Abstract

We have studied the effect of diffraction on the flux received by a detector located in the fully illuminated region cast by a circular aperture which is placed between a point source and the detector. This case was studied previously by Steel, De, and Bell, who have derived a diffraction correction formula based on an intensity distribution approximation due to Focke. We have derived a diffraction correction formula based on a different type of approximation. The corrections predicted by this formula are much greater than those predicted by Steel et al.'s formula. The validity of our formula and reasons for the discrepancy between the two methods are discussed; some supporting experimental evidence is presented. The case of complex radiation is discussed; it is shown that, under certain conditions, the calculation of diffraction corrections is very simple.

© 1975 Optical Society of America

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References

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  1. W. R. Blevin, Metrologia, 6, 39 (1970).
    [CrossRef]
  2. J. Focke, Opt. Acta, 3, 161 (1956).
    [CrossRef]
  3. W. H. Steel, M. De, J. A. Bell, J. Opt. Soc. Am. 62, 1099 (1972).
    [CrossRef]
  4. E. Lommel, Abh. Bayer. Akad. 15, 233 (1885).
  5. E. Wolf, Proc. Roy. Soc. (London) 204A, 533 (1951).
  6. M. Abramowitz, I. A. Stegun, Eds. Handbook of Mathematical Functions (AMS 55, NBS, Washington, D.C., 1964), p. 300.
  7. G. Petiau, “La Théorie des Fonctions de Bessel” (CNRS, Paris, 1955), p. 376.
  8. E. N. Dekanosidze, Tables of Lommel's Functions of Two Variables (Pergamon, Oxford, 1960).
  9. L. P. Boivin, Appl. Opt. 14, 197 (1975).
    [CrossRef] [PubMed]

1975 (1)

1972 (1)

1970 (1)

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

1956 (1)

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

1951 (1)

E. Wolf, Proc. Roy. Soc. (London) 204A, 533 (1951).

1885 (1)

E. Lommel, Abh. Bayer. Akad. 15, 233 (1885).

Bell, J. A.

Blevin, W. R.

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

Boivin, L. P.

De, M.

Dekanosidze, E. N.

E. N. Dekanosidze, Tables of Lommel's Functions of Two Variables (Pergamon, Oxford, 1960).

Focke, J.

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

Lommel, E.

E. Lommel, Abh. Bayer. Akad. 15, 233 (1885).

Petiau, G.

G. Petiau, “La Théorie des Fonctions de Bessel” (CNRS, Paris, 1955), p. 376.

Steel, W. H.

Wolf, E.

E. Wolf, Proc. Roy. Soc. (London) 204A, 533 (1951).

Abh. Bayer. Akad. (1)

E. Lommel, Abh. Bayer. Akad. 15, 233 (1885).

Appl. Opt. (1)

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]

Proc. Roy. Soc. (London) (1)

E. Wolf, Proc. Roy. Soc. (London) 204A, 533 (1951).

Other (3)

M. Abramowitz, I. A. Stegun, Eds. Handbook of Mathematical Functions (AMS 55, NBS, Washington, D.C., 1964), p. 300.

G. Petiau, “La Théorie des Fonctions de Bessel” (CNRS, Paris, 1955), p. 376.

E. N. Dekanosidze, Tables of Lommel's Functions of Two Variables (Pergamon, Oxford, 1960).

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Figures (9)

Fig. 1
Fig. 1

Diffraction effects in radiometry. Case 1: the detector is meant to receive all of the flux transmitted by the aperture. Case 2: the detector lies wholly in the fully illuminated region.

Fig. 2
Fig. 2

The diffraction correction (u,υ) as calculated from Steel et al.'s formula [Eq. (9)] for (a) u = 750π and (b) u = 1500π.

Fig. 3
Fig. 3

The diffraction correction (u,υ) as calculated from the formula derived here [Eq. (15)] for (a) u = 750π and (b) u = 1500π.

Fig. 4
Fig. 4

Photograph of the diffraction pattern of a 1.25-cm diameter aperture located 1 m from a point source and 1 m from the receiving screen. Magnification is about 3×.

Fig. 5
Fig. 5

Densitometer trace along a diameter of the diffraction pattern on the photographic plate corresponding to Fig. 4. The trace shows the complete scan from the center to the edge in two contiguous regions (a) and (b).

Fig. 6
Fig. 6

Intensity distribution calculated from the asymptotic form of Focke's expression [Eq. (7)] for the same case as Figs. 4 and 5: (a) central region and (b) edge region.

Fig. 7
Fig. 7

Intensity distribution calculated from the approximate form of Lommel's expression [Eq. (16)] for the same case as Figs. 4 and 5: (a) central region and (b) edge region.

Fig. 8
Fig. 8

Boundary curves and median curve corresponding to the diffraction correction (u,υ) calculated from Eq. (15).

Fig. 9
Fig. 9

Typical absolute radiometer configuration, from Ref. 9.

Tables (4)

Tables Icon

Table I Calculation of Intensity Distribution I(u,υ): Comparison with Tabulated Values

Tables Icon

Table II Calculation of F2(u,υ): Comparison Between Approximate Formula and Numerical Integration

Tables Icon

Table III Calculation of F2(u,υ): Comparison of Integration Techniques

Tables Icon

Table IV Comparison of Diffraction Corrections ¯ ( u 0 , υ 0 ) Averaged over Different Wavelength Intervals

Equations (31)

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F 2 ( u , υ 0 ) = u 2 2 υ 0 2 0 υ 0 I ( u , υ ) υ d υ ,
u = 2 π λ ( a + b ) a b R 2 , υ = 2 π λ R r b ,
I ( u , υ ) = 4 u 2 [ U 1 2 ( u , υ ) + U 2 2 ( u , υ ) ]
U n ( u , υ ) = s = 0 ( 1 ) s ( u υ ) n + 2 s J n + 2 s ( υ ) .
I ( u , υ ) = 4 u 2 [ 1 + V 0 2 ( u , υ ) + V 1 2 ( u , υ ) 2 V 0 ( u , υ ) cos ( u 2 + υ 2 2 u ) 2 V 1 ( u , υ ) sin ( u 2 + υ 2 2 u ) ]
V n ( u , υ ) = s = 0 ( 1 ) s ( υ u ) n + 2 s J n + 2 s ( υ ) .
I ( u , υ ) 2 u 2 { [ 1 2 C ( t ) ] 2 + [ 1 2 S ( t ) ] 2 }
I ( u , υ ) 2 u 2 { 2 + u π ( u υ ) 2 + 2 ( u υ ) ( u / π ) [ sin ( u υ ) 2 2 u cos ( u υ ) 2 2 u ] } .
f ( t ) 1 t , g ( t ) 0 .
F 2 ( u , υ 0 ) 1 + u π υ 0 ( u υ 0 ) + u π υ 0 2 ln ( u υ 0 ) u + ( 8 u π ) 1 / 2 u υ 0 ( u υ 0 ) 2 cos [ ( u υ 0 ) 2 2 u π 4 ] .
u = 10 υ ( υ / u = a a + b r R ) .
I ( u , υ ) 4 u 2 [ 1 + J 0 2 ( υ ) + υ 2 u 2 J 1 2 ( υ ) 2 J 0 ( υ ) cos ( u 2 + υ 2 2 u ) 2 υ u J 1 ( υ ) sin ( u 2 + υ 2 2 u ) ] .
0 x 0 x J 0 ( x ) cos α x 2 d x = 1 2 α [ U 1 ( 2 α x 0 2 , x 0 ) cos α x 0 2 + U 2 ( 2 α x 0 2 , x 0 ) sin α x 0 2 ] 0 x 0 x J 0 ( x ) sin α x 2 d x = 1 2 α [ U 1 ( 2 α x 0 2 , x 0 ) sin α x 0 2 U 2 ( 2 α x 0 2 , x 0 ) cos α x 0 2 ] .
U 1 ( υ 0 2 u , υ 0 ) υ 0 u J 1 ( υ 0 ) , U 2 ( υ 0 2 u , υ 0 ) υ 0 2 u 2 J 2 ( υ 0 )
F 2 ( u , υ 0 ) 1 + J 0 2 ( υ 0 ) + J 1 2 ( υ 0 ) + υ 0 2 3 u 2 [ J 1 2 ( υ 0 ) + J 2 2 ( υ 0 ) ] 4 υ 0 J 1 ( υ 0 ) cos ( u 2 + υ 0 2 2 u ) 8 u J 2 ( υ 0 ) sin ( u 2 + υ 0 2 2 u ) .
F 2 ( u , υ 0 ) 1 + J 0 2 ( υ 0 ) + J 1 2 ( υ 0 ) 4 υ 0 J 1 ( υ 0 ) cos ( u 2 + υ 0 2 2 u ) .
F 2 ( u , 0 ) = 2 2 cos u / 2 .
F 2 ( u , υ 0 ) 1 + 2 π υ 0 4 υ 0 ( 2 π υ 0 ) sin ( υ 0 π 4 ) cos ( u 2 + υ 0 2 2 u ) .
I ( u , υ ) 4 u 2 { 1 + 2 π υ cos 2 ( υ π 4 ) 2 ( 2 π υ ) [ cos ( υ π 4 ) cos ( u 2 + υ 2 2 u ) + υ u sin ( υ π 4 ) sin ( u 2 + υ 2 2 u ) ] } .
F 1 ( u , υ 0 ) = 1 2 0 υ 0 I ( u , υ ) υ d υ
F 1 ( u , υ 0 ) 1 2 π υ 0 .
I ( u , υ ) 4 u 2 [ u 2 υ 2 J 1 2 ( υ ) + u 4 υ 4 J 2 2 ( υ ) ] .
F 1 ( u , υ 0 ) 1 J 0 2 ( υ 0 ) J 1 2 ( υ 0 ) u 2 3 υ 0 2 [ J 1 2 ( υ 0 ) + J 2 2 ( υ 0 ) ] .
F 1 ( u , υ 0 ) 1 2 π υ 0
¯ = λ 1 λ 2 ( u , υ ) ϕ ( λ ) d λ / λ 1 λ 2 ϕ ( λ ) d λ
¯ ( u 0 , υ 0 ) = λ 1 λ 2 ( u , υ ) d λ / ( λ 2 λ 1 )
¯ ( u 0 , υ 0 ) 2 π υ 0 .
¯ ( λ ) = ( b λ / π 2 R r )
¯ = ( b λ e / π 2 R r )
¯ ( u 0 , υ 0 )
2 π υ 0

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