Abstract

The explanation of wave behavior upon total internal reflection from a gainy medium has defied consensus for 40 years. We examine this question using both the finite-difference time-domain (FDTD) method and theoretical analyses. FDTD simulations of a localized wave impinging on a gainy half space are based directly on Maxwell’s equations and make no underlying assumptions. They reveal that amplification occurs upon total internal reflection from a gainy medium; conversely, amplification does not occur for incidence below the critical angle. Excellent agreement is obtained between the FDTD results and an analytical formulation that employs a new branch cut in the complex “propagation-constant” plane.

© 2008 Optical Society of America

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References

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  1. C. J. Koester, "Laser action by enhanced total internal reflection," IEEE J. Quantum Electron. 2, 580-584 (1966).
    [CrossRef]
  2. M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1964).
  3. G. N. Romanov and S. S. Shakhidzhanov, "Amplification of electromagnetic field in total internal reflection from a region of inverted population," JETP 16, 298-301 (1972).
  4. S. A. Lebedev, V. M. Volkov, and B. Ya. Kogan, "Value of the gain for light internally reflected from a medium with inverted population," Opt. Spectrosc. 35, 976-977 (1973).
  5. P. R. Callary and C. K. Carniglia, "Internal reflection from an amplifying layer," J. Opt. Soc. Am. 66, 775-779 (1976).
    [CrossRef]
  6. R. F. Cybulski and C. K. Carniglia, "Internal reflection from an exponential amplifying region," J. Opt. Soc. Am. 67, 1620-1627 (1977).
    [CrossRef]
  7. C.-W. Lee, K. Kim, J. Noh, and W. Jhe, "Quantum theory of amplified total internal reflection due to evanescentmode coupling," Phys. Rev. A 62, 053805 (2000).
    [CrossRef]
  8. J. Fan, A. Dogariu, and L. J. Wang, "Amplified total internal reflection," Opt. Express 11, 299-308 (2003).
    [CrossRef] [PubMed]
  9. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, 2000).
  10. Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, "Space-time profiles of an ultrashort pulsed Gaussian beam," IEEE J. Quantum Electron. 33, 566-573 (1997).
    [CrossRef]
  11. A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice Hall, 1991).
  12. D. J. Robinson and J. B. Schneider, "On the use of the geometric mean in FDTD near-to-far-field transformations" to appear in IEEE Trans. Antennas Propag. (2007).
    [CrossRef]
  13. J. B. Schneider, "Plane waves in FDTD simulations and a nearly perfect total-field/scattered-field boundary" IEEE Trans. Antennas Propag. 52, 3280-3287 (2004).
    [CrossRef]

2004 (1)

J. B. Schneider, "Plane waves in FDTD simulations and a nearly perfect total-field/scattered-field boundary" IEEE Trans. Antennas Propag. 52, 3280-3287 (2004).
[CrossRef]

2003 (1)

2000 (1)

C.-W. Lee, K. Kim, J. Noh, and W. Jhe, "Quantum theory of amplified total internal reflection due to evanescentmode coupling," Phys. Rev. A 62, 053805 (2000).
[CrossRef]

1997 (1)

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, "Space-time profiles of an ultrashort pulsed Gaussian beam," IEEE J. Quantum Electron. 33, 566-573 (1997).
[CrossRef]

1977 (1)

1976 (1)

1973 (1)

S. A. Lebedev, V. M. Volkov, and B. Ya. Kogan, "Value of the gain for light internally reflected from a medium with inverted population," Opt. Spectrosc. 35, 976-977 (1973).

1972 (1)

G. N. Romanov and S. S. Shakhidzhanov, "Amplification of electromagnetic field in total internal reflection from a region of inverted population," JETP 16, 298-301 (1972).

1966 (1)

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

IEEE J. Quantum Electron. (2)

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

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, "Space-time profiles of an ultrashort pulsed Gaussian beam," IEEE J. Quantum Electron. 33, 566-573 (1997).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

J. B. Schneider, "Plane waves in FDTD simulations and a nearly perfect total-field/scattered-field boundary" IEEE Trans. Antennas Propag. 52, 3280-3287 (2004).
[CrossRef]

J. Opt. Soc. Am. (2)

JETP (1)

G. N. Romanov and S. S. Shakhidzhanov, "Amplification of electromagnetic field in total internal reflection from a region of inverted population," JETP 16, 298-301 (1972).

Opt. Express (1)

Opt. Spectrosc. (1)

S. A. Lebedev, V. M. Volkov, and B. Ya. Kogan, "Value of the gain for light internally reflected from a medium with inverted population," Opt. Spectrosc. 35, 976-977 (1973).

Phys. Rev. A (1)

C.-W. Lee, K. Kim, J. Noh, and W. Jhe, "Quantum theory of amplified total internal reflection due to evanescentmode coupling," Phys. Rev. A 62, 053805 (2000).
[CrossRef]

Other (4)

A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice Hall, 1991).

D. J. Robinson and J. B. Schneider, "On the use of the geometric mean in FDTD near-to-far-field transformations" to appear in IEEE Trans. Antennas Propag. (2007).
[CrossRef]

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, 2000).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1964).

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Figures (8)

Fig. 1.
Fig. 1.

System under consideration: dielectric half spaces. Medium 1 corresponds to the lossless core of a positive index-step optical waveguide, and medium 2 corresponds to either the gainy or lossy cladding.

Fig. 2.
Fig. 2.

Snapshots of a pulsed-in-time Gaussian beam launched into the FDTD grid, where red indicates positive values of Ez and blue indicates negative values of Ez . (a) shows the incident pulse, and (b) shows the pulse after reflection from the dielectric interface.

Fig. 3.
Fig. 3.

Gaussian beam time-modulated with a Gaussian pulse. The diagonal line where Ez is pre-calculated corresponds to the bottom-most row of Ez field components in the FDTD grid.

Fig. 4.
Fig. 4.

Comparison of three FDTD-computed beam reflection coefficient magnitudes for a gainy half space with a critical angle of 45°. The three different beam waists W 0 are given in terms of λ, the wavelength at the carrier frequency in medium 1. The plane wave reference curve is computed using Fresnel theory, taking k x2 to be negative for θi >θc .

Fig. 5.
Fig. 5.

The (θi ,σ 2)-plane. Each point in the plane maps to a unique value of ξ as given by Eq. (9). The point A corresponds to (θi =θc ,σ 2=0), i.e., ξ=0 where the field is neither propagating nor decaying in the x direction. The incident angle θi is between zero and π/2. The conductivity σ 2 can take on any real value. The four quadrants are distinguished by whether they representing a propagating or evanescent field in either a lossy or a gainy medium. These quadrants also correspond to the different signs for the real and imaginary parts of ξ.

Fig. 6.
Fig. 6.

The mapping of the ξ-plane to the (gx ,k x2)-plane. Points in the (gx ,k x2)-plane are obtained from the square root of points in the ξ -plane scaled by ω μ 0 ε 0 . In the (gx ,k x2)- plane points below and to the right of the heavy diagonal line are not realizable. The sector labels shown in the (gx ,k x2)-plane are taken from the corresponding quadrant labels in the ξ-plane.

Fig. 7.
Fig. 7.

Decomposition of an incident Gaussian beam to a spectrum of plane waves. The reflected beam is similar to the incident beam except it is weighted by the reflection coefficient and propagates to the left.

Fig. 8.
Fig. 8.

Comparison of reflection coefficient magnitudes computed for the incident beam using the analytical and FDTD formulations. (a) shows results for reflection from the lossy half space, and (b) shows results for reflection from the gainy half space.

Equations (19)

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R = k 1 cos θ i ( k x 2 + jg x ) k 1 cos θ i + ( k x 2 + jg x ) ,
      = Γ exp ( j ϕ ) .
R = n 1 cos θ i ( u 2 + jv 2 ) n 1 cos θ i + ( u 2 + jv 2 )
2 u 2 2 = n 2 2 ( 1 κ 2 2 ) n 1 2 sin 2 θ i + [ n 2 2 ( 1 κ 2 2 ) n 1 2 sin 2 θ i ] 2 + 4 n 2 4 κ 2 2 ,
2 v 2 2 = n 2 2 ( 1 κ 2 2 ) + n 1 2 sin 2 θ i + [ n 2 2 ( 1 κ 2 2 ) n 1 2 sin 2 θ i ] 2 + 4 n 2 4 κ 2 2 .
n ̂ 2 = n 2 ( 1 + j κ 2 ) = ε r 2 j σ 2 ω ε 0 .
exp ( ( g j k t ) · r ) = exp ( j k y 1 y [ g x + j k x 2 ] x )
( g x + j k x 2 ) 2 k y 1 2 j ω μ 0 ( σ 2 + j ω ε 2 ) = 0 .
g x + j k x 2 = k 0 ( ε r 1 sin 2 θ i ε r 2 + j σ 2 ω ε 0 ) 1 2
ξ = ξ r + j ξ i = ( ε r 1 sin 2 θ i ε r 2 ) + j ( σ 2 ω ε 0 ) .
g x + j k x 2 = k 0 ξ 1 2 .
θ c = sin 1 ( ε r 2 ε r 1 ) .
ξ = ξ e j θ ξ           where         π 2 θ ξ < 3 π 2 .
g x + j k x 2 = k 0 ξ 1 2 = k 0 ( ξ r 2 + ξ i 2 ) 1 4 e j θ ξ 2 .
g x = k 0 ( ξ r 2 + ξ i 2 ) 1 4 cos ( θ ξ 2 ) ,
k x 2 = k 0 ( ξ r 2 + ξ i 2 ) 1 4 sin ( θ ξ 2 ) .
E z i ( χ , ζ ) = 1 2 π A ( k ζ ) e jk ζ ζ j k χ χ d k ζ ,
A ( k ζ ) = π W 0 2 e w 0 2 k ζ 2 4
E z r ( χ , ζ ) = 1 2 π R ( k ζ , θ i ) e j k ζ ζ + j k χ χ d k ζ .

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