Abstract

Totally internal reflected beams can be amplified if the lower-index medium has gain. We analyze the reflection and refraction of light, and analytically derive the expression for the Goos-Hänchen shifts of a Gaussian beam incident on a lower-index medium, both active and absorptive. We examine the energy flow and the Goos-Hänchen shifts for various cases. The analytical results are consistent with the numerical results. For the TE mode, the Goos-Hänchen shift for the transmitted beam is exactly half of that of the reflected beam, resulting in a “1/2” rule.

© 2002 Optical Society of America

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References

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  1. F. Goos and H. Hänchen, �??Ein neuer und fundamentaler versuch zur totalreflexion,�?? Ann. Phys. 1, 23 (1947).
  2. Von Kurt Artmann, �??Berechnung der Seitenversetzung des totalreflektierten strahles,�?? Ann. Phys. 1, 87 (1948).
    [CrossRef]
  3. Helmut K. V. Lotsch, �??Beam displacement at total reflection: the Goos Hänchen shift,�?? OPTIK 32, 116 (1970).
  4. M. McGuirk and C. K. Carniglia, �??An angular spectrum representation approach to the Goos Hänchen shift,�?? J. Opt. Soc. Am. 67, 103 (1976).
    [CrossRef]
  5. F. Bretenaker, A. L. Floch, and L. Dutriaux, �??Direct measurement of the optical Goos-Hänchen effect in lasers,�?? Phys. Rev. Lett. 17, 931 (1992).
    [CrossRef]
  6. S. Kozaki and H. Sakurai, �??Characteristics of a Gaussian beam at a dielectric interface,�?? J. Opt. Soc. Am. 68, 508 (1978).
    [CrossRef]
  7. C. J. Koester, �??9A4-Laser action by enhanced total internal reflection,�?? IEEE J. Quantum Electron. QE-2, 580 (1966).
    [CrossRef]
  8. E. P. Ippen and C. V. Shank, �??Evanescent-field-pumped dye laser,�?? Appl. Phys. Lett. 21, 301 (1972).
    [CrossRef]
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  10. W. Y. Liu and O. M. Stafsudd, �??Optical amplification of a multimode evanescently active planar optical waveguide,�?? Appl. Opt. 29, 3114 (1990).
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  11. E. Pfleghaar, A. Marseille, and A. Weis, �??Quantative investigation of the effect of resonant absorpers on the Goos-Hänchen shift,�?? Phys. Rev. Lett. 12, 2281 (1993).
    [CrossRef]
  12. B. Ya. Kogan, V. M. Volkov, and S. A. Lebedev, �??Superluminescence and generation of stimulated radiation under internal-reflection conditions,�?? JETP Lett. 16, 100 (1972).
  13. G. N. Romanov and S. S. Shakhidzhanov, �??Amplification of electromagnetic field in total internal reflection from a regeion of inverted population,�?? JETP Lett. 16, 209 (1972).
  14. 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, 565 (1973).
  15. P. R. Callary and C. K. Carniglia, �??Internal reflection from an amplifying layer,�?? J. Opt. Soc. Am. 66, 775 (1976).
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  16. W. Lukosz and P. P. Herrmann, �??Amplification by reflection from an active medium,�?? Opt. Commun. 17, 192 (1976).
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  17. R. F. Cybulski, Jr. and C. K. Carniglia, �??Internal reflection from an exponential amplifying region,�?? J. Opt. Soc. Am. 67, 1620 (1977).
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  18. S. A. Lebedev and B. Ya. Kogan, �??Light amplification by internal reflection from an inverted medium,�?? Opt. Spectrosc. 48, 564 (1980).
  19. R. F. Cybulski and M. P. Silverman, �??Investigation of light amplification by enhanced internal reflection. I. Theoretical reflectance and transmittance of an exponentially nonuniform gain region,�?? J. Opt. Soc. Am. 73, 1732 (1983).
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  20. M. P. Silverman and R. F. Cybulski, �??Investigation of light amplification by enhanced internal reflection. II. Experimental determination of the single-pass reflectance of an optically pumped gain region,�?? J. Opt. Soc. Am. 73, 1739 (1983).
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  21. Max Born and Emil Wolf, Principles of Optics, 6th edition (Cambridge, 1997).
  22. J. W. Goodman, Introduction to fourier optics (McGraw-Hill, 1968), Chap. 3.

Ann. Phys. (2)

F. Goos and H. Hänchen, �??Ein neuer und fundamentaler versuch zur totalreflexion,�?? Ann. Phys. 1, 23 (1947).

Von Kurt Artmann, �??Berechnung der Seitenversetzung des totalreflektierten strahles,�?? Ann. Phys. 1, 87 (1948).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

E. P. Ippen and C. V. Shank, �??Evanescent-field-pumped dye laser,�?? Appl. Phys. Lett. 21, 301 (1972).
[CrossRef]

IEEE J. Quantum Electron. (1)

C. J. Koester, �??9A4-Laser action by enhanced total internal reflection,�?? IEEE J. Quantum Electron. QE-2, 580 (1966).
[CrossRef]

J. Opt. Soc. Am. (6)

JETP Lett. (2)

B. Ya. Kogan, V. M. Volkov, and S. A. Lebedev, �??Superluminescence and generation of stimulated radiation under internal-reflection conditions,�?? JETP Lett. 16, 100 (1972).

G. N. Romanov and S. S. Shakhidzhanov, �??Amplification of electromagnetic field in total internal reflection from a regeion of inverted population,�?? JETP Lett. 16, 209 (1972).

Opt. Commun. (1)

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

Opt. Spectrosc. (2)

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, 565 (1973).

S. A. Lebedev and B. Ya. Kogan, �??Light amplification by internal reflection from an inverted medium,�?? Opt. Spectrosc. 48, 564 (1980).

OPTIK (1)

Helmut K. V. Lotsch, �??Beam displacement at total reflection: the Goos Hänchen shift,�?? OPTIK 32, 116 (1970).

Phys. Rev. Lett. (2)

F. Bretenaker, A. L. Floch, and L. Dutriaux, �??Direct measurement of the optical Goos-Hänchen effect in lasers,�?? Phys. Rev. Lett. 17, 931 (1992).
[CrossRef]

E. Pfleghaar, A. Marseille, and A. Weis, �??Quantative investigation of the effect of resonant absorpers on the Goos-Hänchen shift,�?? Phys. Rev. Lett. 12, 2281 (1993).
[CrossRef]

Other (2)

Max Born and Emil Wolf, Principles of Optics, 6th edition (Cambridge, 1997).

J. W. Goodman, Introduction to fourier optics (McGraw-Hill, 1968), Chap. 3.

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Figures (6)

Fig. 1.
Fig. 1.

(a) The geometry and coordinate system of the reflection of a plane wave on a planar interface between two different media; (b) illustration of ktz as a root solution in a complex coordinate plane.

Fig. 2.
Fig. 2.

ktz and energy flux across the interface vs εI for TE mode, ε1=3.24, ε2=1.44±iεI, , θc=41.8°. (a) kR vs. εI, smooth line: θ=43°>θc, dashed line: θ=40°<θc; (b) κ vs. εI, smooth line: θ=43°, dashed line: θ=40°; (c)z-component Poynting vectors of the reflected and transmitted beam normalized to that of the incident beam, θ=43°. Smooth line: reflected beam, dashed line: transmitted beam; (d) same as in (c), θ=40°. Quantities are evaluated at the interface.

Fig. 3.
Fig. 3.

Goos Hänchen shifts vs εI. Smooth lines and dashed lines are shifts for the reflected and transmitted beams, respectively, obtained using Eq. (7). Dotted lines are from the numerical calculation using angular spectrum method [22]. ε1=3.24, ε2=1.44±iεI, θc=41.8°. (a) and (c) are for the TE mode, (b) and (d) are for the TM mode. Incident angle is θ=43° in (a) and (b), θ=40° in (c) and (d).

Fig. 4.
Fig. 4.

Projection of z-component energy flux for the incident Siz, reflected Srz and refracted beams Stz at x-z plane. ε1=3.24, ε1=1.44±εI, θc=41.8°. Gaussian beam: a=17µm, with peak amplitude 0.2V/m for the electric field in the TE mode and 0.2A/m for the magnetic field in the TM modes. Sz is along +z if positive, -z if negative, as shown by the arrows. Dotted line shows the center of the incident Gaussian beam.

Fig. 5.
Fig. 5.

Electric fields (a-c) and beam centers (d-f) for the TE mode. (a)(d), εI=+0.5; (b)(e), εI=0; (c)(f), εI=-0.5; other conditions are the same as in Fig. 4. The arrows in (d)-(f) show the propagation directions of different beams.

Fig. 6.
Fig. 6.

Energy flux of the refracted beam Stz, as a function of x at the interface. In each subplot, from top to bottom, curves correspond to εI=0.1, 0.01, 0, -0.01, -0.1, respectively. Other conditions are the same as in Fig. 4.

Equations (22)

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r k ix ε 1 ε 2 = η k iz k tz η k iz + k tz
t k ix ε 1 ε 2 = k iz η k iz + k tz
k ix 2 + k R 2 κ 2 = ε R k 2 ,
2 k R κ = ε I k 2 .
r * k ix ε 1 ε 2 = r k ix ε 1 ε 2 *
t * k ix ε 1 ε 2 = t k ix ε 1 ε 2 *
k 2 = k 2 [ ( ε R ε 1 sin 2 θ ) 2 + ε I 2 + ( ε R ε 1 sin 2 θ ) ] 1 2 ,
κ = k 2 [ ( ε R ε 1 sin 2 θ ) 2 + ε I 2 ( ε R ε 1 sin 2 θ ) ] 1 2 .
x = ( ϕ ω k k x ) k x = k x 0 .
ϕ r ( k ix ) = Im [ ln ( r ) ]
ϕ t ( k ix ) = Im [ ln ( t ) ] ,
x ¯ r = ϕ ( k ix ) k ix = 2 tan θ Im ( 1 k tz η ( k iz 2 k tz 2 ) ( η 2 k iz 2 k tz 2 ) )
x ¯ t = tan θ Im ( 1 k iz k tz k iz 2 k tz 2 η k iz + k tz ) ,
x ¯ r = 2 tan θ Im ( 1 k tz ) ,
x ¯ t = tan θ Im ( 1 k tz 1 k iz ) .
x ¯ t = 1 2 x ¯ r = tan θ Im ( 1 k tz ) = tan θ κ k tz 2 .
E x 1 z 1 = + d k 1 x E k 1 x z 0 e i ( k 1 x x 1 + k 1 z z 1 ) ,
< k 1 x > = + E ( k 1 x ) k 1 x E * ( k 1 x ) d k 1 x = 0
E i x z = E x 1 z 1 = + d k 1 x E k 1 x z 0 e i ( k 1 z x + k 1 z z 1 ) = + d k ix f ( k ix ) e i ( k ix x + k iz z ) ,
E r x z = + d k ix r TE ( k ix ) f ( k ix ) e i ( k ix x k iz z ) = + d k 1 x r TE ( k ix ) E k 1 x z 0 e i ( k ix x k iz z ) ,
E t x z = + d k ix t TE ( k ix ) f ( k ix ) e i ( k ix x k tiz z ) = + d k 1 x t TE ( k ix ) E k 1 x z 0 e i ( k ix x + k tz z ) ,
< x > p = < E p x z x E p * x z > x < E p x z E p * x z > x ,

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