Abstract

The concept of the Fresnel number is generalized to make it possible to describe the regular polygon and slit illuminated by a homogeneous plane wave. The generalization is based on nonlinear regression of the axial intensity distribution curve. A useful analytical expression for the Fresnel number is presented. A simple experiment to show the different Fresnel numbers in one observation plane is illustrated.

© 2000 Optical Society of America

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References

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  1. S. Wang, D. Zhao, Principles of Matrix Optics (Hangzhou University, Hangzhou, China, 1994).
  2. A. J. Campillo, J. E. Pearson, S. L. Shapire, N. J. Terrell, “Fresnel diffraction effects in the design of high-power laser systems,” Appl. Phys. Lett. 23, 85–87 (1973).
    [CrossRef]
  3. D. Fan, “Fresnel number in terms of ray transfer matrix elements,” Opt. Acta 3, 319–325 (1983).
  4. S. Wang, E. Bernabeu, J. Alda, “Unified and generalized Fresnel numbers,” Opt. Quantum Electron. 24, 1351–1358 (1992).
    [CrossRef]
  5. S. Wang, Q. Lin, E. Bernabeu, J. Alda, Introduction to Apertured Optics (Hangzhou University, Hangzhou, China, 1991).
  6. Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–214 (1981).
    [CrossRef]
  7. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
  8. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

1992 (1)

S. Wang, E. Bernabeu, J. Alda, “Unified and generalized Fresnel numbers,” Opt. Quantum Electron. 24, 1351–1358 (1992).
[CrossRef]

1983 (1)

D. Fan, “Fresnel number in terms of ray transfer matrix elements,” Opt. Acta 3, 319–325 (1983).

1981 (1)

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–214 (1981).
[CrossRef]

1973 (1)

A. J. Campillo, J. E. Pearson, S. L. Shapire, N. J. Terrell, “Fresnel diffraction effects in the design of high-power laser systems,” Appl. Phys. Lett. 23, 85–87 (1973).
[CrossRef]

Alda, J.

S. Wang, E. Bernabeu, J. Alda, “Unified and generalized Fresnel numbers,” Opt. Quantum Electron. 24, 1351–1358 (1992).
[CrossRef]

S. Wang, Q. Lin, E. Bernabeu, J. Alda, Introduction to Apertured Optics (Hangzhou University, Hangzhou, China, 1991).

Bernabeu, E.

S. Wang, E. Bernabeu, J. Alda, “Unified and generalized Fresnel numbers,” Opt. Quantum Electron. 24, 1351–1358 (1992).
[CrossRef]

S. Wang, Q. Lin, E. Bernabeu, J. Alda, Introduction to Apertured Optics (Hangzhou University, Hangzhou, China, 1991).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Campillo, A. J.

A. J. Campillo, J. E. Pearson, S. L. Shapire, N. J. Terrell, “Fresnel diffraction effects in the design of high-power laser systems,” Appl. Phys. Lett. 23, 85–87 (1973).
[CrossRef]

Fan, D.

D. Fan, “Fresnel number in terms of ray transfer matrix elements,” Opt. Acta 3, 319–325 (1983).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Li, Y.

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–214 (1981).
[CrossRef]

Lin, Q.

S. Wang, Q. Lin, E. Bernabeu, J. Alda, Introduction to Apertured Optics (Hangzhou University, Hangzhou, China, 1991).

Pearson, J. E.

A. J. Campillo, J. E. Pearson, S. L. Shapire, N. J. Terrell, “Fresnel diffraction effects in the design of high-power laser systems,” Appl. Phys. Lett. 23, 85–87 (1973).
[CrossRef]

Shapire, S. L.

A. J. Campillo, J. E. Pearson, S. L. Shapire, N. J. Terrell, “Fresnel diffraction effects in the design of high-power laser systems,” Appl. Phys. Lett. 23, 85–87 (1973).
[CrossRef]

Terrell, N. J.

A. J. Campillo, J. E. Pearson, S. L. Shapire, N. J. Terrell, “Fresnel diffraction effects in the design of high-power laser systems,” Appl. Phys. Lett. 23, 85–87 (1973).
[CrossRef]

Wang, S.

S. Wang, E. Bernabeu, J. Alda, “Unified and generalized Fresnel numbers,” Opt. Quantum Electron. 24, 1351–1358 (1992).
[CrossRef]

S. Wang, D. Zhao, Principles of Matrix Optics (Hangzhou University, Hangzhou, China, 1994).

S. Wang, Q. Lin, E. Bernabeu, J. Alda, Introduction to Apertured Optics (Hangzhou University, Hangzhou, China, 1991).

Wolf, E.

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–214 (1981).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Zhao, D.

S. Wang, D. Zhao, Principles of Matrix Optics (Hangzhou University, Hangzhou, China, 1994).

Appl. Phys. Lett. (1)

A. J. Campillo, J. E. Pearson, S. L. Shapire, N. J. Terrell, “Fresnel diffraction effects in the design of high-power laser systems,” Appl. Phys. Lett. 23, 85–87 (1973).
[CrossRef]

Opt. Acta (1)

D. Fan, “Fresnel number in terms of ray transfer matrix elements,” Opt. Acta 3, 319–325 (1983).

Opt. Commun. (1)

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–214 (1981).
[CrossRef]

Opt. Quantum Electron. (1)

S. Wang, E. Bernabeu, J. Alda, “Unified and generalized Fresnel numbers,” Opt. Quantum Electron. 24, 1351–1358 (1992).
[CrossRef]

Other (4)

S. Wang, Q. Lin, E. Bernabeu, J. Alda, Introduction to Apertured Optics (Hangzhou University, Hangzhou, China, 1991).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

S. Wang, D. Zhao, Principles of Matrix Optics (Hangzhou University, Hangzhou, China, 1994).

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Figures (6)

Fig. 1
Fig. 1

Schematic of Fresnel diffraction for polygon.

Fig. 2
Fig. 2

Axial intensity of uniform plane wave diffracted by circular and square apertures.

Fig. 3
Fig. 3

Relation between axial distance l and Fresnel number N for rectangular aperture.

Fig. 4
Fig. 4

Axial intensity distributions of uniform plane wave diffracted by regular triangle, regular hexagon, and regular octagon apertures.

Fig. 5
Fig. 5

Schematic of experimental setup.

Fig. 6
Fig. 6

Diffraction pattern for Fresnel number varying from ∞ to 0 formed by cuneiform aperture.

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

E2x2, y2=1iλlexpikl  E1x1, y1expik/2l×x1-x22+y1-y22dx1dy1,
E20, 0=1iλlexpikl  E1x1, y1×expik/2lx12+y12dx1dy1.
N=a2λl+13.80.
N=a2λl+1fm,
fm=0.30618m2-0.10533m-0.68095.
Esq=1iλlexpikl-aa-aaexpik/2lx12+y12dx1dy1=1iλl1/2 expikl/2-aaexpik/2lx12dx12=Esl2,

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