Abstract

We propose a procedure for the comparison of two ultrashort optical pulses based on the root-mean-square field error. This procedure supplies a practical criterion that allows one to test the accuracy of pulse characterization techniques. We apply this procedure to spectral phase interferometry for direct electric field reconstruction (SPIDER), a method that uses shearing interferometry in the frequency domain to measure ultrashort light pulses. Two novel reconstruction algorithms are described in detail, one based on the reconstruction of the spectral phase by concatenation, the other by integration. The range of validity of these algorithms and the influence of noise on the accuracy of the reconstruction are studied using the proposed criterion.

© 2002 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics 6th ed. (Cambridge U. Press, Cambridge, UK, 1980).
  2. V. Wong and I. A. Walmsley, “Linear filter analysis of methods for ultrashort pulse shape measurements,” J. Opt. Soc. Am. B 12, 1491–1499 (1995).
    [CrossRef]
  3. A. Eskicioglu and P. Fisher, “Image quality measures and their performance,” IEEE Trans. Commun. 43, 2959–2965 (1995).
    [CrossRef]
  4. D. N. Fittinghoff, K. W. Delong, R. Trebino, and C. L. Ladera, “Noise sensitivity in frequency-resolved-optical-gating measurements of ultrashort optical pulses,” J. Opt. Soc. Am. B 12, 1955–1967 (1995).
    [CrossRef]
  5. S. Yeremenko, A. Baltuska, M. S. Pshenichnikov, and D. A. Wiersma, “The criterion of pulse reconstruction quality based on Wigner representation,” Appl. Phys. B 70, S109–S117 (2000).
    [CrossRef]
  6. 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]
  7. L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, U. Keller, C. Iaconis, and I. A. Walmsley, “Characterization of sub-6-fs optical pulses with spectral phase interferometry for direct electric field reconstruction,” Opt. Lett. 24, 1314–1316 (1999).
    [CrossRef]
  8. C. Dorrer, B. de Beauvoir, C. Le Blanc, S. Ranc, J. P. Rousseau, P. Rousseau, and J. P. Chambaret, “Single-shot real-time characterization of chirped-pulse amplification systems by spectral phase interferometry for direct electric-field reconstruction,” Opt. Lett. 24, 1644–1646 (1999).
    [CrossRef]
  9. C. Dorrer and I. A. Walmsley, “Precision and consistency criteria for spectral phase interferometry for direct electric-field reconstruction,” J. Opt. Soc. Am. B 19, 1030–1038 (2002).
    [CrossRef]
  10. L. Gallmann, D. H. 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]
  11. P. Dietrich, F. Krausz, and P. B. Corkum, “Determining the absolute carrier phase of a few-cycle laser pulse,” Opt. Lett. 25, 16–18 (2000).
    [CrossRef]
  12. M. Kakehata, H. Takada, Y. Kobayashi, K. Torizuka, Y. Fujihira, T. Homma, and H. Takahashi, “Single-shot measurement of carrier-envelope phase changes by spectral interferometry,” Opt. Lett. 26, 1436–1438 (2001).
    [CrossRef]
  13. J. R. Fienup, “Invariant error metrics for image reconstruction,” Appl. Opt. 36, 8352–8357 (1997).
    [CrossRef]
  14. 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 spectral phase interferometry for direct electric field reconstruction,” Appl. Phys. B 70, S85–S93 (2000).
    [CrossRef]
  15. J. Paye, “The chronocyclic representation of ultrashort light pulses,” J. Opt. Soc. Am. B 28, 2262–2273 (1992).
  16. C. Dorrer, N. Belabas, J. P. Likforman, and M. Joffre, “Spectral resolution and sampling issues in Fourier-transform spectral interferometry,” J. Opt. Soc. Am. B 17, 1795–1802 (2000).
    [CrossRef]
  17. L. Lepetit, G. Cheriaux, and M. Joffre, “Linear techniques of phase measurement by femtosecond spectral interferometry for applications in spectroscopy,” J. Opt. Soc. Am. B 12, 2467–2474 (1995).
    [CrossRef]
  18. C. Dorrer, “Influence of the calibration of the detector on spectral interferometry,” J. Opt. Soc. Am. B 16, 1160–1168 (1999).
    [CrossRef]
  19. C. Dorrer, “Implementation of spectral phase interferometry for direct electric-field reconstruction with a simultaneously recorded reference interferogram,” Opt. Lett. 24, 1532–1534 (1999).
    [CrossRef]
  20. M. Murty, “Lateral shearing interferometers” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978).

2002 (1)

2001 (1)

2000 (5)

C. Dorrer, N. Belabas, J. P. Likforman, and M. Joffre, “Spectral resolution and sampling issues in Fourier-transform spectral interferometry,” J. Opt. Soc. Am. B 17, 1795–1802 (2000).
[CrossRef]

S. Yeremenko, A. Baltuska, M. S. Pshenichnikov, and D. A. Wiersma, “The criterion of pulse reconstruction quality based on Wigner representation,” Appl. Phys. B 70, S109–S117 (2000).
[CrossRef]

L. Gallmann, D. H. 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]

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 spectral phase interferometry for direct electric field reconstruction,” Appl. Phys. B 70, S85–S93 (2000).
[CrossRef]

P. Dietrich, F. Krausz, and P. B. Corkum, “Determining the absolute carrier phase of a few-cycle laser pulse,” Opt. Lett. 25, 16–18 (2000).
[CrossRef]

1999 (4)

1998 (1)

1997 (1)

1995 (4)

1992 (1)

J. Paye, “The chronocyclic representation of ultrashort light pulses,” J. Opt. Soc. Am. B 28, 2262–2273 (1992).

Anderson, M. E.

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 spectral phase interferometry for direct electric field reconstruction,” Appl. Phys. B 70, S85–S93 (2000).
[CrossRef]

Baltuska, A.

S. Yeremenko, A. Baltuska, M. S. Pshenichnikov, and D. A. Wiersma, “The criterion of pulse reconstruction quality based on Wigner representation,” Appl. Phys. B 70, S109–S117 (2000).
[CrossRef]

Belabas, N.

Chambaret, J. P.

Cheriaux, G.

Corkum, P. B.

de Araujo, L. E. E.

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 spectral phase interferometry for direct electric field reconstruction,” Appl. Phys. B 70, S85–S93 (2000).
[CrossRef]

de Beauvoir, B.

Delong, K. W.

Dietrich, P.

Dorrer, C.

Eskicioglu, A.

A. Eskicioglu and P. Fisher, “Image quality measures and their performance,” IEEE Trans. Commun. 43, 2959–2965 (1995).
[CrossRef]

Fienup, J. R.

Fisher, P.

A. Eskicioglu and P. Fisher, “Image quality measures and their performance,” IEEE Trans. Commun. 43, 2959–2965 (1995).
[CrossRef]

Fittinghoff, D. N.

Fujihira, Y.

Gallmann, L.

L. Gallmann, D. H. 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]

L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, U. Keller, C. Iaconis, and I. A. Walmsley, “Characterization of sub-6-fs optical pulses with spectral phase interferometry for direct electric field reconstruction,” Opt. Lett. 24, 1314–1316 (1999).
[CrossRef]

Homma, T.

Iaconis, C.

Joffre, M.

Kakehata, M.

Keller, U.

L. Gallmann, D. H. 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]

L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, U. Keller, C. Iaconis, and I. A. Walmsley, “Characterization of sub-6-fs optical pulses with spectral phase interferometry for direct electric field reconstruction,” Opt. Lett. 24, 1314–1316 (1999).
[CrossRef]

Kobayashi, Y.

Kosik, E. M.

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 spectral phase interferometry for direct electric field reconstruction,” Appl. Phys. B 70, S85–S93 (2000).
[CrossRef]

Krausz, F.

Ladera, C. L.

Le Blanc, C.

Lepetit, L.

Likforman, J. P.

Matuschek, N.

L. Gallmann, D. H. 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]

L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, U. Keller, C. Iaconis, and I. A. Walmsley, “Characterization of sub-6-fs optical pulses with spectral phase interferometry for direct electric field reconstruction,” Opt. Lett. 24, 1314–1316 (1999).
[CrossRef]

Paye, J.

J. Paye, “The chronocyclic representation of ultrashort light pulses,” J. Opt. Soc. Am. B 28, 2262–2273 (1992).

Pshenichnikov, M. S.

S. Yeremenko, A. Baltuska, M. S. Pshenichnikov, and D. A. Wiersma, “The criterion of pulse reconstruction quality based on Wigner representation,” Appl. Phys. B 70, S109–S117 (2000).
[CrossRef]

Ranc, S.

Rousseau, J. P.

Rousseau, P.

Steinmeyer, G.

L. Gallmann, D. H. 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]

L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, U. Keller, C. Iaconis, and I. A. Walmsley, “Characterization of sub-6-fs optical pulses with spectral phase interferometry for direct electric field reconstruction,” Opt. Lett. 24, 1314–1316 (1999).
[CrossRef]

Sutter, D. H.

L. Gallmann, D. H. 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]

L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, U. Keller, C. Iaconis, and I. A. Walmsley, “Characterization of sub-6-fs optical pulses with spectral phase interferometry for direct electric field reconstruction,” Opt. Lett. 24, 1314–1316 (1999).
[CrossRef]

Takada, H.

Takahashi, H.

Torizuka, K.

Trebino, R.

Walmsley, I. A.

Wiersma, D. A.

S. Yeremenko, A. Baltuska, M. S. Pshenichnikov, and D. A. Wiersma, “The criterion of pulse reconstruction quality based on Wigner representation,” Appl. Phys. B 70, S109–S117 (2000).
[CrossRef]

Wong, V.

Yeremenko, S.

S. Yeremenko, A. Baltuska, M. S. Pshenichnikov, and D. A. Wiersma, “The criterion of pulse reconstruction quality based on Wigner representation,” Appl. Phys. B 70, S109–S117 (2000).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (3)

S. Yeremenko, A. Baltuska, M. S. Pshenichnikov, and D. A. Wiersma, “The criterion of pulse reconstruction quality based on Wigner representation,” Appl. Phys. B 70, S109–S117 (2000).
[CrossRef]

L. Gallmann, D. H. 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]

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 spectral phase interferometry for direct electric field reconstruction,” Appl. Phys. B 70, S85–S93 (2000).
[CrossRef]

IEEE Trans. Commun. (1)

A. Eskicioglu and P. Fisher, “Image quality measures and their performance,” IEEE Trans. Commun. 43, 2959–2965 (1995).
[CrossRef]

J. Opt. Soc. Am. B (7)

Opt. Lett. (6)

Other (2)

M. Born and E. Wolf, Principles of Optics 6th ed. (Cambridge U. Press, Cambridge, UK, 1980).

M. Murty, “Lateral shearing interferometers” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978).

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Figures (8)

Fig. 1
Fig. 1

(a) Spectrum (solid curve) and spectral phases (dashed curves) used in the simulations. (b) Temporal intensity for the flat phase, the parabolic phase, the cubic phase, and the phase jump, from top to bottom; in the time domain, the plotted window is the Nyquist window corresponding to the SPIDER shear.

Fig. 2
Fig. 2

Difference between the test and the reconstructed phases in the case of the flat phase (three lower curves), the parabolic phase (three middle curves), and the cubic phase (three upper curves). In each case, the dashed curve corresponds to I1, the dotted curve to I2 and the solid curve to the concatenation technique.

Fig. 3
Fig. 3

Difference between the test and the reconstructed phases by the concatenation technique (solid curves) and the integration technique I1 (dashed curves) in the case of the second- and third-order phases corresponding to fields not satisfying the Nyquist limit.

Fig. 4
Fig. 4

Difference between the test and the reconstructed phases in the case of a fourth-order polynomial phase corresponding to a field not satisfying the Nyquist limit for the concatenation technique (solid curve), the integration technique I1 (dashed curve) and the integration technique I2 (dotted curve). The small systematic error present when I1 is used disappears when I2 is used.

Fig. 5
Fig. 5

Difference between the test and the reconstructed phases in the case of a phase jump for the concatenation technique (solid curve), the integration technique I1 (dashed curve) and the integration technique I2 (dotted curve).

Fig. 6
Fig. 6

RMS field error of the reconstruction of a flat phase by the integration technique I1 as a function of the noise fraction. The error for I2 and C exactly overlap with the plotted curve. The higher left-hand inset represents a simulated interferogram with a noise fraction of 10-4; the lower right-hand inset represents a simulated interferogram with a noise fraction of 10-1.

Fig. 7
Fig. 7

Reconstructed phases for a flat test phase for levels of noise of 10-4, 10-3, 10-2, and 10-1, from bottom to top. The corresponding values of the RMS field error are respectively 8×10-3, 3×10-2, 10-2 and 4×10-1.

Fig. 8
Fig. 8

Accuracy of the reconstruction of the phase jump by the integration technique I1 (solid curve), the integration technique I2 (dotted curve), and the concatenation technique (dashed curve) as a function of the noise fraction.

Equations (40)

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E˜(ω)=-+dtE(t)exp(iωt),
E(t)=12π 0+dωE˜(ω)exp(-iωt).
E˜(ω)=-+dtE(t)exp(iωt),
E(t)=12π -+dωE˜(ω)exp(-iωt).
E2=-+dt|E(t)|2=12π -+dω|E˜(ω)|2.
ε=E1-E2=-+dt|E1(t)-E2(t)|21/2=12π -+dω|E˜1(ω)-E˜2(ω)|21/2.
E˜(ω)=|E˜(ω)|exp[iφ(ω)]=[|E˜(ω)|+δ|E˜(ω)|]exp[i(φ(ω)+δφ(ω)],
E˜(ω)=E˜(ω)+[δ|E˜(ω)|+iδφ(ω)|E˜(ω)|]exp[i(φ(ω)].
ε=12π -+dω|δE˜|2+|δφ|2|E˜(ω)|21/2=(δ|E˜|2+δφ|E˜|2)1/2.
εI=-+dt |I(t)-I(t)|21/2,
εφ=-+dt|φ(t)-φ(t)|2I2(t)1/2-+dtI2(t)1/2.
W(t, ω)=-+dtEt+t2E*t-t2exp(iωt)dt=12π -+dωE˜ω+ω2×E˜*ω-ω2exp(-iωt)dω.
Δ(ω0)=-ω0Γ(ω)dω=-ω0[φ(ω+Ω)-φ(ω)]dω=ω0ω0+Ωφ(ω)dω.
ω0ω0+Ωdωφ(ω)jajφ(ω0+αjΩ),
Δ=φm,wherem=jajδ(αjΩ).
ω0ω0+Ωdωφ(ω)=Ωφω0+Ω2.
φ(ω0)=(1/Ω)Δ(ω0-Ω/2),
ω0ω0+Ωdωφ(ω)=ω0ω0+Ωdωφω0+Ω2+ω-ω0-Ω2 φω ω0+Ω2+12 ω-ω0-Ω22 2φω2(f(ω)),
ω0ω0+Ωdωφ(ω)=Ω*φ(ω0+Ω/2)+Ω224O2φω2,
ω0ω0+Ωd ωφ(ω)=Ω6 φ(ω0)+4φω0+Ω2+φ(ω0+Ω).
m(ω)=Ω6 δ(ω)+4δω+Ω2+δ(ω+Ω),
m˜(t)=Ω3 exp-i Ωt2*2+cosΩt2.
φ(0)=0,
φ[(n+1)Ω]=φ(nΩ)+Γ(nΩ).
φ(mδω)=0,
φ[mδω+(n+1)Ω]=φ(mδω+nΩ)+Γ(mδω+nΩ).
Em(nT)=1NJ jE˜(mδω+jΩ)×exp[-i(mδω+jΩ)nT]=exp(-imnδnT)1N jE˜(mδω+jΩ)×exp-i 2πjnN,
E(nT)=1M mEm(nT).
E˜(nΩ)=jE(jT)exp-i 2πnjN.
ε2=E2+E2-12π -+dωE˜(ω)E˜*(ω)+c.c.=2-2 Re12π -+dωE˜(ω)E˜*(ω).
maxφ0,t0 Re12π -+dωE˜(ω)exp[i(φ0+ωt0)]E˜*(ω)
=maxt012π -+dωE˜(ω)exp(iωt0)E˜*(ω)
dW(E, E)=12π -+-+dtdω|W(t, ω)-W(t, ω)|21/2.
dW(E, E)2=12π -+-+dtdω[W(t, ω)2-2W(t, ω)W(t, ω)+W(t, ω)2].
12π-+-+dtdωW(t, ω)W(t, ω)
=-+dtE(t)E*(t)2,
dW(E, E)2=-+dt|E(t)|22-2-+dtE(t)E*(t)2+-+dt|E(t)|22.
dW(E, E)2=2-2-+dtE(t)E*(t)2.
ε=2-212π -+dωE˜(ω)E˜*(ω)1/2=2-2-+dtE(t)E*(t)1/2.
dW(E, E)2=2ε2[1-(1/4)ε2].

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