Abstract

The symplectic wavelet transform, which is related to the quantum optical Fresnel transform, is developed to the symplectic-dilation mixed wavelet transform (SDWT). The SDWT involves both a real-variable dilation-transform and a complex-variable symplectic transform and possesses well-behaved properties such as the Parseval theorem and the inversion formula. The entangled-coherent state representation not only underlies the SDWT but also helps to derive the corresponding quantum transform operator whose counterpart in classical optics is the lens-Fresnel mixed transform.

© 2009 Optical Society of America

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  1. S. Jaffard, Y. Meyer, and R. D. Ryan, Wavelets, Tools for Science & Technology (Society for Industrial and Applied Mathematics, 2001).
  2. C. K. Chui, Introduction to Wavelets (Academic, 1992).
  3. I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Series in Applied Mathematics (Society for Industrial and Applied Mathematics, 1992).
    [CrossRef]
  4. M. A. Pinsky, Introduction to Fourier Analysis and Wavelets (Book/Cole, 2002).
  5. H.-Y. Fan and H.-L. Lu, Opt. Lett. 31, 407 (2006).
    [CrossRef] [PubMed]
  6. H.-Y. Fan and H.-L. Lu, Opt. Lett. 31, 3432 (2006).
    [CrossRef] [PubMed]
  7. R. J. Glauber, Phys. Rev. 130, 2529 (1963).
    [CrossRef]
  8. J. R. Klauder and B. S. Skargerstam, Coherent States (World Scientific, 1985).
  9. H.-Y. Fan and H.-L. Lu, Opt. Commun. 258, 51 (2006).
    [CrossRef]
  10. D. F. V. James and G. S. Agarwal, Opt. Commun. 126, 207 (1996).
    [CrossRef]
  11. H.-Y. Fan and H.-L. Lu, J. Phys. A 37, 10993 (2004).
    [CrossRef]
  12. H.-Y. Fan, X.-B. Tang, and H.-L. Lu, Phys. Lett. A 357, 163 (2006).
    [CrossRef]
  13. H.-Y. Fan and X.-B. Tang, J. Opt. B 7, S765 (2005).
    [CrossRef]
  14. H.-Y. Fan, H.-L. Lu, and Y. Fan, Ann. Phys. (N.Y.) 321, 480 (2006).
    [CrossRef]
  15. A. Wünsche, J. Opt. B 1, R11 (1999).
    [CrossRef]
  16. H.-Y. Fan and J. R. Klauder, Phys. Rev. A 49, 704 (1994).
    [CrossRef]
  17. A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935).
    [CrossRef]
  18. H.-Y. Fan and H.-L. Lu, Opt. Lett. 32, 554 (2007).
    [CrossRef] [PubMed]
  19. H.-Y. Fan and S.-G. Liu, Opt. Lett. 32, 1507 (2007).
    [CrossRef] [PubMed]

2007

2006

H.-Y. Fan, X.-B. Tang, and H.-L. Lu, Phys. Lett. A 357, 163 (2006).
[CrossRef]

H.-Y. Fan and H.-L. Lu, Opt. Lett. 31, 407 (2006).
[CrossRef] [PubMed]

H.-Y. Fan and H.-L. Lu, Opt. Lett. 31, 3432 (2006).
[CrossRef] [PubMed]

H.-Y. Fan and H.-L. Lu, Opt. Commun. 258, 51 (2006).
[CrossRef]

H.-Y. Fan, H.-L. Lu, and Y. Fan, Ann. Phys. (N.Y.) 321, 480 (2006).
[CrossRef]

2005

H.-Y. Fan and X.-B. Tang, J. Opt. B 7, S765 (2005).
[CrossRef]

2004

H.-Y. Fan and H.-L. Lu, J. Phys. A 37, 10993 (2004).
[CrossRef]

1999

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

1996

D. F. V. James and G. S. Agarwal, Opt. Commun. 126, 207 (1996).
[CrossRef]

1994

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

1963

R. J. Glauber, Phys. Rev. 130, 2529 (1963).
[CrossRef]

1935

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

Agarwal, G. S.

D. F. V. James and G. S. Agarwal, Opt. Commun. 126, 207 (1996).
[CrossRef]

Chui, C. K.

C. K. Chui, Introduction to Wavelets (Academic, 1992).

Daubechies, I.

I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Series in Applied Mathematics (Society for Industrial and Applied Mathematics, 1992).
[CrossRef]

Einstein, A.

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

Fan, H.-Y.

H.-Y. Fan and H.-L. Lu, Opt. Lett. 32, 554 (2007).
[CrossRef] [PubMed]

H.-Y. Fan and S.-G. Liu, Opt. Lett. 32, 1507 (2007).
[CrossRef] [PubMed]

H.-Y. Fan, H.-L. Lu, and Y. Fan, Ann. Phys. (N.Y.) 321, 480 (2006).
[CrossRef]

H.-Y. Fan and H.-L. Lu, Opt. Lett. 31, 407 (2006).
[CrossRef] [PubMed]

H.-Y. Fan and H.-L. Lu, Opt. Lett. 31, 3432 (2006).
[CrossRef] [PubMed]

H.-Y. Fan, X.-B. Tang, and H.-L. Lu, Phys. Lett. A 357, 163 (2006).
[CrossRef]

H.-Y. Fan and H.-L. Lu, Opt. Commun. 258, 51 (2006).
[CrossRef]

H.-Y. Fan and X.-B. Tang, J. Opt. B 7, S765 (2005).
[CrossRef]

H.-Y. Fan and H.-L. Lu, J. Phys. A 37, 10993 (2004).
[CrossRef]

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

Fan, Y.

H.-Y. Fan, H.-L. Lu, and Y. Fan, Ann. Phys. (N.Y.) 321, 480 (2006).
[CrossRef]

Glauber, R. J.

R. J. Glauber, Phys. Rev. 130, 2529 (1963).
[CrossRef]

Jaffard, S.

S. Jaffard, Y. Meyer, and R. D. Ryan, Wavelets, Tools for Science & Technology (Society for Industrial and Applied Mathematics, 2001).

James, D. F. V.

D. F. V. James and G. S. Agarwal, Opt. Commun. 126, 207 (1996).
[CrossRef]

Klauder, J. R.

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

J. R. Klauder and B. S. Skargerstam, Coherent States (World Scientific, 1985).

Liu, S.-G.

Lu, H.-L.

H.-Y. Fan and H.-L. Lu, Opt. Lett. 32, 554 (2007).
[CrossRef] [PubMed]

H.-Y. Fan, X.-B. Tang, and H.-L. Lu, Phys. Lett. A 357, 163 (2006).
[CrossRef]

H.-Y. Fan, H.-L. Lu, and Y. Fan, Ann. Phys. (N.Y.) 321, 480 (2006).
[CrossRef]

H.-Y. Fan and H.-L. Lu, Opt. Commun. 258, 51 (2006).
[CrossRef]

H.-Y. Fan and H.-L. Lu, Opt. Lett. 31, 3432 (2006).
[CrossRef] [PubMed]

H.-Y. Fan and H.-L. Lu, Opt. Lett. 31, 407 (2006).
[CrossRef] [PubMed]

H.-Y. Fan and H.-L. Lu, J. Phys. A 37, 10993 (2004).
[CrossRef]

Meyer, Y.

S. Jaffard, Y. Meyer, and R. D. Ryan, Wavelets, Tools for Science & Technology (Society for Industrial and Applied Mathematics, 2001).

Pinsky, M. A.

M. A. Pinsky, Introduction to Fourier Analysis and Wavelets (Book/Cole, 2002).

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]

Ryan, R. D.

S. Jaffard, Y. Meyer, and R. D. Ryan, Wavelets, Tools for Science & Technology (Society for Industrial and Applied Mathematics, 2001).

Skargerstam, B. S.

J. R. Klauder and B. S. Skargerstam, Coherent States (World Scientific, 1985).

Tang, X.-B.

H.-Y. Fan, X.-B. Tang, and H.-L. Lu, Phys. Lett. A 357, 163 (2006).
[CrossRef]

H.-Y. Fan and X.-B. Tang, J. Opt. B 7, S765 (2005).
[CrossRef]

Wünsche, A.

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

Ann. Phys. (N.Y.)

H.-Y. Fan, H.-L. Lu, and Y. Fan, Ann. Phys. (N.Y.) 321, 480 (2006).
[CrossRef]

J. Opt. B

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

H.-Y. Fan and X.-B. Tang, J. Opt. B 7, S765 (2005).
[CrossRef]

J. Phys. A

H.-Y. Fan and H.-L. Lu, J. Phys. A 37, 10993 (2004).
[CrossRef]

Opt. Commun.

H.-Y. Fan and H.-L. Lu, Opt. Commun. 258, 51 (2006).
[CrossRef]

D. F. V. James and G. S. Agarwal, Opt. Commun. 126, 207 (1996).
[CrossRef]

Opt. Lett.

Phys. Lett. A

H.-Y. Fan, X.-B. Tang, and H.-L. Lu, Phys. Lett. A 357, 163 (2006).
[CrossRef]

Phys. Rev.

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

R. J. Glauber, Phys. Rev. 130, 2529 (1963).
[CrossRef]

Phys. Rev. A

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

Other

J. R. Klauder and B. S. Skargerstam, Coherent States (World Scientific, 1985).

S. Jaffard, Y. Meyer, and R. D. Ryan, Wavelets, Tools for Science & Technology (Society for Industrial and Applied Mathematics, 2001).

C. K. Chui, Introduction to Wavelets (Academic, 1992).

I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Series in Applied Mathematics (Society for Industrial and Applied Mathematics, 1992).
[CrossRef]

M. A. Pinsky, Introduction to Fourier Analysis and Wavelets (Book/Cole, 2002).

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Equations (24)

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W φ f ( a , b ) = 1 a f ( x ) φ * ( x b a ) d x ,
W ϕ f ( r , s ; κ ) = d 2 z π f ( z ) ϕ r , s * ( z κ ) ,
α , x = exp [ 1 2 x 2 1 4 α 2 + ( x + α 2 ) a 1 + ( x α 2 ) a 2 1 4 ( a 1 + a 2 ) 2 ] 00 ,
d x π d 2 α 2 π α , x α , x = 1 , d 2 α = d α 1 d α 2 ,
W ψ g ( s , r , κ ; a , b ) d x π d 2 α 2 π g ( α , x ) ψ s , r , κ ; a , b * ( α , x ) .
ψ s , r , κ ; a , b ( α , x ) = s * a ψ [ s ( α κ ) r ( α * κ * ) , x b a ] .
W ψ g ( s , r , κ ; a , b ) = ψ U ( s , r , κ ; a , b ) g ,
U ( s , r , κ ; a , b ) = s a d x π d 2 α 2 π × s α r α * , x b a α + κ , x .
W ψ g 1 = s a e κ * β κ β * i p b d x π d 2 α 2 π × ψ * ( s α r α * , x a ) e α * β α β * i p x .
W ψ g 1 = s a Φ * ( s * β * r * β , a p ) e κ * β κ β * i p b ,
Φ * ( s * β * r * β , a p ) = d x π d 2 w 2 π ψ * ( w , x ) × e w * ( s β r β * ) w ( s * β * r * β ) i a p x .
W ψ * ( W ψ g 1 ) ( α , x ) = s a g 1 ( α , x ) Φ * ( s * β * r * β , a p ) d b π d 2 κ 2 π e κ β * κ * β + i a p b ψ ( s κ r κ * , b ) = s a g 1 ( α , x ) Φ ( s * β * r * β , a p ) 2 .
d a a 2 d 2 s s 2 W ψ * ( W ψ g 1 ) ( α , x ) = g 1 ( α , x ) d a a d 2 s s Φ ( s * β * r * β , a p ) 2 ,
g 1 ( α , x ) = d a a 2 d 2 s s 2 W ψ * ( W ψ g 1 ) ( α , x ) d a a d 2 s s Φ ( s * β * r * β , a p ) 2 .
d a a d 2 s s Φ ( s * β * r * β , a p ) 2 = 1 ,
g 1 ( α , x ) = d a a 2 d 2 s s 2 W ψ * ( W ψ g 1 ) ( α , x ) .
d a d b a 2 d 2 κ d 2 s s 2 W ψ g ( s , r , κ ; a , b ) W ψ * g ( s , r , κ ; a , b ) = d x d 2 α g ( α , x ) g * ( α , x ) .
F ( β , p ) = d x 2 π d 2 α π g ( α , x ) e α β * α * β + i p x .
W ψ g ( s , r , κ ; a , b ) = s a d p 2 π d 2 β π F ( β , p ) d x π d 2 α 2 π × e α * β α β * i p x ψ * [ s ( α κ ) r ( α * κ * ) , x b a ] = s a d p 2 π d 2 β π F ( β , p ) × Φ * ( s * β * r * β , a p ) e κ * β κ β * i p b .
d b d 2 κ W ψ g ( s , r , κ ; a , b ) W ψ * g ( s , r , κ ; a , b ) = a s d p d p d 2 β d 2 β F ( β , p ) F * ( β , p ) × Φ * ( s * β * r * β , a p ) Φ ( s * β * r * β , a p ) × d b 2 π d 2 κ π 2 e κ * ( β β ) κ ( β * β * ) + i ( p p ) b = a s d p d 2 β F ( β , p ) F * ( β , p ) Φ ( s * β * r * β , a p ) 2 .
LHS of ( 18 ) = d p d 2 β F ( β , p ) F * ( β , p ) × d a a d 2 s s Φ * ( s * β * r * β , a p ) 2 = d p d 2 β F ( β , p ) F * ( β , p ) .
g ( α , x ) = d a d b π a 2 d 2 κ d 2 s 2 π s 2 W ψ g ( s , r , κ ; a , b ) ψ s , r , κ ; a , b ( α , x ) ,
η U ( s , r , 0 ; a , 0 ) η = s a d x π d 2 α 2 π η s α r α * , x a α , x η .
η U ( s , r , 0 ; a , 0 ) η = π a δ ( η 2 η 2 a ) 1 2 i π B exp [ i 2 B ( A η 1 2 2 η 1 η 1 + D η 1 2 ) ] ,

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