Abstract

A study is made of the directional transmittance of an isothermal dielectric surrounded by air. The dielectric sheet studied is assumed to be infinite in extent but of finite thickness much greater than one wavelength with plane parallel reflecting surfaces. The dielectric–air interfaces are assumed to be smooth, and therefore, reflect specularly according to Fresnel’s law. Interference is not considered in the solution. The approach to the problem is through the solution of the transport equation with boundary conditions considering refraction, total reflection beyond the critical angle, and directional irradiance. The results are plotted as the directional monochromatic transmittance for various optical depths with diffuse irradiance. The same curves may be read at a discrete direction to determine the monochromatic transmittance for mon-directional irradiance. The same procedure is followed to determine the directional emittance.

© 1966 Optical Society of America

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References

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  1. H. O. McMahon, J. Opt. Soc. Am. 40, 376, (1950).
    [CrossRef]
  2. J. E. Francis and T. J. Love, AIAA paper No. 65-119.
  3. J. P. Funk, J. Opt. Soc. Am. 50, 986 (1960).
    [CrossRef]
  4. J. C. Richmond, J. Res. Natl. Bur. Std. 67C, 217 (1963).
    [CrossRef]
  5. Robert Gardon, J. Am. Ceram. Soc. 39, 278 (1956).
    [CrossRef]
  6. E. U. Condon and Hugh Odishaw, Handbook of Physics (McGraw-Hill Book Co., Inc., New York, 1958).
  7. V. Kourganoff, Basic Methods in Transfer Problems (Oxford University Press, London, England, 1952).
  8. T. J. Love, Aeronautical Research Laboratories Publication No. ARL 63-3, U. S. Air Force, Wright-Patterson AFB, Ohio.
  9. M. Planck, The Theory of Heat Radiation (Dover Publications, Inc., New York, 1959).

1963 (1)

J. C. Richmond, J. Res. Natl. Bur. Std. 67C, 217 (1963).
[CrossRef]

1960 (1)

1956 (1)

Robert Gardon, J. Am. Ceram. Soc. 39, 278 (1956).
[CrossRef]

1950 (1)

Condon, E. U.

E. U. Condon and Hugh Odishaw, Handbook of Physics (McGraw-Hill Book Co., Inc., New York, 1958).

Francis, J. E.

J. E. Francis and T. J. Love, AIAA paper No. 65-119.

Funk, J. P.

Gardon, Robert

Robert Gardon, J. Am. Ceram. Soc. 39, 278 (1956).
[CrossRef]

Kourganoff, V.

V. Kourganoff, Basic Methods in Transfer Problems (Oxford University Press, London, England, 1952).

Love, T. J.

T. J. Love, Aeronautical Research Laboratories Publication No. ARL 63-3, U. S. Air Force, Wright-Patterson AFB, Ohio.

J. E. Francis and T. J. Love, AIAA paper No. 65-119.

McMahon, H. O.

Odishaw, Hugh

E. U. Condon and Hugh Odishaw, Handbook of Physics (McGraw-Hill Book Co., Inc., New York, 1958).

Planck, M.

M. Planck, The Theory of Heat Radiation (Dover Publications, Inc., New York, 1959).

Richmond, J. C.

J. C. Richmond, J. Res. Natl. Bur. Std. 67C, 217 (1963).
[CrossRef]

J. Am. Ceram. Soc. (1)

Robert Gardon, J. Am. Ceram. Soc. 39, 278 (1956).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Res. Natl. Bur. Std. (1)

J. C. Richmond, J. Res. Natl. Bur. Std. 67C, 217 (1963).
[CrossRef]

Other (5)

J. E. Francis and T. J. Love, AIAA paper No. 65-119.

E. U. Condon and Hugh Odishaw, Handbook of Physics (McGraw-Hill Book Co., Inc., New York, 1958).

V. Kourganoff, Basic Methods in Transfer Problems (Oxford University Press, London, England, 1952).

T. J. Love, Aeronautical Research Laboratories Publication No. ARL 63-3, U. S. Air Force, Wright-Patterson AFB, Ohio.

M. Planck, The Theory of Heat Radiation (Dover Publications, Inc., New York, 1959).

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Figures (9)

F. 1
F. 1

Coordinate scheme for the dielectric interface.

F. 2
F. 2

Sheet of dielectric material with parallel specularly reflecting surfaces.

F. 3
F. 3

Coordinate system for axial symmetry nomenclature: cosθ=μ, μds=dx, and ds=dx/μ.

F. 4
F. 4

Monochromatic directional transmittance τν(θ′) for various optical depths of a dielectric with a refractive index of 1.50.

F. 5
F. 5

Monochromatic directional transmittance for various optical depths of a dielectric with a refractive index of 3.50.

F. 6
F. 6

Monochromatic directional transmittance for various optical depths of a dielectric with a refractive index of 5.50.

F. 7
F. 7

Monochromatic directional emittance for various optical depths of an isothermal dielectric with a refractive index of 1.50.

F. 8
F. 8

Monochromatic directional emittance for various optical depths of an isothermal dielectric with a refractive index of 3.50.

F. 9
F. 9

Monochromatic directional emittance for various optical depths of an isothermal dielectric with a refractive index of 5.50.

Equations (31)

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d I υ ( s ) / d s = ρ k υ I υ ( s ) + ρ k υ I bb υ ( T ) .
k υ = n 2 k υ .
d I υ ( s ) / d s = ρ k υ [ I υ ( s ) + n 2 I bb υ ( T ) ] .
τ = x 1 x 2 ρ k υ d x .
μ d I υ ( τ , μ ) / d τ = I υ ( τ , μ ) + n 2 I bb υ ( T ) .
μ d I υ + ( τ , μ ) / d τ = I υ + ( τ , μ ) + n 2 I bb υ ( T )
μ d I υ ( τ , μ ) / d τ = I υ ( τ , μ ) + n 2 I bb υ ( T ) .
I υ + ( τ , μ ) = c 1 e ( τ / μ ) + n 2 I bb υ ( T ) ,
I υ ( τ , μ ) = c 2 e ( τ τ 0 ) / μ + n 2 I bb υ ( T ) .
I υ + ( 0 , μ ) = ρ υ ( θ , θ ) I υ ( 0 , μ )
I υ ( τ 0 , μ ) = ρ υ ( θ , θ ) I υ + ( τ 0 , μ ) + [ 1 ρ υ ( θ , θ ) ] I i υ ( θ ) n 2 .
c 1 = c 2 e ( τ 0 / μ ) ρ υ ( θ , θ )
c 2 = ρ υ ( θ , θ ) c 1 e ( τ 0 / μ ) + [ 1 ρ υ ( θ , θ ) ] n 2 I i υ ( θ ) .
c 1 = n 2 [ 1 ρ υ ( θ , θ ) ] ρ υ ( θ , θ ) e ( τ 0 / μ ) I i υ ( θ ) 1 ρ υ 2 ( θ , θ ) e ( 2 τ 0 / μ ) .
I net υ ( τ 0 , μ ) = I υ ( τ 0 , μ ) I υ + ( τ 0 , μ ) = c 2 c 1 e ( τ 0 / μ )
I net υ ( 0 , μ ) = c 2 e ( τ 0 / μ ) c 1 = c 1 [ 1 / ρ υ ( θ , θ ) 1 ] .
τ υ = I net υ ( 0 , μ ) / n 2 I i υ ( θ ) .
τ υ ( θ ) = [ 1 ρ υ ( θ , θ ) ] [ 1 ρ υ ( θ , θ ) ] e ( τ 0 / μ ) 1 ρ υ ( θ , θ ) 2 e ( 2 τ 0 / μ ) .
I υ + ( 0 . μ ) = ρ υ ( θ , θ ) I υ ( 0 . μ ) at τ = 0
I υ ( τ 0 , μ ) = ρ υ ( θ , θ ) I υ + ( τ 0 , μ ) at τ = τ 0 .
I υ + ( 0 , ) μ = c 1 + n 2 I bb υ ( T )
I υ ( 0 , μ ) = c 2 e ( τ 0 / μ ) + n 2 I bb υ ( T ) .
c 1 = ρ υ ( θ , θ ) [ c 2 e ( τ 0 / μ ) + n 2 I bb υ ( T ) ] n 2 I bb υ ( T ) .
I υ + ( τ 0 , μ ) = c 1 e ( τ 0 / μ ) + n 2 I bb υ ( T )
I υ ( τ 0 , μ ) = c 2 + n 2 I bb υ ( T ) .
c 2 = ρ υ ( θ , θ ) [ c 1 e ( τ 0 / μ ) + n 2 I bb υ ( T ) ] n 2 I bb υ ( T ) .
c 2 = c 1 = n 2 I bb υ ( T ) [ ρ υ ( θ , θ ) 1 ] 1 ρ υ ( θ , θ ) e ( τ 0 / μ ) .
υ ( θ ) = [ I υ + ( τ 0 , μ ) I υ ( τ 0 , μ ) ] / n 2 I bb υ ( T ) ,
υ ( θ ) = [ ρ υ ( θ , θ ) 1 ] [ e ( τ 0 / μ ) 1 ] 1 ρ υ ( θ , θ ) e ( τ 0 / μ ) .
τ ( μ ) = 1 2 [ τ υ ( μ ) + τ υ ( μ ) ]
υ ( θ ) = 1 2 [ υ ( θ ) + υ ( θ ) ] .