Abstract

The diffraction corrections associated with a circular aperture in the case of an extended source and a detector located in the fully illuminated region have been calculated in 1972 by Steel, De, and Bell. We have shown recently that the intensity distribution formula that they have used is not accurate in the central region of the diffraction pattern and greatly underestimates the diffraction corrections associated with a point source. In this paper, we take up the case of an extended source; we have followed Steel et al.’s method of calculation, but have used an intensity distribution which we have shown to be valid in the central region. We show that in the case of complex radiation, the variation of the diffraction correction with the source radius ρ takes a very simple form: the diffraction correction remains approximately constant as ρ increases, until the source and detector subtend equal angles at the center of the aperture; if ρ is increased further, the diffraction correction decreases linearly with 1/ρ over a certain range of ρ. Experimental results are presented that confirm these theoretical predictions.

© 1976 Optical Society of America

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References

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  1. L. P. Boivin, Appl. Opt. 14, 2002 (1975).
    [CrossRef] [PubMed]
  2. W. H. Steel, M. De, J. A. Bell, J. Opt. Soc. Am. 62, 1099 (1972).
    [CrossRef]
  3. W. R. Blevin, Metrologia 6, 39 (1970).
    [CrossRef]
  4. J. Focke, Opt. Acta 3, 161 (1956).
    [CrossRef]
  5. These curves move rapidly above and below the ∊ axis as w′ is varied due to the presence of an oscillatory term; in each case, we have given the curve corresponding to the value of w′ closest to the one indicated in Fig. 2 which would give maximum values of ∊ (in absolute value). The actual values of w′ differ by no more than 2–3% from the ones shown.
  6. L. P. Boivin, Appl. Opt. 14, 197 (1975).
    [CrossRef] [PubMed]

1975 (2)

1972 (1)

1970 (1)

W. R. Blevin, Metrologia 6, 39 (1970).
[CrossRef]

1956 (1)

J. Focke, Opt. Acta 3, 161 (1956).
[CrossRef]

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

Metrologia (1)

W. R. Blevin, Metrologia 6, 39 (1970).
[CrossRef]

Opt. Acta (1)

J. Focke, Opt. Acta 3, 161 (1956).
[CrossRef]

Other (1)

These curves move rapidly above and below the ∊ axis as w′ is varied due to the presence of an oscillatory term; in each case, we have given the curve corresponding to the value of w′ closest to the one indicated in Fig. 2 which would give maximum values of ∊ (in absolute value). The actual values of w′ differ by no more than 2–3% from the ones shown.

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Figures (9)

Fig. 1
Fig. 1

Diffraction effects in the case of an extended source: definition of symbols.

Fig. 2
Fig. 2

Diffraction correction (= F2 − 1) calculated by numerical integration (for comparison, the values of predicted by Steel et al.’s calculations2 are also shown; these curves are labeled S.D.B.): (a) u = 500, w′ = 100, w = 0–100; (b) u = 1000, w′ = 100, w = 0–100; (c) u = 2500, w′ = 400, w = 0–400.

Fig. 3
Fig. 3

Diffraction correction for u = 500, w′ = 100, and for the same geometrical parameters, but a 1% larger wavelength (u = 495, w′ = 99). Abscissa is independent of wavelength, but proportional to source diameter.

Fig. 4
Fig. 4

Theoretical prediction for the variation of with the source radius ρ in the case of complex radiation.

Fig. 5
Fig. 5

Photograph of the experimental setup for the measurement of diffraction corrections corresponding to extended sources of complex radiation. Some baffles for the elimination of stray light have been omitted for clarity.

Fig. 6
Fig. 6

Mathematical model for the spectral sensitivity S(λ) of a U.D.T. P-I-N 10 silicon diode (Schottky). The plotted points are typical actual values obtained from the manufacturer’s specifications.

Fig. 7
Fig. 7

Effective wavelengths as a function of color temperature for a blackbody or tungsten source and a detector having the spectral sensitivity shown in Fig. 6.

Fig. 8
Fig. 8

Experimentally measured diffraction corrections for the parameters shown. The geometrical parameters were selected to show mainly the flat portion of the theoretical variation (solid line).

Fig. 9
Fig. 9

Experimentally measured diffraction corrections for the parameters shown. The geometrical parameters were selected to show mainly the sloping portion of the theoretical variation (solid line).

Equations (21)

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F 2 ( u , w , w ) = 1 2 ( u w ) 2 0 w + w I ( v , w , w ) J ( u , v ) v d v ,
w = 2 π λ R ρ a , w = 2 π λ R r b , u = 2 π λ ( a + b ) R 2 a b ,
I ( v , w , w ) = 1 2 T ( m 1 ) + 1 2 ( w 2 w 2 ) T ( m 2 ) ,
m 1 = ( v 2 + w 2 - w 2 ) / 2 w v , m 2 = ( v 2 + w 2 - w 2 ) / 2 w v ,
T ( x ) = 2 [ cos - 1 x - x ( 1 - x 2 ) 1 / 2 ] / π .
I = 0 for v w + w ; I = 1 for v w - w .
J ( u , v ) ( 4 u 2 ) [ 1 + g ( u , v ) ] ,
g ( u , v ) = J 0 2 ( v ) + v 2 u 2 J 1 2 ( v ) - 2 J 0 ( v ) cos ( u 2 + v 2 2 u ) - 2 v u J 1 ( v ) sin ( u 2 + v 2 2 u ) .
F 2 ( u , w , w ) = 2 w 2 0 w + w I ( v , w , w ) v d v + 2 w 2 0 w + w I ( v , w , w ) g ( u , v ) v d v .
F 2 ( u , w , w ) = 1 + 2 w 2 0 w + w I ( v , w , w ) g ( u , v ) v d v = 1 + 2 w 2 0 w - w g ( u , v ) v d v + 2 w 2 w - w w + w I ( v , w , w ) g ( u , v ) v d v .
I ( v , w , w ) ( w + w - v ) / 2 w .
f ( x ) = 0 x g ( u , v ) v d v .
F 2 ( u , w , w ) = 1 + 2 w 2 f ( w - w ) + ( w + w ) w w 2 [ f ( w + w ) - f ( w - w ) ] - 1 w w 2 w - w w + w v 2 g ( u , v ) d v .
w - w w + w v 2 g ( u , v ) d v = [ v f ( v ) ] w - w w + w - w - w w + w f ( v ) d v .
F 2 ( u , w , w ) = 1 + 1 w w 2 w - w w + w f ( v ) d v .
f ( v ) v 2 2 [ J 0 2 ( v ) + J 1 2 ( v ) - 4 v J 1 ( v ) cos ( u 2 + v 2 2 u ) ] .
f ( v ) ~ v 2 2 [ 2 π v - 4 v ( 2 π v ) 1 / 2 sin ( v - π 4 ) cos ( u 2 + v 2 2 u ) ] .
F 2 ( u , w , w ) ~ 1 + 1 π w w 2 w - w w + w v d v 1 + 2 π w .
= 2 π w = λ e b π 2 R r for w < w ,
λ e = λ 1 λ 2 λ S ( λ ) ϕ ( λ ) d λ / λ 1 λ 2 S ( λ ) ϕ ( λ ) d λ ,
= 2 π w = λ e a π 2 R ρ for w w .

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