Abstract

The reflectance from an amplifying region in which the gain decreases exponentially with distance from the surface is calculated as a function of incident angle. The results are similar in some respects to the reflectance from a uniform amplifying layer reported previously. In particular, a singular point occurs in the reflectance for certain values of the thickness of the amplifying region. However, the results for the exponential case do not explain the previously reported observation of large reflectance from a pumped laser dye. An asymptotic form of the reflectance for a thick amplifying region is presented. The results for asymptotic case are compared with various other theories of reflection from an amplifying region.

© 1977 Optical Society of America

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References

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  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, 565–566 (1973).
  2. B. Ya. Kogan, V. M. Volkov, and S. A. Lebedev, “Superluminescence and generation of stimulated radiation under internal-reflection conditions,” JETP Lett. 16, 100–101 (1972).
  3. P. R. Callary and C. K. Carniglia, “Internal reflection from an amplifying layer,” J. Opt. Soc. Am. 66, 775–779 (1976).
    [Crossref]
  4. S. F. Monaco, “Reflectance of an inhomogeneous thin film,” J. Opt. Soc. Am. 51, 280–282 (1961).
    [Crossref]
  5. E. M. Conwell, “Modes in optical waveguides formed by diffusion,” Appl. Phys. Lett. 23, 328–329 (1973).
    [Crossref]
  6. 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).
  7. A. A. Kolokolov, “Reflection of plane waves from an amplifying medium,” JETP Lett. 21, 312–313 (1975).
  8. G. N. Watson, Theory of Bessel Functions, 2nd ed. (Macmillan, New York, 1944), pp. 262–267.
  9. F. A. Jenkins and H. E. White, Fundamentals of Optics4th ed. (McGraw-Hill, New York, 1976), p. 485.
  10. P. S. Epstein, “Reflection of waves in an inhomogeneous absorbing medium,” Proc. Nat. Acad. Sci. 16, 627–637 (1930).

1976 (1)

1975 (1)

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

1973 (2)

E. M. Conwell, “Modes in optical waveguides formed by diffusion,” Appl. Phys. Lett. 23, 328–329 (1973).
[Crossref]

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, 565–566 (1973).

1972 (2)

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

1961 (1)

1930 (1)

P. S. Epstein, “Reflection of waves in an inhomogeneous absorbing medium,” Proc. Nat. Acad. Sci. 16, 627–637 (1930).

Callary, P. R.

Carniglia, C. K.

Conwell, E. M.

E. M. Conwell, “Modes in optical waveguides formed by diffusion,” Appl. Phys. Lett. 23, 328–329 (1973).
[Crossref]

Epstein, P. S.

P. S. Epstein, “Reflection of waves in an inhomogeneous absorbing medium,” Proc. Nat. Acad. Sci. 16, 627–637 (1930).

Jenkins, F. A.

F. A. Jenkins and H. E. White, Fundamentals of Optics4th ed. (McGraw-Hill, New York, 1976), p. 485.

Kogan, B. Ya.

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, 565–566 (1973).

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

Lebedev, S. A.

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, 565–566 (1973).

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

Monaco, S. F.

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).

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).

Volkov, V. M.

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, 565–566 (1973).

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

Watson, G. N.

G. N. Watson, Theory of Bessel Functions, 2nd ed. (Macmillan, New York, 1944), pp. 262–267.

White, H. E.

F. A. Jenkins and H. E. White, Fundamentals of Optics4th ed. (McGraw-Hill, New York, 1976), p. 485.

Appl. Phys. Lett. (1)

E. M. Conwell, “Modes in optical waveguides formed by diffusion,” Appl. Phys. Lett. 23, 328–329 (1973).
[Crossref]

J. Opt. Soc. Am. (2)

JETP Lett. (3)

B. Ya. 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. 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, 565–566 (1973).

Proc. Nat. Acad. Sci. (1)

P. S. Epstein, “Reflection of waves in an inhomogeneous absorbing medium,” Proc. Nat. Acad. Sci. 16, 627–637 (1930).

Other (2)

G. N. Watson, Theory of Bessel Functions, 2nd ed. (Macmillan, New York, 1944), pp. 262–267.

F. A. Jenkins and H. E. White, Fundamentals of Optics4th ed. (McGraw-Hill, New York, 1976), p. 485.

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Figures (4)

FIG. 1
FIG. 1

Reflection from an exponential region. Rays 1 and 2 represent the incident and reflected waves. Ray 3 is the refracted wave in the transparent region far from the interface. Ray 4 is the refracted wave in the amplifying region at the interface discussed in Sec. IV.

FIG. 2
FIG. 2

Reflectance vs incident angle for an exponential amplifying region. The value of d/λ is indicated by each curve. For all curves θc = 89° and γ = 0. 00015.

FIG. 3
FIG. 3

Reflectance vs incident angle. The solid curve is for d/λ = 246. The lower and upper dashed curves are the solutions corresponding to regions I and III, respectively, discussed in the text. For all curves, θc = 89°, γ = 0. 00015.

FIG. 4
FIG. 4

The complex plane ρ = σ + . The dashed curve is the locus of ρ for the case of amplification. The solid curve divides the dashed curve into three parts corresponding to the three regions of the incident angle discussed in Sec. II.

Tables (1)

Tables Icon

TABLE I Singular points of the reflectance for θc = 89° and γ = 0. 00015.

Equations (60)

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n = 1 + Δ n e z / d .
sin θ c = 1 / n .
E y i = E e i ( k x x + k z z ω t ) ,
k x = n k sin θ
k z = n k cos θ
E y r = R e i ( k x x k z z ω t ) .
E y = F ( z ) e i ( k x x ω t ) ,
d 2 F / d z 2 + [ ( z ) k 2 k x 2 ] F = 0 .
( z ) 1 + 2 Δ n e z / d .
ξ α e z / 2 d ,
α = 2 k d ( 2 Δ n ) 1 / 2 ,
d 2 F d ξ 2 + 1 ξ d F d ξ + ( ( 2 k d cos ψ ) 2 ξ 2 + 1 ) F = 0 .
cos ψ = ( 1 n 2 sin 2 θ ) 1 / 2 .
F ( z ) = A J ν ( ξ ) ,
ν = i 2 k d cos ψ .
J ν ( ξ ) e ik cos ψ z .
r = n cos θ i ( 2 Δ n ) 1 / 2 J ν ( α ) / J ν ( α ) n cos θ + i ( 2 Δ n ) 1 / 2 J ν ( α ) / J ν ( α ) ,
r = n cos θ cos ψ + i ( 2 Δ n ) 1 / 2 J ν + 1 ( α ) / J ν ( α ) n cos θ + cos ψ i ( 2 Δ n ) 1 / 2 J ν + 1 ( α ) / J ν ( α ) .
r = n cos θ cos ψ n cos θ + cos ψ ,
J ν ( α ) = ( α / 2 ) ν Γ ( ν ) m = 0 ( 1 ) m ( α / 2 ) 2 m m ! ( m + ν ) ( m + ν 1 ) ( ν ) .
| arg ξ | < π / 2 .
ρ = ln { ν / ξ + [ ( ν / ξ ) 2 1 ] 1 / 2 } .
0 < Im [ ρ ] < π
S ν ± ( ξ ) ~ exp { ± [ ν ( tanh ρ ρ ) i π / 4 ] } ( 1 2 i π ν tanh ρ ) 1 / 2 m = 0 B m ( ± ν tanh ρ ) m ,
S ν ± ( ξ ) { ν / ξ + [ ( ν / ξ ) 2 1 ] 1 / 2 } ν .
ν / ξ + [ ( ν / ξ ) 2 1 ] 1 / 2 1 2 ξ / ν ,
S ν ± ( ξ ) e i k cos ψ z .
S ν ± ( ξ ) c ± i k cos ψ z .
J ν ( α ) = S ν ( α ) .
J ν ( α ) = 1 2 [ S ν + ( α ) + S ν ( α ) ] .
J ν ( α ) = S + ( α ) ,
n = 1 i γ e z / d ,
n = 1 i γ for z < d
n = 1 for z > d .
d / λ 23. 48 ( m 1 ) + 11. 25 ,
n = 1 i γ .
r ± = 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 .
lim d S ν ± ( ξ ) = S ν ± ( α ) e i k ( 1 i γ ) cos ϕ z ,
lim d S ν + 1 ± ( α ) = ( i 2 k d / α ) S ν ± ( α ) [ cos ψ ± ( 1 i γ ) cos ϕ ]
B m = [ 2 m Γ ( m + 1 2 ) / Γ ( 1 2 ) ] A m ,
A 0 = 1 ,
A 1 = 1 8 5 24 coth 2 ρ ,
A 2 = 3 128 77 576 coth 2 ρ + 385 3456 coth 4 ρ ,
A 3 = 5 1024 1521 25 600 coth 2 ρ + 17 017 138 240 coth 4 ρ 17 017 248 832 coth 6 ρ .
cosh ρ = | cos ψ | γ 1 / 2 e z / 2 d ( 1 i ) / 2 ,
1 τ cot τ σ tanh σ = 0 .
| cos ψ | = 1. 24 γ 1 / 2 e z / 2 d .
ν tanh ρ = ( ν 2 α 2 ) 1 / 2 ,
ν tanh ρ = i 2 k d ( 1 i γ ) cos ϕ ,
( 1 i γ ) cos ϕ = ( cos 2 ψ 2 i γ ) 1 / 2 .
ξ α ( 1 z / d ) .
ν tanh ρ ν tanh ρ + ( α 2 z / d ν tanh ρ ) .
( ν + ν tanh ρ ) ( ν ν tanh ρ ) = α 2 ,
e ρ e ρ ( 1 + z ν / d ν tanh ρ ) ,
lim d e ν ρ = e ν ρ e z ν 2 / d ν tanh ρ
( ν + 1 ) tanh ρ ν tanh ρ + 1 / tanh ρ ,
e ρ e ρ ( 1 + 1 / ν tanh ρ ) ,
lim d e ν ρ = e ν ρ e 1 / tanh ρ .
lim d S ν + 1 ± ( ξ ) = ( 1 / ξ ) ( ν ν tanh ρ ) S ν ± ( ξ )