Abstract

Revised formulas to estimate diffraction effects in radiometry for point and extended sources are derived. They are found to work as well as or better than previous formulas. In some instances the formulas can be written in closed form; otherwise their evaluation entails performing simple integrations as indicated. Formulas have been found for nonlimiting apertures, large defining apertures, and pinhole apertures. Examples of all three types of application are presented.

© 1998 Optical Society of America

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References

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  1. W. R. Blevin, “Diffraction losses in radiometry and photometry,” Metrologia 6, 39–44 (1970).
    [CrossRef]
  2. J. Focke, “Total illumination in an aberration-free diffraction image,” Opt. Acta 3, 161–163 (1956).
    [CrossRef]
  3. W. H. Steel, M. De, J. A. Bell, “Diffraction corrections in radiometry,” J. Opt. Sci. Am. 62, 1099–1103 (1972).
    [CrossRef]
  4. L. P. Boivin, “Diffraction corrections in radiometry: comparison of two different methods of calculation,” Appl. Opt. 14, 2002–2009 (1975).
    [CrossRef] [PubMed]
  5. E. Lommel, “Die Beugungserscheinungen einer kreisrunden Oeffnung und eines kreisrunden Schirmschens theoretisch und experimentell Bearbeitet,” Abh. Bayer. Akad. 15, 233–328 (1885).
  6. L. P. Boivin, “Diffraction corrections in the radiometry of extended sources,” Appl. Opt. 15, 1204–1209 (1976).
    [CrossRef] [PubMed]
  7. M. Born, E. Wolf, Principles of Optics, 5th. ed. (Pergamon, New York, 1975).
  8. L. P. Boivin, “Reduction of diffraction errors in radiometry by means of toothed apertures,” Appl. Opt. 17, 3323–3328 (1978).
    [CrossRef] [PubMed]
  9. E. L. Shirley, R. U. Datla, “Optimally toothed apertures for reduced diffraction,” J. Res. Natl. Inst. Stand. Technol. 101, 745–753 (1996).
    [CrossRef]
  10. W. R. Blevin, W. J. Brown, “A precise measurement of the Stefan–Boltzmann constant,” Metrologia 7, 15–29 (1971).
    [CrossRef]
  11. F. C. Witteborn, Mail Stop 245-6, NASA Ames Research Center, Moffett Field, Palo Alto, Calif. 94035 (personal communication, 1997).
  12. J. B. Keller, “Diffraction by an aperture. I,” J. Appl. Phys. 28, 426–444 (1957);A. Mohsen, M. A. K. Hamid, “Higher order asymptotic terms of the two-dimensional diffraction by a small aperture,” Radio Sci. 3, 1105–1108 (1968).
    [CrossRef]

1996 (1)

E. L. Shirley, R. U. Datla, “Optimally toothed apertures for reduced diffraction,” J. Res. Natl. Inst. Stand. Technol. 101, 745–753 (1996).
[CrossRef]

1978 (1)

1976 (1)

1975 (1)

1972 (1)

W. H. Steel, M. De, J. A. Bell, “Diffraction corrections in radiometry,” J. Opt. Sci. Am. 62, 1099–1103 (1972).
[CrossRef]

1971 (1)

W. R. Blevin, W. J. Brown, “A precise measurement of the Stefan–Boltzmann constant,” Metrologia 7, 15–29 (1971).
[CrossRef]

1970 (1)

W. R. Blevin, “Diffraction losses in radiometry and photometry,” Metrologia 6, 39–44 (1970).
[CrossRef]

1957 (1)

J. B. Keller, “Diffraction by an aperture. I,” J. Appl. Phys. 28, 426–444 (1957);A. Mohsen, M. A. K. Hamid, “Higher order asymptotic terms of the two-dimensional diffraction by a small aperture,” Radio Sci. 3, 1105–1108 (1968).
[CrossRef]

1956 (1)

J. Focke, “Total illumination in an aberration-free diffraction image,” Opt. Acta 3, 161–163 (1956).
[CrossRef]

1885 (1)

E. Lommel, “Die Beugungserscheinungen einer kreisrunden Oeffnung und eines kreisrunden Schirmschens theoretisch und experimentell Bearbeitet,” Abh. Bayer. Akad. 15, 233–328 (1885).

Bell, J. A.

W. H. Steel, M. De, J. A. Bell, “Diffraction corrections in radiometry,” J. Opt. Sci. Am. 62, 1099–1103 (1972).
[CrossRef]

Blevin, W. R.

W. R. Blevin, W. J. Brown, “A precise measurement of the Stefan–Boltzmann constant,” Metrologia 7, 15–29 (1971).
[CrossRef]

W. R. Blevin, “Diffraction losses in radiometry and photometry,” Metrologia 6, 39–44 (1970).
[CrossRef]

Boivin, L. P.

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th. ed. (Pergamon, New York, 1975).

Brown, W. J.

W. R. Blevin, W. J. Brown, “A precise measurement of the Stefan–Boltzmann constant,” Metrologia 7, 15–29 (1971).
[CrossRef]

Datla, R. U.

E. L. Shirley, R. U. Datla, “Optimally toothed apertures for reduced diffraction,” J. Res. Natl. Inst. Stand. Technol. 101, 745–753 (1996).
[CrossRef]

De, M.

W. H. Steel, M. De, J. A. Bell, “Diffraction corrections in radiometry,” J. Opt. Sci. Am. 62, 1099–1103 (1972).
[CrossRef]

Focke, J.

J. Focke, “Total illumination in an aberration-free diffraction image,” Opt. Acta 3, 161–163 (1956).
[CrossRef]

Keller, J. B.

J. B. Keller, “Diffraction by an aperture. I,” J. Appl. Phys. 28, 426–444 (1957);A. Mohsen, M. A. K. Hamid, “Higher order asymptotic terms of the two-dimensional diffraction by a small aperture,” Radio Sci. 3, 1105–1108 (1968).
[CrossRef]

Lommel, E.

E. Lommel, “Die Beugungserscheinungen einer kreisrunden Oeffnung und eines kreisrunden Schirmschens theoretisch und experimentell Bearbeitet,” Abh. Bayer. Akad. 15, 233–328 (1885).

Shirley, E. L.

E. L. Shirley, R. U. Datla, “Optimally toothed apertures for reduced diffraction,” J. Res. Natl. Inst. Stand. Technol. 101, 745–753 (1996).
[CrossRef]

Steel, W. H.

W. H. Steel, M. De, J. A. Bell, “Diffraction corrections in radiometry,” J. Opt. Sci. Am. 62, 1099–1103 (1972).
[CrossRef]

Witteborn, F. C.

F. C. Witteborn, Mail Stop 245-6, NASA Ames Research Center, Moffett Field, Palo Alto, Calif. 94035 (personal communication, 1997).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th. ed. (Pergamon, New York, 1975).

Abh. Bayer. Akad. (1)

E. Lommel, “Die Beugungserscheinungen einer kreisrunden Oeffnung und eines kreisrunden Schirmschens theoretisch und experimentell Bearbeitet,” Abh. Bayer. Akad. 15, 233–328 (1885).

Appl. Opt. (3)

J. Appl. Phys. (1)

J. B. Keller, “Diffraction by an aperture. I,” J. Appl. Phys. 28, 426–444 (1957);A. Mohsen, M. A. K. Hamid, “Higher order asymptotic terms of the two-dimensional diffraction by a small aperture,” Radio Sci. 3, 1105–1108 (1968).
[CrossRef]

J. Opt. Sci. Am. (1)

W. H. Steel, M. De, J. A. Bell, “Diffraction corrections in radiometry,” J. Opt. Sci. Am. 62, 1099–1103 (1972).
[CrossRef]

J. Res. Natl. Inst. Stand. Technol. (1)

E. L. Shirley, R. U. Datla, “Optimally toothed apertures for reduced diffraction,” J. Res. Natl. Inst. Stand. Technol. 101, 745–753 (1996).
[CrossRef]

Metrologia (2)

W. R. Blevin, W. J. Brown, “A precise measurement of the Stefan–Boltzmann constant,” Metrologia 7, 15–29 (1971).
[CrossRef]

W. R. Blevin, “Diffraction losses in radiometry and photometry,” Metrologia 6, 39–44 (1970).
[CrossRef]

Opt. Acta (1)

J. Focke, “Total illumination in an aberration-free diffraction image,” Opt. Acta 3, 161–163 (1956).
[CrossRef]

Other (2)

M. Born, E. Wolf, Principles of Optics, 5th. ed. (Pergamon, New York, 1975).

F. C. Witteborn, Mail Stop 245-6, NASA Ames Research Center, Moffett Field, Palo Alto, Calif. 94035 (personal communication, 1997).

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Figures (2)

Fig. 1
Fig. 1

Schematic diagram of the source–aperture–detector setup with relevant dimensions labeled. Angles subtended by the source and the detector are indicated. The particular setup shown features a nonlimiting aperture.

Fig. 2
Fig. 2

λ(λ) - Φλ (0)(λ)]/[(λ/μm)Φλ (0)(λ)], or the relative excess spectral flux divided by wavelength that results from two baffles discussed in the text, as found by numerical calculations carried out in the Fresnel approximation (solid curves), from relation (34) (long-dashed curve), and from 12-point quadrature (short-dashed curve).

Tables (4)

Tables Icon

Table 1 Parameters That Specify the Geometries Used in Ref. 8

Tables Icon

Table 2 F2 for Geometries Indicated in Table 1 at λ = 0.58 μma

Tables Icon

Table 3 Geometry Parameters and F1 for Nine Examples of Limiting Aperturesa

Tables Icon

Table 4 F1 for Two Apertures Found from Simplified Formulas Discussed in the Text and Detailed Calculations Using the Fresnel Approximation for Two Different Pinhole Apertures R

Equations (67)

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u = 2 π λ   R 2 1 d s + 1 d d ,     v s = 2 π λ Rr s d s , v d = 2 π λ Rr d d d .
v 0 = Max v s ,   v d ,     σ = Min v s ,   v d Max v s ,   v d .
Ψ = λ v 0 π R .
u < ( 1 - σ ) v 0 = | v s - v d | .
u > 1 + σ v 0 = v s + v d .
f r s ,   r d = x M 2 + y M 2 R .
Φ λ λ = L λ λ d s + d d 2 Source d 2 r s Detector d 2 r d × | α u ,   f r s ,   r d u | 2 .
| α u ,   v | 2 = 1 + V 0 2 u ,   v + V 1 2 u ,   v   - 2 V 0 u ,   v cos u + v 2 u 2   - 2 V 1 u ,   v sin u + v 2 u 2   v < u U 1 2 u ,   v + U 2 2 u ,   v   v > u .
U n ( u ,   v ) = s = 0 ( - ) s u v n + 2 s J n + 2 s v , V n u ,   v = s = 0 ( - ) s v u n + 2 s J n + 2 s v ,
Φ λ λ = L λ λ d d + d s 2   0 r s d r s r s   0 2 π d θ s 0 r d d r d r d × 0 2 π d θ d | α u ,   f r s ,   r d u | 2 = 2 π r s 2 r d 2 d s + d d 2 L λ λ σ 2   0 σ d σ σ   0 1 d ζ ζ × 0 2 π d θ | α u ,   v 0 σ 2 + ζ 2 - 2 ζ σ   cos   θ 1 / 2 | 2 .
I u ,   v ,   τ = 0 τ d τ τ | α u ,   v τ | 2 = 1 v 2   0 v τ d v v | α u ,   v | 2 ,
Φ λ λ = 2 π r s 2 r d 2 L λ λ d s + d d 2   - 1 1 d x   I u ,   v 0 ,   1 + σ x 1 + σ x × 1 - x 2 2 + σ x 2 - σ 2 1 / 2 ,
Φ λ ( 0 ) ( λ ) | nonlim . = π r s 2 r d 2 L λ ( λ ) ( d s + d d ) 2 Φ λ ( 0 ) ( λ ) | lim . = π 2 σ 2 R 2 ψ 2 L λ ( λ ) 4 = u 2 υ 0 2 π 2 r s 2 r d 2 L λ ( λ ) ( d s + d d ) 2 }
J m v 2 π v cos v - m π 2 - π 4 ,
V 0 u ,   v 2 π v   cos v - π / 4 1 - v 2 / u 2 ,     1     v < u , V 1 u ,   v 2 π v v u sin v - π / 4 1 - v 2 / u 2 ,     1     v < u .
I u ,   v 0 ,   τ | 1 v 0 τ < u 1 v 0 2   0 v 0 τ d v v × 1 + 1 π v 1 + v 2 / u 2 1 - v 2 / u 2 2 + 1 π v sin 2 v 1 - v 2 / u 2 - 2 π v cos u + v 2 / 2 u - π / 4 u + v / u - 2 π v cos u - v 2 / 2 u + π / 4 u - v / u .
I u ,   v 0 ,   τ | 1 v 0 τ < u τ 2 2 + u 2 τ π v 0 u 2 - v 0 2 τ 2 - u 2 v 0 2 2 v 0 τ π sin u + v 0 τ 2 / 2 u - π / 4 u + v 0 τ 2 + u 3 v 0 2 2 v 0 τ π cos u + v 0 τ 2 2 u - π 4 × + 2 u + v 0 τ 4 - 1 2 v 0 τ u + v 0 τ 3 + u 2 v 0 2 2 v 0 τ π sin u - v 0 τ 2 / 2 u + π / 4 u - v 0 τ 2 + u 3 v 0 2 2 v 0 τ π × cos u - v 0 τ 2 2 u + π 4 - 2 u - v 0 τ 4 - 1 2 v 0 τ u - v 0 τ 3 ,
I u ,   v 0 ,   τ | 1 u < v 0 τ = 1 v 0 2 0 d v v | α u ,   v | 2 - v 0 τ d v v | α u ,   v | 2 1 v 0 2 u 2 2 - v 0 τ d v v | α u ,   v | 2 .
U 1 u ,   v 2 π v u v sin v - π / 4 1 - u 2 / v 2 , U 2 u ,   v - 2 π v u v 2 cos v - π / 4 1 - u 2 / v 2 ,
I u ,   v 0 ,   τ | 1 u < v 0 τ u 2 2 v 0 2 1 - 2 v 0 τ π v 0 2 τ 2 - u 2 + cos 2 v 0 τ π v 0 2 τ 2 - u 2 + .
I u ,   v 0 ,   τ | pinhole u 2 2 v 0 2 1 - J 0 2 v 0 τ - J 1 2 v 0 τ .
1 - J 0 2 x - J 1 2 x x 2 4 - x 4 32 + 5 x 6 2304 - 7 x 8 73728 + 7 x 10 2457600 x < 2.67 1 - 2 π x + cos 2 x π x 2 x 2.67 .
F 1 u ,   v 0 ,   σ = 1 π   - 1 1 d x   1 - x 2 2 + σ x 2 - σ 2 1 / 2 1 + σ x × 2 v 0 2 I u ,   v 0 ,   1 + σ x u 2 ,
F 2 u ,   v 0 ,   σ = 1 π   - 1 1 d x   1 - x 2 2 + σ x 2 - σ 2 1 / 2 1 + σ x × 2 I u ,   v 0 ,   1 + σ x .
1 π   - 1 1 d x 1 - x 2   f x 1 N k = 1 N   f cos π   2 k - 1 2 N × sin 2 π   2 k - 1 2 N .
1 - x 2 2 + σ x 2 - σ 2 1 / 2 1 + σ x   2 I u ,   v 0 ,   1 + σ x = 1 - x 2 2 + σ x 2 - σ 2 1 / 2 1 + σ x + 2 u 2 π v 0 1 - x 2 2 + σ x 2 - σ 2 1 / 2 u 2 - v 0 2 1 + σ x 2 + 1 - x 2 A 1 x sin   θ + + A 2 x cos   θ + + A 3 x sin   θ - + A 4 x cos   θ - .
θ ± = u ± v 0 2 u ± v 0 σ 2 u   x 2 ± π 4 ,
A 1 x = - 2 u 2 v 0 2 2 v 0 1 + σ x π 1 / 2 × 2 + σ x 2 - σ 2 1 / 2 1 + σ x u + v 0 1 + σ x 2
a = - v 0 σ u + v 0 ,     b = v 0 σ u - v 0 , α = 1 - 1 - a 2 a ,     β = 1 - 1 - b 2 b , s = σ 4 - σ 2 ,     Σ = - 1 + 1 - 4 s 2 2 s ,
I 2 A ,   a ,   b ,   c = 1 π   - 1 1 d x 1 - x 2 × exp i a + bx 2 + c A x
I 3 u ,   v 0 ,   σ = 1 π   - 1 1 d x   1 - x 2 2 + σ x 2 - σ 2 1 / 2 u 2 - v 0 2 1 + σ x 2 .
I 2 A ,   a ,   b ,   c exp i a 2 + b 2 + c 4 π | ab | 3 / 2 × sin 2 | ab | - π 4 a a + b 3 / 2 A 1 + a a - b 3 / 2 A - 1 - i   ab | ab | cos 2 | ab | - π 4 a a + b 3 / 2 A 1 - a a - b 3 / 2 × A - 1 .
I 3 u ,   v 0 ,   σ - 4 - σ 2 2 uv 0 σ α - β 1 + σ 2 8 + s α 2 - β 2 1 - σ 2 8 - σ 2 1 + Σ 2 8 × α 1 - α Σ - β 1 - β Σ .
F 2 u ,   v 0 ,   σ 1 + 2 u 2 π v 0   I 3 u ,   v 0 ,   σ + Im I 2 A 1 ,   u + v 0 2 u ,   + v 0 σ 2 u ,   - π 4 + Re I 2 A 2 ,   u + v 0 2 u ,   + v 0 σ 2 u ,   - π 4 + Im I 2 A 3 ,   u - v 0 2 u ,   - v 0 σ 2 u ,   + π 4 + Re I 2 A 4 ,   u - v 0 2 u ,   - v 0 σ 2 u , + π 4 .
1 - x 2 2 + σ x 2 - σ 2 1 / 2 1 + σ x 2 v 0 2 I u ,   v 0 ,   1 + σ x u 2 1 - x 2 2 + σ x 2 - σ 2 1 / 2 1 + σ x + 2 v 0 1 - x 2 2 + σ x 2 - σ 2 1 / 2 π u 2 - v 0 2 1 + σ x 2 + 1 - x 2   B x cos 2 v 0 + 2 v 0 σ x ,
B x = 2 + σ x 2 - σ 2 1 / 2 π v 0 2 1 + σ x 2 - u 2 1 + σ x .
F 1 u ,   v 0 ,   σ 1 + 2 v 0 π   I 3 u ,   v 0 ,   σ + cos 2 v 0 Re I 1 B ,   2 v 0 σ - sin 2 v 0 Im I 1 B ,   2 v 0 σ .
F 2 1 + 2 π v 0 .
F 1 u ,   v 0 ,   v 0 σ 1 - 1 2 π v 0 σ ln v 0 2 1 + σ 2 - u 2 v 0 2 1 - σ 2 - u 2
Φ λ λ = 2 π r s 2 r d 2 d s + d d 2 L λ λ σ 2   0 d τ τ | α u ,   v 0 τ | 2 E σ ,   τ ,
E σ ,   τ = 2 π   0 σ d σ σ C σ ,   τ ,
E σ ,   τ = π σ 2 τ < 1 - σ π 1 - τ 2 + 2   1 - τ σ d σ σ   arccos σ 2 + τ 2 - 1 2 τ σ 1 - σ τ < 1 2   τ - 1 σ d σ σ   arccos σ 2 + τ 2 - 1 2 τ σ 1 τ < 1 + σ 0 τ 1 + σ .
0 d τ τ | α u ,   v 0 τ | 2 E σ ,   τ = 0 d τ d I u ,   v 0 ,   τ d τ E σ ,   τ = 0 d τ I u ,   v 0 ,   τ - d E σ ,   τ d τ ,
I u ,   v 0 ,   0 = E σ ,   = 0 .
- d E σ ,   τ d τ = σ τ 1 - x 2 2 + σ x 2 - σ 2 1 / 2 1 - σ < τ < 1 + σ 0 otherwise ,
I 1 f ,   a = 1 π   - 1 1 d x 1 - x 2 exp iax f x = 1 π m = 0 2 - δ m 0 0 π d θ   cos m θ i m J m a × 1 - x 2 f x ,
D θ = 1 - x 2 f x = f x sin 2 θ = m = 0 cos m θ D m ,
I 1 f ,   a = m = 0   i m J m a D m .
m = 0 ± 1 m D m = 1 - x 2 f x x = ± 1 = 0 ,
m = 0 ± 1 m m 2 D m = - 2 θ 2 f x sin 2 θ x = ± 1 = - 2 f ± 1 .
J m | a | 2 π | a | cos | a | - m π 2 - π 4   - m 2 - 1 / 4 2 | a | sin | a | - m π 2 - π 4 ;
I 1 f ,   a 1 2 | a | 2 π | a | sin | a | - π 4 f 1 + f - 1 - i   a | a | cos | a | - π 4 f 1 - f - 1 .
I 2 g ,   a ,   b ,   c = 1 π   - 1 1 d x 1 - x 2 × exp i a + bx 2 + c g x .
α = a + bx 2 ,
x = α 1 / 2 - a b ,     d x = d α 2 b α 1 / 2 ,
I 2 g ,   a ,   b ,   c = e ic 2 π b   a - b 2 a + b 2 d α α 1 / 2 1 - x 2   e i α g x .
α = a 2 + b 2 + 2 abx ,     x = α - a 2 - b 2 2 ab .
I 2 g ,   a ,   b ,   c = exp i a 2 + b 2 + c 1 π   - 1 1 d x 1 - x 2 × exp 2 iabx f x = exp i a 2 + b 2 + c I 1 f ,   2 ab ,
f x = ag x α 1 / 2 1 - x 2 1 - x 2 1 / 2 .
f ± 1 = a a ± b 3 / 2 g ± 1 .
1 u 2 - v 0 2 1 + σ x 2 = 1 u 2 - v 0 2 a - b × a 1 - a x - b 1 - b x = - 1 2 uv 0 σ a 1 - a x - b 1 - b x ,
I 3 u ,   v 0 ,   σ = 1 π   - 1 1 d x   1 - x 2 2 + σ x 2 - σ 2 1 / 2 u 2 - v 0 2 1 + σ x 2 = - K a - K b 2 uv 0 σ ,
K a = 1 π   - 1 1 d x   a 1 - x 2 2 + σ x 2 - σ 2 1 / 2 1 - a x = 2 α π   - 1 1 d x 1 - x 2 2 + σ x 2 - σ 2 1 / 2 × n = 0   α n Ξ n x .
y = sx = σ x 4 - σ 2 ,
2 + σ x 2 - σ 2 1 / 2 = 4 - σ 2 1 + 2 y 2 - σ 2 y 2 1 / 2 = 4 - σ 2 × 1 + 2 y - σ 2 y 2 / 2 1 + 2 y + .
2 + σ x 2 - σ 2 1 / 2 4 - σ 2 1 + σ 2 8   Ξ 0 x + s 1 - σ 2 8   Ξ 1 x - σ 2 8 × 1 + Σ 2 n = 0   Σ n Ξ n x ,
K a α 4 - σ 2 1 + σ 2 / 8 + s α 1 - σ 2 / 8 - σ 2 1 + Σ 2 / 8 1 - α Σ .

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