Abstract

Measurements to determine the radiance of one portion of a nonuniform source can be strongly influenced by radiation from other portions of the source scattered by the optical system used to transmit the radiation to the detector. Conditions under which such scattering is important and procedures to correct for it are discussed. An illustration is provided from measurements made on a nitrogen arc source.

© 1970 Optical Society of America

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References

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  1. R. D. Lee, Metrologia 2, 150 (1966).
    [CrossRef]
  2. E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals (Oxford University Press, New York, 1948).
  3. J. S. Rollet, L. A. Higgs, Proc. Phys. Soc. London 79, 87 (1962).
    [CrossRef]
  4. A. F. Jones, D. L. Misell, Brit. J. Appl. Phys. 18, 1479 (1967).
    [CrossRef]
  5. H. C. Burger, P. H. vanCittert, Z. Physik 79, 722 (1932).
    [CrossRef]
  6. H. W. Drawin, Z. Physik 172, 181 (1963).
    [CrossRef]
  7. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan Company, New York, 1963).
  8. Richard Blazey, Appl. Opt. 6, 831 (1967).
    [CrossRef] [PubMed]
  9. W. H. Venable, J. B. Shumaker, J. Quant. Spect. Radiative Transfer 9, 1215 (1969).
    [CrossRef]
  10. C. C. Eaglesfield, Proc. Phys. Soc. London 82, 406 (1963).
    [CrossRef]
  11. C. C. Eaglesfield, Electron. Lett. 1, 237 (1965).
    [CrossRef]
  12. M. P. Freeman, S. Katz, J. Opt. Soc. Amer. 53, 1172 (1963).
    [CrossRef]
  13. J. G. Kemeny, T. E. Kurtz, Basic (Dartmouth Publications, Hanover, New Hampshire).

1969 (1)

W. H. Venable, J. B. Shumaker, J. Quant. Spect. Radiative Transfer 9, 1215 (1969).
[CrossRef]

1967 (2)

A. F. Jones, D. L. Misell, Brit. J. Appl. Phys. 18, 1479 (1967).
[CrossRef]

Richard Blazey, Appl. Opt. 6, 831 (1967).
[CrossRef] [PubMed]

1966 (1)

R. D. Lee, Metrologia 2, 150 (1966).
[CrossRef]

1965 (1)

C. C. Eaglesfield, Electron. Lett. 1, 237 (1965).
[CrossRef]

1963 (3)

M. P. Freeman, S. Katz, J. Opt. Soc. Amer. 53, 1172 (1963).
[CrossRef]

C. C. Eaglesfield, Proc. Phys. Soc. London 82, 406 (1963).
[CrossRef]

H. W. Drawin, Z. Physik 172, 181 (1963).
[CrossRef]

1962 (1)

J. S. Rollet, L. A. Higgs, Proc. Phys. Soc. London 79, 87 (1962).
[CrossRef]

1932 (1)

H. C. Burger, P. H. vanCittert, Z. Physik 79, 722 (1932).
[CrossRef]

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan Company, New York, 1963).

Blazey, Richard

Burger, H. C.

H. C. Burger, P. H. vanCittert, Z. Physik 79, 722 (1932).
[CrossRef]

Drawin, H. W.

H. W. Drawin, Z. Physik 172, 181 (1963).
[CrossRef]

Eaglesfield, C. C.

C. C. Eaglesfield, Electron. Lett. 1, 237 (1965).
[CrossRef]

C. C. Eaglesfield, Proc. Phys. Soc. London 82, 406 (1963).
[CrossRef]

Freeman, M. P.

M. P. Freeman, S. Katz, J. Opt. Soc. Amer. 53, 1172 (1963).
[CrossRef]

Higgs, L. A.

J. S. Rollet, L. A. Higgs, Proc. Phys. Soc. London 79, 87 (1962).
[CrossRef]

Jones, A. F.

A. F. Jones, D. L. Misell, Brit. J. Appl. Phys. 18, 1479 (1967).
[CrossRef]

Katz, S.

M. P. Freeman, S. Katz, J. Opt. Soc. Amer. 53, 1172 (1963).
[CrossRef]

Kemeny, J. G.

J. G. Kemeny, T. E. Kurtz, Basic (Dartmouth Publications, Hanover, New Hampshire).

Kurtz, T. E.

J. G. Kemeny, T. E. Kurtz, Basic (Dartmouth Publications, Hanover, New Hampshire).

Lee, R. D.

R. D. Lee, Metrologia 2, 150 (1966).
[CrossRef]

Misell, D. L.

A. F. Jones, D. L. Misell, Brit. J. Appl. Phys. 18, 1479 (1967).
[CrossRef]

Rollet, J. S.

J. S. Rollet, L. A. Higgs, Proc. Phys. Soc. London 79, 87 (1962).
[CrossRef]

Shumaker, J. B.

W. H. Venable, J. B. Shumaker, J. Quant. Spect. Radiative Transfer 9, 1215 (1969).
[CrossRef]

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan Company, New York, 1963).

Titchmarsh, E. C.

E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals (Oxford University Press, New York, 1948).

vanCittert, P. H.

H. C. Burger, P. H. vanCittert, Z. Physik 79, 722 (1932).
[CrossRef]

Venable, W. H.

W. H. Venable, J. B. Shumaker, J. Quant. Spect. Radiative Transfer 9, 1215 (1969).
[CrossRef]

Appl. Opt. (1)

Brit. J. Appl. Phys. (1)

A. F. Jones, D. L. Misell, Brit. J. Appl. Phys. 18, 1479 (1967).
[CrossRef]

Electron. Lett. (1)

C. C. Eaglesfield, Electron. Lett. 1, 237 (1965).
[CrossRef]

J. Opt. Soc. Amer. (1)

M. P. Freeman, S. Katz, J. Opt. Soc. Amer. 53, 1172 (1963).
[CrossRef]

J. Quant. Spect. Radiative Transfer (1)

W. H. Venable, J. B. Shumaker, J. Quant. Spect. Radiative Transfer 9, 1215 (1969).
[CrossRef]

Metrologia (1)

R. D. Lee, Metrologia 2, 150 (1966).
[CrossRef]

Proc. Phys. Soc. London (2)

C. C. Eaglesfield, Proc. Phys. Soc. London 82, 406 (1963).
[CrossRef]

J. S. Rollet, L. A. Higgs, Proc. Phys. Soc. London 79, 87 (1962).
[CrossRef]

Z. Physik (2)

H. C. Burger, P. H. vanCittert, Z. Physik 79, 722 (1932).
[CrossRef]

H. W. Drawin, Z. Physik 172, 181 (1963).
[CrossRef]

Other (3)

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan Company, New York, 1963).

E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals (Oxford University Press, New York, 1948).

J. G. Kemeny, T. E. Kurtz, Basic (Dartmouth Publications, Hanover, New Hampshire).

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Figures (9)

Fig. 1
Fig. 1

A schematic illustration of a uniform source of length H being scanned by a slit detector of length h.

Fig. 2
Fig. 2

An illustration of the distorted mapping F of a radiance profile T produced by an instrument with scattering function S. The F and T functions are drawn to the same arbitrary scale. The vertical scale is numbered to indicate the values of the normalized S function.

Fig. 3
Fig. 3

An illustration of the convergence toward a known solution for S. The Oth approximation is F, scaled to a unity integral. Curves 1 and 4 represent the solution after 1 and 4 iterations, respectively, and the circles indicate values taken from the solution after 7 iterations. The vertical scale indicates the value of the normalized functions.

Fig. 4
Fig. 4

An illustration of convergence toward a solution for T. Each curve is numbered according to the number of terms in the series of Eq. (11). The solution with an infinite number of terms would fall within the shaded region. The functions are plotted to the same arbitrary scale as that of F and T in Fig. 2.

Fig. 5
Fig. 5

A schematic representation of the optical system used for measuring arc profiles. All mirrors are front surface.

Fig. 6
Fig. 6

The scattering function per unit displacement for the optical system illustrated in Fig. 5 as obtained at two different wavelengths. The dashed lines represent a best fit inverse square function. (1 displacement unit = 95 μ m).

Fig. 7
Fig. 7

The effect of scattering upon the observed profile of a 1-atm nitrogen arc for an atomic line and a molecular band. Where the scale permits, the uncorrected profile is shown as a dotted line. The lower curve indicates the percentage error in the ratio of the band intensity to the line intensity caused by scattering.

Fig. 8
Fig. 8

A basic program for determining the scattering function.

Fig. 9
Fig. 9

A basic program for solving for the actual radiance.

Equations (18)

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F ( r ) = S ( r , r ) T ( r ) d A ( r ) ,
S ( r , r ) d A ( r ) = 1.
F ( r ) d A ( r ) = T ( r ) d A ( r ) .
S ( r , r ) = S ( | r r | ) .
F ( x ) = S ( x x ) T ( x ) d x .
S ( x x ) = { π [ 1 + ( x x ) ] 2 } 1 ,
T ( x ) d x = F ( x ) d x .
S ( x x ) d x = 1.
A = B D + B d ( D > d )
B = A / D B d / D
B = i = 0 N ( A / D ) ( d / D ) i + B ( d / D ) N + 1 .
B = A / D ( d / D ) ( A / D ) ( d / D ) [ ( d / D ) ( A / D ) ] .
x x + S ( x x ) T ( x ) d x [ S ( z ) d z ] T ( x ) s T ( x ) .
F ( x ) = x x + S ( x x ) T ( x ) d x + S ( x x ) T ( x ) d x ,
T ( x ) = F ( x ) / s ( 1 / s ) S ( x x ) T ( x ) d x .
T ( x ) = F ( x ) / s ( 1 / s ) S ( x x ) [ F ( x ) / s ] d x ( 1 / s ) S ( x x ) { ( 1 / s ) S ( x x ) [ F ( x ) / s ] d x } d x .
S ( x x ) = 1.49 ( x x ) 2 λ 0.89 per mm ,
S ( z ) d z + S ( z ) d z = 1 s .

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