Abstract

We provide a detailed discussion of our recently developed pulse characterization technique for few-cycle pulses, called two-dimensional spectral shearing interferometry (2DSI). Based on spectral phase interferometry for direct electric-field reconstruction (SPIDER), 2DSI is relatively simple to implement, but requires no interferometer delay or scan calibration. We show simulations of 2DSI in the presence of noise and find that it retains the favorable noise performance of spectral shearing methods, performing identically to standard SPIDER for a given measurement time. The optimal choice of experimental parameters is discussed, with results applicable to any spectral shearing method. Experimental considerations when building and operating a 2DSI are provided, with results shown for a 4.9 fs pulse, verifying the accuracy and precision of the 2DSI.

© 2010 Optical Society of America

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  1. C. Iaconis and I. Walmsley, “Self-referencing spectral interferometry for measuring ultrashort pulses,” IEEE J. Quantum Electron. 35, 501–509 (1999).
    [CrossRef]
  2. L. Gallmann, D. Sutter, N. Matuschek, G. Steinmeyer, and U. Keller, “Techniques for the characterization of sub-10-fs optical pulses: a comparison,” Appl. Phys. B 70, S67–S75 (2000).
    [CrossRef]
  3. D. Kane and R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” IEEE J. Quantum Electron. 29, 571–579 (1993).
    [CrossRef]
  4. J. R. Birge and F. X. Kärtner, “Analysis and mitigation of systematic errors in spectral shearing interferometry of pulses approaching the single-cycle limit [invited],” J. Opt. Soc. Am. B 25, A111–A119 (2008).
    [CrossRef]
  5. E. Kosik, A. Radunksy, I. Walmsley, and C. Dorrer, “Interferometric technique for measuring broadband ultrashort pulses at the sampling limit,” Opt. Lett. 30, 326–328 (2005).
    [CrossRef] [PubMed]
  6. A. S. Wyatt, I. A. Walmsley, G. Stibenz, and G. Steinmeyer, “Sub-10 fs pulse characterization using spatially encoded arrangement for spectral phase interferometry for direct electric field reconstruction,” Opt. Lett. 31, 1914–1916 (2006).
    [CrossRef] [PubMed]
  7. J. R. Birge, R. Ell, and F. X. Kärtner, “Two-dimensional spectral shearing interferometry for few-cycle pulse characterization,” Opt. Lett. 31, 2063–2065 (2006).
    [CrossRef] [PubMed]
  8. H. M. Crespo, J. R. Birge, M. Y. Sander, E. L. Falçao-Filho, A. Benedick, and F. X. Kärtner, “Phase stabilization of sub-two-cycle pulses from prismless octave-spanning Ti:sapphire lasers,” J. Opt. Soc. Am. B 25, B147–B154 (2008).
    [CrossRef]
  9. K. F. Lee, K. J. Kubarych, A. Bonvalet, and M. Joffre, “Characterization of mid-infrared femtosecond pulses [invited],” J. Opt. Soc. Am. B 25, A54–A62 (2008).
    [CrossRef]
  10. P. Baum and E. Riedle, “Design and calibration of zero-additional-phase spider,” J. Opt. Soc. Am. B 22, 1875–1883 (2005).
    [CrossRef]
  11. J. R. Birge, “Methods for engineering few-cycle mode-locked lasers,” Ph.D. dissertation (MIT, 2009).
  12. F.X.Kärtner, ed., Few-Cycle Laser Pulse Generation and its Applications (Springer, 2004).
  13. C. Dorrer, P. Londero, and I. A. Walmsley, “Homodyne detection in spectral phase interferometry for direct electric-field reconstruction,” Opt. Lett. 26, 1510–1512 (2001).
    [CrossRef]
  14. S.-P. Gorza, P. Wasylczyk, and I. A. Walmsley, “Spectral shearing interferometry with spatially chirped replicas for measuring ultrashort pulses,” Opt. Express 15, 15168–15174 (2007).
    [CrossRef] [PubMed]
  15. 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]
  16. C. Dorrer, “Influence of the calibration of the detector on spectral interferometry,” J. Opt. Soc. Am. B 16, 1160–1168 (1999).
    [CrossRef]
  17. 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]
  18. V. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals and its applications,” J. Chem. Phys. 107, 6756–6769 (1997).
    [CrossRef]
  19. C. Dorrer and I. 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]
  20. D. R. Austin, T. Witting, and I. A. Walmsley, “High precision self-referenced phase retrieval of complex pulses with multiple-shearing spectral interferometry,” J. Opt. Soc. Am. B 26, 1818–1830 (2009).
    [CrossRef]
  21. H. M. Crespo, J. R. Birge, E. L. Falçao-Filho, M. Y. Sander, A. Benedick, and F. X. Kärtner, “Non-intrusive phase-stabilization of sub-two-cycle pulses from a prismless octave-spanning Ti:sapphire laser,” Opt. Lett. 33, 833–835 (2008).
    [CrossRef] [PubMed]
  22. J. R. Birge, H. M. Crespo, M. Sander, and F. X. Kärtner, “Non-intrusive sub-two-cycle carrier-envelope stabilized pulses using engineered chirped mirrors,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest (CD) (Optical Society of America, 2008), paper CTuC3.
  23. M. Y. Sander, J. R. Birge, A. Benedick, H. M. Crespo, and F. X. Kärtner, “Dynamics of dispersion managed octave-spanning titanium:sapphire lasers,” J. Opt. Soc. Am. B 26, 743–749 (2009).
    [CrossRef]

2009 (2)

2008 (4)

2007 (1)

2006 (2)

2005 (2)

2002 (1)

2001 (1)

2000 (1)

L. Gallmann, D. Sutter, N. Matuschek, G. Steinmeyer, and U. Keller, “Techniques for the characterization of sub-10-fs optical pulses: a comparison,” Appl. Phys. B 70, S67–S75 (2000).
[CrossRef]

1999 (2)

C. Dorrer, “Influence of the calibration of the detector on spectral interferometry,” J. Opt. Soc. Am. B 16, 1160–1168 (1999).
[CrossRef]

C. Iaconis and I. Walmsley, “Self-referencing spectral interferometry for measuring ultrashort pulses,” IEEE J. Quantum Electron. 35, 501–509 (1999).
[CrossRef]

1998 (1)

1997 (1)

V. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals and its applications,” J. Chem. Phys. 107, 6756–6769 (1997).
[CrossRef]

1993 (1)

D. Kane and R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” IEEE J. Quantum Electron. 29, 571–579 (1993).
[CrossRef]

1982 (1)

Austin, D. R.

Baum, P.

Benedick, A.

Birge, J. R.

M. Y. Sander, J. R. Birge, A. Benedick, H. M. Crespo, and F. X. Kärtner, “Dynamics of dispersion managed octave-spanning titanium:sapphire lasers,” J. Opt. Soc. Am. B 26, 743–749 (2009).
[CrossRef]

H. M. Crespo, J. R. Birge, E. L. Falçao-Filho, M. Y. Sander, A. Benedick, and F. X. Kärtner, “Non-intrusive phase-stabilization of sub-two-cycle pulses from a prismless octave-spanning Ti:sapphire laser,” Opt. Lett. 33, 833–835 (2008).
[CrossRef] [PubMed]

J. R. Birge and F. X. Kärtner, “Analysis and mitigation of systematic errors in spectral shearing interferometry of pulses approaching the single-cycle limit [invited],” J. Opt. Soc. Am. B 25, A111–A119 (2008).
[CrossRef]

H. M. Crespo, J. R. Birge, M. Y. Sander, E. L. Falçao-Filho, A. Benedick, and F. X. Kärtner, “Phase stabilization of sub-two-cycle pulses from prismless octave-spanning Ti:sapphire lasers,” J. Opt. Soc. Am. B 25, B147–B154 (2008).
[CrossRef]

J. R. Birge, R. Ell, and F. X. Kärtner, “Two-dimensional spectral shearing interferometry for few-cycle pulse characterization,” Opt. Lett. 31, 2063–2065 (2006).
[CrossRef] [PubMed]

J. R. Birge, “Methods for engineering few-cycle mode-locked lasers,” Ph.D. dissertation (MIT, 2009).

J. R. Birge, H. M. Crespo, M. Sander, and F. X. Kärtner, “Non-intrusive sub-two-cycle carrier-envelope stabilized pulses using engineered chirped mirrors,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest (CD) (Optical Society of America, 2008), paper CTuC3.

Bonvalet, A.

Crespo, H. M.

Dorrer, C.

Ell, R.

Falçao-Filho, E. L.

Gallmann, L.

L. Gallmann, D. Sutter, N. Matuschek, G. Steinmeyer, and U. Keller, “Techniques for the characterization of sub-10-fs optical pulses: a comparison,” Appl. Phys. B 70, S67–S75 (2000).
[CrossRef]

Gorza, S. -P.

Iaconis, C.

C. Iaconis and I. Walmsley, “Self-referencing spectral interferometry for measuring ultrashort pulses,” IEEE J. Quantum Electron. 35, 501–509 (1999).
[CrossRef]

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]

Ina, H.

Joffre, M.

Kane, D.

D. Kane and R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” IEEE J. Quantum Electron. 29, 571–579 (1993).
[CrossRef]

Kärtner, F. X.

M. Y. Sander, J. R. Birge, A. Benedick, H. M. Crespo, and F. X. Kärtner, “Dynamics of dispersion managed octave-spanning titanium:sapphire lasers,” J. Opt. Soc. Am. B 26, 743–749 (2009).
[CrossRef]

H. M. Crespo, J. R. Birge, M. Y. Sander, E. L. Falçao-Filho, A. Benedick, and F. X. Kärtner, “Phase stabilization of sub-two-cycle pulses from prismless octave-spanning Ti:sapphire lasers,” J. Opt. Soc. Am. B 25, B147–B154 (2008).
[CrossRef]

J. R. Birge and F. X. Kärtner, “Analysis and mitigation of systematic errors in spectral shearing interferometry of pulses approaching the single-cycle limit [invited],” J. Opt. Soc. Am. B 25, A111–A119 (2008).
[CrossRef]

H. M. Crespo, J. R. Birge, E. L. Falçao-Filho, M. Y. Sander, A. Benedick, and F. X. Kärtner, “Non-intrusive phase-stabilization of sub-two-cycle pulses from a prismless octave-spanning Ti:sapphire laser,” Opt. Lett. 33, 833–835 (2008).
[CrossRef] [PubMed]

J. R. Birge, R. Ell, and F. X. Kärtner, “Two-dimensional spectral shearing interferometry for few-cycle pulse characterization,” Opt. Lett. 31, 2063–2065 (2006).
[CrossRef] [PubMed]

J. R. Birge, H. M. Crespo, M. Sander, and F. X. Kärtner, “Non-intrusive sub-two-cycle carrier-envelope stabilized pulses using engineered chirped mirrors,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest (CD) (Optical Society of America, 2008), paper CTuC3.

Keller, U.

L. Gallmann, D. Sutter, N. Matuschek, G. Steinmeyer, and U. Keller, “Techniques for the characterization of sub-10-fs optical pulses: a comparison,” Appl. Phys. B 70, S67–S75 (2000).
[CrossRef]

Kobayashi, S.

Kosik, E.

Kubarych, K. J.

Lee, K. F.

Londero, P.

Mandelshtam, V. A.

V. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals and its applications,” J. Chem. Phys. 107, 6756–6769 (1997).
[CrossRef]

Matuschek, N.

L. Gallmann, D. Sutter, N. Matuschek, G. Steinmeyer, and U. Keller, “Techniques for the characterization of sub-10-fs optical pulses: a comparison,” Appl. Phys. B 70, S67–S75 (2000).
[CrossRef]

Radunksy, A.

Riedle, E.

Sander, M.

J. R. Birge, H. M. Crespo, M. Sander, and F. X. Kärtner, “Non-intrusive sub-two-cycle carrier-envelope stabilized pulses using engineered chirped mirrors,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest (CD) (Optical Society of America, 2008), paper CTuC3.

Sander, M. Y.

Steinmeyer, G.

A. S. Wyatt, I. A. Walmsley, G. Stibenz, and G. Steinmeyer, “Sub-10 fs pulse characterization using spatially encoded arrangement for spectral phase interferometry for direct electric field reconstruction,” Opt. Lett. 31, 1914–1916 (2006).
[CrossRef] [PubMed]

L. Gallmann, D. Sutter, N. Matuschek, G. Steinmeyer, and U. Keller, “Techniques for the characterization of sub-10-fs optical pulses: a comparison,” Appl. Phys. B 70, S67–S75 (2000).
[CrossRef]

Stibenz, G.

Sutter, D.

L. Gallmann, D. Sutter, N. Matuschek, G. Steinmeyer, and U. Keller, “Techniques for the characterization of sub-10-fs optical pulses: a comparison,” Appl. Phys. B 70, S67–S75 (2000).
[CrossRef]

Takeda, M.

Taylor, H. S.

V. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals and its applications,” J. Chem. Phys. 107, 6756–6769 (1997).
[CrossRef]

Trebino, R.

D. Kane and R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” IEEE J. Quantum Electron. 29, 571–579 (1993).
[CrossRef]

Walmsley, I.

Walmsley, I. A.

Wasylczyk, P.

Witting, T.

Wyatt, A. S.

Appl. Phys. B (1)

L. Gallmann, D. Sutter, N. Matuschek, G. Steinmeyer, and U. Keller, “Techniques for the characterization of sub-10-fs optical pulses: a comparison,” Appl. Phys. B 70, S67–S75 (2000).
[CrossRef]

IEEE J. Quantum Electron. (2)

D. Kane and R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” IEEE J. Quantum Electron. 29, 571–579 (1993).
[CrossRef]

C. Iaconis and I. Walmsley, “Self-referencing spectral interferometry for measuring ultrashort pulses,” IEEE J. Quantum Electron. 35, 501–509 (1999).
[CrossRef]

J. Chem. Phys. (1)

V. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals and its applications,” J. Chem. Phys. 107, 6756–6769 (1997).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (8)

C. Dorrer and I. 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]

P. Baum and E. Riedle, “Design and calibration of zero-additional-phase spider,” J. Opt. Soc. Am. B 22, 1875–1883 (2005).
[CrossRef]

C. Dorrer, “Influence of the calibration of the detector on spectral interferometry,” J. Opt. Soc. Am. B 16, 1160–1168 (1999).
[CrossRef]

K. F. Lee, K. J. Kubarych, A. Bonvalet, and M. Joffre, “Characterization of mid-infrared femtosecond pulses [invited],” J. Opt. Soc. Am. B 25, A54–A62 (2008).
[CrossRef]

J. R. Birge and F. X. Kärtner, “Analysis and mitigation of systematic errors in spectral shearing interferometry of pulses approaching the single-cycle limit [invited],” J. Opt. Soc. Am. B 25, A111–A119 (2008).
[CrossRef]

H. M. Crespo, J. R. Birge, M. Y. Sander, E. L. Falçao-Filho, A. Benedick, and F. X. Kärtner, “Phase stabilization of sub-two-cycle pulses from prismless octave-spanning Ti:sapphire lasers,” J. Opt. Soc. Am. B 25, B147–B154 (2008).
[CrossRef]

M. Y. Sander, J. R. Birge, A. Benedick, H. M. Crespo, and F. X. Kärtner, “Dynamics of dispersion managed octave-spanning titanium:sapphire lasers,” J. Opt. Soc. Am. B 26, 743–749 (2009).
[CrossRef]

D. R. Austin, T. Witting, and I. A. Walmsley, “High precision self-referenced phase retrieval of complex pulses with multiple-shearing spectral interferometry,” J. Opt. Soc. Am. B 26, 1818–1830 (2009).
[CrossRef]

Opt. Express (1)

Opt. Lett. (6)

Other (3)

J. R. Birge, H. M. Crespo, M. Sander, and F. X. Kärtner, “Non-intrusive sub-two-cycle carrier-envelope stabilized pulses using engineered chirped mirrors,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest (CD) (Optical Society of America, 2008), paper CTuC3.

J. R. Birge, “Methods for engineering few-cycle mode-locked lasers,” Ph.D. dissertation (MIT, 2009).

F.X.Kärtner, ed., Few-Cycle Laser Pulse Generation and its Applications (Springer, 2004).

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Figures (8)

Fig. 1
Fig. 1

Frequency domain block diagram of 2DSI process.

Fig. 2
Fig. 2

Experimental schematic of 2DSI setup.

Fig. 3
Fig. 3

Top: 2DSI phase matching plot for Type II sum frequency generation for a 50 μ m thick BBO cut to measure a typical few-cycle Ti:sapphire laser, assuming a beam crossing angle of 17°. The lined areas denote the phase matched regions, with the lines denoting efficiency differences of 10%. Bottom: Slices of the phase matching curves for two upconversion wavelengths separated by 6 THz, showing the efficiency of upconversion for the two spectrally sheared components.

Fig. 4
Fig. 4

2DSI interferograms from 5 fs laser, plot with respect to the wavelength of the fundamental pulse (i.e., before upconversion): (a) extracted spectral GD overlaid to demonstrate the interpretation of fringe offset, and (b) the same pulse after dispersion by approximately 1 mm of fused silica. The presence of extra dispersion is evident in the interferogram. The color scale is arbitrary. The y-axis is shown calibrated in terms of both the actual mirror translation, as well as the corresponding implied spectral GD computed by dividing the phase of the fringe by the shear frequency Ω (taken from [7]).

Fig. 5
Fig. 5

Illustration of the interference fringes in 2DSI (top left) and SPIDER (bottom left), for a hypothetical Gaussian pulse with second- and third-order dispersions and 1% satellite pulses. The Fourier transforms of both fringes are shown on the right, illustrating that the delay scanning in 2DSI serves to move the usual SPIDER sidebands into a new dimension, thus minimizing required spectrometer resolution.

Fig. 6
Fig. 6

Simulated 2DSI spectrogram (top) measured with 64 phase steps for a sinc pulse with second- and third-order dispersions and a satellite, in the presence of additive and shot noise such that the resulting SNR per sample is 0.5. A line-out showing the fringe at a single frequency is inset. A sample reconstruction, including comparison with SPIDER is shown in the middle frame. The bottom frame shows the standard deviation of the phase measurement for both SPIDER and 2DSI, showing that the lack of delay calibration in 2DSI yields a factor of 2 improvement in noise performance.

Fig. 7
Fig. 7

(a) Spectrum of the 5 fs laser used in the test; (b) extracted GD both with and without glass slide; (c) phase of glass slide as measured by 2DSI and as predicted by known glass dispersion; (d) net phase delay error in glass dispersion measurement.

Fig. 8
Fig. 8

(a) Raw 2DSI data; (b) comparison of IAC and that predicted from the 2DSI measurement; (c) extracted spectral phase (dashed curve); (d) reconstructed pulse (solid curve), simulated pulse (dotted curve), and temporal phase (dotted curve).

Equations (13)

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I ( ω ) = | A ( ω ) | 2 + | A ( ω Ω ) | 2 + 2 | A ( ω ) A ( ω Ω ) | cos [ τ ω + ϕ ( ω ) ϕ ( ω Ω ) ] .
δ τ < 0.64 Ω Δ ω 2 ,
I ( ω , φ ) = | A ( ω ) + A ( ω Ω ) e i φ | 2 = 2 | A ( ω ) A ( ω Ω ) | cos [ φ + ϕ ( ω ) ϕ ( ω Ω ) τ g ( ω Ω / 2 ) Ω + O [ Ω 2 ] ] + D . C . ,
| ϵ r ( ω ) | Δ ν n + Δ ν ,
T 2 π Ω .
D 2 T p Ω .
f n 0 2 π d x   cos [ ( n + Δ ν ) x + ϕ ] e i n x .
ϕ ext = arctan I f n R f n
= arctan [ n   tan [ π Δ ν + ϕ ] n + δ ν ] .
ϵ r ( ω ) = 1 | ϕ ext ϕ | ω
= 1 n ( n + Δ ν ) sec [ π Δ ν + ϕ ] 2 ( n + Δ ν ) 2 + n 2   tan [ π Δ ν + ϕ ( ω ) ] 2 .
2 n Δ ν ( n + Δ ν ) ( 2 n + Δ ν ) sec [ π Δ ν + ϕ ] 2 tan [ π Δ ν + ϕ ] ( ( n + Δ ν ) 2 + n 2   tan [ π Δ ν + ϕ ] 2 ) 2 = 0.
| ϵ r ( ω ) | Δ ν n + Δ ν .

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