## A fast Gabor wavelet transform for high-precision phase retrieval in spectral interferometry

Optics Express, Vol. 15, Issue 22, pp. 14313-14321 (2007)

http://dx.doi.org/10.1364/OE.15.014313

Acrobat PDF (409 KB)

### 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

## 1. Introduction

2. C. Iaconis and I. A. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett. **23**, 792–794 (1998). [CrossRef]

3. 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]

4. T. M. Shuman, M. E. Anderson, J. Bromage, C. Iaconis, L. Waxer, and I. A. Walmsley, “Real-time SPIDER: ultrashort pulse characterization at 20 Hz,” Opt. Express **5**, 134–143 (1999). [CrossRef] [PubMed]

*a posteriori*phase processing has been demonstrated at rates ranging up to 1 kHz [5

5. W. Kornelis, J. Biegert, J. W. G. Tisch, M. Nisoli, G. Sansone, C. Vozzi, S. De Silvestri, and U. Keller, “Single-shot kilohertz characterization of ultrashort pulses by spectral phase interferometry for direct electric-field reconstruction,” Opt. Lett. **28**, 281–283 (2003). [CrossRef] [PubMed]

*S*(

*ω*), arising from the interference between two mutually temporally delayed and spectrally shifted replicas of the pulse under test [2

2. C. Iaconis and I. A. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett. **23**, 792–794 (1998). [CrossRef]

3. 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]

_{0}= 2π/Δ

*t*

_{0}. Introducing a spectral shear between the replicas gives rise to deviations from equidistance. Differences between the spectrally varying Δ

*ω*(

*ω*) and the fringe period Dw0 prior to application of the spectral shear then gives access to the spectral phase of the pulse under test. For determining the spectrally varying modulation period Δ

*ω*(

*ω*) = 2

*π*/Δ

*t*(

*ω*), one typically uses the equation

6. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. **72**, 156–160 (1982). [CrossRef]

*τ*

_{f}. In the absence of noise a maximum passband is obtained with

*τ*

_{f}= Δ

*t*

_{0}/2. As soon as experimental data has to be processed, narrowing of the passband filters out noise in the retrieved phase. However, as filtering must not affect the encoded signal, the frequency modulation depth of the fringe pattern imposes a lower limit for

*τ*

_{f}, i.e., it is impossible to reject noise directly in the signal band. One further problem of Eq. (1) is the use of the complex argument function, which is restricted to a range of 2

*π*and requires phase unwrapping prior to further processing. Moreover, the Fourier formalism yields the phase

*ϕ*

_{mod}rather than Δ

*t*directly, which requires numerical differentiation and makes the Takeda algorithm susceptible to experimental and numerical noise. As we will demonstrate in the subsequent section, this becomes a particular problem at low signal-to-noise ratios, i.e., in the spectral wings of the interferograms and for signals with strongly modulated spectral amplitude. Eventually, the algorithm may not be able to track the modulation signal anymore, a scenario that will be discussed as carrier collapse in the following. If a carrier collapse occurs in between two decodable segments of the spectrum, this may give rise to arbitrary multiple 2

*π*phase jumps in the retrieved spectral phase. Such a cycle slip of the phase then results in an erroneous kink of the reconstructed Δ

*t*(

*ω*).

7. 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]

8. 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]

9. G. Beylkin, R. Coifman, and V. Rokhlin, “Fast Wavelet Transform and Numerical Algorithms 1,” Com-mun. Pure Appl. Math. **44**, 141–183 (1991). [CrossRef]

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

## 2. Performance of Takeda phase retrieval for supercontinuum characterization

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

14. M. Nisoli, S. De Silvestri, O. Svelto, R. Szipo″cs, K. Ferencz, C. Spielmann, S. Sartania, and F. Krausz, “Compression of high-energy laser pulses below 5 fs,” Opt. Lett. **22**, 522–524 (1997). [CrossRef] [PubMed]

15. 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]

16. G. Stibenz and G. Steinmeyer, “High dynamic range characterization of ultrabroadband white-light contiuum pulses,” Opt. Express **12**, 6319–6325 (2004). [CrossRef] [PubMed]

*A*

_{mod}and the dc part

*A*

_{dc}of the trace via Fourier filtering, i.e.,

*A*

_{mod}/(

*A*

_{dc}-

*A*

_{mod}) depicted in Fig. 1(b). In this representation, ∑ = 0dB corresponds to a perfect 100% modulation of the fringes, whereas a value of ∑ = -24 dB indicates the detection limit for an 8-bit camera system. The central spectral drop-outs are also clearly discernible in Fig. 1(b). Using Eq. (1) for an analysis of the measured data, we retrieved the phases

*ϕ*

_{mod}from all 60 interferograms and evaluated their standard deviation, as shown in Fig. 1(c). We removed cycle slips prior to calculation of the standard deviation and plotted their occurence vs. frequency separately in Fig. 1(d). For ∑ > -10dB, we do not observe any cycle slips in our retrieved phases. In this regime, however, the spectral drop-outs become clearly visible giving rise to an increased standard deviation of the phase. This is coupling of amplitude noise into phase noise and has been discussed before, see, e.g. [17

17. C. Dorrer and I. A. Walmsley, “Accuracy criterion for ultrashort pulse characterization techniques: application to spectral phase interferometry for direct electric field reconstruction,” J. Opt. Soc. Am. B **19**, 1019–1029 (2002). [CrossRef]

## 3. Fast wavelet transforms for phase retrieval

*ω*(

*ω*). This is accomplished by choice of the wavelet

*g*(

*ξ*) that allows for localization in the

*ξ*domain. If

*g̃*, i.e. , the Fourier transform of

*g*, is also localized, the parameter δ enables an analysis in the Fourier domain as

*κ*) =

*g̃*(

*κ*-

*δ*), with

*ξ*and

*κ*being a Fourier pair. Localization in both domains is ensured when

*g*is a bell-shaped single maximum function.

*ξ*is a dimensionless variable, which, for our application of the analysis of spectral interferograms

*S*(

*ω*), spells out as

*ω*enables translation (analysis of different parts of the measured spectrum) and Δ

*ω*dilation, i.e., adjustment for detection of particular fringe periods Δ

*ω*in the interferogram. In the following, we will use

*ω*and Δ

*ω*as the fundamental two coordinates for our wavelet-based analysis of SPIDER interferograms.

*σ*= 2log2 and the Heisenberg scaling parameter

*h*, which determines the localization in both domains [8

8. 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]

*S*(

*ω*) as

*S*=

_{m}*S*(

*ω*) is sampled at

_{m}*N*discrete equidistant spectral positions

*ω*=

_{m}*ω*

_{0}+

*mδω*with

*m*= 0…

*N*-1. Here

*δω*is equivalent to the spectral resolution imposed by the used CCD camera and spectrograph, and

*ω*

_{0}defines the grating position of the spectrograph. Similarly, we define Δ

*ω*=

_{n}*nδω*with

*n*= 1…

*N*/2. In this representation the Gabor wavelet is then evaluated as

*N*, i.e., requires evaluation of an

*N*-term sum for

*N*possible values of

*ω*and

_{m}*N*/2 sensible values of Δ

*ω*. Moreover, as each individual calculation requires computation of two real-valued trigonometric functions and one exponential, on the order of 10

_{n}^{12}floating-point operations are necessary at

*N*= 2048 for a full evaluation of the GWT, resulting in several minutes computation time on standard PCs.

*N*

^{2}different independent input values as

*ξ*= (

*j*-

*m*)/

*n*in the discrete representation of the wavelet. This allows for usage of a look-up table, i.e., Ψ

_{mnj}is computed for every possible combination of indices

*m*,

*n*and

*j*at initialization of the program and permanently stored in the computer’s random access memory. For example, considering the grid size of

*N*= 2048 used below in our measurements, this requires 64 MByte of memory at most, which is readily available on modern PCs. This optimization step renders calculation of Eq. (9) into the evaluation of a simple scalar product, roughly gaining a factor 100 over repeated calculation of the transcendental functions, which allows for a complete evaluation of the GWT in a few seconds on a modern PC.

*ω*typically stays within a narrow ±50% range around Δ

*ω*

_{0}. Moreover, SPIDER typically works best in terms of noise rejection when a rather small modulation period is chosen [13

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

*N*evaluations. This is accomplished with the ridge tracking algorithm discussed below.

*W*(Δ

*ω*,

_{s}*ω*), which is easily obtainable from inspection of a single measured interferogram. Initially setting

_{t}*ω*=

_{m}*ω*, we scan the Δ

_{t}*ω*coordinate and determine the center of gravity

_{n}*n*

_{1}<

*s*<

*n*

_{2}. Typically, we find it sufficient to include some ten terms in the search for the ridge position, i.e., |

*n*

_{1}-

*n*

_{2}| ≈ 10. It is important to note that computation of the center of gravity is essential for reasonable precision in ridge tracking as simple determination of the maximum value along the Δ

*ω*coordinate may otherwise repeatedly yield the same integer coordinate

*n*along the ridge and show no variation with

*ω*. The procedure in Eq. (10) is therefore repeated along the

*ω*coordinate, retrieving the complete spectral dependence Δ

_{m}*ω*

_{max}(

*ω*) from the interferogram under test. For this purpose, however, it is typically not necessary to determine the center of gravity for each possible value of

_{m}*ω*. We find it sufficient to sparsely track the ridge, scanning the

_{m}*ω*coordinate in steps of the period Δ

_{m}*ω*

_{max}(

*ω*) or even slightly coarser. In the SPIDER method, variations of the spectral phase on scales finer than the spectral shear are meaniningless. Therefore rather coarse scanning can be used. We use the setup described in [13

_{m}13. G. Stibenz and G. Steinmeyer, “Optimizing spectral phase interferometry for direct electric-field reconstruction,” Rev. Sci. Instrum **77**, 073105 1–9 (2006). [CrossRef]

_{shear}= 81 × 10

^{12}rad/s= 12.9 THz, which is to be compared to a camera pixel spacing corresponding to only 0.15THz. Therefore, even sparsely tracking the ridge at a rate of 10 pixels ≈ 1.5 THz still oversamples the data by a factor 10.

*ω*coordinate becomes essentially a linear algorithm, requiring computation of two multiplications and two additions per point carried into the sum of Eq. (9), which then has to be evaluated about

_{m}*N*times for a complete phase retrieval. Depending on the chosen truncation and sparsity, this allows reduction of the computational time to 10 to 100 ms, which is to be compared with about 1000 s for a complete unoptimized GWT on the same computer.

## 4. Comparison of the fast GWT with the Takeda algorithm

3. 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]

*τ*

_{f}in Eq. (1), the wavelet is localized in

*ω*, which allows for adaptive noise rejection along this coordinate. The GWT therefore allows for partial rejection of noise in the signal band. For the case of an equidistant fringe pattern, one could certainly set the limits in Eq. (1) reasonably tight without suppressing any information, yielding the same noise suppression as with the GWT. However, as soon as information is encoded in the fringe spacing, one has to use less selective filtering with the Takeda algorithm. We therefore expect the relative noise immunity of the GWT to be most pronounced for a SPIDER apparatus with strongly modulated fringe spacings, i.e., in particular for broadband set-ups and those using relatively large amounts of dispersion for generating the reference pulse.

7. 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]

## 5. Optimum Heisenberg box

*h*= 0.7 and

*h*= 3.0, respectively. In this representation, the central drop-outs and their consequences on ridge tracking are also clearly visible, and ultimately the ridge tracking algorithm also collapses in the far wings of the spectrum at above 440 THz and below 335 THz.

*g*and

*g̃*in either Fourier domain, i.e.,

*σ*

_{rms}and

σ ˜

_{rms}, respectively. For our choice of unchirped Gaussian gating functions, the product of the two is fixed via

*ω*axis because the signal is strongly oversampled in

*ω*, and ultimately SPIDER cannot resolve features of the spectral phase that are finer than the spectral shear. To investigate the influence of the aspect ratio of the Heisenberg box, we reconstructed the spectral phase with different

*h*parameters, yielding the results shown in Fig. 3(a). For comparison, the phase error is also shown as a function of the width of the Heisenberg box along the

*ω*coordinate in Fig. 3(b). In both representations, a minimum phase error is observed for particular values of

*h*≈ 2 and

*σ*

_{rms}≈ 0.4THz. While the phase error changes only by little in a broad plateau region at 1 <

*h*< 3, it does exhibit a sharp rise at

*σ*

_{rms}< 0.3THz. This behavior is caused by the discretization of the signal and by the determination of the ridge via center of gravity formation. Once the localization in the Δ

*ω*direction becomes smaller than the discretization along this coordinate, only one significant term enters into computation of the center of gravity [Eq.(10)], and noise enters everywhere else. The other way around, when the signal is too strongly localized in

*ω*, the signal extends too far along the Δ

*ω*domain, with little difference between the individual elements of the sum and poor rejection of spurious signals at other fringe periods Δ

*ω*. This makes it quite understandable that there is an optimum value of

*h*≈ 2, which corresponds to equal localization in both domains at about the discretization in these domains.

## 6. Conclusion

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

20. S. Üzender, Ü. Kocahan, E. Coşkun, and H. Güktaş, “Optical phase distribution evaluation by using an S-transform,” Opt. Lett. **32**, 591–593 (2007). [CrossRef]

## Acknowledgment

## References and links

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

2. | C. Iaconis and I. A. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett. |

3. | 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. |

4. | T. M. Shuman, M. E. Anderson, J. Bromage, C. Iaconis, L. Waxer, and I. A. Walmsley, “Real-time SPIDER: ultrashort pulse characterization at 20 Hz,” Opt. Express |

5. | W. Kornelis, J. Biegert, J. W. G. Tisch, M. Nisoli, G. Sansone, C. Vozzi, S. De Silvestri, and U. Keller, “Single-shot kilohertz characterization of ultrashort pulses by spectral phase interferometry for direct electric-field reconstruction,” Opt. Lett. |

6. | M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. |

7. | Y. Deng, Z. Wu, and C. Wang, “Wavelet-transform analysis of spectral shearing interferometry for phase reconstruction of femtosecond optical pulses,” Opt. Express |

8. | 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 |

9. | G. Beylkin, R. Coifman, and V. Rokhlin, “Fast Wavelet Transform and Numerical Algorithms 1,” Com-mun. Pure Appl. Math. |

10. | S. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Pattern Anal. and Machine Intell. |

11. | S. Mallat, “A wavelet tour of signal processing,” 2nd Edition, Academic Press, San Diego, CA, 2004. |

12. | D. Gabor, “Theory of Communication,” J. IEE |

13. | G. Stibenz and G. Steinmeyer, “Optimizing spectral phase interferometry for direct electric-field reconstruction,” Rev. Sci. Instrum |

14. | M. Nisoli, S. De Silvestri, O. Svelto, R. Szipo″cs, K. Ferencz, C. Spielmann, S. Sartania, and F. Krausz, “Compression of high-energy laser pulses below 5 fs,” Opt. Lett. |

15. | 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 |

16. | G. Stibenz and G. Steinmeyer, “High dynamic range characterization of ultrabroadband white-light contiuum pulses,” Opt. Express |

17. | C. Dorrer and I. A. Walmsley, “Accuracy criterion for ultrashort pulse characterization techniques: application to spectral phase interferometry for direct electric field reconstruction,” J. Opt. Soc. Am. B |

18. | 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 |

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

20. | S. Üzender, Ü. Kocahan, E. Coşkun, and H. Güktaş, “Optical phase distribution evaluation by using an S-transform,” Opt. Lett. |

**OCIS Codes**

(100.5070) Image processing : Phase retrieval

(100.7410) Image processing : Wavelets

(120.2650) Instrumentation, measurement, and metrology : Fringe analysis

(320.7100) Ultrafast optics : Ultrafast measurements

**ToC Category:**

Image Processing

**History**

Original Manuscript: July 26, 2007

Revised Manuscript: September 27, 2007

Manuscript Accepted: October 2, 2007

Published: October 15, 2007

**Citation**

J. Bethge, C. Grebing, and G. Steinmeyer, "A fast Gabor wavelet transform for high-precision phase retrieval in spectral
interferometry," Opt. Express **15**, 14313-14321 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-22-14313

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### References

- R. Trebino, "Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses," (Kluwer Academic Publishers, Boston, MA, 2000).
- C. Iaconis and I. A. Walmsley, "Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses," Opt. Lett. 23, 792-794 (1998). [CrossRef]
- 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]
- T. M. Shuman, M. E. Anderson, J. Bromage, C. Iaconis, L. Waxer, and I. A. Walmsley, "Real-time SPIDER: ultrashort pulse characterization at 20 Hz," Opt. Express 5, 134-143 (1999). [CrossRef] [PubMed]
- W. Kornelis, J. Biegert, J. W. G. Tisch, M. Nisoli, G. Sansone, C. Vozzi, S. De Silvestri, and U. Keller, "Singleshot kilohertz characterization of ultrashort pulses by spectral phase interferometry for direct electric-field reconstruction," Opt. Lett. 28, 281-283 (2003). [CrossRef] [PubMed]
- M. Takeda, H. Ina, and S. Kobayashi, "Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry," J. Opt. Soc. Am. 72, 156-160 (1982). [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]
- 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. Beylkin, R. Coifman, and V. Rokhlin, "Fast Wavelet Transform and Numerical Algorithms 1," Commun. Pure Appl. Math. 44, 141-183 (1991). [CrossRef]
- S. Mallat, "A theory for multiresolution signal decomposition: the wavelet representation," IEEE Pattern Anal. and Machine Intell. 11, 674-693 (1989). [CrossRef]
- S. Mallat, "A wavelet tour of signal processing," 2nd Edition, Academic Press, San Diego, CA, 2004.
- D. Gabor, "Theory of Communication," J. IEE 93, 429-457 (1946).
- G. Stibenz and G. Steinmeyer, "Optimizing spectral phase interferometry for direct electric-field reconstruction," Rev. Sci. Instrum 77, 073105 1-9 (2006). [CrossRef]
- M. Nisoli, S. De Silvestri, O. Svelto, R. Szip cs, K. Ferencz, C. Spielmann, S. Sartania, and F. Krausz, "Compression of high-energy laser pulses below 5 fs," Opt. Lett. 22, 522-524 (1997). [CrossRef] [PubMed]
- 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]
- G. Stibenz and G. Steinmeyer, "High dynamic range characterization of ultrabroadband white-light continuum pulses," Opt. Express 12, 6319-6325 (2004). [CrossRef] [PubMed]
- C. Dorrer and I. A. Walmsley, "Accuracy criterion for ultrashort pulse characterization techniques: application to spectral phase interferometry for direct electric field reconstruction," J. Opt. Soc. Am. B 19, 1019-1029 (2002). [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]
- 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]
- S. Özender, Ö. Kocahan, E. Coskun and H. Göktas, "Optical phase distribution evaluation by using an Stransform," Opt. Lett. 32, 591-593 (2007). [CrossRef]

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