Abstract

A set of equations is presented for the reflection and transmission of normally incident radiation by absorbing (metal) films on nonabsorbing backings.

Using these equations, the results of two different methods of calculation of the reflection and transmission coefficients of a thick nonabsorbing plate are shown to be practically identical, both with and without an absorbing (metal) film on one surface of the plate. One of these methods involves the addition of intensities of the multiple reflections from the two surfaces of the plate. The other method is a rigorous calculation involving solution of the boundary condition equations (amplitude addition) for small areas followed by intensity averaging over the whole plate.

In order to calculate optical constants of a metal film on a thick nonabsorbing backing, the much simpler method of intensity addition of internal reflections in the backing may usually be applied.

© 1951 Optical Society of America

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Corrections

Louis Harris, John F. Beasley, and Arthur L. Loeb, "Errata*: Reflection and Transmission of Radiation by Metal Films, and the Influence of Nonabsorbing Backings.," J. Opt. Soc. Am. 41, 1071_4-1071 (1951)
https://www.osapublishing.org/josa/abstract.cfm?uri=josa-41-12-1071_4

References

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  1. J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941), pp. 511–515.
  2. K. Koller, Z. Physik 110, 661 (1938).
    [CrossRef]
  3. F. Matossi, J. Opt. Soc. Am. 39, 928 (1949).
    [CrossRef]
  4. H. Goldschmidt, Ann. Physik 82, 947 (1927).
    [CrossRef]
  5. H. Murmann, Z. Physik 80, 191 (1933).
  6. J. Krautkrämer, Ann. Physik 32, 537 (1938).
    [CrossRef]
  7. F. Goos, Z. Physik 106, 606 (1937).
    [CrossRef]
  8. L. N. Hadley and D. M. Dennison, J. Opt. Soc. Am. 37, 451 (1947).
    [CrossRef]
  9. R. L. Mooney, J. Opt. Soc. Am. 35, 574 (1945).
    [CrossRef]
  10. F. A. Jenkins and H. E. White, Fundamentals of Physical Optics (McGraw-Hill Book Company, Inc., New York, 1937), p. 36.
  11. L. A. Pipes, Applied Mathematics for Engineers and Physicists (McGraw-Hill Book Company, Inc., New York, 1946), p. 461.

1949 (1)

1947 (1)

1945 (1)

1938 (2)

K. Koller, Z. Physik 110, 661 (1938).
[CrossRef]

J. Krautkrämer, Ann. Physik 32, 537 (1938).
[CrossRef]

1937 (1)

F. Goos, Z. Physik 106, 606 (1937).
[CrossRef]

1933 (1)

H. Murmann, Z. Physik 80, 191 (1933).

1927 (1)

H. Goldschmidt, Ann. Physik 82, 947 (1927).
[CrossRef]

Dennison, D. M.

Goldschmidt, H.

H. Goldschmidt, Ann. Physik 82, 947 (1927).
[CrossRef]

Goos, F.

F. Goos, Z. Physik 106, 606 (1937).
[CrossRef]

Hadley, L. N.

Jenkins, F. A.

F. A. Jenkins and H. E. White, Fundamentals of Physical Optics (McGraw-Hill Book Company, Inc., New York, 1937), p. 36.

Koller, K.

K. Koller, Z. Physik 110, 661 (1938).
[CrossRef]

Krautkrämer, J.

J. Krautkrämer, Ann. Physik 32, 537 (1938).
[CrossRef]

Matossi, F.

Mooney, R. L.

Murmann, H.

H. Murmann, Z. Physik 80, 191 (1933).

Pipes, L. A.

L. A. Pipes, Applied Mathematics for Engineers and Physicists (McGraw-Hill Book Company, Inc., New York, 1946), p. 461.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941), pp. 511–515.

White, H. E.

F. A. Jenkins and H. E. White, Fundamentals of Physical Optics (McGraw-Hill Book Company, Inc., New York, 1937), p. 36.

Ann. Physik (2)

H. Goldschmidt, Ann. Physik 82, 947 (1927).
[CrossRef]

J. Krautkrämer, Ann. Physik 32, 537 (1938).
[CrossRef]

J. Opt. Soc. Am. (3)

Z. Physik (3)

H. Murmann, Z. Physik 80, 191 (1933).

K. Koller, Z. Physik 110, 661 (1938).
[CrossRef]

F. Goos, Z. Physik 106, 606 (1937).
[CrossRef]

Other (3)

J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941), pp. 511–515.

F. A. Jenkins and H. E. White, Fundamentals of Physical Optics (McGraw-Hill Book Company, Inc., New York, 1937), p. 36.

L. A. Pipes, Applied Mathematics for Engineers and Physicists (McGraw-Hill Book Company, Inc., New York, 1946), p. 461.

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Figures (6)

Figs 1 and 1′
Figs 1 and 1′

Internal reflections in a thick nonabsorbing plate with a thin absorbing film on one surface.

Fig. 2
Fig. 2

Reflection and transmission at the surface of a semi-infinite nonabsorbing medium.

Fig. 3
Fig. 3

Reflection and transmission of a nonabsorbing layer in air.

Fig. 4
Fig. 4

Reflection and transmission of a thin absorbing layer between air and a nonabsorbing plate.

Fig. 5
Fig. 5

Reflection and transmission of two adjacent layers with plane parallel surfaces.

Fig. 6
Fig. 6

Distribution function of θ.

Tables (3)

Tables Icon

Table I Reflection and transmission of radiation (λ=6000A) by a thin, absorbing film on a thick, nonabsorbing backing, for various values of the optical constants na and ka and of the thickness a of the absorbing material, and of the index of refraction nb of the backing: (1) as found by intensity addition of internal reflections; (2) as found by amplitude addition followed by averaging; (3) as reported by Koller.

Tables Icon

Table II Reflection and transmission coefficients as a function of the angle of incidence as calculated from the equations of Hadley and Dennison, and the coefficients obtained from Eqs. (20) and (21).

Tables Icon

Table III Discrepancy (Δ) between reflections (expressed in fractions) obtained by intensity and by amplitude addition.

Equations (67)

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R 1 = R o a T 1 = T o a T 2 = T o a ( 1 - R o b ) R 2 = T o a R o b T 3 = T o a 2 R o b R 3 = T o a R o b R b a T 4 = T o a R o b R b a ( 1 - R 4 b ) R 4 = T o a R o b 2 R b a T 5 = T o a 2 R o b 2 R b a R 5 = T o a R o b 2 R b a 2 T 6 = T o a R o b 2 R b a 2 ( 1 - R o b ) R = R 1 + T 3 + T 5 + = R o a + R o b T o a 2 / ( 1 - R o b R b a )
T = T 2 + T 4 + T 6 + = T o a ( 1 - R o b ) / ( 1 - R 4 b R b a ) .
R = R 1 + T 3 + T 5 + = R o b + R b a ( 1 - R o b ) 2 / ( 1 - R b a R o b )
T = T o a ( 1 - R o b ) / ( 1 - R o b R b a ) T = T , but R R .
R ¯ = R ( θ ) f ( θ ) d θ ;             T ¯ = T ( θ ) f ( θ ) d θ .
f ( θ ) = ( φ ( λ ) ψ ( θ / 2 π n b ) d λ ) ( φ ( λ ) ψ ( λ / 2 π n b θ ) d θ d λ ) .
f ( θ ) d θ = 1.
f ( θ ) = 1 /             for             θ ¯ - / 2 < θ < θ ¯ + / 2 = 0             elsewhere , being a constant .
R ¯ = 1 θ ¯ - / 2 θ ¯ + / 2 R ( θ ) d θ , T ¯ = 1 θ ¯ - / 2 θ ¯ + / 2 T ( θ ) d θ . }
E m = ( E m + exp [ i K m z ] + E m - exp [ - i K m z ] ) exp [ - i ω t ] ,
H m = Z m - 1 ( E m + exp [ i K m z ] - E m - exp [ - i K m z ] ) exp [ - i ω t ] ,
K m = a m + i β m = ( 2 π / λ ) ( n m + i k m ) = ( 2 π / λ ) n m ( 1 + i κ m ) , K o = α o = ( 2 π / λ ) n o ,
Z m , m + 1 = Z m / Z m + 1 = K m + 1 / K m .
E m ( at z = m ) = E m + 1 ( at z = m )
H m ( at z = m ) = H m + 1 ( at z = m ) .
E m + exp [ i K m m ] + E m - exp [ - i K m m ] = E m + 1 + exp [ i K m + 1 m ] + E m + 1 - exp [ - i K m + 1 m ] , E m + exp [ i K m m ] - E m - exp [ - i K m m ] = Z m , m + 1 ( E m + 1 + exp [ i K m + 1 m ] - E m + 1 - exp [ - i K m + 1 m ] ) .
T m = E m × H m E o × H o = Z o E m 2 Z m E o 2 = K m K o E m 2 E o 2 = Z o m E m 2 E o 2 .
( 1 )             E o + + E o - = E b , ( 2 )             E o + - E o - = Z o b E b .
R o b = E o - × H o - E o + × H o + = E o - 2 E o + 2 = ( n b - 1 ) 2 ( n b + 1 ) 2 ,
T o b = E b × H b E o + × H o + = Z o b E b 2 E o + 2 = 4 n b ( n b + 1 ) 2 . R o b + T o b = 1.
R = E o - / E o + 2 = ( Z o b - Z b o ) 2 sin 2 K b b 4 cos 2 K b b + ( Z o b + Z b o ) 2 sin 2 K b b = sin 2 K b b [ 4 / ( Z o b - Z b o ) 2 ] + sin 2 K b b ,
T = E o t / E o + 2 = 4 4 cos 2 K b b + ( Z o b + Z b o ) 2 sin 2 K b b = 4 / ( Z o b - Z b o ) 2 [ 4 / ( Z o b - Z b o ) 2 ] + sin 2 K b b , R + T = 1.
R = ( Z o a - Z a o ) i sin K a a 2 2 cos K a a - i ( Z o a + Z a o ) sin K a a 2 ,
T = 4 2 cos K a a - i ( Z o a + Z a o ) sin K a a 2 .
z < 0 is air , z > a is a transparent medium .
R o a = E o - 2 E o + 2 = ( Z o b - 1 ) cos K a a + ( Z a b - Z o a ) i sin K a a 2 ( Z o b + 1 ) cos K a a - ( Z a b + Z o a ) i sin K a a 2 ,
T o a = Z o b E b t 2 E o + 2 = 4 Z o b ( Z o b + 1 ) cos K a a - ( Z a b + Z o a ) i sin K a a 2 .
R b a = ( Z o b - 1 ) cos K a a - ( Z a b - Z o a ) i sin K a a 2 ( Z o b + 1 ) cos K a a - ( Z a b + Z o a ) i sin K a a 2 ,
T b a = 4 Z o b ( Z o b + 1 ) cos K a a - ( Z a b + Z o a ) i sin K a a 2 ,
T o a = T b a ,             but             R o a R b a .
p = Z o b - Z b o ; s = Z o b + Z b o , q = Z a b - Z b a ; t = Z a b + Z b a , r = Z a o - Z o a ; u = Z a o + Z o a , R = q sin K a a sin K b b - i ( p cos K a a sin K b b - r sin K a a cos K b b ) 2 ( 2 cos K a a cos K b b - t sin K a a sin K b b ) - i ( s cos K a a sin K b b + u sin K a a cos K b b ) 2 ,
T = 4 ( 2 cos K a a cos K b b - t sin K a a sin K b b ) - i ( s cos K a a sin K b b + u sin K a a cos K b b ) 2 .
R = - q sin K a a sin K b b - i ( p cos K a a sin K b b - r sin K a a cos K b b ) 2 ( 2 cos K a a cos K b b - t sin K a a sin K b b ) - i ( s cos K a a sin K b b + u sin K a a cos K b b ) 2 ,
T = 4 ( 2 cos K a a cos K b b - t sin K a a sin K b b ) - i ( s cos K a a sin K b b + u sin K a a cos K b b ) . T = T , but R R .
R ¯ = 1 θ ¯ - / 2 θ ¯ + / 2 sin 2 θ [ 4 n b 2 / ( n b 2 - 1 ) 2 ] + sin 2 θ d θ = 1 - T ¯ ,
T ¯ = 1 θ ¯ - / 2 θ ¯ + / 2 [ 4 n b 2 / ( n b 2 - 1 ) 2 ] [ 4 n b 2 / ( n b 2 - 1 ) 2 ] + sin 2 θ d θ = 2 n b ( n b 2 + 1 ) [ tan - 1 { n b 2 + 1 2 n b sin cos [ ( θ ¯ + ( / 2 ) ] cos [ ( θ ¯ - ( / 2 ) ] + ( n b 2 + 1 / 2 n b ) 2 sin [ θ ¯ + ( / 2 ) ] sin [ θ ¯ - ( / 2 ) ] } ] .
R I = ( n b - 1 ) 2 / n b 2 + 1 ,
T I = 2 n b / n b 2 + 1.
R = ( A tan θ + B ) ( A * tan θ + B * ) ( C tan θ + D ) ( C * tan θ + D * ) ; T = 4 sec 2 θ ( C tan θ + D ) ( C * tan θ + D * ) ,
x = tan θ .
R ¯ = 1 π - ( A x + B ) ( A * x + B * ) ( C x + D ) ( C * x + D * ) ( 1 + x 2 ) d x ,
T ¯ = 4 π - d x ( C x + D ) ( C * x + D * ) .
- Q ( x ) d x             ( x is real ) ,
- Q ( x ) d x = 2 π i Res + .
Res at z = z 0             Q ( z ) = lim z z 0             ( z - z 0 ) Q ( z ) .
Q ( z ) = ( A z + B ) ( A * z + B * ) ( C z + D ) ( C * z + D * ) ( 1 + z 2 )
Q ( z ) = 1 ( C z + D ) ( C * z + D )
R ¯ = ( B + i A ) ( B * + i A * ) ( D + i C ) ( D * + i C * ) + 2 i ( B C - A D ) ( B * C - A * D ) ( C D * - C * D ) ( C 2 + D 2 ) if imaginary part of ( - D C ) > 0 = ( B + i A ) ( B * + i A * ) ( D + i C ) ( D * + i C * ) + 2 i ( B C * - A D * ) ( B * C * - A * D * ) ( C * D - C D * ) ( C * 2 + D * 2 ) if imaginary part of ( - D C ) < 0 ,
T ¯ = 8 i C D * - C * D if imaginary part of ( - D C ) > 0 = - 8 i C D * - C * D if imaginary part of ( - D C ) < 0.
- D C = - ( C r D r + C i D i ) + i ( C i D r - C r D i ) C 2
B C - A D = - 2 i p - 2 i ( Z 0 b - Z b 0 ) .
R ( a = 0 , θ ) ¯ = R ( a = 0 , θ b )
T ( a = 0 , θ ) ¯ = T ( a = 0 , θ b )
R ( a , θ ) ¯ = R ( a , θ b )             and             T ( a , θ ) ¯ = T ( a , θ b ) .
cos θ ¯ = sin θ ¯ = 0             and             cos 2 θ ¯ = sin 2 θ ¯ = 1 2 .
F ( sin θ ) , cos θ ¯ F ( sin θ ¯ ) , ( cos θ ¯ ) ,
R = ( n b 2 - 1 ) 2 ( n b 2 + 1 ) 2 + 4 n b 2 ;             T = 8 n b 2 ( n b 2 + 1 ) 2 + 4 n b 2 .
φ ( λ ) = 1 / γ when λ ¯ - γ / 2 < λ < λ ¯ + γ / 2 φ ( λ ) = 0 elsewhere .
ψ ( b ) = 1 / ζ when b ¯ - ζ / 2 < b < b ¯ + ζ / 2 ψ ( b ) = 0 elsewhere .
f ( θ ) = 0 in the interval θ θ 1 , f ( θ ) = 1 ζ ln λ ¯ + γ / 2 λ ¯ - γ / 2 [ λ ¯ + γ / 2 2 π n b - b ¯ - ζ / 2 θ ] in the interval θ 1 θ θ 2 , f ( θ ) = { 1 θ ln λ ¯ + γ / 2 λ ¯ - γ / 2 when γ λ ¯ ζ b ¯ γ 2 π n b ζ ln λ ¯ + γ / 2 λ ¯ - γ / 2 when γ λ ¯ ζ b ¯ } in the interval θ 2 θ θ 3 , f ( θ ) = 1 ζ ln λ ¯ + γ / 2 λ ¯ - γ / 2 [ b ¯ + ζ / 2 θ - λ ¯ - γ / 2 2 π n b ] in the interval θ 3 θ θ 4 , f ( θ ) = 0 in the interval θ 4 θ ,
θ 1 = 2 π n b b ¯ - ζ / 2 λ ¯ + γ / 2 , θ 2 = { 2 π n b b ¯ - ζ / 2 λ ¯ + γ / 2 when γ / λ ¯ ζ / b ¯ 2 π n b b ¯ + ζ / 2 λ ¯ - γ / 2 when γ / λ ¯ ζ / b ¯ , θ 3 = { 2 π n b b ¯ + ζ / 2 λ ¯ - γ / 2 when γ / λ ¯ ζ / b ¯ 2 π n b b ¯ - ζ / 2 λ ¯ + γ / 2 when γ / λ ¯ ζ / b ¯ θ 4 = 2 π n b b ¯ + ζ / 2 λ ¯ - γ / 2 .
δ ( x - x ¯ ) = 0 when x x ¯ , δ ( x - x ¯ ) = when x = x ¯ , - δ ( x - x ¯ ) d κ = 1.
R ¯ = - R ( θ ) δ ( θ - b ¯ / λ ¯ ) d θ = R ( b ¯ / λ ¯ ) ;             T ¯ = T ( b ¯ / λ ¯ ) .
R ¯ = R I + Δ ( , δ ) ;             T ¯ = T I - Δ ( , δ ) ,
Δ max 0 = ( 4 N - 3 ) / 8 N 2 when δ = 0 ,
Δ max ( 4 N - 3 ) / 8 N 2 δ when δ π / 2 ,
N = 4 n b 2 ( n b 2 - 1 ) 2 .