Abstract

Fractional-order Fourier transforms are adapted to the mathematical expression of Fresnel diffraction, just as the standard Fourier transform is adapted to Fraunhofer diffraction. The continuity of fractional Fourier transforms with respect to their orders corresponds to the continuity of wave propagation, and their composition is in accordance with the Huygens principle.

© 1994 Optical Society of America

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References

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  1. V. Namias, J. Inst. Math. Appl. 25, 241 (1980).
    [CrossRef]
  2. A. C. McBride, F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
    [CrossRef]
  3. H. M. Ozaktas, D. Mendlovic, Opt. Commun. 101, 163 (1993).
    [CrossRef]
  4. D. Mendlovic, H. M. Ozaktas, J. Opt. Soc. Am. A 10, 1875 (1993).
    [CrossRef]
  5. H. M. Ozaktas, D. Mendlovic, J. Opt. Soc. Am. A 10, 2522 (1993).
    [CrossRef]
  6. A. W. Lohmann, J. Opt. Soc. Am. A 10, 2181 (1993).
    [CrossRef]
  7. H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, J. Opt. Soc. Am. A 11, 547 (1994).
    [CrossRef]
  8. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 4,p. 60.

1994 (1)

1993 (4)

1987 (1)

A. C. McBride, F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[CrossRef]

1980 (1)

V. Namias, J. Inst. Math. Appl. 25, 241 (1980).
[CrossRef]

IMA J. Appl. Math. (1)

A. C. McBride, F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[CrossRef]

J. Inst. Math. Appl. (1)

V. Namias, J. Inst. Math. Appl. 25, 241 (1980).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Commun. (1)

H. M. Ozaktas, D. Mendlovic, Opt. Commun. 101, 163 (1993).
[CrossRef]

Other (1)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 4,p. 60.

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Figures (3)

Fig. 1.
Fig. 1.

Fresnel diffraction from plane A observed at distance D. Sphere S α is tangent to plane D with radius Rα = −D/sin2 α. The complex amplitude on S α is related to the field amplitude on A by a fractional Fourier transform of order α.

Fig. 2.
Fig. 2.

The field on sphere S α is observed on the plane ℬ if the object focus of the lens is at center of curvature of S α . The field amplitude on ℬ is obtained from the field amplitude on A with a fractional Fourier transform of order α.

Fig. 3.
Fig. 3.

According to the Huygens principle, the field transfer from A to S can be separated into the field transfer from A to S followed by the field transfer from S to S . The property holds true for the corresponding fractional Fourier transforms.

Equations (16)

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α [ f ] ( s ) = i exp ( i α ) 2 π sin α exp ( i 2 s 2 cot α ) × + exp ( i 2 r 2 cot α ) × exp ( i r s sin α ) f ( r ) d r ,
U D ( s ) = i λ D exp ( i π λ D s 2 ) + exp ( i π λ D r 2 ) × exp ( 2 i π λ D r s ) U A ( r ) d r .
V D ( σ ) = i 2 π K 2 exp ( i σ 2 2 L 2 ) + exp ( i ρ 2 2 K 2 ) × exp ( i ρ σ K L ) V A ( ρ ) d ρ .
V D ( σ ) = exp ( i α ) cos α exp ( i 2 σ 2 tan α ) α [ V A ] ( σ ) .
V S ( σ ) = exp ( i α ) cos α α [ V A ] ( σ ) .
U S ( s ) = i λ D exp [ i π λ ( 1 R + 1 D ) s 2 ] × + exp ( i π λ D r 2 ) × exp ( 2 i π λ D r s ) U A ( r ) d r .
U S ( s ) = i λ D exp [ i π λ ( 1 R + 1 D ) s 2 ] × + exp ( i π λ D r 2 ) × exp ( 2 i π λ D r s ) U A ( r ) d r .
V S ( σ ) = exp ( i α ) cos α α [ V A ] ( σ ) ,
U S ( s ) = i λ Δ exp [ i π λ ( 1 R + 1 Δ ) s 2 ] × + exp [ i π λ ( 1 Δ 1 R s 2 ) ] × exp ( 2 i π λ Δ s s ) U S ( s ) d s .
D tan α = D tan α .
i λ Δ d s = i D 2 π Δ L 2 d σ = i cos α 2 π cos α sin β d σ .
π λ ( 1 Δ 1 R ) s 2 = ½ σ 2 cot β , π λ ( 1 R + 1 Δ ) s 2 = ½ σ 2 cot β .
V S ( σ ) = cos α cos α exp ( i β ) β [ V S ] ( σ ) .
W ( θ ) = + exp ( 2 i π λ r θ ) U A ( r ) d r ,
W ( θ ) = λ 2 2 π π / 2 [ W A ] ( θ ) ,
W D ( θ ) = i λ D exp ( i α ) cos α α [ V A ] [ ( 2 π D / λ ) 1 / 2 L θ ] .

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