Abstract

Starting from a complex fractional Fourier transformation [Opt. Lett. 28, 680 (2003)], it is shown that the integral kernel of a fractional Hankel transformation is equivalent to the matrix element of an appropriate operator in the charge-amplitude state representations; i.e., the fractional Hankel transformation is endowed with a definite physical meaning (definite quantum-mechanical representation transform).

© 2003 Optical Society of America

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References

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  1. V. Namias, J. Inst. Math. Appl. 25, 241 (1980).
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  2. A. W. Lohmann, J. Opt. Soc. Am. A 10, 2181 (1993).
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  3. D. Mendlovic and H. M. Ozaktas, J. Opt. Soc. Am. A 10, 1875 (1993).
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  4. A. C. McBride and F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
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  5. P. Pellat-Finet, Opt. Lett. 19, 388 (1994).
    [CrossRef]
  6. D. Mendlovic, H. M. Ozaktas, and A. W. Lohmann, Appl. Opt. 33, 6188 (1994).
    [CrossRef] [PubMed]
  7. L. Bernardo and O. D. D. Soares, Opt. Commun. 110, 517 (1994).
    [CrossRef]
  8. H.-Y. Fan and H.-L. Lu, Opt. Lett. 28, 680 (2003).
    [CrossRef] [PubMed]
  9. H.-Y. Fan and J. R. Klauder, Phys. Rev. A 49, 704 (1994).
    [CrossRef]
  10. H.-Y. Fan and Y. Fan, Phys. Rev. A 54, 958 (1996).
    [CrossRef]
  11. H.-Y. Fan and X. Ye, Phys. Rev. A 51, 3343 (1995).
    [CrossRef]
  12. A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935).
    [CrossRef]
  13. L. Yu, Y. Lu, X. Zeng, M. Huang, M. Chen, W. Huang, and Z. Zhu, Opt. Lett. 23, 1158 (1998).
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  14. A. Erdelryi, Higher Transcendental Functions, The Batemann Manuscript Project (McGraw-Hill, New York, 1953).
  15. W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd ed. (Springer-Verlag, Berlin, 1966).
    [CrossRef]

2003 (1)

1998 (1)

1996 (1)

H.-Y. Fan and Y. Fan, Phys. Rev. A 54, 958 (1996).
[CrossRef]

1995 (1)

H.-Y. Fan and X. Ye, Phys. Rev. A 51, 3343 (1995).
[CrossRef]

1994 (4)

L. Bernardo and O. D. D. Soares, Opt. Commun. 110, 517 (1994).
[CrossRef]

P. Pellat-Finet, Opt. Lett. 19, 388 (1994).
[CrossRef]

H.-Y. Fan and J. R. Klauder, Phys. Rev. A 49, 704 (1994).
[CrossRef]

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

1993 (2)

1987 (1)

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

1980 (1)

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

1966 (1)

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd ed. (Springer-Verlag, Berlin, 1966).
[CrossRef]

1953 (1)

A. Erdelryi, Higher Transcendental Functions, The Batemann Manuscript Project (McGraw-Hill, New York, 1953).

1935 (1)

A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935).
[CrossRef]

Bernardo, L.

L. Bernardo and O. D. D. Soares, Opt. Commun. 110, 517 (1994).
[CrossRef]

Chen, M.

Einstein, A.

A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935).
[CrossRef]

Erdelryi, A.

A. Erdelryi, Higher Transcendental Functions, The Batemann Manuscript Project (McGraw-Hill, New York, 1953).

Fan, H.-Y.

H.-Y. Fan and H.-L. Lu, Opt. Lett. 28, 680 (2003).
[CrossRef] [PubMed]

H.-Y. Fan and Y. Fan, Phys. Rev. A 54, 958 (1996).
[CrossRef]

H.-Y. Fan and X. Ye, Phys. Rev. A 51, 3343 (1995).
[CrossRef]

H.-Y. Fan and J. R. Klauder, Phys. Rev. A 49, 704 (1994).
[CrossRef]

Fan, Y.

H.-Y. Fan and Y. Fan, Phys. Rev. A 54, 958 (1996).
[CrossRef]

Huang, M.

Huang, W.

Kerr, F. H.

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

Klauder, J. R.

H.-Y. Fan and J. R. Klauder, Phys. Rev. A 49, 704 (1994).
[CrossRef]

Lohmann, A. W.

Lu, H.-L.

Lu, Y.

Magnus, W.

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd ed. (Springer-Verlag, Berlin, 1966).
[CrossRef]

McBride, A. C.

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

Mendlovic, D.

Namias, V.

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

Oberhettinger, F.

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd ed. (Springer-Verlag, Berlin, 1966).
[CrossRef]

Ozaktas, H. M.

Pellat-Finet, P.

P. Pellat-Finet, Opt. Lett. 19, 388 (1994).
[CrossRef]

Podolsky, B.

A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935).
[CrossRef]

Rosen, N.

A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935).
[CrossRef]

Soares, O. D. D.

L. Bernardo and O. D. D. Soares, Opt. Commun. 110, 517 (1994).
[CrossRef]

Soni, R. P.

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd ed. (Springer-Verlag, Berlin, 1966).
[CrossRef]

Ye, X.

H.-Y. Fan and X. Ye, Phys. Rev. A 51, 3343 (1995).
[CrossRef]

Yu, L.

Zeng, X.

Zhu, Z.

Appl. Opt. (1)

IMA J. Appl. Math. (1)

A. C. McBride and 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 (2)

Opt. Commun. (1)

L. Bernardo and O. D. D. Soares, Opt. Commun. 110, 517 (1994).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. A (1)

H.-Y. Fan and X. Ye, Phys. Rev. A 51, 3343 (1995).
[CrossRef]

Phys. Rev. (1)

A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935).
[CrossRef]

Phys. Rev. A (2)

H.-Y. Fan and J. R. Klauder, Phys. Rev. A 49, 704 (1994).
[CrossRef]

H.-Y. Fan and Y. Fan, Phys. Rev. A 54, 958 (1996).
[CrossRef]

Other (2)

A. Erdelryi, Higher Transcendental Functions, The Batemann Manuscript Project (McGraw-Hill, New York, 1953).

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd ed. (Springer-Verlag, Berlin, 1966).
[CrossRef]

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Equations (29)

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Fαgξ=12 sin α expiα-π/2d2ξπ×expiη2+ξ22 tan α+ξη*-ξ*η2 sin αgξGη,
Kαξ,ηexpiα-π/22 sin α×expiη2+ξ22 tan α+ξη*-ξ*η2 sin α=η|expiπ2-αa1a1+a2a2+1×|ξ,
|η=exp-12η2+ηa1-η*a2+a1a2|00,
|ξ=exp-12ξ2+ξa1+ξ*a2-a1a2|00,
d2ηπ|ηη=1,  d2ξπξξ|=1.
d2ξπη|expiπ2-αa1a1+a2a2+1×|ξξg=ηG,
|G=expiπ2-αa1a1+a2a2|g.
14π202πdθ expisθη=ηexpiθ|×expiπ/2-αa1a1+a2a2+1×02π|ξ=|ξ|expiφexp-iqφ=14π212 sin α02π02π expisθexp-iqφdφ×expiη2+ξ22 tan α+ηξ{exp[iφ-θ-exp-iφ-θ}2 sin αdθdφ12 sin α expiη2+ξ22 tan αI.
expix sint=m=-Jmxexpimt,  Jmx=k=0-1mk!m+k!x2m+2k,
I=14π202π02π expisθexp-iqφ×expirr sinφ-θsin αdθdφ=14π202π02π expisθexp-iqφm=-Jmrrsin α×expimφ-θdθdφ=m=-δm,qδm,sJmrrsin α=δs,qJsrrsin α.
12π02πdθ|ξ=r expiφexp-iqφ|q,r,
12π02πdθ expisθη=r expiθ|s,r|.
s,rexpiπ/2-αa1a1+a2a2+1q,r=δs,q12 sin α expir2+r22 tan αJsrrsin α.
s,rq,r=12δs,qJsrr.
Hm,nξ,ξ*=l=0minm,nm!n!l!n-l!m-l!×-1lξm-lξ*n-l,
exp-tt+tξ+tξ*=m,n=0tmtnm!n!Hm,nξ,ξ*.
|q,r=12π02πdφ exp-r2/2+ξa1+ξ*a2-iqφ-a1a2|00=12π exp-r2/202πdφm,n=0a1ma2nm!n!×Hm,nξ,ξ*exp-iqφ|00=12π exp-r2/202πdφm,n=01m!n!×Hm,nr,rexpiφm-n-q|m,n=exp-r2/2n=01n+q!n!×Hn+q,nr,r|n+q,n.
a1a1-a2a2|q,r=q|q,r.
a1+a2a1+a2|q,r=r2|q,r,
a1+a2a1+a2,a1a1-a2a2=0.
d2ξπ exp-ξ21m!n!m!n!Hm,nξ,ξ*×Hm,n*ξ,ξ*=δm,mδn,n,
δm-n,m-n0dr2exp-r21m!n!m!n!Hm,nr,r×Hm,nr,r=δm,mδn,n,
q=-0dr2|q,rq,r|=q=-0dr2exp-r2n,n=0×1n+q!n!n+q!n!1/2Hn+q,nr,rHn+q,nr,r×|n+q,nn+q,n|=q=-n=max0,-q|n+q,nn+q,n|=1.
s=-0dr2|s,rs,r|=1.
q,rg=gq,r,  s,rG=Gs,r,
s,r|expiπ/2-αa1a1+a2a2+1|g=q=-0dr2s,r|×expiπ/2-αa1a1+a2a2+1×|q,rq,rg.
Hαgq,r12 sin αq=-δs,q0dr2×expir2+r22 tan αJsrrsin αgq,rGs,r.
q,rg=q,r|exp-iπ/2-αa1a1+a2a2+1×|G=s=-0dr2q,r|×exp-iπ/2-αa1a1+a2a2+1×|s,rs,rG.
0dr2exp-λr2JsvrJswr=1λ exp-14λv2+w2-isπ2Jsivw2λ

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