Abstract

For an absorbing plane parallel plate, a simple reciprocal transformation exists between the expressions for the reflectance R and transmittance T, enabling easy passage from one to the other. Then it is shown that the expression A=1RT is always positive and, further, may be identified with the energy computed for the dissipation by the Joule effect in the plate, provided that the resultant electric field is calculated at each point inside the plate.

© 1968 Optical Society of America

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References

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  1. O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths Scientific Publications Ltd., London, 1955).
  2. M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1965), p. 627.
  3. B. Salzberg, Am. J. Phys. 16, 444 (1948).
    [Crossref]
  4. A. Vašíček, Optics of Thin Films (North-Holland Publishing Co., Amsterdam, 1960) p. 301.
  5. The choice of this convention is dictated by the simplicity of the transformation demonstrated later, −α is equal to β, the phase defined by J. M. Bennett in J. Opt. Soc. Am. 54, 612 (1964).
    [Crossref]
  6. A. R. Cownie, J. Opt. Soc. Am. 47, 132 (1957).
    [Crossref]
  7. R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. II (Addison–Wesley Publishing Co., Reading, Massachusetts, 1964), Ch. 27.
  8. w0 is a part of the total energy density, the other part corresponds to the potential and kinetic energy of the dipoles.
  9. This definition of w0and S, as pointed out by many authors, is not unique, but the expressions adopted are the simplest and no disagreement with experiment has ever been found.
  10. J. A. Stratton, Electromagnetic Theory (McGraw–Hill Book Co., New York, 1941), p. 131.
  11. P. H. Berning, in Physics of Thin Films, Vol. 1, G. Hass and R. Tbun, Eds. (Academic Press Inc., New York and London, 1963), p. 76.

1964 (1)

1957 (1)

1948 (1)

B. Salzberg, Am. J. Phys. 16, 444 (1948).
[Crossref]

Bennett, J. M.

Berning, P. H.

P. H. Berning, in Physics of Thin Films, Vol. 1, G. Hass and R. Tbun, Eds. (Academic Press Inc., New York and London, 1963), p. 76.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1965), p. 627.

Cownie, A. R.

Feynman, R. P.

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. II (Addison–Wesley Publishing Co., Reading, Massachusetts, 1964), Ch. 27.

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths Scientific Publications Ltd., London, 1955).

Leighton, R. B.

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. II (Addison–Wesley Publishing Co., Reading, Massachusetts, 1964), Ch. 27.

Salzberg, B.

B. Salzberg, Am. J. Phys. 16, 444 (1948).
[Crossref]

Sands, M.

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. II (Addison–Wesley Publishing Co., Reading, Massachusetts, 1964), Ch. 27.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw–Hill Book Co., New York, 1941), p. 131.

Vašícek, A.

A. Vašíček, Optics of Thin Films (North-Holland Publishing Co., Amsterdam, 1960) p. 301.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1965), p. 627.

Am. J. Phys. (1)

B. Salzberg, Am. J. Phys. 16, 444 (1948).
[Crossref]

J. Opt. Soc. Am. (2)

Other (8)

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths Scientific Publications Ltd., London, 1955).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1965), p. 627.

A. Vašíček, Optics of Thin Films (North-Holland Publishing Co., Amsterdam, 1960) p. 301.

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. II (Addison–Wesley Publishing Co., Reading, Massachusetts, 1964), Ch. 27.

w0 is a part of the total energy density, the other part corresponds to the potential and kinetic energy of the dipoles.

This definition of w0and S, as pointed out by many authors, is not unique, but the expressions adopted are the simplest and no disagreement with experiment has ever been found.

J. A. Stratton, Electromagnetic Theory (McGraw–Hill Book Co., New York, 1941), p. 131.

P. H. Berning, in Physics of Thin Films, Vol. 1, G. Hass and R. Tbun, Eds. (Academic Press Inc., New York and London, 1963), p. 76.

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Equations (41)

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r = ( 1 N ) / ( 1 + N ) = ( R ) e i α , 5 F 1 = exp ( i 2 π N l / λ ) = ( A ) e i φ , A = exp ( 4 π k l / λ ) , φ = 2 π n l / λ .
ρ = ( R ) e i Φ r = ( 1 F 2 ) r / ( 1 r 2 F 2 ) ,
τ = ( T ) e i Φ t = ( 1 r 2 ) F 1 / ( 1 r 2 F 2 ) ,
ρ τ ( or vice versa ) if r F 1 .
R A α φ .
R = [ ( 1 A ) / ( 1 A R ) ] 2 R [ 1 + 4 A sin 2 φ / ( 1 A ) 2 ] / [ 1 + 4 A R sin 2 ( φ + α ) / ( 1 A R ) 2 ] ,
T = [ ( 1 R ) / ( 1 A R ) ] 2 A [ 1 + 4 R sin 2 α / ( 1 R ) 2 ] / [ 1 + 4 A R sin 2 ( φ + α ) / ( 1 A R ) 2 ] ,
tan Φ r = [ ( 1 A 2 R ) sin α ( 1 R ) A sin ( α + 2 φ ) ] / [ ( 1 + A 2 R ) cos α ( 1 + R ) A cos ( α + 2 φ ) ] ,
tan Φ t = [ ( 1 A R 2 ) sin φ ( 1 A ) R sin ( φ + 2 α ) ] / [ ( 1 + A R 2 ) cos φ ( 1 + A ) R cos ( φ + 2 α ) ] .
tan Φ r = [ ( 1 R ) / ( 1 + R ) ] cot φ , tan Φ t = [ ( 1 + R ) / ( 1 R ) ] tan φ ,
tan Φ r tan Φ t = 1.
A = 1 R T = [ ( 1 A ) ( 1 R ) ( 1 + A R ) + 8 A R sin α sin φ cos ( φ + α ) / [ ( 1 A R ) 2 + 4 A R sin 2 ( φ + α ) ] .
[ ( R ) / ( 1 R ) ] sin α = ( log A ) / 4 φ = k / 2 n .
g / ( 1 R ) = ( 1 A ) ( 1 + A R ) + 2 A ( R ) log A ( sin φ / φ ) cos ( φ + α ) > ( 1 A ) ( 1 + A R ) + 2 A ( R ) log A = G , G / R = A ( 1 A ) [ 1 + log A / ( 1 A ) ( R ) ] < 0.
w 0 = ( 0 / 2 ) E · E + ( 0 c 2 / 2 ) B · B ,
S = 0 c 2 E × B ,
E · J = w 0 / t · S .
( ω ) = ( ω ) i ( ω ) = ( ω ) i [ σ ( ω ) / ω ] .
S = Re ( 0 c 2 / 2 ) E 0 × B 0 * ,
· S = ( σ / 2 ) E 0 · E 0 * ,
E i x = 0 , E i y = 0 , E i z = E i exp [ i ( ω t k i x ) ] .
S i x = ( 0 c / 2 ) | E i | 2 , S r x = ( 0 c / 2 ) | E i | 2 [ ( 1 n ) 2 + k 2 ] / [ ( 1 + n ) 2 + k 2 ] , S t x = ( 0 c / 2 ) | E i | 2 4 n / [ ( 1 + n ) 2 + k 2 ] } .
S i x + S r x = S t x .
S i x = ( 0 c / 2 ) n | E i | 2 , S r x = ( 0 c / 2 ) n | E i | 2 [ ( n 1 ) 2 + k 2 ] / [ ( n + 1 ) 2 + k 2 ] , S t x = ( 0 c / 2 ) | E i | 2 4 ( n 2 + k 2 ) / [ ( n + 1 ) 2 + k 2 ] }
w J = ( σ / 2 ) ( E i 0 · E i 0 * + E r 0 · E r 0 * ) .
w J = ( σ / 2 ) Re [ ( E i 0 + E r 0 ) · ( E i 0 + E r 0 ) * ] = w J + ( σ / 2 ) Re ( E i 0 · E r 0 * + E r 0 · E i 0 * ) .
S I x = S I I x .
k 0 = 2 π / λ , k = ( 2 π / λ ) ( n i k ) , = 0 ( n 2 k 2 ) , = σ / ω = 2 0 n k ;
E 1 + = exp [ i ( ω t k 0 x ) ] E 1 = [ ( 1 F 2 ) r / ( 1 r 2 F 2 ) ] exp [ i ( ω t + k 0 x ) ] ;
E 2 + = [ ( 1 + r ) / ( 1 r 2 F 2 ) ] exp [ i ( ω t k x ) ] , E 2 = [ ( 1 r ) r F 2 / ( 1 r 2 F 2 ) ] exp [ i ( ω t + k x ) ] ;
E 3 + = [ ( 1 r 2 ) F 1 / ( 1 r 2 F 2 ) ] exp { i [ ω t k 0 ( x l ) ] } , E 3 = 0.
E = E 0 e i ω t = [ ( 1 + r ) / ( 1 r 2 F 2 ) ] × ( e i k x r F 2 e i k x ) e i ω t .
w J ( x ) = ( σ / 2 ) E 0 E 0 * .
M = e i k x r F 2 e i k x .
( 1 + r ) ( 1 + r ) * = 1 + 2 ( R ) cos α + R , ( 1 r 2 F 2 ) ( 1 r 2 F 2 ) * = ( 1 A R ) 2 + 4 A R sin 2 ( φ + α ) , M M * = e a x + A 2 R e a x 2 A ( R ) × cos [ ( 4 π n x / λ ) 2 φ α ] ,
a = 4 π k / λ .
W J = 0 l W J ( x ) d x ,
W J = ( σ / 2 ) [ 1 + 2 ( R ) cos α + R ] ( 1 A R ) 2 + 4 A R sin 2 ( φ + α ) × [ ( λ / 4 π k ) ( 1 A ) ( 1 + A R ) ( λ / 4 π n ) 4 A ( R ) sin φ cos ( φ + α ) ] .
1 R = n [ 1 + 2 ( R ) cos α + R ]
W J = ( 0 c / 2 ) [ ( 1 A ) ( 1 R ) ( 1 + A R ) + 8 A R sin α sin φ cos ( φ + α ) ] / [ ( 1 A R ) 2 + 4 A R sin 2 ( φ + α ) ] ,
W J = ( 0 c / 2 ) A .