Abstract

We derive the integral representation of a fractional Hankel transform (FRHT) from a fractional Fourier transform. Some basic properties of the FRHT such as Parseval’s theorem and its optical implementation are discussed qualitatively.

© 1998 Optical Society of America

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References

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  1. V. Namias, J. Inst. Math. Its Appl. 25, 241 (1980).
    [CrossRef]
  2. Y. B. Karasik, Opt. Lett. 19, 769 (1993).
    [CrossRef]
  3. A. W. Lohmann, J. Opt. Soc. Am. A 10, 2181 (1993).
    [CrossRef]
  4. D. Mendlovic and H. M. Ozaktas, J. Opt. Soc. Am. A 10, 1875 (1993).
    [CrossRef]
  5. H. M. Ozaktas and D. Mendlovic, J. Opt. Soc. Am. A 10, 2522 (1993).
    [CrossRef]
  6. T. Aleiva, V. Lopez, F. A. Lopez, and L. B. Almeida, J. Mod. Opt. 41, 1035 (1994).
  7. D. Mendlovic, H. M. Ozaktas, and A. W. Lohmann, Appl. Opt. 33, 6188 (1994).
    [CrossRef] [PubMed]
  8. R. G. Dorsch and A. W. Lohmann, Appl. Opt. 34, 4111 (1995).
    [CrossRef] [PubMed]
  9. J. Hua, L. Liu, and G. Li, Opt. Commun. 137, 11 (1997).
    [CrossRef]
  10. After writing this Letter, we were informed that the FRHT had been studied by Namias.11?Using some unusual relations, Namias derived the integral representations of the FRHT directly from the eigenvalue equation of the ordinary HT.11
  11. V. Namias, J. Inst. Math. Its Appl. 26, 187 (1980).
    [CrossRef]
  12. L. Yu, M. Huang, M. Chen, W. Chen, W. Huang, and Z. Zhu, Opt. Lett. 23, 409 (1998).
    [CrossRef]
  13. A. E. Siegman, Opt. Lett. 1, 13 (1977).
    [CrossRef]
  14. W. Magnus and F. Oberhettinger, Formulas and Theorems for the Functions of Mathematical Physics (Chelsea, New York, 1954).

1998 (1)

1997 (1)

J. Hua, L. Liu, and G. Li, Opt. Commun. 137, 11 (1997).
[CrossRef]

1995 (1)

1994 (2)

D. Mendlovic, H. M. Ozaktas, and A. W. Lohmann, Appl. Opt. 33, 6188 (1994).
[CrossRef] [PubMed]

T. Aleiva, V. Lopez, F. A. Lopez, and L. B. Almeida, J. Mod. Opt. 41, 1035 (1994).

1993 (4)

1980 (2)

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

V. Namias, J. Inst. Math. Its Appl. 26, 187 (1980).
[CrossRef]

1977 (1)

Aleiva, T.

T. Aleiva, V. Lopez, F. A. Lopez, and L. B. Almeida, J. Mod. Opt. 41, 1035 (1994).

Almeida, L. B.

T. Aleiva, V. Lopez, F. A. Lopez, and L. B. Almeida, J. Mod. Opt. 41, 1035 (1994).

Chen, M.

Chen, W.

Dorsch, R. G.

Hua, J.

J. Hua, L. Liu, and G. Li, Opt. Commun. 137, 11 (1997).
[CrossRef]

Huang, M.

Huang, W.

Karasik, Y. B.

Li, G.

J. Hua, L. Liu, and G. Li, Opt. Commun. 137, 11 (1997).
[CrossRef]

Liu, L.

J. Hua, L. Liu, and G. Li, Opt. Commun. 137, 11 (1997).
[CrossRef]

Lohmann, A. W.

Lopez, F. A.

T. Aleiva, V. Lopez, F. A. Lopez, and L. B. Almeida, J. Mod. Opt. 41, 1035 (1994).

Lopez, V.

T. Aleiva, V. Lopez, F. A. Lopez, and L. B. Almeida, J. Mod. Opt. 41, 1035 (1994).

Magnus, W.

W. Magnus and F. Oberhettinger, Formulas and Theorems for the Functions of Mathematical Physics (Chelsea, New York, 1954).

Mendlovic, D.

Namias, V.

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

V. Namias, J. Inst. Math. Its Appl. 26, 187 (1980).
[CrossRef]

Oberhettinger, F.

W. Magnus and F. Oberhettinger, Formulas and Theorems for the Functions of Mathematical Physics (Chelsea, New York, 1954).

Ozaktas, H. M.

Siegman, A. E.

Yu, L.

Zhu, Z.

Appl. Opt. (2)

J. Inst. Math. Its Appl. (2)

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

V. Namias, J. Inst. Math. Its Appl. 26, 187 (1980).
[CrossRef]

J. Mod. Opt. (1)

T. Aleiva, V. Lopez, F. A. Lopez, and L. B. Almeida, J. Mod. Opt. 41, 1035 (1994).

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

J. Hua, L. Liu, and G. Li, Opt. Commun. 137, 11 (1997).
[CrossRef]

Opt. Lett. (3)

Other (2)

W. Magnus and F. Oberhettinger, Formulas and Theorems for the Functions of Mathematical Physics (Chelsea, New York, 1954).

After writing this Letter, we were informed that the FRHT had been studied by Namias.11?Using some unusual relations, Namias derived the integral representations of the FRHT directly from the eigenvalue equation of the ordinary HT.11

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Equations (28)

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HmaHmbf=HmaHmbf=HmbHmaf=Hma+bf.
f2u1,u2=Faf1x1,x2=--dx1dx2×f1x1,x2Kax1,u1,x2,u2,
KaFx1,u1,x2,u2=expiπ1-a/22π sinaπ/2×exp-12ix12+u12+x22+u22cotaπ/2×expix1u1+x2u2sinaπ/2,
f2r2,φ=0+dr1r1 exp-12ir12+r22cotaπ/2×expiπ1-a/22π sinaπ/202πdθf1r1,θ×expisinaπ/2r1r2 cosθ-φ.
f1r1,θ=g1r1expimθ.
f2r2,φ=0+dr1r1gr1expiπ1-a/2sinaπ/2×exp-i2r12+r22cotaπ/2×expimφ+π2Jmr1r2sinaπ/2,
Jmx=12π02πdθ expix cos θ+imθ-π2.
f2r2,φ=g2r2expimφ.
g2r2=Hmag1=0+dr1r1g1r1KaHr1,r2,m,
KaHr1,r2,mexpiπ1+m1-a/2sinaπ/2×exp-12ir12+r22cotaπ/2×Jmr1r2sinaπ/2,
Hma=exp-imaπ2Fa.
K-aHr1,r2,m=KaHr1,r2,m*.
RR=0+dr2r2f2r2g2*r2.
RR=0+dr2r2f2r20+dr1r1g1*r1×KaHr1,r2,m*.
RR=0+dr1r1g1*r10+dr2r2f2r2×K-aHr1,r2,m=0+dr1r1f1r1g1*r1.
0+dr1r1f1r1g1*r1=0+dr2r2f2r2g2*r2.
Hmac1f+c2g=c1Hmaf+c2Hmag,
Hmc1a1+c2a2f=Hmc1a1Hmc2a2f=Hmc2a2Hmc1a1f.
Hmc1a1+2f=Hmc1a1f=Hmc1a1Hm2f.
K1Hf1,r2,m=Jmr1,r2.
HaHbg1=0+dr1r1g1r1s1s2,
s1=expim+11-a+im+11-b×exp-12i(r12cot a+r32 cot bsin a sin b-1,
s2=0+dr2r2 exp-12ir22cot a+cot b×Jmr1r2sin aJmr3r2sin b.
12A-2 expimJmp1exp-i2p2+p3×expim+1expim+11-a-bsina+b=Ka+bHr1,r2,m=s1s2,
Ka+bHr1,r2,m=expim+11-a-bsina+b×exp-i2r12+r22cota+b×Jmr1r3sina+b.
HaHbg1=Ha+bg1.
0+dtt exp-A2t2JmBtJmCt=12A-2 exp-14A-2B2+C2×expim2πJmi2BCA-2,
A2=12icot a+cot b,B=r1sin a, C=r3sin b, p1i2BCA-2=-r1r3sina+b-1,p2-14A-2B2+C2,p3r12 cot a+r32 cot b.

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