Abstract

The admissibility condition of a mother wavelet is explored in the context of quantum optics theory. By virtue of Dirac’s representation theory and the coherent state property we derive a general formula for finding qualified mother wavelets. A comparison between a wavelet transform computed with the newly found mother wavelet and one computed with a Mexican hat wavelet is presented.

© 2006 Optical Society of America

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References

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  1. I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, 1992).
    [Crossref]
  2. G. Kaiser, A Friendly Guide to Wavelets (Birkhäuser, 1994).
  3. C. K. Chui , An Introduction to Wavelets (Academic, 1992).
  4. P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed (Oxford U. Press, 1958).
  5. H.-Y. Fan, H. R. Zaidi, and J. R. Klauder, Phys. Rev. D 35, 1831 (1987).
    [Crossref]
  6. H. Fan and H. R. Zaidi, Phys. Rev. A 37, 2985(1988).
    [Crossref]
  7. H. Fan and J. R. Klauder, J. Phys. A 21, L715 (1988).
    [Crossref]
  8. H.-Y. Fan, J. Opt. B: Quantum Semiclassical Opt. 5, R147 (2003).
    [Crossref]
  9. A. Wünsche, J. Opt. B: Quantum Semiclassical Opt. 1, R11 (1999) .
    [Crossref]
  10. D. F. Walls, Nature 324, 210 (1986) .
    [Crossref]
  11. R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987).
    [Crossref]
  12. V. V. Dodonov, J. Opt. B: Quantum Semiclassical Opt. 4, R1 (2002) .
    [Crossref]
  13. R. J. Glauber, Phys. Rev. 130, 2529 (1963) .
    [Crossref]
  14. J. R. Klauder and B. S. Skargerstam, Coherent States (World Scientific, 1985).
  15. D. R. Scifres, R. D. Burnham, and W. Streifer, Appl. Phys. Lett. 12, 33 (1978) .
  16. E. Kapon, J. Katz, and A. Yariv, Opt. Lett. 10, 125 (1984) .
    [Crossref]
  17. M. Oka, H. Masuda, Y. Kaneda, and S. Kubota, IEEE J. Quantum Electron. 28, 1142(1992) .
    [Crossref]
  18. H.-Y. Fan and H.-L. Lu, Opt. Lett. 28, 680 (2003) .
    [Crossref] [PubMed]
  19. H.-Y. Fan, Opt. Lett. 28, 21772003) .
    [Crossref] [PubMed]
  20. H.-Y. Fan and X.-F. Xu, Opt. Lett. 29, 1048 (2004) .
    [Crossref] [PubMed]

2004 (1)

2003 (3)

2002 (1)

V. V. Dodonov, J. Opt. B: Quantum Semiclassical Opt. 4, R1 (2002) .
[Crossref]

1999 (1)

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

1992 (1)

M. Oka, H. Masuda, Y. Kaneda, and S. Kubota, IEEE J. Quantum Electron. 28, 1142(1992) .
[Crossref]

1988 (2)

H. Fan and H. R. Zaidi, Phys. Rev. A 37, 2985(1988).
[Crossref]

H. Fan and J. R. Klauder, J. Phys. A 21, L715 (1988).
[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]

1986 (1)

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

1984 (1)

1978 (1)

D. R. Scifres, R. D. Burnham, and W. Streifer, Appl. Phys. Lett. 12, 33 (1978) .

1963 (1)

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

Burnham, R. D.

D. R. Scifres, R. D. Burnham, and W. Streifer, Appl. Phys. Lett. 12, 33 (1978) .

Chui, C. K.

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

Daubechies, I.

I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, 1992).
[Crossref]

Dirac, P. A.

P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed (Oxford U. Press, 1958).

Dodonov, V. V.

V. V. Dodonov, J. Opt. B: Quantum Semiclassical Opt. 4, R1 (2002) .
[Crossref]

Fan, H.

H. Fan and H. R. Zaidi, Phys. Rev. A 37, 2985(1988).
[Crossref]

H. Fan and J. R. Klauder, J. Phys. A 21, L715 (1988).
[Crossref]

Fan, H.-Y.

Glauber, R. J.

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

Kaiser, G.

G. Kaiser, A Friendly Guide to Wavelets (Birkhäuser, 1994).

Kaneda, Y.

M. Oka, H. Masuda, Y. Kaneda, and S. Kubota, IEEE J. Quantum Electron. 28, 1142(1992) .
[Crossref]

Kapon, E.

Katz, J.

Klauder, J. R.

H. Fan and J. R. Klauder, J. Phys. A 21, L715 (1988).
[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]

Kubota, S.

M. Oka, H. Masuda, Y. Kaneda, and S. Kubota, IEEE J. Quantum Electron. 28, 1142(1992) .
[Crossref]

Loudon, R.

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

Lu, H.-L.

Masuda, H.

M. Oka, H. Masuda, Y. Kaneda, and S. Kubota, IEEE J. Quantum Electron. 28, 1142(1992) .
[Crossref]

Oka, M.

M. Oka, H. Masuda, Y. Kaneda, and S. Kubota, IEEE J. Quantum Electron. 28, 1142(1992) .
[Crossref]

Scifres, D. R.

D. R. Scifres, R. D. Burnham, and W. Streifer, Appl. Phys. Lett. 12, 33 (1978) .

Skargerstam, B. S.

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

Streifer, W.

D. R. Scifres, R. D. Burnham, and W. Streifer, Appl. Phys. Lett. 12, 33 (1978) .

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]

Xu, X.-F.

Yariv, A.

Zaidi, H. R.

H. Fan and H. R. Zaidi, Phys. Rev. A 37, 2985(1988).
[Crossref]

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

Appl. Phys. Lett. (1)

D. R. Scifres, R. D. Burnham, and W. Streifer, Appl. Phys. Lett. 12, 33 (1978) .

IEEE J. Quantum Electron. (1)

M. Oka, H. Masuda, Y. Kaneda, and S. Kubota, IEEE J. Quantum Electron. 28, 1142(1992) .
[Crossref]

J. Mod. Opt. (1)

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

J. Opt. B: Quantum Semiclassical Opt. (3)

V. V. Dodonov, J. Opt. B: Quantum Semiclassical Opt. 4, R1 (2002) .
[Crossref]

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

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

J. Phys. A (1)

H. Fan and J. R. Klauder, J. Phys. A 21, L715 (1988).
[Crossref]

Nature (1)

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

Opt. Lett. (4)

Phys. Rev. (1)

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

Phys. Rev. A (1)

H. Fan and H. R. Zaidi, Phys. Rev. A 37, 2985(1988).
[Crossref]

Phys. Rev. D (1)

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

Other (5)

I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, 1992).
[Crossref]

G. Kaiser, A Friendly Guide to Wavelets (Birkhäuser, 1994).

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

P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed (Oxford U. Press, 1958).

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

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Figures (5)

Fig. 1
Fig. 1

Traditional Mexican hat wavelet ψ 1 ( x ) .

Fig. 2
Fig. 2

Generalized Mexican hat wavelet ψ 2 ( x ) (solid curve), when g 0 = 2 , g 2 = 1 , g 4 = 1 , and g 6 = 0 . The points show the generalized Mexican hat wavelet ψ 3 ( x ) when g 0 = 1 , g 2 = 2 , g 4 = 1 , and g 6 = 0 .

Fig. 3
Fig. 3

Generalized Mexican hat wavelet ψ 4 ( x ) , when g 0 = 1 , g 2 = 2 , g 4 = 4 , and g 6 = 1 .

Fig. 4
Fig. 4

Contour line representation of Ω 1 ( X , T ) .

Fig. 5
Fig. 5

Contour line representation of Ω 2 ( X , T ) .

Equations (15)

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ψ ( x ) d x = 0 ,
W ψ f ( μ , s ) = 1 μ f ( x ) ψ * ( x s μ ) d x .
U ( μ , s ) 1 μ x s μ x d x
x = π 1 4 exp ( 1 2 x 2 + 2 x a a 2 2 ) 0 ;
U ( μ , s ) = exp [ s 2 2 ( 1 + μ 2 ) a 2 2 tanh λ a s 2 sech λ ] exp [ ( a a + 1 2 ) ln sech λ ] × exp [ a 2 2 tanh λ + s a 2 sech λ ] ,
ψ ( x ) d x = 0 p = 0 ψ = 0 .
ψ = G ( a ) 0 = n = 0 g n a n 0 ,
p = 0 ψ = p = 0 d 2 z π z z n g n a n 0 = π 1 4 n g n d 2 z π e z 2 z * n m 1 m ! ( z 2 2 ) m = π 1 4 n ( 2 n ) ! n ! 2 n g 2 n = 0 .
g 0 + g 2 + 3 g 4 + 15 g 6 = 0 ,
ψ = ( g 0 + g 2 a 2 + g 4 a 4 + g 6 a 6 ) 0 .
ψ ( x ) = π 1 4 e x 2 2 [ g 0 + g 2 ( 2 x 2 1 ) + g 4 ( 4 x 4 12 x 2 + 3 ) + g 6 ( 8 x 6 60 x 4 + 90 x 2 15 ) ] ,
ψ 2 ( x ) = 2 π 1 4 e x 2 2 ( 2 x 4 7 x 2 + 1 ) ,
ψ 3 ( x ) = π 1 4 e x 2 2 ( 8 x 6 + 76 x 4 134 x 2 + 26 ) ,
Ω 1 ( X , T ) = 2 3 d x ψ 1 [ T ( x X ) ] cos π x ,
Ω 2 ( X , T ) = 1 30 d x ψ 2 [ T ( x X ) ] cos π x ,

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