Abstract

We find that the Collins diffraction formula in cylindrical coordinates is just the transformation matrix element of a three-parameter two-mode squeezing operator in the deduced entangled state representation. This is a new tie connecting the unitary transform in quantum optics to the generalized Hankel transform in Fourier optics. The group multiplication rule of the squeezing operators maps to the Collins formula related to two successive Hankel transforms.

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References

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2005

H.-y. Fan and H.-I. Lu, Phys. Lett. A 334, 132 (2005).
[CrossRef]

2003

H.-y. Fan, Opt. Lett. 28, 2177 (2003).
[CrossRef] [PubMed]

H.-y. Fan, J. Opt. B: Quantum Semiclassical Opt. 5, R147 (2003).
[CrossRef]

1999

H.-y. Fan, H. Zou, and Y. Fan, Phys. Lett. A 254, 137 (1999).
[CrossRef]

A. Wünsche, J. Opt. B: Quantum Semiclassical Opt. 1, R11 (1999).
[CrossRef]

1994

1993

1987

H.-y. Fan, H. R. Zaidi, and J. R. Klauder, Phys. Rev. D 35, 1831 (1987).
[CrossRef]

1986

D. F. Walls, Nature 324, 210 (1986).
[CrossRef]

1980

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

1976

H. P. Yuen, Phys. Rev. A 13, 2226 (1976).
[CrossRef]

1971

1970

S. A. Collins, J. Opt. Soc. Am. A 60, 1168 (1970).
[CrossRef]

1965

1935

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

Arnaud, J. A.

Collins, S. A.

S. A. Collins, J. Opt. Soc. Am. A 60, 1168 (1970).
[CrossRef]

Einstein, A.

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

Erderyi, A.

A. Erderyi, Higher Transcendental Functions, The Batemann Manuscript Project (McGraw Hill, 1953).

Fan, H.-y.

H.-y. Fan and H.-I. Lu, Phys. Lett. A 334, 132 (2005).
[CrossRef]

H.-y. Fan, Opt. Lett. 28, 2177 (2003).
[CrossRef] [PubMed]

H.-y. Fan, J. Opt. B: Quantum Semiclassical Opt. 5, R147 (2003).
[CrossRef]

H.-y. Fan, H. Zou, and Y. Fan, Phys. Lett. A 254, 137 (1999).
[CrossRef]

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

H.-y. Fan, H. R. Zaidi, and J. R. Klauder, Phys. Rev. D 35, 1831 (1987).
[CrossRef]

Fan, Y.

H.-y. Fan, H. Zou, and Y. Fan, Phys. Lett. A 254, 137 (1999).
[CrossRef]

Glasser, M. L.

M. L. Glasser and E. Montaldi, arXiv.org/math.CA/9307213.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Klauder, J. R.

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

H.-y. Fan, H. R. Zaidi, and J. R. Klauder, Phys. Rev. D 35, 1831 (1987).
[CrossRef]

Kogelnik, H.

Lohmann, A. W.

Lu, H.-I.

H.-y. Fan and H.-I. Lu, Phys. Lett. A 334, 132 (2005).
[CrossRef]

Mendlovic, D.

Montaldi, E.

M. L. Glasser and E. Montaldi, arXiv.org/math.CA/9307213.

Namias, V.

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

Ozakatas, H. M.

Ozaktas, 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]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986).

Walls, D. F.

D. F. Walls, Nature 324, 210 (1986).
[CrossRef]

Wünsche, A.

A. Wünsche, J. Opt. B: Quantum Semiclassical Opt. 1, R11 (1999).
[CrossRef]

Yuen, H. P.

H. P. Yuen, Phys. Rev. A 13, 2226 (1976).
[CrossRef]

Zaidi, H. R.

H.-y. Fan, H. R. Zaidi, and J. R. Klauder, Phys. Rev. D 35, 1831 (1987).
[CrossRef]

Zou, H.

H.-y. Fan, H. Zou, and Y. Fan, Phys. Lett. A 254, 137 (1999).
[CrossRef]

Appl. Opt.

J. Inst. Math. Appl.

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

J. Opt. B: Quantum Semiclassical Opt.

A. Wünsche, J. Opt. B: Quantum Semiclassical Opt. 1, R11 (1999).
[CrossRef]

H.-y. Fan, J. Opt. B: Quantum Semiclassical Opt. 5, R147 (2003).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Nature

D. F. Walls, Nature 324, 210 (1986).
[CrossRef]

Opt. Lett.

Phys. Lett. A

H.-y. Fan, H. Zou, and Y. Fan, Phys. Lett. A 254, 137 (1999).
[CrossRef]

H.-y. Fan and H.-I. Lu, Phys. Lett. A 334, 132 (2005).
[CrossRef]

Phys. Rev.

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

Phys. Rev. A

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

H. P. Yuen, Phys. Rev. A 13, 2226 (1976).
[CrossRef]

Phys. Rev. D

H.-y. Fan, H. R. Zaidi, and J. R. Klauder, Phys. Rev. D 35, 1831 (1987).
[CrossRef]

Other

A. Erderyi, Higher Transcendental Functions, The Batemann Manuscript Project (McGraw Hill, 1953).

M. L. Glasser and E. Montaldi, arXiv.org/math.CA/9307213.

A. E. Siegman, Lasers (University Science, 1986).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

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Equations (28)

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U 2 ( r 2 , φ ) = i κ 2 π B 0 r 1 d r 1 0 2 π d θ exp { κ 2 i B [ A r 1 2 + D r 2 2 2 r 1 r 2 cos ( θ φ ) ] } U 1 ( r 1 , θ ) ,
u 2 ( r 2 ) = i m + 1 B 0 exp [ 1 2 i B ( A r 1 2 + D r 2 2 ) ] J m ( r 1 r 2 B ) u 1 ( r 1 ) r 1 d r 1 ,
s , r exp [ i ( π 2 α ) ( a 1 a 1 + a 2 a 2 + 1 ) ] q , r = δ s , q 1 2 sin α exp [ i ( r 2 + r 2 ) 2 tan α ] J s ( r r sin α ) ,
s , r 1 2 π 0 2 π d θ e i s θ η = r e i θ ,
q , r 1 2 π 0 2 π d φ ξ = r e i φ e i q φ .
η = exp { 1 2 η 2 + η a 1 η * a 2 + a 1 a 2 } 00 ,
ξ = exp { 1 2 ξ 2 + ξ a 1 ξ * a 2 a 1 a 2 } 00 ;
F ( t , k ) = exp ( t k * a 1 a 2 ) exp [ ( a 1 a 1 + a 2 a 2 + 1 ) ln ( k * ) 1 ] exp ( t * k * a 1 a 2 ) ;
d 2 η 2 π η η = 1 , d 2 η d η 1 d η 2 ,
η η = π δ ( η 1 η 1 ) δ ( η 2 η 2 ) .
K ( t , k ) ( η , η ) = d 2 z 1 d 2 z 2 d 2 z 1 d 2 z 2 π 5 η z 1 , z 2 z 1 , z 2 F ( t , k ) z 1 , z 2 z 1 , z 2 η = 1 π ( t k t * + k * ) exp [ ( k t * ) η 2 + ( t k ) η 2 + η η * + η * η t k t * + k * 1 2 ( η 2 + η 2 ) ] .
k = 1 2 [ A + D + i ( B C ) ] , t = 1 2 [ A D i ( B + C ) ] ,
K ( t , k ) ( η , η ) = i 2 B π exp { 1 2 i B [ A η 2 ( η η * + η * η ) + D η 2 ] } ,
s , r = e r 2 2 n = 0 1 ( n + s ) ! n ! H n + s , n ( r , r ) n + s , n ,
N s , r = s s , r , K s , r = r 2 s , r ,
s = 0 d ( r 2 ) s , r s , r = 1 , s , r s , r = δ s , s 1 2 r δ ( r r ) .
s , r F ( t , k ) s , r = 1 4 π 2 0 2 π d θ 0 2 π d φ e i s θ e i s φ η = r e i φ F ( t , k ) η = r e i θ = i 8 B π 2 0 2 π d θ 0 2 π d φ exp ( i s φ i s θ ) exp [ 1 2 i B ( A r 2 + D r 2 ) r r i B cos ( θ φ ) ] = i s + 1 2 B exp [ 1 2 i B ( A r 2 + D r 2 ) ] δ s , s J s ( r r B ) ,
s , r ϕ ϕ s ( r ) , ψ = F ( t , k ) ϕ ,
ψ s ( r ) s , r ψ = s , r F ( t , k ) ϕ = s = 0 d ( r 2 ) s r F ( t , k ) s , r s , r ϕ = i s + 1 2 B 0 d ( r 2 ) exp [ 1 2 i B ( A r 2 + D r 2 ) ] J s ( r r B ) ϕ s ( r ) .
( k t t * k * ) = ( k t t * k * ) ( k t t * k * ) ,
s , r F ( t , k ) F ( t , k ) s , r = s = 0 d ( r 2 ) s , r F ( t , k ) s , r
× s , r F ( t , k ) s , r = s , r F ( t , k ) s , r .
0 d ( r 2 ) J s ( r r B ) J s ( r r B ) exp ( B r 2 2 i B B )
= 2 B B i B exp [ i 2 ( B r 2 B B + B r 2 B B π s ) ] J s ( r r B ) ,
u ( r , z ) 1 q ( z ) exp [ i r 2 2 q ( z ) ] ,
ϕ = 2 q ( z ) + i exp { i q ( z ) i + q ( z ) a 1 a 2 } 00 .
F ( t , k ) ϕ = 2 q ( z ) + i F ( t , k ) exp [ i q ( z ) i + q ( z ) a 1 a 2 ] 00 = 2 [ D + C q ( z ) ] [ i + q ( z ) ] exp [ i q ( z ) i + q ( z ) a 1 a 2 ] 00 ,
q ( z ) = A q ( z ) + B C q ( z ) + D .

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