Abstract

A theoretical treatment is given for the diffraction of a Gaussian beam around an opaque strip mask. Such situations arise frequently in the diffraction of laser beams around wires and fibers. Scalar derivations are given for the Fraunhofer and Fresnel regions with both developments, leading to similar forms of rapidly convergent series for the field at an observation plane. Predictions show good agreement with measurements on the diffraction patterns from wires.

© 1998 Optical Society of America

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References

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  1. E. Hecht, A. Zajac, Optics (Addison-Wesley, New York, 1974).
  2. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).
  3. T. W. Mayes, B. F. Melton, “Fraunhofer diffraction of visible light by a narrow slit,” Am. J. Phys. 62, 397–403 (1994).
    [CrossRef]
  4. H. C. van de Hulst, The Scattering of Light by Small Particles (Dover, New York, 1981).
  5. F. Kuik, F. F. de Haan, J. W. Hovenier, “Single scattering of light by circular cylinders,” Appl. Opt. 33, 4906–4918 (1994).
    [CrossRef] [PubMed]
  6. R. T. Wang, H. C. van de Hulst, “Application of the exact solution for scattering by an infinite cylinder to the estimation of scattering by a finite cylinder,” Appl. Opt. 34, 2811–2821 (1995).
    [CrossRef] [PubMed]
  7. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  8. M. Glass, T. P. Dabbs, P. W. Chudleigh, “The optics of the wool Fiber Diameter Analyser,” Text. Res. J. 65(2), 85–94 (1995).
    [CrossRef]
  9. M. Glass, “Fresnel diffraction from curved fiber snippets with application to fiber diameter measurement,” Appl. Opt. 35, 1605–1616 (1996).
    [CrossRef] [PubMed]
  10. D. Lebrun, S. Belaid, C. Özkul, K. F. Ren, G. Gréhan, “Enhancement of wire diameter measurements: comparison between Fraunhofer diffraction and Lorenz-Mie theory,” Opt. Eng. 35, 946–950 (1996).
    [CrossRef]
  11. H. Wang, R. Valdivia-Hernandez, “Laser scanner and diffraction pattern detection: a novel concept for dynamic gauging of fine wires,” Meas. Sci. Technol. 6, 452–457 (1995).
    [CrossRef]
  12. S. Kozaki, “Scattering of a Gaussian beam by a homogeneous dielectric cylinder,” J. Appl. Phys. 53, 7195–7200 (1982).
    [CrossRef]
  13. E. Zimmermann, R. Dändliker, N. Souli, “Scattering of an off-axis Gaussian beam by a dielectric cylinder compared with a rigorous electromagnetic approach,” J. Opt. Soc. Am. A 12, 398–403 (1995).
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  14. J. E. Pearson, T. C. McGill, S. Curtain, A. Yariv, “Diffraction of Gaussian laser beams by a semi-infinite plane,” J. Opt. Soc. Am. 59, 1440–1445 (1969).
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  16. E. Kreyszig, Advanced Engineering Mathematics (Wiley, New York, 1983).
  17. S. A. Self, “Focusing of spherical Gaussian beams,” Appl. Opt. 22, 658–661 (1983).
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  18. H. Kogelnik, “On the propagation of Gaussian beams of light through lenslike media including those with a loss or gain variation,” Appl. Opt. 4, 1562–1569 (1965).
    [CrossRef]
  19. W. H. Carter, “Focal shift and concept of effective Fresnel number for a Gaussian laser beam,” Appl. Opt. 21, 1989–1994 (1982).
    [CrossRef] [PubMed]
  20. P. Rochon, T. J. Racey, N. Gauthier, “Diffraction from small wires, including surface roughness,” Opt. Acta. 31, 1385–1397 (1984).
    [CrossRef]
  21. R. G. Greenler, J. W. Hable, P. O. Slane, “Diffraction around a fine wire: how good is the single-slit approximation?” Am. J. Phys. 58, 330–331 (1990).
    [CrossRef]

1996 (2)

M. Glass, “Fresnel diffraction from curved fiber snippets with application to fiber diameter measurement,” Appl. Opt. 35, 1605–1616 (1996).
[CrossRef] [PubMed]

D. Lebrun, S. Belaid, C. Özkul, K. F. Ren, G. Gréhan, “Enhancement of wire diameter measurements: comparison between Fraunhofer diffraction and Lorenz-Mie theory,” Opt. Eng. 35, 946–950 (1996).
[CrossRef]

1995 (5)

1994 (2)

T. W. Mayes, B. F. Melton, “Fraunhofer diffraction of visible light by a narrow slit,” Am. J. Phys. 62, 397–403 (1994).
[CrossRef]

F. Kuik, F. F. de Haan, J. W. Hovenier, “Single scattering of light by circular cylinders,” Appl. Opt. 33, 4906–4918 (1994).
[CrossRef] [PubMed]

1990 (1)

R. G. Greenler, J. W. Hable, P. O. Slane, “Diffraction around a fine wire: how good is the single-slit approximation?” Am. J. Phys. 58, 330–331 (1990).
[CrossRef]

1984 (1)

P. Rochon, T. J. Racey, N. Gauthier, “Diffraction from small wires, including surface roughness,” Opt. Acta. 31, 1385–1397 (1984).
[CrossRef]

1983 (1)

1982 (2)

W. H. Carter, “Focal shift and concept of effective Fresnel number for a Gaussian laser beam,” Appl. Opt. 21, 1989–1994 (1982).
[CrossRef] [PubMed]

S. Kozaki, “Scattering of a Gaussian beam by a homogeneous dielectric cylinder,” J. Appl. Phys. 53, 7195–7200 (1982).
[CrossRef]

1969 (1)

1965 (1)

Belaid, S.

D. Lebrun, S. Belaid, C. Özkul, K. F. Ren, G. Gréhan, “Enhancement of wire diameter measurements: comparison between Fraunhofer diffraction and Lorenz-Mie theory,” Opt. Eng. 35, 946–950 (1996).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Carter, W. H.

Chudleigh, P. W.

M. Glass, T. P. Dabbs, P. W. Chudleigh, “The optics of the wool Fiber Diameter Analyser,” Text. Res. J. 65(2), 85–94 (1995).
[CrossRef]

Curtain, S.

Dabbs, T. P.

M. Glass, T. P. Dabbs, P. W. Chudleigh, “The optics of the wool Fiber Diameter Analyser,” Text. Res. J. 65(2), 85–94 (1995).
[CrossRef]

Dändliker, R.

de Haan, F. F.

Gauthier, N.

P. Rochon, T. J. Racey, N. Gauthier, “Diffraction from small wires, including surface roughness,” Opt. Acta. 31, 1385–1397 (1984).
[CrossRef]

Glass, M.

M. Glass, “Fresnel diffraction from curved fiber snippets with application to fiber diameter measurement,” Appl. Opt. 35, 1605–1616 (1996).
[CrossRef] [PubMed]

M. Glass, T. P. Dabbs, P. W. Chudleigh, “The optics of the wool Fiber Diameter Analyser,” Text. Res. J. 65(2), 85–94 (1995).
[CrossRef]

Greenler, R. G.

R. G. Greenler, J. W. Hable, P. O. Slane, “Diffraction around a fine wire: how good is the single-slit approximation?” Am. J. Phys. 58, 330–331 (1990).
[CrossRef]

Gréhan, G.

D. Lebrun, S. Belaid, C. Özkul, K. F. Ren, G. Gréhan, “Enhancement of wire diameter measurements: comparison between Fraunhofer diffraction and Lorenz-Mie theory,” Opt. Eng. 35, 946–950 (1996).
[CrossRef]

Hable, J. W.

R. G. Greenler, J. W. Hable, P. O. Slane, “Diffraction around a fine wire: how good is the single-slit approximation?” Am. J. Phys. 58, 330–331 (1990).
[CrossRef]

Hecht, E.

E. Hecht, A. Zajac, Optics (Addison-Wesley, New York, 1974).

Hovenier, J. W.

Kogelnik, H.

Kozaki, S.

S. Kozaki, “Scattering of a Gaussian beam by a homogeneous dielectric cylinder,” J. Appl. Phys. 53, 7195–7200 (1982).
[CrossRef]

Kreyszig, E.

E. Kreyszig, Advanced Engineering Mathematics (Wiley, New York, 1983).

Kuik, F.

Lebrun, D.

D. Lebrun, S. Belaid, C. Özkul, K. F. Ren, G. Gréhan, “Enhancement of wire diameter measurements: comparison between Fraunhofer diffraction and Lorenz-Mie theory,” Opt. Eng. 35, 946–950 (1996).
[CrossRef]

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Mayes, T. W.

T. W. Mayes, B. F. Melton, “Fraunhofer diffraction of visible light by a narrow slit,” Am. J. Phys. 62, 397–403 (1994).
[CrossRef]

McGill, T. C.

Melton, B. F.

T. W. Mayes, B. F. Melton, “Fraunhofer diffraction of visible light by a narrow slit,” Am. J. Phys. 62, 397–403 (1994).
[CrossRef]

Özkul, C.

D. Lebrun, S. Belaid, C. Özkul, K. F. Ren, G. Gréhan, “Enhancement of wire diameter measurements: comparison between Fraunhofer diffraction and Lorenz-Mie theory,” Opt. Eng. 35, 946–950 (1996).
[CrossRef]

Pearson, J. E.

Racey, T. J.

P. Rochon, T. J. Racey, N. Gauthier, “Diffraction from small wires, including surface roughness,” Opt. Acta. 31, 1385–1397 (1984).
[CrossRef]

Ren, K. F.

D. Lebrun, S. Belaid, C. Özkul, K. F. Ren, G. Gréhan, “Enhancement of wire diameter measurements: comparison between Fraunhofer diffraction and Lorenz-Mie theory,” Opt. Eng. 35, 946–950 (1996).
[CrossRef]

Rochon, P.

P. Rochon, T. J. Racey, N. Gauthier, “Diffraction from small wires, including surface roughness,” Opt. Acta. 31, 1385–1397 (1984).
[CrossRef]

Rose, T. S.

Self, S. A.

Slane, P. O.

R. G. Greenler, J. W. Hable, P. O. Slane, “Diffraction around a fine wire: how good is the single-slit approximation?” Am. J. Phys. 58, 330–331 (1990).
[CrossRef]

Snyder, A. W.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Souli, N.

Valdivia-Hernandez, R.

H. Wang, R. Valdivia-Hernandez, “Laser scanner and diffraction pattern detection: a novel concept for dynamic gauging of fine wires,” Meas. Sci. Technol. 6, 452–457 (1995).
[CrossRef]

van de Hulst, H. C.

Wang, H.

H. Wang, R. Valdivia-Hernandez, “Laser scanner and diffraction pattern detection: a novel concept for dynamic gauging of fine wires,” Meas. Sci. Technol. 6, 452–457 (1995).
[CrossRef]

Wang, R. T.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Yariv, A.

Yura, H. T.

Zajac, A.

E. Hecht, A. Zajac, Optics (Addison-Wesley, New York, 1974).

Zimmermann, E.

Am. J. Phys. (2)

T. W. Mayes, B. F. Melton, “Fraunhofer diffraction of visible light by a narrow slit,” Am. J. Phys. 62, 397–403 (1994).
[CrossRef]

R. G. Greenler, J. W. Hable, P. O. Slane, “Diffraction around a fine wire: how good is the single-slit approximation?” Am. J. Phys. 58, 330–331 (1990).
[CrossRef]

Appl. Opt. (7)

J. Appl. Phys. (1)

S. Kozaki, “Scattering of a Gaussian beam by a homogeneous dielectric cylinder,” J. Appl. Phys. 53, 7195–7200 (1982).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Meas. Sci. Technol. (1)

H. Wang, R. Valdivia-Hernandez, “Laser scanner and diffraction pattern detection: a novel concept for dynamic gauging of fine wires,” Meas. Sci. Technol. 6, 452–457 (1995).
[CrossRef]

Opt. Acta. (1)

P. Rochon, T. J. Racey, N. Gauthier, “Diffraction from small wires, including surface roughness,” Opt. Acta. 31, 1385–1397 (1984).
[CrossRef]

Opt. Eng. (1)

D. Lebrun, S. Belaid, C. Özkul, K. F. Ren, G. Gréhan, “Enhancement of wire diameter measurements: comparison between Fraunhofer diffraction and Lorenz-Mie theory,” Opt. Eng. 35, 946–950 (1996).
[CrossRef]

Text. Res. J. (1)

M. Glass, T. P. Dabbs, P. W. Chudleigh, “The optics of the wool Fiber Diameter Analyser,” Text. Res. J. 65(2), 85–94 (1995).
[CrossRef]

Other (5)

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

H. C. van de Hulst, The Scattering of Light by Small Particles (Dover, New York, 1981).

E. Hecht, A. Zajac, Optics (Addison-Wesley, New York, 1974).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

E. Kreyszig, Advanced Engineering Mathematics (Wiley, New York, 1983).

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Figures (10)

Fig. 1
Fig. 1

Fraunhofer diffraction geometry.

Fig. 2
Fig. 2

Fresnel diffraction geometry with a point source.

Fig. 3
Fig. 3

Geometry for Fresnel diffraction of a Gaussian beam by a strip mask with a detector lens in place.

Fig. 4
Fig. 4

Schematic diagram of the Fraunhofer diffraction experimental arrangement.

Fig. 5
Fig. 5

Unobstructed collimated beam intensity profile at the diffraction plane of the Fraunhofer experiment. A fitted Gaussian curve is shown for comparison.

Fig. 6
Fig. 6

Measured and predicted Fraunhofer diffraction pattern from the 152.3-μm wire.

Fig. 7
Fig. 7

Measured and predicted Fraunhofer diffraction pattern from the 199.2-μm wire.

Fig. 8
Fig. 8

Unobstructed beam intensity profile at the diffraction plane of the Fresnel experiment 100 mm from the laser casing. A fitted Gaussian curve is shown for comparison.

Fig. 9
Fig. 9

Measured and predicted Fresnel diffraction pattern from the 152.3-μm wire.

Fig. 10
Fig. 10

Measured and predicted Fresnel diffraction pattern from the 199.2-μm wire.

Equations (88)

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Θ 2 = α 2 + β 2 .
Ψ α ,   β = L     A   Ψ ξ ,   η exp - ik α ξ + β η d ξ d η ,
Ψ ρ = Ψ 0 exp - ρ / ω 0 2 ,
ρ 2 = ξ 2 + η 2 .
Ψ α ,   β = L   unobstructed diffracting plane - region   of strip mask L I 1 - I 2 .
I 1 = 2 π   0   Ψ 0 exp - ρ / ω 0 2 J 0 k ρ Θ ρ d ρ ,
I 1 = π Ψ 0 ω 0 2 exp - k Θ ω 0 / 2 2 .
I 2 = region   of strip   mask   Ψ ξ ,   η exp - ik α ξ + β η d ξ d η     = Ψ 0 - exp - η / ω 0 2 + ik β η d η × - d / 2 d / 2 exp - ξ / ω 0 2 + ik α ξ d ξ     Ψ 0 I 3 I 4 .
I 3 = ω 0 π exp - k β ω 0 / 2 2 .
I 4 = exp - k α ω 0 / 2 2 - d / 2 d / 2 exp - ξ ω 0 + ik α ω 0 2 2 d ξ
u = ξ ω 0 + ik α ω 0 2 ,
d u = d ξ / ω 0 ,
I 4 = ω 0 exp - k α ω 0 / 2 2 u - u + exp - u 2 d u ,
u + = d 2 ω 0 + ik α ω 0 2 ,
u - = - d 2 ω 0 + ik α ω 0 2
erf u 2 π 0 u exp - ζ 2 d ζ ,
I 4 = ω 0 π exp - k α ω 0 / 2 2 2 erf u + - erf u - .
erf u = 2 π u - u 3 1 ! 3 + u 5 2 ! 5 - u 7 3 ! 7 + ,
u + = δ cos   ϕ + i   sin   ϕ ,
u - = δ - cos   ϕ + i   sin   ϕ ,
δ = d 2 ω 0 2 + k α ω 0 2 2 1 / 2 ,
ϕ = arctan k α ω 0 2 d .
cos   ϕ + i   sin   ϕ n = cos   n ϕ + i   sin   n ϕ ,
I 4 = 2 ω 0 exp - k α ω 0 / 2 2 δ   cos   ϕ - δ 3 1 ! 3 cos   3 ϕ + δ 5 2 ! 5 cos   5 ϕ - .
Ψ α ,   β = L   exp - k Θ ω 0 / 2 2 1 - 2 π δ   cos   ϕ - δ 3 1 ! 3 cos   3 ϕ + .
α = x / F d ,
β = y / F d ,
u + = 2 ξ off + d 2 ω 0 + ik α ω 0 2 , u - = 2 ξ off - d 2 ω 0 + ik α ω 0 2
u + = δ + cos   ϕ + + i   sin   ϕ + ,
u - = δ - cos   ϕ - + i   sin   ϕ - ,
δ + = 2 ξ off + d 2 ω 0 2 + k α ω 0 2 2 1 / 2 ,
δ - = 2 ξ off - d 2 ω 0 2 + k α ω 0 2 2 1 / 2 ,
ϕ + = arctan k α ω 0 2 2 ξ off + d ,
ϕ - = arctan k α ω 0 2 2 ξ off - d .
Ψ α ,   β = L   exp - k Θ ω 0 / 2 2 1 - 1 π δ + cos   ϕ + - δ - cos   ϕ - - δ + 3 cos   3 ϕ + - δ - 3 cos   3 ϕ - 1 ! 3 + - i   1 π δ + sin   ϕ + - δ - sin   ϕ - - δ + 3 sin   3 ϕ + - δ - 3 sin   3 ϕ - 1 ! 3 + .
Ψ P = L   A exp i   π λ 1 a + 1 b ξ 2 + η 2 d ξ d η ,
E ρ ,   z = ω 0 ω exp if z - ρ 2 1 ω 2 - ik 2 a ,
ω 2 = ω 0 2 1 + z z R 2 ,
a = z 1 + z R z 2 , f z = kz + tan - 1 z z R ,
z R = π ω 0 2 λ
ρ 2 = ξ 0 + ξ 2 + η 0 + η 2 .
E ξ ,   η = exp - ξ 0 + ξ 2 + η 0 + η 2 / ω 2 .
Ψ P = L   A exp - ξ 0 + ξ 2 + η 0 + η 2 ω 2 - i   π λ 1 a + 1 b ξ 2 + η 2 d ξ d η .
π 2   p 2 = π λ 1 a + 1 b ξ 2 ,
π 2   q 2 = π λ 1 a + 1 b η 2 ,
d ξ d η = λ 2 1 a + 1 b d p d q .
x ,   y = b a   ξ 0 ,   b a   η 0 ,
π 2   u 2 = π λ 1 a + 1 b ξ 0 2 ,
π 2   v 2 = π λ 1 a + 1 b η 0 2 .
Ψ u ,   v = L     exp - γ v + q 2 - i π q 2 / 2 d q ×   exp - γ u + p 2 - i π p 2 / 2 d p ,
γ = λ 2 ω 2 1 a + 1 b ,
1 a + 1 b 0 .
Ψ u ,   v = L   A = L unobstructed aperture - region   of strip   mask .
Δ u 2 = d 2 2 λ 1 a + 1 b 1 / 2
Ψ u ,   v = L   - exp - γ v + q 2 - i   π 2   q 2 d q × - exp - γ u + p 2 - i   π 2   p 2 d p - - u - u off + Δ u / 2 - u - u off - Δ u / 2 exp - γ u + p 2 - i   π 2   p 2 d p LI 1 I 2 - I 3 .
I 1 = exp - γ v 2 - exp - γ - i π / 2 q 2 + 2 v γ q d q = exp - γ v 2 π γ - i π / 2 1 / 2 exp γ 2 v 2 γ - i π / 2 .
γ - i π / 2 =   exp - i Ω ,
Ω = tan - 1 π 2 γ ,
= γ 1 + π 2 γ 2 1 / 2 ,
I 1 = π exp - γ π / 2 γ 2 v 2 1 + π / 2 γ 2 exp i Ω 2 + π / 2 v 2 1 + π / 2 γ 2 .
Ψ u ,   v = L   exp - Mv 2 π exp - Mu 2 × exp i Nu 2 + Ω / 2 - - U - - U + exp - γ u + p 2 exp i   π 2   p 2 d p ,
U + = u - u off - Δ u / 2 ,
U - = u - u off + Δ u / 2 ,
M = γ π 2 γ 2 1 + π 2 γ 2 ,
N = π 2 1 + π 2 γ 2 .
I 3 = exp - γ u 2 exp γ 2 u 2 γ - i π / 2 × - U - - U + exp - γ - i π / 2   p + γ u γ - i π / 2 2 d p .
ζ = γ - i π / 2   p + γ u γ - i π / 2 ,
d ζ = γ - i π / 2 d p ,
I 3 = exp - γ u 2 1 γ - i π / 2 exp γ 2 u 2 γ - i π / 2 × ζ - ζ + exp - ζ 2 d ζ = 1 2 π exp - Mu 2 exp i Nu 2 + Ω / 2 × erf ζ + - erf ζ - ,
ζ + = γ - i π / 2   U + + γ u γ - i π / 2 ,
ζ - = γ - i π / 2   U - + γ u γ - i π / 2 .
ζ + = - cos   Ω / 2 - i   sin   Ω / 2 U + + γ u cos   Ω / 2 + i   sin   Ω / 2 , = γ u -   U + cos   Ω / 2 + i γ u +   U + sin   Ω / 2 , σ + cos   Φ + + i   sin   Φ + ,
Φ + = tan - 1 γ u + U + tan   Ω / 2 γ u - U + ,
σ + = γ u -   U + cos   Ω / 2 2 + γ u +   U + sin   Ω / 2 2 1 / 2 = γ 2 u 2 - 2 γ uU + cos   Ω + U + 2 1 / 2 .
ζ - σ - cos   Φ - + i   sin   Φ - ,
Φ - = tan - 1 γ u + U - tan   Ω / 2 γ u - U - ,
σ - = γ 2 u 2 - 2 γ uU - cos   Ω + U - 2 1 / 2 .
erf ζ = 2 π ζ - ζ 3 1 ! 3 + ζ 5 2 ! 5 - ζ 7 3 ! 7 + .
Ψ u ,   v = L   exp - M u 2 + v 2 1 - 1 π σ + cos   Φ + - σ - cos   Φ - - σ + 3 cos   3 Φ + - σ - 3 cos   3 Φ - 1 ! 3 + - i   1 π σ + sin   Φ + - σ - sin   Φ - - σ + 3 sin   3 Φ + - σ - 3 sin   3 Φ - 1 ! 3 + .
1 ó + 1 í = 1 F d ,
ó = F d í í - F d , = F d í Δ í ,
= F d 2 Δ í + F d
X X = ó í , = F d Δ í ,
Y Y = F d Δ í .
b = Δ ó - F d 2 Δ í ,
w = z + Δ ó .
Δ í w F d = w F d w F d 2 + z R F d 2 ,
Δ í = Δ í w + Δ í off .

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