Abstract

The problem of noise analysis in measuring the group delay introduced by a dispersive optical element by use of white-light interferometric cross correlation is investigated. Two noise types, detection noise and position noise, are specifically analyzed. Detection noise is shown to be highly sensitive to the spectral content of the white-light source at the frequency considered and to the temporal acquisition window. Position noise, which arises from the finite accuracy of the measurement of the scanning mirror’s position, can severely damage the estimation of the group delay. Such is shown to be the case for fast Fourier transform–based estimation algorithms. A new algorithm that is insensitive to scanning delay errors is proposed, and subfemtosecond accuracy is obtained without any postprocessing.

© 2002 Optical Society of America

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References

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    [CrossRef]
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1999

1998

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuscheck, F. Morier-Genoud, V. Scheuer, M. Tilsch, T. Tschudi, and U. Keller, “Self-starting 6.5-fs pulses from a Ti: sapphire laser using a semiconductor saturable absorber and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

1997

R. Szipöcs and A. Köhzi-Kis, “Theory and design of chirped dielectric laser mirrors,” Appl. Phys. B 65, 115–127 (1997).
[CrossRef]

1994

1990

1988

1982

1973

C. Froehly, A. Lacourt, and J. C. Vienot, “Notions de réponse impulsionelle et de fonction de transfert temporelles des pupilles optiques, justifications expérimentales et applications,” Nouv. Rev. Opt. 4, 183–196 (1973).
[CrossRef]

Dorrer, C.

Ferencz, K.

Froehly, C.

C. Froehly, A. Lacourt, and J. C. Vienot, “Notions de réponse impulsionelle et de fonction de transfert temporelles des pupilles optiques, justifications expérimentales et applications,” Nouv. Rev. Opt. 4, 183–196 (1973).
[CrossRef]

Hirlimann, C. A.

Ina, H.

Jung, I. D.

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuscheck, F. Morier-Genoud, V. Scheuer, M. Tilsch, T. Tschudi, and U. Keller, “Self-starting 6.5-fs pulses from a Ti: sapphire laser using a semiconductor saturable absorber and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

Kärtner, F. X.

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuscheck, F. Morier-Genoud, V. Scheuer, M. Tilsch, T. Tschudi, and U. Keller, “Self-starting 6.5-fs pulses from a Ti: sapphire laser using a semiconductor saturable absorber and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

Keller, U.

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuscheck, F. Morier-Genoud, V. Scheuer, M. Tilsch, T. Tschudi, and U. Keller, “Self-starting 6.5-fs pulses from a Ti: sapphire laser using a semiconductor saturable absorber and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

Knox, W. H.

Kobayashi, S.

Köhzi-Kis, A.

R. Szipöcs and A. Köhzi-Kis, “Theory and design of chirped dielectric laser mirrors,” Appl. Phys. B 65, 115–127 (1997).
[CrossRef]

Krausz, F.

Lacourt, A.

C. Froehly, A. Lacourt, and J. C. Vienot, “Notions de réponse impulsionelle et de fonction de transfert temporelles des pupilles optiques, justifications expérimentales et applications,” Nouv. Rev. Opt. 4, 183–196 (1973).
[CrossRef]

Li, K. D.

Matuscheck, N.

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuscheck, F. Morier-Genoud, V. Scheuer, M. Tilsch, T. Tschudi, and U. Keller, “Self-starting 6.5-fs pulses from a Ti: sapphire laser using a semiconductor saturable absorber and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

Mogi, K.

Morier-Genoud, F.

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuscheck, F. Morier-Genoud, V. Scheuer, M. Tilsch, T. Tschudi, and U. Keller, “Self-starting 6.5-fs pulses from a Ti: sapphire laser using a semiconductor saturable absorber and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

Naganuma, K.

Pearson, N. M.

Scheuer, V.

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuscheck, F. Morier-Genoud, V. Scheuer, M. Tilsch, T. Tschudi, and U. Keller, “Self-starting 6.5-fs pulses from a Ti: sapphire laser using a semiconductor saturable absorber and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

Spielmann, Ch.

Stingl, A.

Sutter, D. H.

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuscheck, F. Morier-Genoud, V. Scheuer, M. Tilsch, T. Tschudi, and U. Keller, “Self-starting 6.5-fs pulses from a Ti: sapphire laser using a semiconductor saturable absorber and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

Szipöcs, R.

Takeda, M.

Tilsch, M.

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuscheck, F. Morier-Genoud, V. Scheuer, M. Tilsch, T. Tschudi, and U. Keller, “Self-starting 6.5-fs pulses from a Ti: sapphire laser using a semiconductor saturable absorber and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

Tschudi, T.

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuscheck, F. Morier-Genoud, V. Scheuer, M. Tilsch, T. Tschudi, and U. Keller, “Self-starting 6.5-fs pulses from a Ti: sapphire laser using a semiconductor saturable absorber and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

Vienot, J. C.

C. Froehly, A. Lacourt, and J. C. Vienot, “Notions de réponse impulsionelle et de fonction de transfert temporelles des pupilles optiques, justifications expérimentales et applications,” Nouv. Rev. Opt. 4, 183–196 (1973).
[CrossRef]

Yamada, H.

Appl. Phys. B

R. Szipöcs and A. Köhzi-Kis, “Theory and design of chirped dielectric laser mirrors,” Appl. Phys. B 65, 115–127 (1997).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuscheck, F. Morier-Genoud, V. Scheuer, M. Tilsch, T. Tschudi, and U. Keller, “Self-starting 6.5-fs pulses from a Ti: sapphire laser using a semiconductor saturable absorber and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

Nouv. Rev. Opt.

C. Froehly, A. Lacourt, and J. C. Vienot, “Notions de réponse impulsionelle et de fonction de transfert temporelles des pupilles optiques, justifications expérimentales et applications,” Nouv. Rev. Opt. 4, 183–196 (1973).
[CrossRef]

Opt. Lett.

Other

G. Engeln-Müllges and F. Uhlig, Numerical Algorithms with C (Springer-Verlag, Berlin, 1996).

G. Chériaux, J.-P. Rousseau, S. Ranc, J.-P. Chambaret, Ph. Balcou, V. Laude, and L. Vigroux, “Compression of terawatt level pulses in the air,” in Conference on Lasers and Electro-Optics (CLEO), Vol. 39 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), p. 541.

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Figures (5)

Fig. 1
Fig. 1

White-light interferometric cross-correlation setup. Photodiode PD1 monitors the white-light dispersion signal and photodiode PD2 monitors the He–Ne. laser position signal (BSs, beam splitters; M1, scanning mirror; M2, mirror).

Fig. 2
Fig. 2

Example of a cross-correlation measurement: (a) Dispersion signal on photodiode PD1 with white-light illumination, (b) closeup of (a), (c) position signal on photodiode PD2 with 632.8-nm He–Ne laser illumination.

Fig. 3
Fig. 3

Calibration example: (a) calibration intensity and group delay, (b) difference between the calibration group delay and a polynomial fit to third order.

Fig. 4
Fig. 4

Group-delay standard deviation caused by detection and position noise.

Fig. 5
Fig. 5

Chirped-mirror measurement. The two thin solid curves are one standard deviation away from the measurement (thicker solid curve).

Equations (56)

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I(τ)=I0+12π - dωS(ω)exp{-i[ωτ-ϕ(ω)]},
S(ω)exp[iϕ(ω)]=FT[I](ω),
FT[f](ω)=1T -T/2+T/2 f(τ)exp(iωτ)dτ.
tg(ω)=dϕ(ω)dω=ϕ(ω).
[S(ω)+iϕ(ω)S(ω)]exp[iϕ(ω)]=iFT[τI](ω).
tg(ω)=R -T/2+T/2dττI(τ)exp(iωτ)-T/2+T/2dτI(τ)exp(iωτ)=R FT[τI](ω)FT[I](ω),
S(ω)S(ω)=-I FT[τI](ω)FT[I](ω),
FT[I](ω)1N p I(τp)exp(iωτp),
J(τp)=I(τp)+n(τp),
n(τp)=0
n(τp)n(τq)=σ2δp-q,
FT[nf](ω)1N pn(τp)f(τp)exp(iωτp).
FT[nf](ω)=0
FT[nf](ω)FT[ng](ω)*=σ2N2 pf(τp)g*(τp),
FT[J](ω)=FT[I](ω)=S(ω)exp[iϕ(ω)],
VAR1=|FT[J](ω)-FT[I](ω)|2=|FT[n](ω)|2=σ2/N.
SNR1=N S2(ω)σ2.
tˆg(ω)=R FT[τJ](ω)FT[J](ω).
tˆg(ω)R FT[τI](ω)FT[I](ω) 1+FT[τn](ω)FT[τI](ω)-FT[n](ω)FT[I](ω).
tˆg(ω)R FT[τI](ω)FT[I](ω)=tg(ω).
VAR2=12N σ2S2(ω) T212+tg2(ω)+S(ω)S(ω)2.
VAR2T26N σS(ω).
S(ω)exp[iϕ(ω)]=p Wpg(τp),
tg(ω)=R p Wpf(τp)p Wpg(τp),
τp=τ¯p+ϑp,
p Wpf(τp)p Wpf(τ¯p)+p Wpϑpf(τ¯p)+p qϑq Wpτqf(τ¯p),
tg(ω)=R p Wpf(τ¯p)pWpg(τ¯p)+R p Wpϑpf(τ¯p)+p q ϑq Wpτqf(τ¯p)p Wpg(τ¯p).
1N p f(τp)1N p f(τ¯p)+1N p ϑpf(τ¯p).
ϑp=0,
ϑpϑq=σϑ2δp-q.
1N p f(τp)1N p f(τ¯p),
VAR3σϑ2N2 p|f(τ¯p)|2=σϑ2h2T-2 p|f(τ¯p)|2,
VAR4σϑ2h2 p|f(τ¯p)|2T2S2(ω).
p Wpf(τp)=12T p=2N(τp-τp-1)[f(τp)+f(τp-1)].
VAR5σϑ216T2 p=2N-1f(τ¯p)(δτp2-δτp+12)-13f(τ¯p)(δτp3+δτp+13)2,
VAR6σϑ2 p=2N-1f(τ¯p)(δτp2-δτp+12)-13f(τ¯p)(δτp3+δτp+13)216T2S2(ω).
VAR5σϑ236T2h6 p=2N-1|f(τ¯p)|2,
VAR6σϑ2h6 p=2N-1|f(τ¯p)|236T2S2(ω),
[tg(ω)-tˆg(ω)]2
R FT[τn](ω)FT[I](ω)-R FT[τI](ω)FT[n](ω)FT[I](ω)22,
[tg(ω)-tˆg(ω)]2
14 FT[τn](ω)FT[I](ω)-FT[τI](ω)FT[n](ω)FT[I](ω)2+c.c.2,
FT[nf](ω)FT[ng](ω)
=σ2N2 p f(τp)g(τp)exp(2iωτp)
FT[nf](ω)FT[ng](ω)
=σ2NT -T/2T/2 f(τ)g(τ)exp(2iωτ)dτ.
FT[τn](ω)FT[n](ω)*=0.
VAR212|FT[T](ω)|2 |FT[τn](ω)|2+|FT[τI](ω)|2|FT[I](ω)|2|FT[n](ω)|2.
VAR2=12N σ2S2(ω) T212+tg2(ω)+S(ω)S(ω)2,
p Wpf(τp)=S¯+12T p=2N[(ϑp-ϑp-1)(fp+fp-1)+(τ¯p-τ¯p-1)(ϑpfp+ϑp-1fp-1)],
S¯=p Wpfp.
p Wpf(τp)=S¯+12T p=2N-1 ϑp×[fp-1-fp+1+(τ¯p+1-τ¯p-1)fp].
f(τ¯p+)fp+fp+22fp+36fp,
p Wpf(τp)
=S¯+14T p=2N-1 ϑp
×fp(δτp2-δτp+12)-13fp(δτp3+δτp+13)

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