
ɛ 2 ( ω ) = ɛ ¯ 2 + Δ ɛ 2 ( ω ) , where | Δ ɛ 2 ( ω ) | ɛ ¯ 2 for real ω .
| d ɛ 2 d ω | < 2 k x c for | ω k x c | < Δ ɛ max 2 k x c .
ɛ 2 ( ω ) ω 2 c 2 = k x 2 ,
ω = ± k x c ( 1 Δ ɛ 2 ( ω ) 2 )
Δ ɛ 2 ( ω a ) Δ ɛ 2 ( ω b ) = 2 k x c ( ω b ω a ) .
ω = k x c + δ + i δ , where δ = Re Δ ɛ 2 ( ω ) 2 k x c and δ = Im Δ ɛ 2 ( ω ) 2 k x c .
δ Δ ɛ max 2 k x c .
ρ ( x , t ) = 1 2 π + i γ + + i γ k 1 z k 2 z k 1 z + k 2 z exp ( i k x x i ω t ) i ω 1 i ω d ω ,
ρ ( 0 , t ) = k 1 z k 2 z k 1 z + k 2 z exp ( i ω 1 t ) + bc ( 0 , t ) , t Γ 1 ,
| bc ( 0 , t ) | const k x c ω 1 exp ( δ t ) .
ɛ 2 ( ω ) = ( ω N ) ( ω + N * ) ( ω P ) ( ω + P * ) + ω 2 2 ω 2 ,
k 2 z 2 ( ω ) = ω 2 ( ω N ) ( ω + N * ) c 2 ( ω P ) ( ω + P * ) + ω 2 2 c 2 k x 2 .
k 2 z 2 ( ω ) = A ( ω ) ( B ( ω ) + i C ( n p ) )
( x , t ) = 1 ( 2 π ) 2 + i γ + i γ d ω e i ω t d k x E ( k x , ω ) e i k x x = 1 ( 2 π ) 2 E ( k x , ω + i γ ) e i ω t + γ t e i k x x d k x d ω .
| ( x , z , t ) | C exp ( γ t ) ,
ɛ 2 ( ω ) = 1 F ω 0 2 ω 0 2 ω 2 i ω Γ .
ρ ( 0 , t ) = 1 2 π + i γ + + i γ k 1 z k 2 z k 1 z + k 2 z exp ( i ω t ) i ω 1 i ω d ω .
ρ ( 0 , t ) = k 1 z k 2 z k 1 z + k 2 z exp ( i ω 1 t ) + bc ( 0 , t ) .
bc + ( 0 , t ) 1 2 π k x c i Γ 2 k x c + i δ ρ 1 ( ω ) exp ( i ω t ) i ω 1 i ω d ω 1 2 π k x c i Γ 2 k x c + i δ ρ r ( ω ) exp ( i ω t ) i ω 1 i ω d ω .
bc + ( 0 , t ) 2 i π k x c i Γ 2 k x c + i δ f 1 ( ω ) exp ( i ω t ) ω 1 ω d ω ,
k 2 z 2 = ( ω ω ω 0 ) ( ω + ω ω 0 * ) ( ω ω k x c ) ( ω + ω k x c * ) c 2 ( ω ω p ) ( ω + ω p * ) .
| f 1 ( k x c + i ω i ) | 2 ( δ ω i ) / ( k x c ( ɛ 1 1 ) ) .
| bc ( 0 , t ) | Γ 3 / 2 ( k x c ω 1 ) k x c ( ɛ 1 1 ) ( e F Γ t e Γ t / 2 )

Metrics