Abstract

The Einstein–Podolsky–Rosen entangled state representation is applied to studying the admissibility condition of mother wavelets for complex wavelet transforms, which leads to a family of new mother wavelets. Mother wavelets thus are classified as the Hermite–Gaussian type for real wavelet transforms and the Laguerre–Gaussian type for the complex case.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. A. Pinsky, Introduction to Fourier Analysis and Wavelets (Brooks/Cole, 2002).
  2. I. Daubechies, in CBMS-NSF Series in Applied Mathematics (SIAM, 1992).
  3. H.-Y. Fan and H.-L. Lu, Opt. Lett. 31, 407 (2006).
    [Crossref] [PubMed]
  4. A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935).
    [Crossref]
  5. H.-Y. Fan and J. R. Klauder, Phys. Rev. A 49, 704 (1994).
    [Crossref]
  6. H.-Y. Fan and Y. Fan, Phys. Rev. A 54, 958 (1996).
    [Crossref]
  7. H.-Y. Fan, H. R. Zaidi, and J. R. Klauder, Phys. Rev. D 35, 1831 (1987).
    [Crossref]
  8. H.-Y. Fan, J. Opt. B Quantum Semiclassical Opt. 5, R147 (2003).
    [Crossref]
  9. H.-Y. Fan, H.-L. Lu, and Y. Fan, Ann. Phys. 321, 480 (2006).
    [Crossref]
  10. R. J. Glauber, Phys. Rev. 130, 2529 (1963).
    [Crossref]
  11. J. R. Klauder and B. S. Skargerstam, Coherent States (World Scientific, 1985).
  12. A. Erdelyi, Higher Transcendental Functions, The Bateman Manuscript Project (McGraw-Hill, 1953).
  13. H.-Y. Fan, Phys. Lett. A 303, 311 (2002).
    [Crossref]
  14. R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987).
    [Crossref]
  15. D. F. Walls, Nature 306, 141 (1983).
    [Crossref]
  16. H. P. Yuen, Phys. Rev. A 13, 2226 (1976).
    [Crossref]

2006 (2)

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

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

2003 (1)

H.-Y. Fan, J. Opt. B Quantum Semiclassical Opt. 5, R147 (2003).
[Crossref]

2002 (1)

H.-Y. Fan, Phys. Lett. A 303, 311 (2002).
[Crossref]

1996 (1)

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

1994 (1)

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

1987 (2)

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

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

1983 (1)

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

1976 (1)

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

1963 (1)

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

1935 (1)

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

Daubechies, I.

I. Daubechies, in CBMS-NSF Series in Applied Mathematics (SIAM, 1992).

Einstein, A.

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

Erdelyi, A.

A. Erdelyi, Higher Transcendental Functions, The Bateman Manuscript Project (McGraw-Hill, 1953).

Fan, H.-Y.

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

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

H.-Y. Fan, J. Opt. B Quantum Semiclassical Opt. 5, R147 (2003).
[Crossref]

H.-Y. Fan, Phys. Lett. A 303, 311 (2002).
[Crossref]

H.-Y. Fan and Y. Fan, Phys. Rev. A 54, 958 (1996).
[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.-L. Lu, and Y. Fan, Ann. Phys. 321, 480 (2006).
[Crossref]

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

Glauber, R. J.

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

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]

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.

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

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

Pinsky, M. A.

M. A. Pinsky, Introduction to Fourier Analysis and Wavelets (Brooks/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]

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]

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]

Ann. Phys. (1)

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

J. Mod. Opt. (1)

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

J. Opt. B Quantum Semiclassical Opt. (1)

H.-Y. Fan, J. Opt. B Quantum Semiclassical Opt. 5, R147 (2003).
[Crossref]

Nature (1)

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

Opt. Lett. (1)

Phys. Lett. A (1)

H.-Y. Fan, Phys. Lett. A 303, 311 (2002).
[Crossref]

Phys. Rev. (2)

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 (3)

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]

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

Phys. Rev. D (1)

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

Other (4)

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

A. Erdelyi, Higher Transcendental Functions, The Bateman Manuscript Project (McGraw-Hill, 1953).

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

I. Daubechies, in CBMS-NSF Series in Applied Mathematics (SIAM, 1992).

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.


Figures (4)

Fig. 1
Fig. 1

Laguerre–Gaussian mother wavelet ϕ 1 ( η ) .

Fig. 2
Fig. 2

2D Mexican hat mother wavelet (Hermite–Gaussian mother wavelet).

Fig. 3
Fig. 3

Laguerre–Gaussian mother wavelet ϕ 2 ( η ) .

Fig. 4
Fig. 4

Laguerre–Gaussian mother wavelet ϕ 3 ( η ) .

Equations (30)

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

ψ = n = 0 g 2 n a 2 n 0 ,
ψ ( x ) = π 1 4 n = 0 2 n g 2 n H 2 n ( x ) e x 2 2 ,
n = 0 ( 2 n ) ! n ! 2 n g 2 n = n = 0 ( 2 n 1 ) ! ! g 2 n = 0 .
d 2 η 2 π g ( η ) e ( η ξ * η * ξ ) 2 = G ( ξ ) ,
η = η 1 + i η 2 , ξ = ξ 1 + i ξ 2 ,
W ϕ g ( μ , κ ) = 1 μ d 2 η π g ( η ) ϕ * ( η κ μ ) , d 2 η = d η 1 d η 2 ,
d 2 η 2 π ϕ ( η ) = 0 .
η = exp [ 1 2 η 2 + η a 1 η * a 2 + a 1 a 2 ] 00 ,
η = η 1 + i η 2 ,
( X 1 X 2 ) η = 2 η 1 η , ( P 1 + P 2 ) η = 2 η 2 η .
[ ( X 1 X 2 ) , ( P 1 P 2 ) ] = 2 i , [ ( X 1 + X 2 ) , ( P 1 + P 2 ) ] = 2 i ,
ξ = exp [ 1 2 ξ 2 + ξ a 1 + ξ * a 2 a 1 a 2 ] 00 ,
d 2 η 2 π η ϕ = 0 .
d 2 η 2 π η = exp { a 1 a 2 } 00 = ξ = 0 ,
ξ = 0 ϕ = 0 .
ϕ = n , m = 0 K n , m a 1 n a 2 m 00 .
ξ = 0 ϕ = ξ = 0 d 2 z 1 d 2 z 2 π 2 z 1 , z 2 z 1 , z 2 n , m = 0 K n , m a 1 n a 2 m 00 = n , m = 0 K n , m d 2 z 1 d 2 z 2 π 2 z 1 * n z 2 * m e z 1 2 z 2 2 z 1 z 2 = n = 0 n ! K n , n ( 1 ) n = 0 .
ϕ = n = 0 n ! K n , n n , n .
H m , n ( η , η * ) = l = 0 min ( m , n ) m ! n ! l ! ( m l ) ! ( n l ) ! ( ) l η m l η * n l ,
m , n = 0 t m t n m ! n ! H m , n ( η , η * ) = exp ( t t + t η + t η * ) ,
η = e ( 1 2 ) η 2 m , n = 0 H m , n ( η , η * ) ( ) n m ! n ! m , n ,
ϕ ( η ) = η ϕ = e ( 1 2 ) η 2 n = 0 K n , n H n , n * ( η , η * ) ( ) n ,
H n , n * ( η , η * ) = H n , n ( η , η ) = n ! ( ) n L n ( η 2 ) ,
ϕ 1 = 1 2 ( 1 + a 1 a 2 ) 00 ,
ϕ 1 ( η ) 1 2 η ( 00 + 11 ) = e ( 1 2 ) η 2 ( 1 1 2 η 2 ) ,
ϕ 2 ( η ) ( 6 7 η 2 + η 4 ) e ( 1 2 ) η 2 .
ϕ 3 ( η ) = ( 14 31 η 2 + 12 η 4 η 6 ) e ( 1 2 ) η 2 .
W ϕ g ( μ , κ ) = d 2 η μ π ϕ η κ μ η g = ϕ U ( μ , κ ) g ,
U ( μ , κ ) d 2 η μ π η κ μ η
U ( μ , κ ) = sech λ : exp { ( a 1 a 2 a 1 a 2 ) tanh λ + ( sech λ 1 ) ( a 1 a 1 + a 2 a 2 ) 1 2 κ a 1 sech λ + 1 2 κ * a 2 sech λ + 1 1 + μ 2 ( κ * a 1 κ a 2 ) κ 2 2 ( 1 + μ 2 ) } : ,

Metrics