Abstract

The reflectance of a homogeneous amplifying layer between two transparent regions is determined theoretically with the help of the Fresnel formulas. The transparent region containing the incident beam has a higher refractive index, corresponding to an internal reflection configuration. In the limit of infinite layer thickness, the reflectance is a continuous monotonic function of incident angle, greater than unity for all angles. For certain finite values of layer thickness, the reflectance has a singular point. The results explain the reported observation of a reflectance of 1000 from an excited laser dye in contact with a quartz prism.

© 1976 Optical Society of America

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References

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  1. G. N. Romanov and S. S. Shakhidzhanov, ZhETF Pis. Red. 16, 298–301 (1972) [JETP Lett. 16,209 (1972)].
  2. S. A. Lebedev, V. M. Volkov, and B. Ya. Kogan, Opt. Spectrosc. 35, 565 (1973).
  3. The sign convention regarding γ assumes waves of the form exp[i(k · r − ωt).
  4. B. Ya Kogan, V. M. Volkov, and S. A. Lebedev, ZhETF Pis. Red. 16, 144–147 (1972) [JETP Lett. 16,100 (1972)].
  5. O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths, London, 1955), p. 57.Note: The sign of δ differs from ours due to the different form of the waves used. See Ref. 3 above.
  6. S. A. Tuccio and F. C. Strome, Appl. Opt. 11, 64 (1972).
    [Crossref] [PubMed]

1973 (1)

S. A. Lebedev, V. M. Volkov, and B. Ya. Kogan, Opt. Spectrosc. 35, 565 (1973).

1972 (3)

B. Ya Kogan, V. M. Volkov, and S. A. Lebedev, ZhETF Pis. Red. 16, 144–147 (1972) [JETP Lett. 16,100 (1972)].

S. A. Tuccio and F. C. Strome, Appl. Opt. 11, 64 (1972).
[Crossref] [PubMed]

G. N. Romanov and S. S. Shakhidzhanov, ZhETF Pis. Red. 16, 298–301 (1972) [JETP Lett. 16,209 (1972)].

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths, London, 1955), p. 57.Note: The sign of δ differs from ours due to the different form of the waves used. See Ref. 3 above.

Kogan, B. Ya.

S. A. Lebedev, V. M. Volkov, and B. Ya. Kogan, Opt. Spectrosc. 35, 565 (1973).

Lebedev, S. A.

S. A. Lebedev, V. M. Volkov, and B. Ya. Kogan, Opt. Spectrosc. 35, 565 (1973).

B. Ya Kogan, V. M. Volkov, and S. A. Lebedev, ZhETF Pis. Red. 16, 144–147 (1972) [JETP Lett. 16,100 (1972)].

Romanov, G. N.

G. N. Romanov and S. S. Shakhidzhanov, ZhETF Pis. Red. 16, 298–301 (1972) [JETP Lett. 16,209 (1972)].

Shakhidzhanov, S. S.

G. N. Romanov and S. S. Shakhidzhanov, ZhETF Pis. Red. 16, 298–301 (1972) [JETP Lett. 16,209 (1972)].

Strome, F. C.

Tuccio, S. A.

Volkov, V. M.

S. A. Lebedev, V. M. Volkov, and B. Ya. Kogan, Opt. Spectrosc. 35, 565 (1973).

B. Ya Kogan, V. M. Volkov, and S. A. Lebedev, ZhETF Pis. Red. 16, 144–147 (1972) [JETP Lett. 16,100 (1972)].

Ya Kogan, B.

B. Ya Kogan, V. M. Volkov, and S. A. Lebedev, ZhETF Pis. Red. 16, 144–147 (1972) [JETP Lett. 16,100 (1972)].

Appl. Opt. (1)

Opt. Spectrosc. (1)

S. A. Lebedev, V. M. Volkov, and B. Ya. Kogan, Opt. Spectrosc. 35, 565 (1973).

ZhETF Pis. Red. (2)

B. Ya Kogan, V. M. Volkov, and S. A. Lebedev, ZhETF Pis. Red. 16, 144–147 (1972) [JETP Lett. 16,100 (1972)].

G. N. Romanov and S. S. Shakhidzhanov, ZhETF Pis. Red. 16, 298–301 (1972) [JETP Lett. 16,209 (1972)].

Other (2)

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths, London, 1955), p. 57.Note: The sign of δ differs from ours due to the different form of the waves used. See Ref. 3 above.

The sign convention regarding γ assumes waves of the form exp[i(k · r − ωt).

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Figures (5)

FIG. 1
FIG. 1

Multiple reflections and propagation across an amplifying layer with thickness d. The numbered arrows represent the various waves and are defined in the text. The refractive indices are given at the bottom. The displacement between two successive reflected waves is denoted x.

FIG. 2
FIG. 2

A graph of reflectance versus incident angle for the two possible roots of Eq. (2) with θc = 89° and γ = 0.00015. The solid curve corresponds to the theory of R & S.

FIG. 3
FIG. 3

A graph of reflectance versus incident angle for various layer thicknesses. The number by each curve is the corresponding value for d/λ. The dashed curve is for d = ∞. For all curves θc = 89° and γ = 0.00015.

FIG. 4
FIG. 4

A graph of θs versus layer thickness with θc = 89° and γ = 0.00015.

FIG. 5
FIG. 5

A graph of reflectance versus incident angle. The solid curve is for d/λ = 257.178. The dashed curve is for d = ∞. For both curves θc = 89° and γ = 0.00015.

Tables (1)

Tables Icon

TABLE I Singular points of the reflectance for θc = 89° and γ = 0.00015.

Equations (13)

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r 1 = n cos θ - ( 1 - i γ ) cos ϕ n cos θ + ( 1 - i γ ) cos ϕ ,
( 1 - i γ ) cos ϕ = ( 1 - γ 2 - n 2 sin 2 θ - 2 i γ ) 1 / 2 .
r = r 1 + ( 1 - r 1 2 ) r 2 e 2 i δ m = 0 ( - v ) m ,
r 2 = ( 1 - i γ ) cos ϕ - cos ψ ( 1 - i γ ) cos ϕ + cos ψ .
cos ψ = ( 1 - n 2 sin 2 θ ) 1 / 2 .
δ = ( 2 π / λ ) d ( 1 - i γ ) cos ϕ ,
v = r 1 r 2 e 2 i δ .
r = ( r 1 + v / r 1 ) / ( 1 + v ) .
θ c = arcsin ( 1 / n ) ,
v = 1.
d λ = ln r 1 + ln r 2 4 π Im [ ( 1 - i γ ) cos ϕ ] ,
γ = λ σ e N / 4 π .
d e = 1 / σ p N .