Abstract

The symplectic wavelet transformation proposed in Opt. Lett. 31, 3432 (2006) , which is related to the optical Fresnel transform in the quantum optics version, is developed into an entangled symplectic wavelet transformation (ESWT) after pointing out the contrast between the single-mode Fresnel operator and the entangled Fresnel operator. The ESWT possesses well-behaved properties and corresponds to the entangled Fresnel transform [Phys. Lett. A 334, 132 (2005) ].

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. Jaffard, Y. Meyer, and R. D. Ryan, Wavelets, Tools for Science & Technology (Society for Industrial and Applied Mathematics, 2001).
  2. C. S. Burrus, R. A. Gopinath, and H. T. Guo, Introduction to Wavelets and Wavelet Transformation (A Primer) (Prentice-Hall, 1998).
  3. C. K. Chiu, Introduction to Wavelets (Academic, 1992).
  4. I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Series in Applied Mathematics (Society for Industrial and Applied Mathematics, 1992).
    [CrossRef]
  5. M. A. Pinsky, Introduction to Fourier Analysis and Wavelets (Book/Cole, 2002).
  6. H.-Y. Fan and H.-L. Lu, Opt. Lett. 31, 407 (2006).
    [CrossRef] [PubMed]
  7. H.-Y. Fan and H.-L. Lu, Opt. Lett. 32, 554 (2007).
    [CrossRef] [PubMed]
  8. H.-Y. Fan and H.-L. Lu, Opt. Lett. 31, 3432 (2006).
    [CrossRef] [PubMed]
  9. H.-Y. Fan and H.-L. Lu, Phys. Lett. A 334, 132 (2005).
    [CrossRef]
  10. D. F. Walls, Nature 306, 141 (1983).
    [CrossRef]
  11. R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987).
    [CrossRef]
  12. D. F. V. James and G. S. Agarwal, Opt. Commun. 126, 207 (1996).
    [CrossRef]
  13. R. J. Glauber, Phys. Rev. 130, 2529 (1963).
    [CrossRef]
  14. R. J. Glauber, Phys. Rev. 131, 2766 (1963).
    [CrossRef]
  15. J. R. Klauder and B. S. Skargerstam, Coherent States (World Scientific, 1985).

2007

2006

2005

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

1996

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

1987

R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987).
[CrossRef]

1983

D. F. Walls, Nature 306, 141 (1983).
[CrossRef]

1963

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

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

Agarwal, G. S.

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

Burrus, C. S.

C. S. Burrus, R. A. Gopinath, and H. T. Guo, Introduction to Wavelets and Wavelet Transformation (A Primer) (Prentice-Hall, 1998).

Chiu, C. K.

C. K. Chiu, 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]

Fan, H.-Y.

Glauber, R. J.

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

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

Gopinath, R. A.

C. S. Burrus, R. A. Gopinath, and H. T. Guo, Introduction to Wavelets and Wavelet Transformation (A Primer) (Prentice-Hall, 1998).

Guo, H. T.

C. S. Burrus, R. A. Gopinath, and H. T. Guo, Introduction to Wavelets and Wavelet Transformation (A Primer) (Prentice-Hall, 1998).

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.

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

Knight, P. L.

R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987).
[CrossRef]

Loudon, R.

R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987).
[CrossRef]

Lu, H.-L.

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).

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).

Walls, D. F.

D. F. Walls, Nature 306, 141 (1983).
[CrossRef]

J. Mod. Opt.

R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987).
[CrossRef]

Nature

D. F. Walls, Nature 306, 141 (1983).
[CrossRef]

Opt. Commun.

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

Opt. Lett.

Phys. Lett. A

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

Phys. Rev.

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

R. J. Glauber, Phys. Rev. 131, 2766 (1963).
[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. S. Burrus, R. A. Gopinath, and H. T. Guo, Introduction to Wavelets and Wavelet Transformation (A Primer) (Prentice-Hall, 1998).

C. K. Chiu, 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).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Equations (31)

Equations on this page are rendered with MathJax. Learn more.

W ψ f ( a , b ) = 1 a + f ( x ) ψ * ( x b a ) d x .
W ψ f ( r , s ; k ) = d 2 z π f ( z ) ψ r , s ; k * ( z ) , d 2 z = d x d y ,
W ψ f ( r , s ; k ) = s d 2 z π f ( z ) ψ * [ s ( z k ) r ( z * k * ) ] = ψ F 1 ( r , s ; k ) f ,
F 1 ( r , s ; k ) = s d 2 z π s z r z * z + k ,
F 1 ( r , s ; k = 0 ) = exp ( r 2 s * a 2 ) exp [ ( a a + 1 2 ) ln 1 s * ] exp ( r * 2 s * a 2 ) .
F 2 ( r , s ) = s d 2 z d 2 z π 2 s z + r z * , s z + r z * z , z = exp ( r s * a 1 a 2 ) exp [ ( a 1 a 1 + a 2 a 2 + 1 ) ln 1 s * ] exp ( r * s * a 1 a 2 ) ,
W ϕ g ( r , s ; k , k ) = d 2 z d 2 z π 2 g ( z , z ) ϕ r , s ; k , k * ( z , z ) ,
ϕ r , s ; k , k ( z , z ) s * ϕ [ s ( z k ) + r ( z * k * ) , s ( z k ) + r ( z * k * ) ] ,
( z k z * k * z k z * k * ) M ( z k z * k * z k z * k * ) , M = [ s 0 0 r 0 s * r * 0 0 r s 0 r * 0 0 s * ]
J = [ 0 I I 0 ] , I = ( 1 0 0 1 ) .
g 1 ( z , z ) = exp ( z β * z * β + z γ * z * γ ) ,
W ϕ g 1 = s d 2 z d 2 z π 2 ϕ * [ s z + r z * , s z + r z * ] × exp [ ( z + k ) β * ( z * + k * ) β + ( z + k ) γ * ( z * + k * ) γ ] .
W ϕ g 1 = s exp ( k β * k * β + k γ * k * γ ) d 2 w d 2 w π 2 ϕ * ( w , w ) × exp [ w ( s * β * + r * γ ) w * ( s β + r γ * ) + w ( s * γ * + r * β ) w * ( s γ + r β * ) ] ,
W ϕ g 1 = s exp ( k β * k * β + k γ * k * γ ) Φ * ( s * β * + r * γ , s * γ * + r * β ) .
W ϕ * ( W ϕ g 1 ) ( z , z ) = s * d 2 k d 2 k π 2 { ( W ϕ g 1 ) ( r , s ; k , k ) } ϕ [ s ( z k ) + r ( z * k * ) , s ( z k ) + r ( z * k * ) ] , = s 2 Φ * ( s * β * + r * γ , s * γ * + r * β ) exp ( z β * z * β + z γ * z * γ ) d 2 v d 2 v π 2 ϕ ( v , v ) × exp [ v ( s * β * + r * γ ) + v * ( s β + r γ * ) v ( s * γ * + r * β ) + v * ( s γ + r β * ) ] ,
W ϕ * ( W ϕ g 1 ) ( z , z ) = s 2 exp ( z β * z * β + z γ * z * γ ) Φ ( s * β * + r * γ , s * γ * + r * β ) 2 .
d 2 s W ϕ * ( W ϕ g 1 ) ( z , z ) s 4 = exp ( z β * z * β + z γ * z * γ ) d 2 s Φ ( s * β * + r * γ , s * γ * + r * β ) 2 s 2 ,
g 1 ( z , z ) = d 2 s W ϕ * ( W ϕ g 1 ) ( z , z ) s 4 d 2 s Φ ( s * β * + r * γ , s * γ * + r * β ) 2 s 2 .
d 2 s Φ ( s * β * + r * γ , s * γ * + r * β ) 2 s 2 = 1 ,
g 1 ( z , z ) = d 2 s W ϕ * ( W ϕ g 1 ) ( z , z ) s 4 .
W ϕ g ( r , s ; k , k ) W ϕ * g ( r , s ; k , k ) d 2 k d 2 k d 2 s s 4 = d 2 z d 2 z g ( z , z ) g * ( z , z ) .
F ( β , γ ) = d 2 z d 2 z π 2 g ( z , z ) exp ( z β * z * β + z γ * z * γ ) ,
d 2 z d 2 z g ( α z , α * z * ; α z , α * z * ) g ( z , z ) = d 2 β d 2 γ F ( β , γ ) F ( β , γ ) exp ( α * β α β * + α * γ α γ * ) ,
d 2 z d 2 z g ( z , z ) ϕ * [ s ( z k ) + r ( z * k * ) , s ( z k ) + r ( z * k * ) ] = d 2 β d 2 γ F ( β , γ ) Φ * ( s * β * + r * γ , s * γ * + r * β ) exp ( k β * k * β + k γ * k * γ ) .
W ϕ g ( r , s ; k , k ) W ϕ * g ( r , s ; k , k ) d 2 k d 2 k = s 2 d 2 β d 2 γ d 2 β d 2 γ F ( β , γ ) Φ * ( s * β * + r * γ , s * γ * + r * β ) × F * ( β , γ ) Φ ( s * β * + r * γ , s * γ * + r * β ) δ ( β β ) δ ( β * β * ) δ ( γ γ ) δ ( γ * γ * ) = s 2 d 2 β d 2 γ F ( β , γ ) F * ( β , γ ) Φ ( s * β * + r * γ , s * γ * + r * β ) 2 .
d 2 s s 4 W ϕ g ( r , s ; k , k ) W ϕ * g ( r , s ; k , k ) d 2 k d 2 k = d 2 β d 2 γ F ( β , γ ) F * ( β , γ ) d 2 s Φ ( s * β * + r * γ , s * γ * + r * β ) 2 s 2 = d 2 z d 2 z g ( z , z ) g * ( z , z ) ,
g ( z , z ) = W ϕ g ( r , s ; k , k ) ϕ r , s ; k , k ( z , z ) d 2 k d 2 k d 2 s π 2 s 4 ,
g ( z , z ) = δ ( z u ) δ ( z * u * ) δ ( z u ) δ ( z * u * ) ,
W ϕ g ( r , s ; k , k ) = 1 π 2 ϕ r , s ; k , k * ( u , u ) .
W ϕ g ( r , s ; k , k ) = s d 2 z d 2 z π 2 ϕ M ( z k z * k * z k z * k * ) z , z g = ϕ F 2 ( r , s ; k , k ) g ,
F 2 ( r , s ; k , k ) = s d 2 z d 2 z π 2 s z + r z * , s z + r z * z + k , z + k .

Metrics