Abstract

An analytic formula for the light field amplitude (and therefore also intensity) behind a circular aperture illuminated by spherical or plane waves in the Fresnel limit is given, and the essential difference between Fresnel diffraction and Fraunhofer diffraction is discussed.

© 1998 Optical Society of America

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References

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  1. J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  2. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [CrossRef]
  3. P. Sprangle, B. Hafizi, “Comment on nondiffracting beams,” Phys. Rev. Lett. 66, 837–838 (1991).
    [CrossRef] [PubMed]
  4. S. Wang, “On principles of diffraction,” Optik (Stuttgart) 100, 143–144 (1995).
  5. S. Ruschin, “Modified Bessel nondiffracting beams,” J. Opt. Soc. Am. A 11, 3224–3228 (1994).
    [CrossRef]
  6. P. L. Overfelt, C. S. Kenney, “Comparison of the propagation characteristics of Bessel, Bessel–Gauss, and Gaussian beams diffracted by a circular aperture,” J. Opt. Soc. Am. A 8, 732–745 (1991).
    [CrossRef]
  7. Z. Jiang, Q. Lu, Z. Liu, “Several questions related to diffraction,” Appl. Lasers (China) 6, 266–270 (1994).
  8. J. Gu, “Discussion on π jump at boundary in diffraction phenomena,” Appl. Lasers (China) 6, 270–272 (1994).
  9. P. Chen, J. Qin, “Discussion about the non-diffraction and super-diffraction,” Appl. Lasers (China) 6, 274–276 (1994).
  10. W. H. Southwell, “Validity of the Fresnel approximation in the near field,” J. Opt. Soc. Am. 71, 7–14 (1981).
    [CrossRef]
  11. A. M. Steane, H. N. Rutt, “Diffraction calculations in the near field and validity of the Fresnel approximation,” J. Opt. Soc. Am. A 6, 1809–1814 (1989).
    [CrossRef]
  12. C. J. R. Sheppard, M. Hrynevych, “Diffraction by a circular aperture: a generalization of Fresnel diffraction theory,” J. Opt. Soc. Am. A 9, 274–281 (1992).
    [CrossRef]
  13. Carl R. Schultheisz, “Numerical solution of the Huygens–Fresnel–Kirchhoff diffraction of spherical waves by a circular aperture,” J. Opt. Soc. Am. A 11, 774–778 (1989).
    [CrossRef]
  14. H. G. Kraus, “Huygens–Fresnel–Kirchoff wave-front diffraction formulation: spherical waves,” J. Opt. Soc. Am. A 6, 1196–1205 (1989).
    [CrossRef]
  15. D. E. Dauger, “Simulation and study of Fresnel diffraction for arbitrary two-dimensional apertures,” Comput. Phys. 10, 591–604 (1996).
    [CrossRef]
  16. C. de Jong, “Precise mathematical description of the diffraction pattern beyond a square aperture in an absorbing medium,” Opt. Eng. 30, 641–645 (1991).
    [CrossRef]
  17. H. Haman, “Efficient Fresnel-transform algorithm based on fractional Fresnel diffraction,” J. Opt. Soc. Am. A 12, 1920–1931 (1995).
    [CrossRef]
  18. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 8.
  19. Q. Liang, Physical Optics (Publishing House of the Mechanics Industry, Beijing, China, 1986), p. 167.
  20. K. Zhao, X. Chan, Optics (Publishing House of Beijing University, Beijing, China, 1982), p. 192.

1996 (1)

D. E. Dauger, “Simulation and study of Fresnel diffraction for arbitrary two-dimensional apertures,” Comput. Phys. 10, 591–604 (1996).
[CrossRef]

1995 (2)

1994 (4)

Z. Jiang, Q. Lu, Z. Liu, “Several questions related to diffraction,” Appl. Lasers (China) 6, 266–270 (1994).

J. Gu, “Discussion on π jump at boundary in diffraction phenomena,” Appl. Lasers (China) 6, 270–272 (1994).

P. Chen, J. Qin, “Discussion about the non-diffraction and super-diffraction,” Appl. Lasers (China) 6, 274–276 (1994).

S. Ruschin, “Modified Bessel nondiffracting beams,” J. Opt. Soc. Am. A 11, 3224–3228 (1994).
[CrossRef]

1992 (1)

1991 (3)

P. Sprangle, B. Hafizi, “Comment on nondiffracting beams,” Phys. Rev. Lett. 66, 837–838 (1991).
[CrossRef] [PubMed]

P. L. Overfelt, C. S. Kenney, “Comparison of the propagation characteristics of Bessel, Bessel–Gauss, and Gaussian beams diffracted by a circular aperture,” J. Opt. Soc. Am. A 8, 732–745 (1991).
[CrossRef]

C. de Jong, “Precise mathematical description of the diffraction pattern beyond a square aperture in an absorbing medium,” Opt. Eng. 30, 641–645 (1991).
[CrossRef]

1989 (3)

1987 (2)

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

1981 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 8.

Chan, X.

K. Zhao, X. Chan, Optics (Publishing House of Beijing University, Beijing, China, 1982), p. 192.

Chen, P.

P. Chen, J. Qin, “Discussion about the non-diffraction and super-diffraction,” Appl. Lasers (China) 6, 274–276 (1994).

Dauger, D. E.

D. E. Dauger, “Simulation and study of Fresnel diffraction for arbitrary two-dimensional apertures,” Comput. Phys. 10, 591–604 (1996).
[CrossRef]

de Jong, C.

C. de Jong, “Precise mathematical description of the diffraction pattern beyond a square aperture in an absorbing medium,” Opt. Eng. 30, 641–645 (1991).
[CrossRef]

Durnin, J.

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Gu, J.

J. Gu, “Discussion on π jump at boundary in diffraction phenomena,” Appl. Lasers (China) 6, 270–272 (1994).

Hafizi, B.

P. Sprangle, B. Hafizi, “Comment on nondiffracting beams,” Phys. Rev. Lett. 66, 837–838 (1991).
[CrossRef] [PubMed]

Haman, H.

Hrynevych, M.

Jiang, Z.

Z. Jiang, Q. Lu, Z. Liu, “Several questions related to diffraction,” Appl. Lasers (China) 6, 266–270 (1994).

Kenney, C. S.

Kraus, H. G.

Liang, Q.

Q. Liang, Physical Optics (Publishing House of the Mechanics Industry, Beijing, China, 1986), p. 167.

Liu, Z.

Z. Jiang, Q. Lu, Z. Liu, “Several questions related to diffraction,” Appl. Lasers (China) 6, 266–270 (1994).

Lu, Q.

Z. Jiang, Q. Lu, Z. Liu, “Several questions related to diffraction,” Appl. Lasers (China) 6, 266–270 (1994).

Miceli, J. J.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Overfelt, P. L.

Qin, J.

P. Chen, J. Qin, “Discussion about the non-diffraction and super-diffraction,” Appl. Lasers (China) 6, 274–276 (1994).

Ruschin, S.

Rutt, H. N.

Schultheisz, Carl R.

Sheppard, C. J. R.

Southwell, W. H.

Sprangle, P.

P. Sprangle, B. Hafizi, “Comment on nondiffracting beams,” Phys. Rev. Lett. 66, 837–838 (1991).
[CrossRef] [PubMed]

Steane, A. M.

Wang, S.

S. Wang, “On principles of diffraction,” Optik (Stuttgart) 100, 143–144 (1995).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 8.

Zhao, K.

K. Zhao, X. Chan, Optics (Publishing House of Beijing University, Beijing, China, 1982), p. 192.

Appl. Lasers (China) (3)

Z. Jiang, Q. Lu, Z. Liu, “Several questions related to diffraction,” Appl. Lasers (China) 6, 266–270 (1994).

J. Gu, “Discussion on π jump at boundary in diffraction phenomena,” Appl. Lasers (China) 6, 270–272 (1994).

P. Chen, J. Qin, “Discussion about the non-diffraction and super-diffraction,” Appl. Lasers (China) 6, 274–276 (1994).

Comput. Phys. (1)

D. E. Dauger, “Simulation and study of Fresnel diffraction for arbitrary two-dimensional apertures,” Comput. Phys. 10, 591–604 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (8)

Opt. Eng. (1)

C. de Jong, “Precise mathematical description of the diffraction pattern beyond a square aperture in an absorbing medium,” Opt. Eng. 30, 641–645 (1991).
[CrossRef]

Optik (Stuttgart) (1)

S. Wang, “On principles of diffraction,” Optik (Stuttgart) 100, 143–144 (1995).

Phys. Rev. Lett. (2)

P. Sprangle, B. Hafizi, “Comment on nondiffracting beams,” Phys. Rev. Lett. 66, 837–838 (1991).
[CrossRef] [PubMed]

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Other (3)

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 8.

Q. Liang, Physical Optics (Publishing House of the Mechanics Industry, Beijing, China, 1986), p. 167.

K. Zhao, X. Chan, Optics (Publishing House of Beijing University, Beijing, China, 1982), p. 192.

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Figures (5)

Fig. 1
Fig. 1

Divergent-wave Fresnel diffraction.

Fig. 2
Fig. 2

Convergent-wave Fresnel diffraction.

Fig. 3
Fig. 3

Intensity distribution of Fresnel diffraction.

Fig. 4
Fig. 4

Intensity distribution of Fraunhofer diffraction.

Fig. 5
Fig. 5

Diagrammatic sketch of M value.

Equations (33)

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E˜(p)=-iλf(θ, θ0)E˜ exp(ikr)rdσ,
f(θ, θ0)=12(cos θ+cos θ0),
E˜=A exp(ikR)R.
E˜(p)=-iλE˜ exp(ikr)rdσ.
rb+R+b2Rbq2+ρ22b-qρbcos α.
M=R+bR.
r=b+M2bq2+ρ22b-qρbcos α.
dσ=R2 sin ududα.
dσqdqdu.
E˜(p)=-iλE˜ expikb+ρ22b0a×expik M2bq2qdq02π exp-ik qpbcos αdα.
E˜(p)=E˜0 expik 12b(ρ2+Ma2)×n=1-i MaρnJnk qab,
E˜0=A exp[ik(R+b)]R+b
N=a2λb,
E˜(p)=E˜0 expiMNπ1+ρ2Ma2×n=1-iM aρnJn2Nπ ρa.
u=expx2t-1t=n=-Jn(x)tn.
x=2Nπ ρa,t=-iM aρ.
u=J02Nπ ρa+n=1-iM aρnJn2Nπ ρa+n=1-i ρManJn2Nπ ρa.
p=n=1-i ρManJn2Nπ ρan=1-iM aρnJn2Nπ ρa=expρ2M2a2-1e-1.
u=J02Nπ ρa+(1+p)n=1-iM aρnJn2Nπ ρa.
u=expNπ ρa-iM aρ--iM aρ-1=exp-iMNπ1+ρ2M2a2.
E˜(p)=DE˜0expiNπ ρ2a21-1M-expiMNπ1+ρ2Ma2J02Nπ ρa,
D=11+p=e-1e+expρ2M2a2-2,
I(p)=D2I01-2J02Nπ ρacos1+ρ2M2a2MNπ+J022Nπ ρa.
I(p0)=4I0 sin2MN2π=4I0 sin2R+b2Rba2 πλ,
a=mRbR+bλ(brighthalf-periodzone),
a=2mRbR+bλ(darkwhole-periodzone),
E˜=A exp(-ikR)R; E˜0=A exp[-ik(R-b)]R-b;
M=R-bR.
I(p)=D2I01-2J02Nπ ρacos1+ρ2a2Nπ+J022Nπ ρa.
E˜(p)=-iλE˜ expikb+ρ22b0aqdq02π×exp-ik qρbcos αdα.
E˜(p)=πa2E˜02J12Nπ ρa2Nπ ρa=πa2E02J1(x)x.
I(p)=I02J1(x)x2,
I0=π2a4(E˜0E˜0*).

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