Abstract

An exploration of wavelet transforms for ultrashort optical pulse characterization is given. Some of the most common wavelets are examined to determine the advantages of using the causal quasi-wavelet suggested in Proceedings of the LEOS 15th Annual Meeting (IEEE, 2002), Vol. 2, p. 592, in terms of pulse analysis and, in particular, chirp extraction. Owing to its ability to distinguish between past and future pulse information, the causal quasi-wavelet is found to be highly suitable for optical pulse characterization.

© 2005 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  13. J. Molina Vázquez, J.-Z. Zhang, I. Galbraith, “Quantum dot versus quantum well semiconductor optical amplifiers for subpicosecond pulse amplification,” Opt. Quantum Electron. 36, 539–549 (2004).
    [CrossRef]
  14. C. K. Chui, Wavelet Analysis and Its Applications Volume 1—An Introduction to Wavelets (Academic, 1992).
  15. C. Torrence, G. P. Compo, “A practical guide to wavelet analysis,” Bull. Am. Meteorol. Soc. 79, 61–78 (1998).
    [CrossRef]
  16. M. Mazilu, A. Miller, “Wavelet transformations for ultrashort pulse characterization,” Proceedings of the LEOS 15th Annual Meeting (IEEE, 2002), Vol. 2, p. 592.
  17. J.-Z. Zhang, J. Molina Vázquez, M. Mazilu, A. Miller, I. Galbraith, “Dispersion induced ultrafast pulse re-shaping in 1.55-μmInGaAs∕InGaAsP optical amplifiers,” IEEE J. Quantum Electron. 39, 1388–1393 (2003).
    [CrossRef]
  18. J. S. Geronimo, D. P. Hardin, P. R. Massopust, “Fractal functions and wavelet expansions based on several scaling functions,” J. Approx. Theory 78, 373–401 (1994).
    [CrossRef]
  19. It is sometimes desirable to use the phase of the wavelet transform. In our case, we found that this provided no further information and has therefore been omitted.
  20. Intel Pentium 4 processor with 3.0 GHz clock speed running on a Linux platform.

2004 (2)

P. Addison, “Beyond Fourier: wavelets make the most of data,” Phys. World 3, 35–39 (2004).

J. Molina Vázquez, J.-Z. Zhang, I. Galbraith, “Quantum dot versus quantum well semiconductor optical amplifiers for subpicosecond pulse amplification,” Opt. Quantum Electron. 36, 539–549 (2004).
[CrossRef]

2003 (1)

J.-Z. Zhang, J. Molina Vázquez, M. Mazilu, A. Miller, I. Galbraith, “Dispersion induced ultrafast pulse re-shaping in 1.55-μmInGaAs∕InGaAsP optical amplifiers,” IEEE J. Quantum Electron. 39, 1388–1393 (2003).
[CrossRef]

2001 (1)

1998 (2)

1997 (1)

1994 (1)

J. S. Geronimo, D. P. Hardin, P. R. Massopust, “Fractal functions and wavelet expansions based on several scaling functions,” J. Approx. Theory 78, 373–401 (1994).
[CrossRef]

1993 (1)

1992 (1)

M. Farge, “Wavelet transforms and their applications to turbulence,” Annu. Rev. Fluid Mech. 24, 395–457 (1992).
[CrossRef]

1986 (1)

I. Daubechies, A. Grossmann, Y. Meyer, “Painless nonorthogonal expansions,” J. Math. Phys. 27, 1271–1283 (1986).
[CrossRef]

1982 (2)

J. Morlet, G. Arens, E. Fourgeau, D. Giard, “Wave propagation and sampling theory—Part I: Complex signal and scattering in multilayered media,” Geophysics 47, 203–221 (1982).
[CrossRef]

J. Morlet, G. Arens, E. Fourgeau, D. Giard, “Wave propagation and sampling theory—Part II: Sampling theory and complex waves,” Geophysics 47, 222–236 (1982).
[CrossRef]

Addison, P.

P. Addison, “Beyond Fourier: wavelets make the most of data,” Phys. World 3, 35–39 (2004).

Addison, P. S.

P. S. Addison, The Illustrated Wavelet Transform Handbook: Introductory Theory and Applications in Science, Engineering, Medicine and Finance (Institute of Physics, 2002).
[CrossRef]

Arens, G.

J. Morlet, G. Arens, E. Fourgeau, D. Giard, “Wave propagation and sampling theory—Part I: Complex signal and scattering in multilayered media,” Geophysics 47, 203–221 (1982).
[CrossRef]

J. Morlet, G. Arens, E. Fourgeau, D. Giard, “Wave propagation and sampling theory—Part II: Sampling theory and complex waves,” Geophysics 47, 222–236 (1982).
[CrossRef]

Chui, C. K.

C. K. Chui, Wavelet Analysis and Its Applications Volume 1—An Introduction to Wavelets (Academic, 1992).

Compo, G. P.

C. Torrence, G. P. Compo, “A practical guide to wavelet analysis,” Bull. Am. Meteorol. Soc. 79, 61–78 (1998).
[CrossRef]

Daubechies, I.

I. Daubechies, A. Grossmann, Y. Meyer, “Painless nonorthogonal expansions,” J. Math. Phys. 27, 1271–1283 (1986).
[CrossRef]

Farge, M.

M. Farge, “Wavelet transforms and their applications to turbulence,” Annu. Rev. Fluid Mech. 24, 395–457 (1992).
[CrossRef]

Fourgeau, E.

J. Morlet, G. Arens, E. Fourgeau, D. Giard, “Wave propagation and sampling theory—Part I: Complex signal and scattering in multilayered media,” Geophysics 47, 203–221 (1982).
[CrossRef]

J. Morlet, G. Arens, E. Fourgeau, D. Giard, “Wave propagation and sampling theory—Part II: Sampling theory and complex waves,” Geophysics 47, 222–236 (1982).
[CrossRef]

Galbraith, I.

J. Molina Vázquez, J.-Z. Zhang, I. Galbraith, “Quantum dot versus quantum well semiconductor optical amplifiers for subpicosecond pulse amplification,” Opt. Quantum Electron. 36, 539–549 (2004).
[CrossRef]

J.-Z. Zhang, J. Molina Vázquez, M. Mazilu, A. Miller, I. Galbraith, “Dispersion induced ultrafast pulse re-shaping in 1.55-μmInGaAs∕InGaAsP optical amplifiers,” IEEE J. Quantum Electron. 39, 1388–1393 (2003).
[CrossRef]

Gallmann, L.

Geronimo, J. S.

J. S. Geronimo, D. P. Hardin, P. R. Massopust, “Fractal functions and wavelet expansions based on several scaling functions,” J. Approx. Theory 78, 373–401 (1994).
[CrossRef]

Giard, D.

J. Morlet, G. Arens, E. Fourgeau, D. Giard, “Wave propagation and sampling theory—Part II: Sampling theory and complex waves,” Geophysics 47, 222–236 (1982).
[CrossRef]

J. Morlet, G. Arens, E. Fourgeau, D. Giard, “Wave propagation and sampling theory—Part I: Complex signal and scattering in multilayered media,” Geophysics 47, 203–221 (1982).
[CrossRef]

Grossmann, A.

I. Daubechies, A. Grossmann, Y. Meyer, “Painless nonorthogonal expansions,” J. Math. Phys. 27, 1271–1283 (1986).
[CrossRef]

Hardin, D. P.

J. S. Geronimo, D. P. Hardin, P. R. Massopust, “Fractal functions and wavelet expansions based on several scaling functions,” J. Approx. Theory 78, 373–401 (1994).
[CrossRef]

Iaconis, C.

Kane, D. J.

Keller, U.

Massopust, P. R.

J. S. Geronimo, D. P. Hardin, P. R. Massopust, “Fractal functions and wavelet expansions based on several scaling functions,” J. Approx. Theory 78, 373–401 (1994).
[CrossRef]

Mazilu, M.

J.-Z. Zhang, J. Molina Vázquez, M. Mazilu, A. Miller, I. Galbraith, “Dispersion induced ultrafast pulse re-shaping in 1.55-μmInGaAs∕InGaAsP optical amplifiers,” IEEE J. Quantum Electron. 39, 1388–1393 (2003).
[CrossRef]

M. Mazilu, A. Miller, “Wavelet transformations for ultrashort pulse characterization,” Proceedings of the LEOS 15th Annual Meeting (IEEE, 2002), Vol. 2, p. 592.

Meyer, Y.

I. Daubechies, A. Grossmann, Y. Meyer, “Painless nonorthogonal expansions,” J. Math. Phys. 27, 1271–1283 (1986).
[CrossRef]

Miller, A.

J.-Z. Zhang, J. Molina Vázquez, M. Mazilu, A. Miller, I. Galbraith, “Dispersion induced ultrafast pulse re-shaping in 1.55-μmInGaAs∕InGaAsP optical amplifiers,” IEEE J. Quantum Electron. 39, 1388–1393 (2003).
[CrossRef]

M. Mazilu, A. Miller, “Wavelet transformations for ultrashort pulse characterization,” Proceedings of the LEOS 15th Annual Meeting (IEEE, 2002), Vol. 2, p. 592.

Molina Vázquez, J.

J. Molina Vázquez, J.-Z. Zhang, I. Galbraith, “Quantum dot versus quantum well semiconductor optical amplifiers for subpicosecond pulse amplification,” Opt. Quantum Electron. 36, 539–549 (2004).
[CrossRef]

J.-Z. Zhang, J. Molina Vázquez, M. Mazilu, A. Miller, I. Galbraith, “Dispersion induced ultrafast pulse re-shaping in 1.55-μmInGaAs∕InGaAsP optical amplifiers,” IEEE J. Quantum Electron. 39, 1388–1393 (2003).
[CrossRef]

Morlet, J.

J. Morlet, G. Arens, E. Fourgeau, D. Giard, “Wave propagation and sampling theory—Part II: Sampling theory and complex waves,” Geophysics 47, 222–236 (1982).
[CrossRef]

J. Morlet, G. Arens, E. Fourgeau, D. Giard, “Wave propagation and sampling theory—Part I: Complex signal and scattering in multilayered media,” Geophysics 47, 203–221 (1982).
[CrossRef]

Reid, D. T.

D. T. Reid, Ultrafast Photonics, A. Miller, D. T. Reid, and D. M. Finlayson, eds. (Institute of Physics, 2004), pp. 59–71.
[CrossRef]

Rupp, T.

Steinmeyer, G.

Sutter, D. H.

Torrence, C.

C. Torrence, G. P. Compo, “A practical guide to wavelet analysis,” Bull. Am. Meteorol. Soc. 79, 61–78 (1998).
[CrossRef]

Trebino, R.

Walmsley, I. A.

Wickerhauser, M. V.

M. V. Wickerhauser, Adapted Wavelet Analysis from Theory to Software (IEEE, 1994).

Wong, V.

Zhang, J.-Z.

J. Molina Vázquez, J.-Z. Zhang, I. Galbraith, “Quantum dot versus quantum well semiconductor optical amplifiers for subpicosecond pulse amplification,” Opt. Quantum Electron. 36, 539–549 (2004).
[CrossRef]

J.-Z. Zhang, J. Molina Vázquez, M. Mazilu, A. Miller, I. Galbraith, “Dispersion induced ultrafast pulse re-shaping in 1.55-μmInGaAs∕InGaAsP optical amplifiers,” IEEE J. Quantum Electron. 39, 1388–1393 (2003).
[CrossRef]

Annu. Rev. Fluid Mech. (1)

M. Farge, “Wavelet transforms and their applications to turbulence,” Annu. Rev. Fluid Mech. 24, 395–457 (1992).
[CrossRef]

Bull. Am. Meteorol. Soc. (1)

C. Torrence, G. P. Compo, “A practical guide to wavelet analysis,” Bull. Am. Meteorol. Soc. 79, 61–78 (1998).
[CrossRef]

Geophysics (2)

J. Morlet, G. Arens, E. Fourgeau, D. Giard, “Wave propagation and sampling theory—Part I: Complex signal and scattering in multilayered media,” Geophysics 47, 203–221 (1982).
[CrossRef]

J. Morlet, G. Arens, E. Fourgeau, D. Giard, “Wave propagation and sampling theory—Part II: Sampling theory and complex waves,” Geophysics 47, 222–236 (1982).
[CrossRef]

IEEE J. Quantum Electron. (1)

J.-Z. Zhang, J. Molina Vázquez, M. Mazilu, A. Miller, I. Galbraith, “Dispersion induced ultrafast pulse re-shaping in 1.55-μmInGaAs∕InGaAsP optical amplifiers,” IEEE J. Quantum Electron. 39, 1388–1393 (2003).
[CrossRef]

J. Approx. Theory (1)

J. S. Geronimo, D. P. Hardin, P. R. Massopust, “Fractal functions and wavelet expansions based on several scaling functions,” J. Approx. Theory 78, 373–401 (1994).
[CrossRef]

J. Math. Phys. (1)

I. Daubechies, A. Grossmann, Y. Meyer, “Painless nonorthogonal expansions,” J. Math. Phys. 27, 1271–1283 (1986).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Lett. (3)

Opt. Quantum Electron. (1)

J. Molina Vázquez, J.-Z. Zhang, I. Galbraith, “Quantum dot versus quantum well semiconductor optical amplifiers for subpicosecond pulse amplification,” Opt. Quantum Electron. 36, 539–549 (2004).
[CrossRef]

Phys. World (1)

P. Addison, “Beyond Fourier: wavelets make the most of data,” Phys. World 3, 35–39 (2004).

Other (7)

D. T. Reid, Ultrafast Photonics, A. Miller, D. T. Reid, and D. M. Finlayson, eds. (Institute of Physics, 2004), pp. 59–71.
[CrossRef]

P. S. Addison, The Illustrated Wavelet Transform Handbook: Introductory Theory and Applications in Science, Engineering, Medicine and Finance (Institute of Physics, 2002).
[CrossRef]

M. V. Wickerhauser, Adapted Wavelet Analysis from Theory to Software (IEEE, 1994).

C. K. Chui, Wavelet Analysis and Its Applications Volume 1—An Introduction to Wavelets (Academic, 1992).

It is sometimes desirable to use the phase of the wavelet transform. In our case, we found that this provided no further information and has therefore been omitted.

Intel Pentium 4 processor with 3.0 GHz clock speed running on a Linux platform.

M. Mazilu, A. Miller, “Wavelet transformations for ultrashort pulse characterization,” Proceedings of the LEOS 15th Annual Meeting (IEEE, 2002), Vol. 2, p. 592.

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Figures (13)

Fig. 1
Fig. 1

Real (dashed curves), imaginary (dotted-dashed curves), and magnitude (solid curves) parts of the (a)–(c) input Gaussian pulse field and (d)–(f) spectrum with (a), (d) negative linear chirp, β = 1 ; (b), (e) no chirp, β = 0 ; and (c), (f) positive linear chirp, β = 1 .

Fig. 2
Fig. 2

Real (dashed curves), imaginary (dotted–dashed curves), and magnitude (solid curves) parts of a pulse after propagating through a quantum-dot optical amplifier.

Fig. 3
Fig. 3

Mexican hat wavelet for different (a) frequency ω and (b) position t.

Fig. 4
Fig. 4

Real (solid curves) and imaginary (dashed curves) (where applicable) parts of the (a) Haar, (b) Mexican hat, (c) Morlet, (d) Paul ( m = 4 ) , (e) linearly chirped gaussian, and (f) causal wavelets.

Fig. 5
Fig. 5

Causal quasi-wavelet for different (a) positions in time t, (b) frequency ω, and (c) response α.

Fig. 6
Fig. 6

Haar wavelet transform for a Gaussian pulse with (a) negative linear chirp, β = 1 ; (b) no chirp, β = 0 ; and (c) positive linear chirp; β = 1 .

Fig. 7
Fig. 7

Mexican hat wavelet transform for a Gaussian pulse with (a) negative linear chirp, β = 1 ; (b) no chirp, β = 0 ; and (c) positive linear chirp, β = 1 .

Fig. 8
Fig. 8

Morlet wavelet transform for a Gaussian pulse with (a) negative linear chirp, β = 1 ; (b) no chirp, β = 0 ; and (c) positive linear chirp, β = 1 .

Fig. 9
Fig. 9

Paul wavelet ( m = 4 ) transform for a Gaussian pulse with (a) negative linear chirp, β = 1 ; (b) no chirp, β = 0 ; and (c) positive linear chirp, β = 1 .

Fig. 10
Fig. 10

Linearly chirped ( β = 1 ) Gaussian wavelet transform for a Gaussian pulse with (a) negative linear chirp, β = 1 ; (b) no chirp, β = 0 ; and (c) positive linear chirp, β = 1 .

Fig. 11
Fig. 11

Causal quasi-wavelet transform for a Gaussian pulse with (a) negative linear chirp, β = 1 ; (b) no chirp, β = 0 ; and (c) positive linear chirp, β = 1 .

Fig. 12
Fig. 12

(a) Haar, (b) Mexican hat, (c) Morlet, (d) Paul ( m = 4 ) , (e) linearly chirped Gaussian, and (f) causal wavelet transforms of a linearly chirped ( β = 1 ) Gaussian pulse with a t 0 = 0.5 ps pulse delay and Δ ω 0 = 10 meV pulse center frequency shift.

Fig. 13
Fig. 13

(a) Haar, (b) Mexican hat, (c) Morlet, (d) Paul ( m = 4 ) , (e) chirped Gaussian ( β = 1 ) , and (f) causal wavelet transform magnitudes of a pulse after propagating through a quantum-dot amplifier.

Tables (2)

Tables Icon

Table 1 Summary of the Causal Quasi-Wavelet’s Properties

Tables Icon

Table 2 Summary of Wavelets’ Ability to Extract Pulse Information

Equations (17)

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E ( t ) = ϵ 0 exp [ ( t t 0 ) 2 τ 2 ] exp { i [ β t 2 τ 2 + ω 0 ( t t 0 ) ] } ,
E ̂ ( ω ) = ϵ 0 τ π ( 1 + β 2 ) 1 4 exp { [ ( ω ω 0 ) τ 2 2 β t 0 ] 2 4 τ 2 ( 1 + β 2 ) } exp ( i { 1 2 arctan ( β ) β [ ( ω ω 0 ) τ 2 2 β t 0 ] 2 4 τ 2 ( 1 + β 2 ) + β t 0 2 τ 2 } ) ,
W Δ t , t ( E ( t ) ) = ψ Δ t , t * ( t ) E ( t ) d t ,
ψ Δ t , t ( t ) d t = 0 .
E ( t ) = 1 C ψ 0 ψ Δ t , t ( t ) W Δ t , t d Δ t d t Δ t 2 ,
C ψ = ψ ̂ ( ω ) ω d ω < .
ψ Δ t , t ( t ) = { 0 : t t + Δ t ; t < t Δ t 1 : t Δ t t < t + 1 : t t < t + Δ t } .
ψ ( t ) = 2 ( 1 2 t 2 ) exp ( t 2 ) .
ψ ( t ) = ( 2 π ) 1 4 exp ( i γ t ) exp ( t 2 ) .
ψ ( t ) = 2 m m ! i m π ( 2 m ) ! ( 1 i t ) ( m + 1 ) ,
ψ α , ω , t ( t ) = [ 1 Θ ( t ) ] exp [ ( α + i ω ) ( t t ) ] .
Θ ( t t ) E ( t ) = 1 2 π exp [ ( α + i ω ) ( t t ) ] W α , ω , t d ω ,
E ( t ) = W α , ω , t t + ( α + i ω ) W α , ω , t .
1 2 π W α , ω , t e i ω t d ω = F T 1 { [ 1 Θ ( t t ) ] e ( α + i ω ) ( t t ) E ( t ) d t } = F T 1 { [ Θ ( t t ) e ( α + i ω ) t e α t E ( t ) ] e i ω t d t } = Θ ( t t ) e ( α + i ω ) t e α t E ( t ) ,
Θ ( t t ) E ( t ) = 1 2 π e ( α + i ω ) ( t t ) W α , ω , t d ω .
d d t ( W α , ω , t ) = d d t { [ 1 Θ ( t t ) ] e ( α + i ω ) ( t t ) E ( t ) d t } = d d t [ e ( α + i ω ) t t e ( α + i ω ) t E ( t ) d t ] = ( α + i ω ) e ( α + i ω ) t t e ( α + i ω ) t E ( t ) d t + e ( α + i ω ) t e ( α + i ω ) t E ( t ) = ( α + i ω ) W α , ω , t + E ( t ) ,
E ( t ) = d W α , ω , t d t + ( α + i ω ) W α , ω , t .

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