Abstract

Using the entangled-state method in quantum mechanics, we find that the eigenmodes of the fractional Hankel transform are two-variable Hermite–Gaussian functions that can be rewritten in a clearer form as Laguerre polynominals.

© 2003 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2002

H. Fan and M. Sun, J. Opt. B 4, 228 (2002).
[CrossRef]

1998

1996

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

1995

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

1994

1993

1980

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

1963

R. J. Glauber, Phys. Rev. 131, 2766 (1963).
[CrossRef]

1935

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

Chen, M.

Einstein, A.

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

Erdelyi, A.

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

Fan, H.

H. Fan and M. Sun, J. Opt. B 4, 228 (2002).
[CrossRef]

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

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

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

Fan, Y.

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

Glauber, R. J.

R. J. Glauber, Phys. Rev. 131, 2766 (1963).
[CrossRef]

Huang, M.

Huang, W.

Klauder, J. R.

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

Lohmann, A. W.

Louisell, W. H.

W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973).

Lu, Y. Y.

Mendlovic, D.

Namias, V.

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

Ozakatas, H. M.

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]

Sun, M.

H. Fan and M. Sun, J. Opt. B 4, 228 (2002).
[CrossRef]

Ye, X.

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

Yu, L.

Zeng, X. M.

Zhu, Z.

Appl. Opt.

J. Inst. Math. Its Appl.

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

J. Opt. B

H. Fan and M. Sun, J. Opt. B 4, 228 (2002).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Lett.

Phys. Rev.

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

R. J. Glauber, Phys. Rev. 131, 2766 (1963).
[CrossRef]

Phys. Rev. A

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

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

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

Other

W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973).

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

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Equations (31)

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HmαR=0r1dr1Rr1KαHr1,r2,m,
KαHr1,r2,m=expiπ1+m1-α/2sinπ2α×exp-ir12+r222 tanπ2αJmr1r2sinπ2α,
Jmx=12π02πdθ expix cos θ+imθ-π2.
|n=exp-η22+ηa1-η*a2+a1a2|00,|ξ=exp-ξ22+ξa1+ξ*a2-a1a2|00,
d2ηπ|ηη=1, d2η=dη1dη2,d2ξπ|ξξ|=1, d2ξ=dξ1dξ2.
X1-X2|η=2η1|η, P1+P2|η=2η2|η, X1+X2|ξ=2ξ1|ξ, P1-P2|ξ=2ξ2|ξ,
Xi=ai+ai/2, Pi=ai-ai/i2.
ξ|η=12expξ*η-ξη*2
|η=d2ξπ|ξξ|η=d2ξ2πexpξ*η-ξη*2|ξ,
Kexpi1-απ22 sinπ2α×exp-iη2+ξ22 tanπ2α+ξ*η-ξη*2 sinπ2α.
ηexpi1+αa1a1+a2a2π2ξ=expi1-απ22 sinπ2α×exp-iη2+ξ22 tanπ2α+ξ*η-ξη*2sinπ2α.
expλaα=:expeλ-1aa:,
d2γd2βπ2|γ,βγ,β|=1;
|γ,β=exp-γ22-β22+γa1+βa2|00,
ηexpλa1a1+a2a2ξ=d2γd2βπ2d2γd2βπ2η|γ,βγ,β:×expeλ-1a1a1+a2a2:γ,βγ,β|ξ=d2γd2βπ2d2γd2βπ2exp-η2/2+η*γ-ηβ+γβ-γ2-β2-ξ2/2×ξγ*+ξ*β*-γ*β*-γ2-β2+eλγ*γ+β*β=11+e2λexp-η2+ξ22+efη2+ξ2+ξη*-ηξ*eλ+e-λ.
Fαfξ=expi1-απ22 sinπ2αd2ξπ×exp-iη2+ξ22 tanπ2α+ξ*η-ξη*2 sinπ2α×fξ=Gη.
Fαfx1,y1=expi1-απ22 sinπ2αexp-ix22+y222 tanπ2α×dx1dy1πexp-ix12+y122 tanπ2α+ix1y2-y1x22 sinπ2α×fx1,y1.
Fα[ξ|m,n]=d2ξπηexpi1+α×a1a1+a2a2π2ξξ|m,n=ηexpi1+αa1a1+a2a2π2m,n=expi1+αm+nπ2η|m,n.
Hm,nλ,λ*=l=0minm,nm!n!l!m-l!n-l!-1lλm-lλ*n-l,
m,n=0tmtnm!n!Hm,nλ,λ*=exp-tt+tλ+tλ*.
η|=00|m,n=0im+na1ma2nm!n!Hm,n-iη*,iη×exp-η2/2,ξ|=00|m,n=0a1ma2nm!n!Hm,nξ*,ξexp-ξ2/2.
η|m,n=im+nHm,n-iη*,iηexp-η2/2/m!n!,ξ|m,n=Hm,nξ*,ξexp-ξ2/2/m!n!.
Fαexp-ξ2/2Hm,nξ*,ξ=-expiαπ2m+n×exp-η2/2Hm,n-iη*,iη,
exp-ξ2/2Hm,nξ*,ξ=exp-im-nθ×exp-r12/2Hm,nr1,r1exp-η2/2×Hm,n-iη*,iη=exp-im-nπ2+φ×exp-r22/2Hm,nr2,r2,
expi1-απ22 sinπ2αr1dr1dθπexp-ir12+r222 tanπ2α+ir1r2 sinφ-θsinπ2αexp-r122exp-im-nθ×Hm,nr1,r1=-expiαπ2m+nexp-r222×exp-im-nπ2+φHm,nr2,r2.
Jmx=12π-ππexpix sin ζ-imζdζ,
expiπ1+n-m1-α/2sinαπ/20r1dr1×exp-ir12+r222 tanπ2αJn-mr1r2sinπ2α×exp-r122Hm,nr1,r1=-1mαexp-r222Hm,nr2,r2.
Hn-mαexp-r122Hm,nr1,r1=-1mαexp-r222Hm,nr2,r2.
Hm,nr,r=p!-prlLplr2p=minm,nl=m-n,
Lnμx=l=onn+μn-l-xll!.
Hl,pαexp-r122r1lLplr12=-1αpexp-r222r2lLplr22;

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