Abstract

A fast implementation of the Gabor wavelet transform for phase retrieval in spectral interferometry is discussed. This algorithm is experimentally demonstrated for the characterization of a supercontinuum, using spectral phase interferometry for direct electric-field reconstruction (SPIDER). The performance of wavelet based ridge tracking for frequency demodulation is evaluated and compared to traditional Fourier filtering techniques. It is found that the wavelet based strategy is significantly less susceptible toward experimental noise and does not exhibit cycle slip artifacts. Optimum performance of the Gabor transform is observed for a Heisenberg box with unity aspect ratio. As a result, the phase jitter of 60 individual measurements is reduced by about a factor 2 compared to Fourier filtering, and the detection window increases by 20%. With an optimized implementation, retrieval rates of several 10 Hz can be reached, which makes the fast Gabor transform a superior one-to-one replacement even in applications that require video-rate update, such as a real-time SPIDER apparatus.

© 2007 Optical Society of America

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References

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2007

2006

G. Stibenz and G. Steinmeyer, "Optimizing spectral phase interferometry for direct electric-field reconstruction," Rev. Sci. Instrum 77, 073105 1-9 (2006).
[CrossRef]

G. Steinmeyer and G. Stibenz, "Generation of sub-4-fs pulses via compression of a white-light continuum using only chirped mirros," Appl. Phys. B 82, 175-181 (2006).
[CrossRef]

2005

Y. Deng, C. Wang, L. Chai, and Z. Zhang, "Determination of Gabor wavelet shaping factor for accurate phase retrieval with wavelet-transform," Appl. Phys. B 81, 1107-1111 (2005).
[CrossRef]

Y. Deng, Z. Wu and C. Wang, "Wavelet-transform analysis of spectral shearing interferometry for phase reconstruction of femtosecond optical pulses," Opt. Express 13, 2120-2126 (2005).
[CrossRef] [PubMed]

2004

G. Stibenz and G. Steinmeyer, "High dynamic range characterization of ultrabroadband white-light continuum pulses," Opt. Express 12, 6319-6325 (2004).
[CrossRef] [PubMed]

I. A. Walmsley, "Characterization of Ultrashort Optical Pulses in the Few-Cycle Regime Using Spectral Phase Interferometry for Direct Electric-Field Reconstruction," Top. Appl. Phys. 95, 265-292 (2004).
[CrossRef]

2003

2002

2000

M.E. Anderson, L.E.E. de Araujo, E.M. Kosik and I.A. Walmsley, "The effects of noise on ultrashort-optical pulse measurement using SPIDER," Appl. Phys. B 70, S85-S93 (2000).
[CrossRef]

1999

1998

1997

1991

G. Beylkin, R. Coifman, and V. Rokhlin, "Fast Wavelet Transform and Numerical Algorithms 1," Commun. Pure Appl. Math. 44, 141-183 (1991).
[CrossRef]

1989

S. Mallat, "A theory for multiresolution signal decomposition: the wavelet representation," IEEE Pattern Anal. and Machine Intell. 11, 674-693 (1989).
[CrossRef]

1982

1946

D. Gabor, "Theory of Communication," J. IEE 93, 429-457 (1946).

Appl. Phys. B

Y. Deng, C. Wang, L. Chai, and Z. Zhang, "Determination of Gabor wavelet shaping factor for accurate phase retrieval with wavelet-transform," Appl. Phys. B 81, 1107-1111 (2005).
[CrossRef]

G. Steinmeyer and G. Stibenz, "Generation of sub-4-fs pulses via compression of a white-light continuum using only chirped mirros," Appl. Phys. B 82, 175-181 (2006).
[CrossRef]

M.E. Anderson, L.E.E. de Araujo, E.M. Kosik and I.A. Walmsley, "The effects of noise on ultrashort-optical pulse measurement using SPIDER," Appl. Phys. B 70, S85-S93 (2000).
[CrossRef]

Commun. Pure Appl. Math.

G. Beylkin, R. Coifman, and V. Rokhlin, "Fast Wavelet Transform and Numerical Algorithms 1," Commun. Pure Appl. Math. 44, 141-183 (1991).
[CrossRef]

IEEE Pattern Anal. and Machine Intell.

S. Mallat, "A theory for multiresolution signal decomposition: the wavelet representation," IEEE Pattern Anal. and Machine Intell. 11, 674-693 (1989).
[CrossRef]

J. IEE

D. Gabor, "Theory of Communication," J. IEE 93, 429-457 (1946).

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

Opt. Express

Opt. Lett.

Rev. Sci. Instrum

G. Stibenz and G. Steinmeyer, "Optimizing spectral phase interferometry for direct electric-field reconstruction," Rev. Sci. Instrum 77, 073105 1-9 (2006).
[CrossRef]

Top. Appl. Phys.

I. A. Walmsley, "Characterization of Ultrashort Optical Pulses in the Few-Cycle Regime Using Spectral Phase Interferometry for Direct Electric-Field Reconstruction," Top. Appl. Phys. 95, 265-292 (2004).
[CrossRef]

Other

R. Trebino, "Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses," (Kluwer Academic Publishers, Boston, MA, 2000).

S. Mallat, "A wavelet tour of signal processing," 2nd Edition, Academic Press, San Diego, CA, 2004.

T. Hansel, C. von Kopylow, J. Müller, C. Falldorf, W. Jüptner, R. Grunwald, G. Steinmeyer, and U. Griebner, "Ultrashort pulse dual-wavelength source for digital holographic two-wavelength contouring," submitted to Appl. Phys. B (2007).
[CrossRef]

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Figures (3)

Fig. 1.
Fig. 1.

a) Example of a SPIDER interferogram of the hollow-fiber continuum out of a series of 60 measurements. (b) Average signal-to-noise ratio ∑ observed in the series. (c) Standard deviation of phases retrieved by the Takeda algorithm. Cycle slips have been eliminated in the evaluation. (d) Relative occurrence of cycle slips in Takeda-type phase retrieval.

Fig. 2.
Fig. 2.

a) Average phases retrieved with the GWT (red curve) and the Takeda algorithm (black curve) and spectral power density (blue curve) of the fundamental. Zero phase was arbitrarily chosen at 390 THz. Error bars indicate the standard deviation in 60 individual measurements. (b) Spectral dependence of the standard deviation for the GWT (red curve) and the Takeda algorithm (black curve). The data from Fig. 1(a) is shown for comparison as the blue curve.

Fig. 3.
Fig. 3.

a) Dependence of the average phase error on the Heisenberg parameter h. The shape of the Heisenberg box is illustrated for three different values of h. b) same as a function of the width of the Heisenberg box in ω direction. c) Complete GWT shown as logarithmic hues, with the ridge position indicated by a white line for h = 0.7. d) same but with h = 3.0.

Equations (11)

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Δ t ( ω ) = d d ω φ mod ( ω ) = d d ω arg [ Δ t 0 - τ f Δ t 0 + τ f 0 S ( ω ´ ) exp ( ´ t ´ ) d ω ´ exp ( ´ t ´ ) d t ´ ] .
A mod = | Δ t 0 - τ f Δ t 0 + τ f 0 S ( ω ´ ) exp ( ´ t ´ ) d ω ´ exp ( t ´ ) d t ´ | ,
A dc = 0 τ f 0 S ( ω ´ ) exp ( ´ t ´ ) d ω ´ exp ( t ´ ) d t ´ .
Ψ ( ξ ) = g ( ξ ) exp ( iξδ )
ξ = ω ´ ω Δ ω ,
Ψ ( ξ ) = h σ 2 π 4 exp ( 2 πiξ ξ 2 h 2 σ 2 ) .
W ( ω , Δ ω ) = 1 Δ ω 0 S ( ω ´ ) Ψ * ( ω ´ ω Δ ω ) d ω ´ .
Ψ mnj = h σ 2 π 4 exp [ 2 πi ( ω j ω m Δ ω n ) h 2 σ 2 ( ω j ω m Δ ω n ) 2 ] .
W mn = 1 Δ ω n j = 1 N S j Ψ mnj * .
Δ ω max ( ω m ) = n = n 1 n 2 Δ ω n W mn n = n 1 n 2 W mn = 2 π [ d d ω φ mod ] 1
σ rms σ ˜ rms = 1 2 ,

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