Abstract

Diffraction loss in radiometry has gained in importance recently because of an increased interest in longer wavelengths and the continuous improvement in experimental accuracy. The deviation from geometrical optics now contributes significantly to the errors of experiments. Previous research has concentrated on geometries classified as F1 and F2, leaving an intermediate case yet to be investigated. This intermediate case has some interesting behavior, as it is in this envelope of geometries that it is possible to have zero diffraction loss. We designate this intermediate geometric regime as F3. We introduce a numerical regime to calculate diffraction loss for intermediate geometries, which is also highly efficient for the F1 and F2 regimes.

© 2003 Optical Society of America

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References

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  1. L. P. Boivin, “Diffraction corrections in radiometry: comparison of two different methods of calculations,” Appl. Opt. 14, 2002–2009 (1975).
    [CrossRef] [PubMed]
  2. W. H. Steel, J. A. Bell, “Diffraction corrections in radiometry,” J. Opt. Soc. Am. 62, 1099–1103 (1972).
    [CrossRef]
  3. E. L. Shirley, “Revised formulas for diffraction effects with point and extended sources,” Appl. Opt. 37, 6581–6590 (1998).
    [CrossRef]
  4. E. L. Shirley, M. L. Teraciano, “Two innovations in diffraction calculations for cylindrically symmetrical systems,” Appl. Opt. 40, 4463–4472 (2001).
    [CrossRef]
  5. E. Wolf, M. Born, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).
  6. K. D. Mielenz, “Algorithms for Fresnel diffraction at rectangular and circular apertures,” J. Res. Natl. Inst. Stand. Technol. 103, 497–509 (1998).
    [CrossRef]
  7. E. Wolf, “Light distribution near focus in an error free diffraction image,” Proc. R. Soc. London Ser. A 204, 533–548 (1951).
    [CrossRef]
  8. E. Shirley, “Accurate efficient evaluation of Lommel functions for arbitrarily large arguments,” Metrologia 40, 5–8 (2003).
    [CrossRef]
  9. J. Focke, “Total illumination in an aberration-free diffraction image,” Opt. Acta 3, 161–163 (1956).
    [CrossRef]

2003 (1)

E. Shirley, “Accurate efficient evaluation of Lommel functions for arbitrarily large arguments,” Metrologia 40, 5–8 (2003).
[CrossRef]

2001 (1)

1998 (2)

E. L. Shirley, “Revised formulas for diffraction effects with point and extended sources,” Appl. Opt. 37, 6581–6590 (1998).
[CrossRef]

K. D. Mielenz, “Algorithms for Fresnel diffraction at rectangular and circular apertures,” J. Res. Natl. Inst. Stand. Technol. 103, 497–509 (1998).
[CrossRef]

1975 (1)

1972 (1)

1956 (1)

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

1951 (1)

E. Wolf, “Light distribution near focus in an error free diffraction image,” Proc. R. Soc. London Ser. A 204, 533–548 (1951).
[CrossRef]

Bell, J. A.

Boivin, L. P.

Born, M.

E. Wolf, M. Born, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

Focke, J.

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

Mielenz, K. D.

K. D. Mielenz, “Algorithms for Fresnel diffraction at rectangular and circular apertures,” J. Res. Natl. Inst. Stand. Technol. 103, 497–509 (1998).
[CrossRef]

Shirley, E.

E. Shirley, “Accurate efficient evaluation of Lommel functions for arbitrarily large arguments,” Metrologia 40, 5–8 (2003).
[CrossRef]

Shirley, E. L.

Steel, W. H.

Teraciano, M. L.

Wolf, E.

E. Wolf, “Light distribution near focus in an error free diffraction image,” Proc. R. Soc. London Ser. A 204, 533–548 (1951).
[CrossRef]

E. Wolf, M. Born, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

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

K. D. Mielenz, “Algorithms for Fresnel diffraction at rectangular and circular apertures,” J. Res. Natl. Inst. Stand. Technol. 103, 497–509 (1998).
[CrossRef]

Metrologia (1)

E. Shirley, “Accurate efficient evaluation of Lommel functions for arbitrarily large arguments,” Metrologia 40, 5–8 (2003).
[CrossRef]

Opt. Acta (1)

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

Proc. R. Soc. London Ser. A (1)

E. Wolf, “Light distribution near focus in an error free diffraction image,” Proc. R. Soc. London Ser. A 204, 533–548 (1951).
[CrossRef]

Other (1)

E. Wolf, M. Born, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

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Figures (7)

Fig. 1
Fig. 1

Diagram of the source, aperture, and detector layout of a simplified radiometric experiment. Parameters describing the geometry are shown.

Fig. 2
Fig. 2

Geometric construction showing the definitions of F1, F2, and F3 geometries. The two end optics (A and C) are the source and detector; B represents the aperture. The geometry is F1 if the aperture impinges on the short dashed lines and it is F2 if the aperture does not impinge on the long dashed lines. If the aperture impinges on the long but not the short dashed lines, indicated by the hatched region, then the geometry is said to be F3. The geometry type shown is F3.

Fig. 3
Fig. 3

Overlap construction used to calculate the total geometric flux in F3 geometries. The shaded region is the area of overlap and is denoted by A.

Fig. 4
Fig. 4

Behavior of the exact Wolf solution (dashed curves) compared with the asymptotic Focke result (solid curve) for the total flux within a prescribed radius that is due to a point source.

Fig. 5
Fig. 5

Comparison between the Focke and Wolf methods to calculate the total flux within a prescribed radius that is due to a point source. A point on the contour plot depicts the percentage difference ϕ between the approximate Focke solution and the Wolf solution. The y axis specifies the value of u. The x axis gives the value of v as a ratio of the currently chosen u value.

Fig. 6
Fig. 6

Diffraction loss contour plots and geometric classification for geometries specified by u, v 0, and σ. (a)–(d) represent geometries with σ values of 0.1, 0.25, 0.5, and 1.0, respectively.

Fig. 7
Fig. 7

(a) Diffraction loss contour plot for σ = 0.05. The dotted curve represents the diffraction loss for a particular geometry when the aperture size is increased. λ = 0.58 × 10-5 m, r s = 0.001 m, r a = 0.001 → 0.0215 m, r d = 0.02 m, d s = 1.0 m, and d d = 1.0 m. (b) Diffraction loss as a function of aperture radius. [Cross section of contour map following the dotted curve in (a).]

Tables (4)

Tables Icon

Table 1 Geometry Classifications for F1, F2, and F3

Tables Icon

Table 2 Geometric Flux Formulas for F1 and F2 Geometries

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Table 3 Angular Formulas for Geometric Fluxa

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Table 4 Error in Diffraction Loss Introduced by Use of the Focke Approximation for Geometries Where σ = 1.0

Equations (50)

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u=2πλ ra21ds+1dd, vs=2πλrarsds, vd=2πλrarddd,
v0=maxvs, vd, σ=minvs, vdmaxvs, vd,
ϕ=maxu, v0, ξ=minu, v0maxu, v0.
F12=λ22v02σϕ2.
u<v0 for F1,
u>v0 for F2.
|v0-u|=ϕ1-ξ.
Φa=λ22v01-ξ2.
Aa, b=1πarccos1+b2-a22b+a2 arccosb2-1+a22ba-b1-1+b2-a224b21/2.
Φb=ϕ1-ξuσv0u Av0u, xxdxϕ1-ξuσv0u xdxλ22v0σ2-ϕ1-ξ.
Φ=λ222u2ϕ-ϕξuσv0u Av0u, xxdx+v021-ξ2.
v0σ-ϕ1-ξ=0
ΦF3 limit=λ22v01-ξ2.
FF3 limit=λ22v02σϕ2.
UFp=-U0pcosθ0αFp,
U0p=A expikds+ddds+dd,
αFp=-iu 01 J0vρexpi 12 uρ2ρdρ.
01 J0vρexpi 12 uρ2ρdρ=12Lu, v+iMu, v=iαFu,
u2 Lu, v=sinv22u+V0u, vsinu2-V1u, vcosu2 =U1u, vcosu2+U2u, vsinu2,
u2 Mu, v=cosv22u-V0u, vcosu2-V1u, vsinu2 =U1u, vsinu2-U2u, vcosu2.
V0u, v=J0v-vu2J2v+vu4J4v-, V1u, v=vuJ1v-vu3J3v+vu5J5v-,
U1u, v=uvJ1v-uv3J3v+uv5J5v-, U2u, v=uv2J2v-uv4J4v+uv6J6v-.
n>-mlogminv, umaxv, u,
nv, Jnv<12 10-m.
Φ=12πλd0ra20v0 |UFv|2vdv,
Φ1v  0v0-v |UFv|2vdv,
Φ2v  v0-vv+v0 Bv, v|UFv|2vdv,
Bv, ξ=1πarccosv2+ξ2-v022vξ.
Φextλ=λ2dsdd2πra220σv0 vΦ1v+Φ2vdv.
Φ=2πrs2rd2ds+dd2-11Iu, v0, 1+σx1+σx×dx1-x22+σx2-σ21/2,
Iu, v, τ=0τ τ|αFu, vτ|2dτ.
Lu, v=1-s=0-1s2s+1uv2sAQ2svB,
Q2sv=p=02s-1pJpvJ2s-pv+Jp+1vJ2s+1-pv.
Lu, v=vu21+s=0-1s2s+1vu2sQ2svC -4uY1u, vcos12u+v2u+Y2u, vsin12u+v2u,
Ynu, v=s=0-1sn+2svun+2sJn+2svD
=12v2u Vn-1u, v+uVn+1u, v.
Lu=1-J0ucos u-J1usin u.
Lv0=1-J02v0-J12v0.
s>v, Jsv<10-m.
L1v0, u=1-2πv0v02-u2.
L2v0, u=1-πu gtv0, u,
tv0, u=v0-uπu,
gx=2π12-Sxcosπ2 x2-2π12-Cxsinπ2 x2
-x12-Cx2+12-Sx2,
fx=x2
L3v0, u=1-πu gtv0, u+πu ftv0, u.
L4v0, uv0u2.
L5=1-1πu.
λ22v02σϕ2Lλ
λ222u2ϕ1-ξuσv0u Av0u, xxdx+v021-ξ2Lλ

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