## Unambiguous chirp characterization using modified-spectrum auto-interferometric correlation and pulse spectrum

Optics Express, Vol. 14, Issue 19, pp. 8890-8899 (2006)

http://dx.doi.org/10.1364/OE.14.008890

Acrobat PDF (135 KB)

### Abstract

Modified-spectrum auto-interferometric correlation (MOSAIC), derived from a conventional second order interferometric autocorrelation trace, is a sensitive and visual chirp diagnostic method for ultrashort laser pulses. We construct several pairs of example pulse shapes that have nearly identical MOSAIC traces and demonstrate that chirp ambiguity can result when the field amplitude or spectrum are not known, thus making MOSAIC a qualitative tool for chirped pulses. However, when the pulse spectrum is known, a unique chirp reconstruction is possible. With the help of a new reconstruction technique, we experimentally demonstrate complete pulse characterization using MOSAIC envelopes and the pulse spectrum.

© 2006 Optical Society of America

## 1. Introduction

1. J. C. M. Diels, J. J. Fontaine, I. C. McMichael, and F. Simoni, “Control and measurement of ultrashort pulse shapes (in amplitude and phase) with femtosecond accuracy.” Appl. Opt. **24**, 1270–82 (1985). [CrossRef] [PubMed]

2. C. Yan and J. C. Diels, “Amplitude and phase recording of ultrashort pulses.” J. Opt. Soc. Am. B **8**, 1259–63 (1991). [CrossRef]

3. C. Spielmann, L. Xu, and F. Krausz, “Measurement of interferometric autocorrelations: comment.” Appl. Opt. **36**, 2523–5 (1997). [CrossRef] [PubMed]

4. T. Hirayama and M. Sheik-Bahae, “Real-time chirp diagnostic for ultrashort laser pulses.” Opt. Lett. **27**, 860–2 (2002). [CrossRef]

5. D. J. Kane and R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating.” IEEE J. Quantum Electron. **29**, 571–9 (1993). [CrossRef]

6. P. O’Shea, M. Kimmel, X. Gu, and R. Trebino, “Highly simplified device for ultrashort-pulse measurement.” Opt. Lett. **26**, 932–4 (2001). [CrossRef]

7. G. Stibenz and G. Steinmeyer, “Interferometric frequency-resolved optical gating,” Opt. Express **13**, 2617–2626 (2005). [CrossRef] [PubMed]

8. C. Iaconis and I. A. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses.” Opt. Lett. **23**, 792–4 (1998). [CrossRef]

9. J. K. Rhee, T. S. Sosnowski, A. C. Tien, and T. B. Norris, “Real-time dispersion analyzer of femtosecond laser pulses with use of a spectrally and temporally resolved upconversion technique,” J. Opt. Soc. Am. B **13**, 1780–1785 (1996). [CrossRef]

10. J. W. Nicholson, J. Jasapara, W. Rudolph, F. G. Omenetto, and A. J. Taylor, “Full-field characterization of femtosecond pulses by spectrum and cross-correlation measurements.” Opt. Lett. **24**, 1774–6 (1999). [CrossRef]

11. R. G. M. P. Koumans and A. Yariv, “Time-resolved optical gating based on dispersive propagation: a new method to characterize optical pulses.” IEEE J. Quantum Electron. **36**, 137–44 (2000). [CrossRef]

*ω*

_{0}and 2

*ω*

_{0}, where

*ω*

_{0}is the pulse center frequency. MOSAIC traces are obtained by filtering the

*ω*

_{0}fringe-band and multiplying the 2

*ω*

_{0}band by a factor of two. The shape of the resultant fringe resolved trace depends on the pulse chirp and one can easily distinguish a chirped pulse from a chirp-free pulse by examining their MOSAIC traces [4

4. T. Hirayama and M. Sheik-Bahae, “Real-time chirp diagnostic for ultrashort laser pulses.” Opt. Lett. **27**, 860–2 (2002). [CrossRef]

4. T. Hirayama and M. Sheik-Bahae, “Real-time chirp diagnostic for ultrashort laser pulses.” Opt. Lett. **27**, 860–2 (2002). [CrossRef]

12. A. K. Sharma, P. A. Naik, and P. D. Gupta, “Estimation of higher order chirp in ultrashort laser pulses using modified spectrum auto-interferometric correlation,” Opt. Commun. **233**, 431–437 (2004). [CrossRef]

13. A. K. Sharma, M. Raghuramaiah, P. A. Naik, and P. D. Gupta, “Use of commercial grade light emitting diode in auto-correlation measurements of femtosecond and picosecond laser pulses at 1054 nm.” Opt. Commun. **246**, 195–204 (2005). [CrossRef]

14. D. A. Bender, M. P. Hasselbeck, and M. Sheik-Bahae, “Sensitive ultrashort pulse chirp measurement.” Opt. Lett. **31**, 122–4 (2006). [CrossRef] [PubMed]

14. D. A. Bender, M. P. Hasselbeck, and M. Sheik-Bahae, “Sensitive ultrashort pulse chirp measurement.” Opt. Lett. **31**, 122–4 (2006). [CrossRef] [PubMed]

15. K. Naganuma, K. Modi, and H. Yamada, “General method for ultrashort light pulse chirp measurement.” IEEE J. Quantum Electron. **25**, 1225–33 (1989). [CrossRef]

## 2. Chirp ambiguity from MOSAIC traces

### 2.1. MOSAIC traces

*S*

_{IAC}(

*τ*)) between a pulse

*E*(

*t*) and its delayed (by τ) replica can be written as [2

2. C. Yan and J. C. Diels, “Amplitude and phase recording of ultrashort pulses.” J. Opt. Soc. Am. B **8**, 1259–63 (1991). [CrossRef]

*I*(

*t*)

*dt*=1. The MOSAIC trace (

*S*

_{MOSAIC}(

*τ*)) is obtained by filtering

*A*

_{1}(

*τ*) and multiplying

*A*

_{2}(

*τ*) by a factor of two [4

**27**, 860–2 (2002). [CrossRef]

**27**, 860–2 (2002). [CrossRef]

*τ*) and 2

*A*

_{0}(

*τ*)) are plotted in part (c) and part (d) of Fig. 1.

*τ*), is only present when the pulse is chirped. It is easy to see this as follows; when there is no chirp, the second harmonic field and intensity envelopes are the same (

*E*

^{2}(

*t*)=

*I*(

*t*)) implying that the envelopes

*A*

_{0}(

*τ*) and

*A*

_{2}(

*τ*) are identical and hence the minima envelope

*τ*) is zero. It is this property of the minima envelope that makes it a powerful qualitative visualization tool for distinguishing chirped pulses from chirp-free pulses.

### 2.2. Pulse construction algorithm

*A*

_{0}(

*τ*) and second harmonic correlation

*A*

_{2}(

*τ*) envelopes. The Fourier transformed counter-parts of these correlations,

*Ã*

_{0,2}(Ω), are real functions [15

15. K. Naganuma, K. Modi, and H. Yamada, “General method for ultrashort light pulse chirp measurement.” IEEE J. Quantum Electron. **25**, 1225–33 (1989). [CrossRef]

*τ*. Therefore the MOSAIC traces are entirely determined by the magnitudes of intensity spectrum (|

*Ĩ*(Ω)|) and second harmonic spectrum (|

*Ẽ*

^{2}(Ω)|). In general, it is known [15

15. K. Naganuma, K. Modi, and H. Yamada, “General method for ultrashort light pulse chirp measurement.” IEEE J. Quantum Electron. **25**, 1225–33 (1989). [CrossRef]

*E*(

*τ*) (except for the time direction ambiguity). Therefore, it is likely that multiple field shapes are constrained to have same magnitude of the intensity spectrum and same magnitude of the second harmonic spectrum. Our algorithm is based on finding a field

*E*(

*τ*) given the intensity and second harmonic spectral magnitudes.

*E*

_{i}(

*τ*), and compute the magnitudes |

*Ĩ*

_{i}(Ω) | and |

*E*(

*τ*), that satisfies the spectral magnitude constraints. This algorithm is very similar to Naganuma’s [15

**25**, 1225–33 (1989). [CrossRef]

*E*(

*τ*) is used to compute the intensity,

*I*(

*τ*) and second harmonic envelopes,

*E*

^{2}(

*τ*). These time domain fields are Fourier transformed and their Fourier magnitudes are replaced by |

*Ĩ*

_{i}(Ω)| and |

*I*(

*τ*) and

*E*

^{2}(

*τ*). The magnitude of second harmonic field is replaced by |

*I*(

*τ*)| while retaining its phase. From this updated

*E*

^{2}(

*τ*), a new estimate for the field envelope

*E*(

*τ*) is obtained which is used in the next iteration of the algorithm. After nearly 100 iterations, the adapted field

*E*(

*τ*) has the same magnitudes for its intensity and second harmonic spectra as the input field

*E*

_{i}(

*τ*). By varying the initial condition, namely the guess field

*E*

_{g}(

*τ*) in the first epoch, the algorithm can be expected to result in different output pulse shapes.

### 2.3. Reconstruction results

*Ĩ*(Ω)| and the input spectral magnitude |

*Ĩ*

_{i}(Ω) | decreased monotonically, whereas the error for the second harmonic spectral magnitude did not decrease monotonically. After adaptation using this approach, output fields have two trivial ambiguities, namely temporal shifts, and time direction ambiguity (

*E*(

*τ*) vs

*E**(-

*τ*)). We plot the reconstruction results after choosing a time direction and correcting for the time shift such that the resultant output field is a better comparison to the input field.

*E*

_{i}(

*τ*), for this simulation is derived by applying a phase,

*ϕ*(

*ω*)=10.4

*ω*

^{2}+50

*ω*

^{3}-0.4

*ω*

^{4}(

*ω*is in units of pHz), to the Fourier transform of sech(

*τ*/

*τ*

_{p}), where

*τ*

_{p}=5 fs. The initial condition for the algorithm is a secant field

*E*

_{g}(

*τ*)=sech(

*τ*/

*τ*

_{p}). The intensity and phase of the reconstruction

*E*(

*τ*), shown in blue, in Fig. 3(b) are obtained after 150 iterations and they differ considerably from the input. The field spectrum is also quite different from the input field spectrum (Fig. 3(c)). However, the adapted spectra |

*Ĩ*(Ω)|, |

*Ẽ*

^{2}(Ω) | match very closely with the corresponding input spectra as seen from the error plots shown in Fig. 3(d) and (e). It is clear from Fig. 3(a) that output field

*E*(

*τ*) has a MOSAIC trace that closely matches the original to one part in 10

^{3}, which is similar to a typical experimental accuracy of an average MOSAIC envelope measurement [14

14. D. A. Bender, M. P. Hasselbeck, and M. Sheik-Bahae, “Sensitive ultrashort pulse chirp measurement.” Opt. Lett. **31**, 122–4 (2006). [CrossRef] [PubMed]

*E*

_{i}(

*τ*)=sech(

*τ*/

*τ*

_{p})exp[

*i*(0.18(

*τ*/

*τ*

_{p})

^{2}+0.2(

*τ*/

*τ*

_{p})

^{3}-0.82(

*τ*/

*τ*

_{p})

^{4})], where

*τ*

_{p}=100 fs. This expression is an experimental fit obtained to a MOSAIC trace in [14

**31**, 122–4 (2006). [CrossRef] [PubMed]

*τ*/

*τ*

_{p}), a fit can be obtained for the chirp. However, we want to see if there are any other pulse shapes that have nearly the same MOSAIC trace. We use the algorithm with an initial condition

*E*

_{g}(

*τ*)=sech(

*τ*/

*τ*

^{p})exp[

*πiαϕ*

_{r}], where

*ϕ*

_{r}is a vector of uniformly distributed random numbers between zero and one, and

*α*is a real constant. Two different reconstructions, color coded green and blue, are shown in Fig. 4. The two reconstructions differ in the values of

*α*and the random vector

*ϕ*

_{r}. The values of

*α*for the blue and green reconstructions are 0.1 and 0.8 respectively. Both reconstructions have very similar MOSAIC traces as the input trace and they are nearly identical over four orders of magnitude. Such MOSAIC traces would be indiscernible experimentally.

*ϕ*

_{r}(and keeping

*α*=0.1), several pulses with different higher order chirp but nearly the same intensity profiles can be generated. This example suggests that there can be significant errors in the chirp measurement even when the pulse amplitude used to fit the MOSAIC is incorrect by only a few percent. We wish to point out that the differences in chirp between the original and the reconstructions are significant compared to the phase accuracy (typically 0.02 rad within the intensity full width half maximum of the pulse) of techniques like FROG.

## 3. Ambiguity free reconstruction

**25**, 1225–33 (1989). [CrossRef]

*Ẽ*(Ω)|. When the pulse spectrum is known, reconstructing the unknown phase of the pulse spectrum completely characterizes the pulse. A Gerchberg-Saxton like approach to recover the phase was already presented in Ref. [15

**25**, 1225–33 (1989). [CrossRef]

*ϕ*

_{g}(Ω), in place of the missing phase of |

*Ẽ*(Ω)| and obtain the time domain electric field

*E*

_{g}(

*τ*). Using this field, we obtain estimates of intensity, |

*Ĩ*

_{g}(Ω)|, and second harmonic, |

*Ẽ*

^{2}(Ω)|, spectra. The guess phase,

*ϕ*

_{g}(Ω), is updated using Powell’s minimization [17] approach such that the error,

*E*

_{i}(

*τ*)). This reconstruction was performed using 128 element vectors for |

*Ẽ*(Ω)|, |

*Ĩ*(Ω)|, and |

*Ẽ*

^{2}(Ω)|.

*η*, similar to Ref. [14

**31**, 122–4 (2006). [CrossRef] [PubMed]

*Ẽ*(Ω) is plotted in Fig. 6(c). The reconstruction does not resolve the time direction and hence

*Ẽ**(Ω) (that is

*E**(-

*τ*)) is also a possible solution.

## 4. Conclusions

## References and links

1. | J. C. M. Diels, J. J. Fontaine, I. C. McMichael, and F. Simoni, “Control and measurement of ultrashort pulse shapes (in amplitude and phase) with femtosecond accuracy.” Appl. Opt. |

2. | C. Yan and J. C. Diels, “Amplitude and phase recording of ultrashort pulses.” J. Opt. Soc. Am. B |

3. | C. Spielmann, L. Xu, and F. Krausz, “Measurement of interferometric autocorrelations: comment.” Appl. Opt. |

4. | T. Hirayama and M. Sheik-Bahae, “Real-time chirp diagnostic for ultrashort laser pulses.” Opt. Lett. |

5. | D. J. Kane and R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating.” IEEE J. Quantum Electron. |

6. | P. O’Shea, M. Kimmel, X. Gu, and R. Trebino, “Highly simplified device for ultrashort-pulse measurement.” Opt. Lett. |

7. | G. Stibenz and G. Steinmeyer, “Interferometric frequency-resolved optical gating,” Opt. Express |

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

9. | J. K. Rhee, T. S. Sosnowski, A. C. Tien, and T. B. Norris, “Real-time dispersion analyzer of femtosecond laser pulses with use of a spectrally and temporally resolved upconversion technique,” J. Opt. Soc. Am. B |

10. | J. W. Nicholson, J. Jasapara, W. Rudolph, F. G. Omenetto, and A. J. Taylor, “Full-field characterization of femtosecond pulses by spectrum and cross-correlation measurements.” Opt. Lett. |

11. | R. G. M. P. Koumans and A. Yariv, “Time-resolved optical gating based on dispersive propagation: a new method to characterize optical pulses.” IEEE J. Quantum Electron. |

12. | A. K. Sharma, P. A. Naik, and P. D. Gupta, “Estimation of higher order chirp in ultrashort laser pulses using modified spectrum auto-interferometric correlation,” Opt. Commun. |

13. | A. K. Sharma, M. Raghuramaiah, P. A. Naik, and P. D. Gupta, “Use of commercial grade light emitting diode in auto-correlation measurements of femtosecond and picosecond laser pulses at 1054 nm.” Opt. Commun. |

14. | D. A. Bender, M. P. Hasselbeck, and M. Sheik-Bahae, “Sensitive ultrashort pulse chirp measurement.” Opt. Lett. |

15. | K. Naganuma, K. Modi, and H. Yamada, “General method for ultrashort light pulse chirp measurement.” IEEE J. Quantum Electron. |

16. | R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures.” Optik |

17. | W. Press, B. Flannery, S. Teukosky, and W. Vetterling, |

18. | For a 64 element matrix, the phase reconstruction required six seconds of computation in IDL on a Pentium M 1.6 GHz processor. |

**OCIS Codes**

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(320.7100) Ultrafast optics : Ultrafast measurements

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: June 30, 2006

Revised Manuscript: August 28, 2006

Manuscript Accepted: September 1, 2006

Published: September 18, 2006

**Citation**

B. Yellampalle, R. D. Averitt, and A. J. Taylor, "Unambiguous chirp characterization using modified-spectrum auto-interferometric correlation and pulse spectrum," Opt. Express **14**, 8890-8899 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-19-8890

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### References

- J. C. M. Diels, J. J. Fontaine, I. C. McMichael, and F. Simoni, "Control and measurement of ultrashort pulse shapes (in amplitude and phase) with femtosecond accuracy," Appl. Opt. 24, 1270 - 82 (1985). [CrossRef] [PubMed]
- C. Yan and J. C. Diels, "Amplitude and phase recording of ultrashort pulses," J. Opt. Soc. Am. B 8, 1259 - 1263 (1991). [CrossRef]
- C. Spielmann, L. Xu, and F. Krausz, "Measurement of interferometric autocorrelations: comment," Appl. Opt. 36, 2523 - 2525 (1997). [CrossRef] [PubMed]
- T. Hirayama and M. Sheik-Bahae, "Real-time chirp diagnostic for ultrashort laser pulses," Opt. Lett. 27, 860 -862 (2002). [CrossRef]
- D. J. Kane and R. Trebino, "Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating," IEEE J. Quantum Electron. 29, 571 - 579 (1993). [CrossRef]
- P. O’Shea, M. Kimmel, X. Gu, and R. Trebino, "Highly simplified device for ultrashort-pulse measurement," Opt. Lett. 26, 932 - 934 (2001). [CrossRef]
- G. Stibenz and G. Steinmeyer, "Interferometric frequency-resolved optical gating," Opt. Express 13, 2617 - 2626 (2005). [CrossRef] [PubMed]
- 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]
- J. K. Rhee, T. S. Sosnowski, A. C. Tien, and T. B. Norris, "Real-time dispersion analyzer of femtosecond laser pulses with use of a spectrally and temporally resolved upconversion technique," J. Opt. Soc. Am. B 13, 1780 - 1785 (1996). [CrossRef]
- J. W. Nicholson, J. Jasapara, W. Rudolph, F. G. Omenetto, and A. J. Taylor, "Full-field characterization of femtosecond pulses by spectrum and cross-correlation measurements," Opt. Lett. 24, 1774 - 1776 (1999). [CrossRef]
- R. G. M. P. Koumans and A. Yariv, "Time-resolved optical gating based on dispersive propagation: a new method to characterize optical pulses," IEEE J. Quantum Electron. 36, 137 - 144 (2000). [CrossRef]
- A. K. Sharma, P. A. Naik, and P. D. Gupta, "Estimation of higher order chirp in ultrashort laser pulses using modified spectrum auto-interferometric correlation," Opt. Commun. 233, 431 - 437 (2004). [CrossRef]
- A. K. Sharma, M. Raghuramaiah, P. A. Naik, and P. D. Gupta, "Use of commercial grade light emitting diode in auto-correlation measurements of femtosecond and picosecond laser pulses at 1054 nm," Opt. Commun. 246, 195 - 204 (2005). [CrossRef]
- D. A. Bender, M. P. Hasselbeck, and M. Sheik-Bahae, "Sensitive ultrashort pulse chirp measurement," Opt. Lett. 31, 122 - 124 (2006). [CrossRef] [PubMed]
- K. Naganuma, K. Modi, and H. Yamada, "General method for ultrashort light pulse chirp measurement," IEEE J. Quantum Electron. 25, 1225 - 1233 (1989). [CrossRef]
- R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237 - 246 (1972).
- W. Press, B. Flannery, S. Teukosky, andW. Vetterling, Numerical Recepies in C - The Art of Scientific Computing (Cambridge University Press, Cambridge, 1986).
- For a 64 element matrix, the phase reconstruction required six seconds of computation in IDL on a Pentium M 1.6 GHz processor.

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