Abstract

The theory of electromagnetic fluctuations as developed by Rytov was used to calculate the radiant power as a function of frequency radiated from a homogeneous isothermal sphere. The formula obtained is valid for all radii, frequencies, and complex dielectric constants. It is also shown that the emission coefficient computed from this theory is precisely equal to the absorption coefficient computed from the Mie theory. The formula obtained is readily adaptable to numerical calculations, and results are presented for the case of a good conducting sphere with a wide range of size parameters.

© 1970 Optical Society of America

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References

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  1. S. M. Rytov, Theory of Electric Fluctuations and Thermal Radiation, AD 226765 (Defense Documentation Center, Arlington, Va., 1959).
  2. L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Addison-Wesley, Reading, Mass., 1960), p. 361.
  3. H. C. Van de Hulst, Light Scattering by Small Particles, (Wiley, New York, 1951), p. 452.
  4. G. W. Kattawar, G. N. Plass, Appl Opt. 6, 1377 (1967).
    [CrossRef] [PubMed]
  5. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), p. 415.
  6. Colorimetry, OSA Committee, The Science of Color (Crowell, New York, 1953), p. 192.
  7. H. B. G. Casimir, J. Chim. Phys. 43, 863 (1965).
    [CrossRef]
  8. G. W. Kattawar, M. Eisner, J. Chem Phys. 43, 863 (1965).
    [CrossRef]
  9. G. N. Plass, Appl. Opt. 5, 279 (1966).
    [CrossRef] [PubMed]

1967 (1)

G. W. Kattawar, G. N. Plass, Appl Opt. 6, 1377 (1967).
[CrossRef] [PubMed]

1966 (1)

1965 (2)

H. B. G. Casimir, J. Chim. Phys. 43, 863 (1965).
[CrossRef]

G. W. Kattawar, M. Eisner, J. Chem Phys. 43, 863 (1965).
[CrossRef]

Casimir, H. B. G.

H. B. G. Casimir, J. Chim. Phys. 43, 863 (1965).
[CrossRef]

Eisner, M.

G. W. Kattawar, M. Eisner, J. Chem Phys. 43, 863 (1965).
[CrossRef]

Kattawar, G. W.

G. W. Kattawar, G. N. Plass, Appl Opt. 6, 1377 (1967).
[CrossRef] [PubMed]

G. W. Kattawar, M. Eisner, J. Chem Phys. 43, 863 (1965).
[CrossRef]

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Addison-Wesley, Reading, Mass., 1960), p. 361.

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Addison-Wesley, Reading, Mass., 1960), p. 361.

Plass, G. N.

G. W. Kattawar, G. N. Plass, Appl Opt. 6, 1377 (1967).
[CrossRef] [PubMed]

G. N. Plass, Appl. Opt. 5, 279 (1966).
[CrossRef] [PubMed]

Rytov, S. M.

S. M. Rytov, Theory of Electric Fluctuations and Thermal Radiation, AD 226765 (Defense Documentation Center, Arlington, Va., 1959).

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), p. 415.

Van de Hulst, H. C.

H. C. Van de Hulst, Light Scattering by Small Particles, (Wiley, New York, 1951), p. 452.

Appl Opt. (1)

G. W. Kattawar, G. N. Plass, Appl Opt. 6, 1377 (1967).
[CrossRef] [PubMed]

Appl. Opt. (1)

J. Chem Phys. (1)

G. W. Kattawar, M. Eisner, J. Chem Phys. 43, 863 (1965).
[CrossRef]

J. Chim. Phys. (1)

H. B. G. Casimir, J. Chim. Phys. 43, 863 (1965).
[CrossRef]

Other (5)

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), p. 415.

Colorimetry, OSA Committee, The Science of Color (Crowell, New York, 1953), p. 192.

S. M. Rytov, Theory of Electric Fluctuations and Thermal Radiation, AD 226765 (Defense Documentation Center, Arlington, Va., 1959).

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Addison-Wesley, Reading, Mass., 1960), p. 361.

H. C. Van de Hulst, Light Scattering by Small Particles, (Wiley, New York, 1951), p. 452.

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Figures (2)

Fig. 1
Fig. 1

Emission coefficient vs size parameter for n1 = 1.01; n2 = 10−4, 10−2, 10−1, 1.0.

Fig. 2
Fig. 2

Emission coefficient vs size parameter for a good conducting sphere for n1 = n2 = 200.

Equations (74)

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D = E + ( - 1 ) K ,
B = H .
× E 0 = - i k H 0 ,
× H 0 = i k E 0 + i k ( - 1 ) K ,
k = ω / c ,
= - i .
× E 2 = - i k H 2 ,
× H 2 = i k E 2 ,
L n m = [ Z n ( q r ) r P n m r ^ + Z n ( q r ) r P n m θ ^ + i m r sin θ Z n ( q r ) P n m ϕ ^ ] e i m ϕ ,
M n m = Z n ( q r ) ( i m sin θ P n m θ ^ - P n m ϕ ^ ) e i m ϕ ,
N n m = { n ( n + 1 ) q r Z n ( q r ) P n m r ^ + 1 q r r [ r Z n ( q r ) ] P n m θ ^ + i m q r sin θ r [ r Z n ( q r ) ] P n m ϕ ^ } e i m ϕ ,
P n m P n m ( η ) ,         η = cos θ ,
P n m = P n m θ ,
× M n m = q N n m , × N n m = q M n m , × L n m = 0.
E 0 = n = 0 m = - n n ( A n m M 1 n m + B n m N 1 n m + C n m L 1 n m + A ˜ n m M ˜ 1 n m + B ˜ n m N ˜ 1 n m + C n m L 1 n m ) , H 0 = i 1 2 n = 0 m = - n n A n m N 1 n m + B n m M 1 n m + A ˜ n m N ˜ 1 n m + B ˜ n m M ˜ 1 n m .
E 1 = n = 0 m = - n n ( A 1 n m M 1 n m + B 1 n m M 1 n m ) , H 1 = i 1 2 n = 0 m = - n n ( A 1 n m N 1 n m + B 1 n m M 1 n m ) .
E 2 = n = 0 m = - n n ( Q n m M 2 n m + S n m N 2 n m ) , H 2 = i n = 0 m = - n n ( Q n m N 2 n m + S n m M 2 n m ) ,
E 0 ϕ + E 1 ϕ = E 2 ϕ E 0 θ + E 1 θ = E 2 θ } r = a ,             H 0 ϕ + H 1 ϕ = H 2 ϕ H 0 θ + H 1 θ = H 2 θ } r = a .
n = 0 m = - n n ( A n m ) × N 1 n m + ( B n m ) × M 1 n m + ( A ˜ n m ) × N ˜ 1 n m + ( B n m ) × M ˜ 1 n m = k 1 2 n = 0 m = - n n C n m L 1 n m + C ˜ n m L 1 n m + k ( - 1 ) / 1 2 K ,
n = 0 m = - n n ( A n m ) × M 1 n m + ( B n m ) × N 1 n m + ( A ˜ n m ) × M ˜ 1 n m + ( B ˜ n m ) × N ˜ 1 n m + ( C n m ) × L 1 n m + ( C ˜ n m ) × L ˜ 1 n m = 0.
n = 0 m = - n n ( C n m Z 1 n r + C ˜ n m Z ˜ 1 n r ) P n m e i m ϕ = - ( - 1 ) K r ,
n = 0 m = - n n { [ - A n m r ( r Z 1 n ) - A ˜ n m r ( r Z ˜ 1 n ) ] i m P n m / ( q r sin θ ) + ( B n m Z 1 n + B ˜ n m Z ˜ 1 n ) P n m - ( C n m Z 1 n + C ˜ n m Z ˜ 1 n ) q P n m / r } e i m ϕ = k ( - 1 ) 1 2 K θ ,
n = 0 m = - n n { [ A n m r ( r Z 1 n ) + A ˜ n m r ( r Z ˜ 1 n ) ] P n m / q r + ( B n m Z 1 n + B ˜ n m Z ˜ 1 n ) i m P n m / sin θ - ( C n m Z 1 n + C ˜ n m Z ˜ 1 n ) × i m q P n m / r sin θ } e i m ϕ = k ( - 1 ) 1 2 K ϕ ,
n = 0 m = - n n { ( A n m Z 1 n + A ˜ n m Z ˜ 1 n ) P n m - [ B n m r ( r Z 1 n ) + B ˜ n m r ( r Z ˜ 1 n ) ] i m P n m / ( q r sin θ ) - ( C n m Z 1 n + C ˜ n m Z ˜ 1 n ) i m P n m / ( r sin θ ) } e i m ϕ = 0 ,
n = 0 m = - n n { ( A n m Z 1 n + A ˜ n m Z ˜ 1 n ) i m P n m / sin θ + [ B n m r ( r Z 1 n ) + B ˜ n m r ( r Z ˜ 1 n ) ] P n m / q r + ( C n m Z 1 n + C ˜ n m Z ˜ 1 n ) P n m / r } e i m ϕ = 0 ,
C n m Z 1 n r + C ˜ n m Z ˜ 1 n r = - ( - 1 ) 2 π ρ n m - π π e - i m ϕ d ϕ 0 π K r P n m sin θ d θ ,
ρ n m = 2 ( n + m ) ! / [ ( 2 n + 1 ) ( n - m ) ! ] .
0 π ( P n m P n 1 m + m 2 sin 2 θ P n m P n 1 m ) sin θ d θ = δ n n 1 γ n m ,
γ n m = 2 n ( n + 1 ) ( n + m ) ! / [ ( 2 n + 1 ) ( n - m ) ! ]
- π π exp [ i ( m - m 1 ) ϕ ] d ϕ = 2 π δ m m , 0 π m sin θ ( P n m P n 1 m + P n m P n 1 m ) sin θ d θ = 0.
B n m Z 1 n + B ˜ n m Z ˜ 1 n - q r ( C n m Z 1 n + C n m Z ˜ 1 n ) = k ( - 1 ) 2 π γ n m 1 2 - π π e - i m ϕ d ϕ 0 π ( K θ P n m sin θ - i m K ϕ P n m ) d θ .
A n m θ + A n m θ ˜ = k ( - 1 ) 2 π γ n m 1 2 - π π e - i m ϕ d ϕ 0 π ( i m P n m K θ / sin θ + K ϕ P n m ) sin θ d θ ,
θ = 1 q r r ( r Z 1 n ) ,
θ ˜ = 1 q r r ( r Z ˜ 1 n ) .
B n m θ + B ˜ n m θ ˜ + ( C n m Z 1 n + C ˜ n m Z ˜ 1 n ) / r = 0 ,
A n m Z 1 n + A ˜ n m Z ˜ 1 n = 0.
C n m Z 1 n + C ˜ n m Z ˜ 1 n = 0
C n m Z 1 n + C ˜ n m Z ˜ 1 n = ( - 1 ) 2 π ρ n m - π π e - i m ϕ d ϕ 0 π K r P n m sin θ d θ .
B n m Z 1 n + B ˜ n m Z ˜ 1 n = k ( - 1 ) 2 π γ n m 1 2 - π π e - i m ϕ d ϕ 0 π ( K θ P n m sin θ - i m K ϕ P n m ) d θ ,
B n m ( r Z 1 n ) + B ˜ n m ( r Z ˜ 1 n ) = - ( - 1 ) 2 π ρ n m - π π e - i m ϕ d ϕ 0 π K r P n m sin θ d θ ,
A n m ( r Z 1 n ) + A ˜ n m ( r Z ˜ 1 n ) = k ( - 1 ) q r 2 π γ n m 1 2 - π π e - i m ϕ d ϕ 0 π ( i m sin θ P n m K θ + P n m K ϕ ) sin θ d θ ,
A n m Z 1 n + A ˜ n m Z ˜ 1 n = 0.
( r Z ˜ 1 n ) Z 1 n - Z ˜ 1 n ( r Z 1 n ) = 1 / q r ,
A ˜ m n = k ( - 1 ) ( q r ) 2 Z ln 2 π γ n m 1 2 - π π e - i m ϕ d ϕ 0 π ( i m sin θ P n m K θ + P n m K ϕ ) sin θ d θ ,
B ˜ n m = - q k r ( - 1 ) ( r Z 1 n ) 2 π γ n m 1 2 - π π e - i m ϕ d ϕ 0 π ( K θ P n m sin θ - i m K ϕ P n m ) d θ - ( - 1 ) q 2 r Z 1 n 2 π ρ n m × - π π e - i m ϕ d ϕ 0 π K r P n m sin θ d θ .
Q n m = - A ˜ n m / q a 2 [ h n ( β ) Z 1 n ( α ) - h n ( β ) Z 1 n ( α ) ] ,
S n m = B n m / k a { [ a h n ( β ) ] Z 1 n ( α ) - h n ( β ) [ a Z 1 n ( α ) ] } ,
α = q a , β = k a , h n ( β ) = ( π 2 β ) 1 2 H n + 1 2 ( 2 ) ( β ) , Z 1 n ( α ) = ( π / 2 a ) 1 2 J n + 1 2 ( α )
A ˜ n m ( r ) = 0 r A ˜ n m ( r ) d r ; B ˜ n m ( r ) = 0 r B ˜ n m ( r ) d r ,
P ω = c a 2 4 π - π π 0 π ( E θ H * ϕ - E ϕ H * θ + E * θ H ϕ - E * ϕ H θ ) sin θ d θ d ϕ ,
P ω = c k 2 n = 1 m = - n n λ n m ( Q n m 2 + S n m 2 ) ,
h * n ( β ) d h n ( β ) d β - d h * n ( β ) d β h n ( β ) = - 2 i / ( β ) 2 .
K α ( r , θ , ϕ ) K * β ( r , θ , ϕ ) = τ δ α β δ ( r - r ) δ ( θ - θ ) δ ( ϕ - ϕ ) / r 2 sin θ ,
τ = 2 coth [ ω / 2 K T ) Im ( 1 / ( - 1 ) ) ] ,
P ω = k c π coth ( ω / 2 K T ) d ω n = 1 ( 2 n + 1 ) ( Im [ j * n ( q a ) j n ( q a ) ] D n 2 + Im { m * 2 [ j n ( q a ) 2 a + j n ( q a ) j * n ( q a ) ] } / E n 2 ) ,
D n = a [ h n ( k a ) j n ( q a ) - h n ( k a ) j n ( q a ) ] ,
E n = [ a h n ( k a ) ] j n ( q a ) - h n ( k a ) [ a j n ( q a ) ] ,
x = k a , y = m x , ψ n ( z ) = z j n ( z ) = ( π z / 2 ) 1 2 J n + 1 2 ( z ) , ζ n ( z ) = z h n ( 2 ) ( z ) = ( π z / 2 ) H n + 1 2 ( 2 ) ( z ) , D n ( y ) = [ ln ψ n ( y ) ] , G n ( x ) = [ ln ζ n ( x ) ] ,
P ω = k c π coth ( ω / 2 K T ) d ω n = 1 ( 2 n + 1 ) ζ n ( x ) 2 { Im [ m D n ( y ) ] m D n ( y ) - G n ( x ) 2 + Im [ m * D n ( y ) ] m G n ( x ) - D n ( y ) 2 } .
coth ( ω / 2 K T ) = 2 [ 1 2 + 1 exp ( ω / K T ) - 1 ] .
P λ = 8 π 2 a 2 h c 2 d λ λ 5 [ 1 2 + 1 exp ( h c / λ K T ) - 1 ] 2 x 2 n = 1 ( 2 n + 1 ) ζ n ( x ) 2 { Im [ m D n ( y ) ] m D n ( y ) - G n ( x ) 2 + Im [ m * D n ( x ) ] m G n ( x ) - D n ( y ) 2 } .
ϕ λ = 8 π 2 a 2 h c 2 e [ 1 exp ( h c / λ K T ) - 1 ] d λ λ 5 .
e = 2 x 2 n = 1 ( 2 n + 1 ) ζ n ( x ) 2 { Im [ m D n ( y ) ] m D n ( y ) - G n ( x ) 2 + Im [ m * D n ( y ) ] m G n ( x ) - D n ( y ) 2 } .
Q a b s = 2 x 2 n = 1 ( 2 n + 1 ) [ Re ( a n + b n ) - a n 2 - b n 2 ] ,
a n = ψ n ( x ) ζ n ( x ) D n ( y ) - m D n ( x ) D n ( y ) - m G n ( x ) ,
b n = ψ n ( x ) ζ n ( x ) m D n ( y ) - D n ( x ) m D n ( y ) - G n ( x ) .
ψ n ( x ) ζ n ( x ) - ψ n ( x ) ζ n ( x ) = i .
ψ n ( x ) ζ n ( x ) [ D n ( x ) - G n ( x ) ] = i .
D n ( x ) [ ζ n ( x ) + ζ * n ( x ) ] = G n ( x ) ζ n ( x ) + G * n ( x ) ζ * n ( x ) .
α n = Re ( α n ) - a n 2 = ( a n + a * n ) / 2 - a n 2 .
α n = { ζ * n ( x ) [ D n ( y ) - m D n ( x ) ] [ D * n ( y ) - m * G * n ( x ) ] + ζ n ( x ) [ D * n ( y ) - m * D n ( x ) ] [ D n ( y ) - m G n ( x ) ] - ( ζ n + ζ * n ) [ D n ( y ) 2 - m D n ( x ) D * n ( y ) - m * D n ( x ) D n ( y ) + m 2 D n 2 ( x ) ] } ψ n ( x ) / ( 2 F n 2 ) ,
ψ n ( x ) = [ ζ n ( x ) + ζ * n ( x ) ] / 2
F n = ζ n ( x ) [ D n ( y ) - m G n ( x ) ] .
α n = Im [ m * D n ( y ) ] / F n 2 .

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