Abstract

The treatment by Blevin of radiometric errors caused by diffraction at a circular aperture is extended to the case where both the source and the detector are of finite sizes. Simple asymptotic expressions are derived, valid when the diffraction corrections are small.

© 1972 Optical Society of America

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References

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  1. W. R. Blevin, Metrologia 6, 39 (1970).
    [Crossref]
  2. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. P., Cambridge, England, 1922), p. 537.
  3. J. Focke, Opt. Acta 3, 161 (1956).
    [Crossref]
  4. M. Cagnet, M. Françon, and J. C. Thrierr, Atlas of Optical Phenomena (Springer, Berlin, 1962), p. 33.
  5. M. Born and E. Wolf, Principles of Optics (Pergamon, New York1959), p. 434.
  6. W. R. Blevin and W. J. Brown, Metrologia 7, 15 (1971).
    [Crossref]
  7. C. L. Sanders and O. C. Jones, J. Opt. Soc. Am. 52, 731 (1962).
    [Crossref]

1971 (1)

W. R. Blevin and W. J. Brown, Metrologia 7, 15 (1971).
[Crossref]

1970 (1)

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

1962 (1)

1956 (1)

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

Blevin, W. R.

W. R. Blevin and W. J. Brown, Metrologia 7, 15 (1971).
[Crossref]

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

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York1959), p. 434.

Brown, W. J.

W. R. Blevin and W. J. Brown, Metrologia 7, 15 (1971).
[Crossref]

Cagnet, M.

M. Cagnet, M. Françon, and J. C. Thrierr, Atlas of Optical Phenomena (Springer, Berlin, 1962), p. 33.

Focke, J.

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

Françon, M.

M. Cagnet, M. Françon, and J. C. Thrierr, Atlas of Optical Phenomena (Springer, Berlin, 1962), p. 33.

Jones, O. C.

Sanders, C. L.

Thrierr, J. C.

M. Cagnet, M. Françon, and J. C. Thrierr, Atlas of Optical Phenomena (Springer, Berlin, 1962), p. 33.

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. P., Cambridge, England, 1922), p. 537.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York1959), p. 434.

J. Opt. Soc. Am. (1)

Metrologia (2)

W. R. Blevin and W. J. Brown, Metrologia 7, 15 (1971).
[Crossref]

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

Opt. Acta (1)

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

Other (3)

M. Cagnet, M. Françon, and J. C. Thrierr, Atlas of Optical Phenomena (Springer, Berlin, 1962), p. 33.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York1959), p. 434.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. P., Cambridge, England, 1922), p. 537.

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Figures (2)

F. 1
F. 1

Notation used. A source S of radius a is at a distance s from a circular aperture A of radius b. Beyond this at a distance s′ is a receiver D of radius a′. Geometrical shadow lines are shown; O′Q′ is the radius of the inner edge of the full shadow, O′P′ of the penumbra.

F. 2
F. 2

The function I(υ,w,w′) plotted against υ for w/w′ = 1 and 1 2 , and for the limit w/w′ → 0. This is the full range of variation of this function.

Tables (1)

Tables Icon

Table I Recalculation of Blevin–Brown corrections (Ref. 6).

Equations (53)

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O P = b ( s + s ) / s a s / s ,
O Q = b ( s + s ) / s + a s / s ,
β < α α .
β > α + α .
Φ 1 = π 2 α 2 b 2 L
Φ 2 = π 2 a 2 a 2 L / ( s + s ) 2
Φ 2 = π 2 α 2 α 2 b 2 L / β 2 .
E ( α x α x , α y α y )
+ + L ( x , y ) E ( α x α x , α y α y ) a 2 d x d y ,
Φ = a 2 a 2 + + + + L ( x , y ) D ( x , y ) × E ( α x α x , α y α y ) d x d y d x d y .
w = k b α , w = k b α , and u = k b β ,
υ i = w x w x , υ j = w y w y , υ 2 = υ i 2 + υ j 2 .
E = π 2 b 4 λ 2 s 2 s 2 I ( u , υ ) ,
I ( u , υ ) = 4 [ U 1 2 ( u , υ ) + U 2 2 ( u , υ ) ] / u 2 ,
L ( x , y ) = L , x 2 + y 2 1 = 0 , x 2 + y 2 > 1 ; D ( x , y ) = 1 , x 2 + y 2 1 = 0 , x 2 + y 2 > 1 .
x = ( υ i + w x ) / w , y = ( υ j + w y ) / w ,
Φ = π 4 α 2 b 2 L + + I ( υ , w , w ) I ( u , υ ) d υ i d υ j ,
I ( υ , w , w ) = 1 π + + L ( x , y ) D ( w x + υ i w , w y + υ j w ) d x d y ,
Φ = π 2 2 α 2 b 2 L 0 w + w I ( υ , w , w ) I ( u , υ ) υ d υ ,
F 1 ( u , w , w ) = 1 2 0 w + w I ( υ , w , w ) I ( u , υ ) υ d υ ,
F 2 ( u , w , w ) = 1 2 ( u w ) 2 0 w + w I ( υ , w , w ) I ( u , υ ) υ d υ .
I ( υ , w , w ) = 1 2 T ( m 1 ) + 1 2 ( w 2 / w 2 ) T ( m 2 ) ,
m 1 = ( υ 2 + w 2 w 2 ) / 2 w υ , m 2 = ( υ 2 + w 2 w 2 ) / 2 w υ ,
T ( x ) = 2 [ arccos x x ( 1 x 2 ) 1 2 ] / π .
F 1 ( u , w , w ) = F 1 ( u , 0 , w w ) + 1 2 w w w + w I ( υ , w , w ) I ( u , υ ) υ d υ
F 1 ( u , w , w ) > F 1 ( u , 0 , w w ) for w > w ,
F 1 ( u , w , w ) = F 1 ( u , 0 , w + w ) 1 2 w w w + w [ 1 I ( υ , w , w ) ] I ( u , υ ) υ d υ
F 1 ( u , w , w ) < F 1 ( u , 0 , w + w ) .
F 1 ( u , w , w ) = F 1 ( u , 0 , w ) 1 2 w w w [ 1 I ( υ ) ] υ I ( u , υ ) d υ + 1 2 w w + w I ( υ ) υ I ( u , υ ) d υ = F 1 ( u , 0 , w ) 1 2 w w w { [ 1 I ( υ ) ] υ I ( u , υ ) ( 2 w υ ) I ( 2 w υ ) I ( u , 2 w υ ) } d υ .
2 w υ > υ for w υ υ w , 1 I ( υ ) > I ( 2 w υ ) ,
υ I ( u , υ ) > ( 2 w υ ) I ( u , 2 w υ ) .
F 1 ( u , w , w ) < F 1 ( u , 0 , w ) .
1 > F 1 ( u , 0 , w ) > F 1 ( u , w , w ) > F 1 ( u , 0 , w w ) .
F 1 ( u , 0 , υ ) 1 2 υ π ( υ 2 u 2 ) + 1 π cos 2 υ υ 2 u 2 + .
υ 2 u 2 > 6 × 10 3 ,
I ( u , υ ) 2 π υ ( 1 ( υ + u ) 2 + 1 ( υ u ) 2 ) .
I ( υ , w , w ) ( w + w υ ) / 2 w for w w υ w + w
F 1 ( u , w , w ) 1 1 2 π w ln ( w + w ) 2 u 2 ( w w ) 2 u 2 ,
F 2 ( u , 0 , υ ) 1 + u π υ ( u υ ) + u π υ 2 ln u υ u + ( 8 u π ) 1 2 u υ ( u υ ) 2 cos ( ( u υ ) 2 2 u π 4 ) + .
υ ( u υ ) 2 u 3 2 > 3 × 10 4 .
I ( u , υ ) 2 u 2 [ 2 + u π ( u υ ) 2 + 2 ( u π ) 1 2 1 u υ ( cos ( u υ ) 2 2 u sin ( u υ ) 2 2 u ) ] + .
( 1 / w 2 ) 0 I ( υ , w , w ) υ d υ ,
F 2 ( u , w , w ) F 2 ( u , 0 , w ) u ( 2 u w ) π w 2 ( u w ) + u 2 π w w 2 [ ( 2 u w w ) ln ( u w w ) ( 2 u w + w ) ln ( u w + w ) 2 w ln ( u w ) ] .
F 2 ( 0 , 0 , υ ) = 1 J 0 2 ( υ ) J 1 2 ( υ )
F 2 ( 0 , 0 , υ ) 1 2 π υ cos 2 υ 2 π υ 2 ,
υ > 55 .
F 2 ( 0 , w , w ) 1 1 π w ln w + w w w .
F 1 ( β , α , α ) 1 λ 4 π 2 α b ln ( α + α ) 2 β 2 ( α α ) 2 β 2
F 2 ( 0 , α , α ) 1 λ 2 π 2 α b ln α + α α α .
F ( λ ) S ( λ ) d λ / S ( λ ) d λ ,
λ e = S ( λ ) λ d λ / S ( λ ) d λ .
F 2 ( 0 , 0 , w ) = 0.00314 ,
F 2 ( 0 , w , w ) = 0.00320 .