Abstract

In time-dependent optical spectra, fast dynamics of narrow transitions give rise to artifacts consisting of spectral oscillations. In contrast with spectrally-integrated experiments, the interpretation of spectrally-resolved pump-probe experiments is thus far from straight-forward in regimes where such coherent effects dominate, i.e. when the timescales of pump-induced processes are shorter than the medium dephasing times. Analyzing the two-dimensional Fourier transform of the time-dependent spectrum, we identify the components responsible for these artifacts and we introduce a simple Fourier-filtering procedure for their systematic removal. This technique is theoretically demonstrated on the time-dependent spectrum of a transition undergoing a frequency shift. The technique is also shown to have the additional benefit of efficient noise filtering of time-dependent spectra.

© 2006 Optical Society of America

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References

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  1. Z. Vardeny and J. Tauc, "Picosecond coherence coupling in the pump and probe technique," Opt. Commun. 39, 396-400 (1981).
    [CrossRef]
  2. C. H. B. Cruz, R. L. Fork, W. H. Knox, and C. V. Shank, "Spectral hole burning in large molecules probed with 10 Fs optical pulses," Chem. Phys. Lett. 132, 341-344 (1986).
    [CrossRef]
  3. B. Fluegel, N. Peyghambarian, G. Olbright, M. Lindberg, S. W. Koch, M. Joffre, D. Hulin, A. Migus, and A. Antonetti, "Femtosecond studies of coherent transients in semiconductors," Phys. Rev. Lett. 59, 2588-2591 (1987).
    [CrossRef] [PubMed]
  4. M. Joffre, D. Hulin, A. Migus, A. Antonetti, C. Benoit`a la Guillaume, N. Peyghambarian, M. Lindberg, and S. W. Koch, "Coherent effects in pump-probe Spectroscopy of Excitons," Opt. Lett. 13, 276-278 (1988).
    [CrossRef] [PubMed]
  5. M. Joffre, D. Hulin, J.-P. Foing, J.-P. Chambaret, A. Migus, and A. Antonetti, "Dynamics and fourier transform studies of the Excitonic Optical Stark Effect," IEEE J. Quantum Electron. 25, 2505-2515 (1989).
    [CrossRef]
  6. C. H. B. Cruz, J. P. Gordon, P. C. Becker, R. L. Fork, and C. V. Shank, "Dynamics of spectral hole burning," IEEE J. Quantum Electron. 24, 261-266 (1988).
    [CrossRef]
  7. S. Maiti, G. C. Walker, B. R. Cowen, R. Pippenger, C. C. Moser, P. L. Dutton, and R. M. Hochstrasser, "Femtosecond coherent transient infrared Spectroscopy of Reaction Centers from Rhodobacter Sphaeroides," Proc. Natl. Acad. Sci. 91, 10360-10364 (1994).
    [CrossRef]
  8. P. Hamm, "Coherent effects in Femtosecond infrared Spectroscopy," Chem. Phys. 200, 415-429 (1995).
    [CrossRef]
  9. K. Wynne and R. Hochstrasser, "The Theory of ultrafast vibrational Spectroscopy," Chem. Phys. 193, 211-236 (1995).
    [CrossRef]
  10. M. Joffre, "Comment on" Coherent nonlinear optical response of single quantum dots studied by Ultrafast Near-Field Spectroscopy," Phys. Rev. Lett. 90, 139,701 (2003) and references therein.
    [CrossRef]
  11. T. Polack, J. P. Ogilvie, S. Franzen, M. H. Vos, M. Joffre, J.-L. Martin, and A. Alexandrou, "CO Vibration as a probe of Ligand dissociation and transfer in Myoglobin," Phys. Rev. Lett. 93, 018,102 (2004).
    [CrossRef]
  12. D. M. Jonas, "Two-dimensional Femtosecond Spectroscopy," Annu. Rev. Phys. Chem. 54, 425-463 (2003).
    [CrossRef] [PubMed]
  13. S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University Press, Oxford, 1995).
  14. J. G. Kirkwood, "Quantum statistics of almost classical assemblies," Phys. Rev. 44, 31-37 (1933).
    [CrossRef]
  15. J. H. Eberly and K. Wódkiewicz, "The time-dependent physical spectrum of light," J. Opt. Soc. Am. 67, 1252-1261 (1977).
    [CrossRef]

2003 (1)

D. M. Jonas, "Two-dimensional Femtosecond Spectroscopy," Annu. Rev. Phys. Chem. 54, 425-463 (2003).
[CrossRef] [PubMed]

1995 (2)

P. Hamm, "Coherent effects in Femtosecond infrared Spectroscopy," Chem. Phys. 200, 415-429 (1995).
[CrossRef]

K. Wynne and R. Hochstrasser, "The Theory of ultrafast vibrational Spectroscopy," Chem. Phys. 193, 211-236 (1995).
[CrossRef]

1994 (1)

S. Maiti, G. C. Walker, B. R. Cowen, R. Pippenger, C. C. Moser, P. L. Dutton, and R. M. Hochstrasser, "Femtosecond coherent transient infrared Spectroscopy of Reaction Centers from Rhodobacter Sphaeroides," Proc. Natl. Acad. Sci. 91, 10360-10364 (1994).
[CrossRef]

1989 (1)

M. Joffre, D. Hulin, J.-P. Foing, J.-P. Chambaret, A. Migus, and A. Antonetti, "Dynamics and fourier transform studies of the Excitonic Optical Stark Effect," IEEE J. Quantum Electron. 25, 2505-2515 (1989).
[CrossRef]

1988 (2)

1987 (1)

B. Fluegel, N. Peyghambarian, G. Olbright, M. Lindberg, S. W. Koch, M. Joffre, D. Hulin, A. Migus, and A. Antonetti, "Femtosecond studies of coherent transients in semiconductors," Phys. Rev. Lett. 59, 2588-2591 (1987).
[CrossRef] [PubMed]

1986 (1)

C. H. B. Cruz, R. L. Fork, W. H. Knox, and C. V. Shank, "Spectral hole burning in large molecules probed with 10 Fs optical pulses," Chem. Phys. Lett. 132, 341-344 (1986).
[CrossRef]

1981 (1)

Z. Vardeny and J. Tauc, "Picosecond coherence coupling in the pump and probe technique," Opt. Commun. 39, 396-400 (1981).
[CrossRef]

1977 (1)

1933 (1)

J. G. Kirkwood, "Quantum statistics of almost classical assemblies," Phys. Rev. 44, 31-37 (1933).
[CrossRef]

Antonetti, A.

M. Joffre, D. Hulin, J.-P. Foing, J.-P. Chambaret, A. Migus, and A. Antonetti, "Dynamics and fourier transform studies of the Excitonic Optical Stark Effect," IEEE J. Quantum Electron. 25, 2505-2515 (1989).
[CrossRef]

M. Joffre, D. Hulin, A. Migus, A. Antonetti, C. Benoit`a la Guillaume, N. Peyghambarian, M. Lindberg, and S. W. Koch, "Coherent effects in pump-probe Spectroscopy of Excitons," Opt. Lett. 13, 276-278 (1988).
[CrossRef] [PubMed]

B. Fluegel, N. Peyghambarian, G. Olbright, M. Lindberg, S. W. Koch, M. Joffre, D. Hulin, A. Migus, and A. Antonetti, "Femtosecond studies of coherent transients in semiconductors," Phys. Rev. Lett. 59, 2588-2591 (1987).
[CrossRef] [PubMed]

Becker, P. C.

C. H. B. Cruz, J. P. Gordon, P. C. Becker, R. L. Fork, and C. V. Shank, "Dynamics of spectral hole burning," IEEE J. Quantum Electron. 24, 261-266 (1988).
[CrossRef]

Benoit, C.

Chambaret, J.-P.

M. Joffre, D. Hulin, J.-P. Foing, J.-P. Chambaret, A. Migus, and A. Antonetti, "Dynamics and fourier transform studies of the Excitonic Optical Stark Effect," IEEE J. Quantum Electron. 25, 2505-2515 (1989).
[CrossRef]

Cowen, B. R.

S. Maiti, G. C. Walker, B. R. Cowen, R. Pippenger, C. C. Moser, P. L. Dutton, and R. M. Hochstrasser, "Femtosecond coherent transient infrared Spectroscopy of Reaction Centers from Rhodobacter Sphaeroides," Proc. Natl. Acad. Sci. 91, 10360-10364 (1994).
[CrossRef]

Cruz, C. H. B.

C. H. B. Cruz, J. P. Gordon, P. C. Becker, R. L. Fork, and C. V. Shank, "Dynamics of spectral hole burning," IEEE J. Quantum Electron. 24, 261-266 (1988).
[CrossRef]

C. H. B. Cruz, R. L. Fork, W. H. Knox, and C. V. Shank, "Spectral hole burning in large molecules probed with 10 Fs optical pulses," Chem. Phys. Lett. 132, 341-344 (1986).
[CrossRef]

Dutton, P. L.

S. Maiti, G. C. Walker, B. R. Cowen, R. Pippenger, C. C. Moser, P. L. Dutton, and R. M. Hochstrasser, "Femtosecond coherent transient infrared Spectroscopy of Reaction Centers from Rhodobacter Sphaeroides," Proc. Natl. Acad. Sci. 91, 10360-10364 (1994).
[CrossRef]

Eberly, J. H.

Fluegel, B.

B. Fluegel, N. Peyghambarian, G. Olbright, M. Lindberg, S. W. Koch, M. Joffre, D. Hulin, A. Migus, and A. Antonetti, "Femtosecond studies of coherent transients in semiconductors," Phys. Rev. Lett. 59, 2588-2591 (1987).
[CrossRef] [PubMed]

Foing, J.-P.

M. Joffre, D. Hulin, J.-P. Foing, J.-P. Chambaret, A. Migus, and A. Antonetti, "Dynamics and fourier transform studies of the Excitonic Optical Stark Effect," IEEE J. Quantum Electron. 25, 2505-2515 (1989).
[CrossRef]

Fork, R. L.

C. H. B. Cruz, J. P. Gordon, P. C. Becker, R. L. Fork, and C. V. Shank, "Dynamics of spectral hole burning," IEEE J. Quantum Electron. 24, 261-266 (1988).
[CrossRef]

C. H. B. Cruz, R. L. Fork, W. H. Knox, and C. V. Shank, "Spectral hole burning in large molecules probed with 10 Fs optical pulses," Chem. Phys. Lett. 132, 341-344 (1986).
[CrossRef]

Gordon, J. P.

C. H. B. Cruz, J. P. Gordon, P. C. Becker, R. L. Fork, and C. V. Shank, "Dynamics of spectral hole burning," IEEE J. Quantum Electron. 24, 261-266 (1988).
[CrossRef]

Hamm, P.

P. Hamm, "Coherent effects in Femtosecond infrared Spectroscopy," Chem. Phys. 200, 415-429 (1995).
[CrossRef]

Hochstrasser, R.

K. Wynne and R. Hochstrasser, "The Theory of ultrafast vibrational Spectroscopy," Chem. Phys. 193, 211-236 (1995).
[CrossRef]

Hochstrasser, R. M.

S. Maiti, G. C. Walker, B. R. Cowen, R. Pippenger, C. C. Moser, P. L. Dutton, and R. M. Hochstrasser, "Femtosecond coherent transient infrared Spectroscopy of Reaction Centers from Rhodobacter Sphaeroides," Proc. Natl. Acad. Sci. 91, 10360-10364 (1994).
[CrossRef]

Hulin, D.

M. Joffre, D. Hulin, J.-P. Foing, J.-P. Chambaret, A. Migus, and A. Antonetti, "Dynamics and fourier transform studies of the Excitonic Optical Stark Effect," IEEE J. Quantum Electron. 25, 2505-2515 (1989).
[CrossRef]

M. Joffre, D. Hulin, A. Migus, A. Antonetti, C. Benoit`a la Guillaume, N. Peyghambarian, M. Lindberg, and S. W. Koch, "Coherent effects in pump-probe Spectroscopy of Excitons," Opt. Lett. 13, 276-278 (1988).
[CrossRef] [PubMed]

B. Fluegel, N. Peyghambarian, G. Olbright, M. Lindberg, S. W. Koch, M. Joffre, D. Hulin, A. Migus, and A. Antonetti, "Femtosecond studies of coherent transients in semiconductors," Phys. Rev. Lett. 59, 2588-2591 (1987).
[CrossRef] [PubMed]

Joffre, M.

M. Joffre, D. Hulin, J.-P. Foing, J.-P. Chambaret, A. Migus, and A. Antonetti, "Dynamics and fourier transform studies of the Excitonic Optical Stark Effect," IEEE J. Quantum Electron. 25, 2505-2515 (1989).
[CrossRef]

M. Joffre, D. Hulin, A. Migus, A. Antonetti, C. Benoit`a la Guillaume, N. Peyghambarian, M. Lindberg, and S. W. Koch, "Coherent effects in pump-probe Spectroscopy of Excitons," Opt. Lett. 13, 276-278 (1988).
[CrossRef] [PubMed]

B. Fluegel, N. Peyghambarian, G. Olbright, M. Lindberg, S. W. Koch, M. Joffre, D. Hulin, A. Migus, and A. Antonetti, "Femtosecond studies of coherent transients in semiconductors," Phys. Rev. Lett. 59, 2588-2591 (1987).
[CrossRef] [PubMed]

Jonas, D. M.

D. M. Jonas, "Two-dimensional Femtosecond Spectroscopy," Annu. Rev. Phys. Chem. 54, 425-463 (2003).
[CrossRef] [PubMed]

Kirkwood, J. G.

J. G. Kirkwood, "Quantum statistics of almost classical assemblies," Phys. Rev. 44, 31-37 (1933).
[CrossRef]

Knox, W. H.

C. H. B. Cruz, R. L. Fork, W. H. Knox, and C. V. Shank, "Spectral hole burning in large molecules probed with 10 Fs optical pulses," Chem. Phys. Lett. 132, 341-344 (1986).
[CrossRef]

Koch, S. W.

B. Fluegel, N. Peyghambarian, G. Olbright, M. Lindberg, S. W. Koch, M. Joffre, D. Hulin, A. Migus, and A. Antonetti, "Femtosecond studies of coherent transients in semiconductors," Phys. Rev. Lett. 59, 2588-2591 (1987).
[CrossRef] [PubMed]

Lindberg, M.

B. Fluegel, N. Peyghambarian, G. Olbright, M. Lindberg, S. W. Koch, M. Joffre, D. Hulin, A. Migus, and A. Antonetti, "Femtosecond studies of coherent transients in semiconductors," Phys. Rev. Lett. 59, 2588-2591 (1987).
[CrossRef] [PubMed]

Maiti, S.

S. Maiti, G. C. Walker, B. R. Cowen, R. Pippenger, C. C. Moser, P. L. Dutton, and R. M. Hochstrasser, "Femtosecond coherent transient infrared Spectroscopy of Reaction Centers from Rhodobacter Sphaeroides," Proc. Natl. Acad. Sci. 91, 10360-10364 (1994).
[CrossRef]

Migus, A.

M. Joffre, D. Hulin, J.-P. Foing, J.-P. Chambaret, A. Migus, and A. Antonetti, "Dynamics and fourier transform studies of the Excitonic Optical Stark Effect," IEEE J. Quantum Electron. 25, 2505-2515 (1989).
[CrossRef]

M. Joffre, D. Hulin, A. Migus, A. Antonetti, C. Benoit`a la Guillaume, N. Peyghambarian, M. Lindberg, and S. W. Koch, "Coherent effects in pump-probe Spectroscopy of Excitons," Opt. Lett. 13, 276-278 (1988).
[CrossRef] [PubMed]

B. Fluegel, N. Peyghambarian, G. Olbright, M. Lindberg, S. W. Koch, M. Joffre, D. Hulin, A. Migus, and A. Antonetti, "Femtosecond studies of coherent transients in semiconductors," Phys. Rev. Lett. 59, 2588-2591 (1987).
[CrossRef] [PubMed]

Moser, C. C.

S. Maiti, G. C. Walker, B. R. Cowen, R. Pippenger, C. C. Moser, P. L. Dutton, and R. M. Hochstrasser, "Femtosecond coherent transient infrared Spectroscopy of Reaction Centers from Rhodobacter Sphaeroides," Proc. Natl. Acad. Sci. 91, 10360-10364 (1994).
[CrossRef]

Olbright, G.

B. Fluegel, N. Peyghambarian, G. Olbright, M. Lindberg, S. W. Koch, M. Joffre, D. Hulin, A. Migus, and A. Antonetti, "Femtosecond studies of coherent transients in semiconductors," Phys. Rev. Lett. 59, 2588-2591 (1987).
[CrossRef] [PubMed]

Peyghambarian, N.

B. Fluegel, N. Peyghambarian, G. Olbright, M. Lindberg, S. W. Koch, M. Joffre, D. Hulin, A. Migus, and A. Antonetti, "Femtosecond studies of coherent transients in semiconductors," Phys. Rev. Lett. 59, 2588-2591 (1987).
[CrossRef] [PubMed]

Pippenger, R.

S. Maiti, G. C. Walker, B. R. Cowen, R. Pippenger, C. C. Moser, P. L. Dutton, and R. M. Hochstrasser, "Femtosecond coherent transient infrared Spectroscopy of Reaction Centers from Rhodobacter Sphaeroides," Proc. Natl. Acad. Sci. 91, 10360-10364 (1994).
[CrossRef]

Shank, C. V.

C. H. B. Cruz, J. P. Gordon, P. C. Becker, R. L. Fork, and C. V. Shank, "Dynamics of spectral hole burning," IEEE J. Quantum Electron. 24, 261-266 (1988).
[CrossRef]

C. H. B. Cruz, R. L. Fork, W. H. Knox, and C. V. Shank, "Spectral hole burning in large molecules probed with 10 Fs optical pulses," Chem. Phys. Lett. 132, 341-344 (1986).
[CrossRef]

Tauc, J.

Z. Vardeny and J. Tauc, "Picosecond coherence coupling in the pump and probe technique," Opt. Commun. 39, 396-400 (1981).
[CrossRef]

Vardeny, Z.

Z. Vardeny and J. Tauc, "Picosecond coherence coupling in the pump and probe technique," Opt. Commun. 39, 396-400 (1981).
[CrossRef]

Walker, G. C.

S. Maiti, G. C. Walker, B. R. Cowen, R. Pippenger, C. C. Moser, P. L. Dutton, and R. M. Hochstrasser, "Femtosecond coherent transient infrared Spectroscopy of Reaction Centers from Rhodobacter Sphaeroides," Proc. Natl. Acad. Sci. 91, 10360-10364 (1994).
[CrossRef]

Wódkiewicz, K.

Wynne, K.

K. Wynne and R. Hochstrasser, "The Theory of ultrafast vibrational Spectroscopy," Chem. Phys. 193, 211-236 (1995).
[CrossRef]

Annu. Rev. Phys. Chem. (1)

D. M. Jonas, "Two-dimensional Femtosecond Spectroscopy," Annu. Rev. Phys. Chem. 54, 425-463 (2003).
[CrossRef] [PubMed]

Chem. Phys. (2)

P. Hamm, "Coherent effects in Femtosecond infrared Spectroscopy," Chem. Phys. 200, 415-429 (1995).
[CrossRef]

K. Wynne and R. Hochstrasser, "The Theory of ultrafast vibrational Spectroscopy," Chem. Phys. 193, 211-236 (1995).
[CrossRef]

Chem. Phys. Lett. (1)

C. H. B. Cruz, R. L. Fork, W. H. Knox, and C. V. Shank, "Spectral hole burning in large molecules probed with 10 Fs optical pulses," Chem. Phys. Lett. 132, 341-344 (1986).
[CrossRef]

IEEE J. Quantum Electron. (2)

M. Joffre, D. Hulin, J.-P. Foing, J.-P. Chambaret, A. Migus, and A. Antonetti, "Dynamics and fourier transform studies of the Excitonic Optical Stark Effect," IEEE J. Quantum Electron. 25, 2505-2515 (1989).
[CrossRef]

C. H. B. Cruz, J. P. Gordon, P. C. Becker, R. L. Fork, and C. V. Shank, "Dynamics of spectral hole burning," IEEE J. Quantum Electron. 24, 261-266 (1988).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

Z. Vardeny and J. Tauc, "Picosecond coherence coupling in the pump and probe technique," Opt. Commun. 39, 396-400 (1981).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. (1)

J. G. Kirkwood, "Quantum statistics of almost classical assemblies," Phys. Rev. 44, 31-37 (1933).
[CrossRef]

Phys. Rev. Lett. (1)

B. Fluegel, N. Peyghambarian, G. Olbright, M. Lindberg, S. W. Koch, M. Joffre, D. Hulin, A. Migus, and A. Antonetti, "Femtosecond studies of coherent transients in semiconductors," Phys. Rev. Lett. 59, 2588-2591 (1987).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. (1)

S. Maiti, G. C. Walker, B. R. Cowen, R. Pippenger, C. C. Moser, P. L. Dutton, and R. M. Hochstrasser, "Femtosecond coherent transient infrared Spectroscopy of Reaction Centers from Rhodobacter Sphaeroides," Proc. Natl. Acad. Sci. 91, 10360-10364 (1994).
[CrossRef]

Other (3)

S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University Press, Oxford, 1995).

M. Joffre, "Comment on" Coherent nonlinear optical response of single quantum dots studied by Ultrafast Near-Field Spectroscopy," Phys. Rev. Lett. 90, 139,701 (2003) and references therein.
[CrossRef]

T. Polack, J. P. Ogilvie, S. Franzen, M. H. Vos, M. Joffre, J.-L. Martin, and A. Alexandrou, "CO Vibration as a probe of Ligand dissociation and transfer in Myoglobin," Phys. Rev. Lett. 93, 018,102 (2004).
[CrossRef]

Supplementary Material (1)

» Media 1: MPG (2205 KB)     

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Figures (3)

Fig. 1.
Fig. 1.

(a) Simulation of a Im[α(t,ω)] time-dependent spectrum in the case of a transition (dephasing time 1/Γ=1ps) going through an instantaneous frequency change of Δν=6THz at time t=0. Artefacts appear at negative pump-probe delays. Around the initial frequency (set to 0), the spectrum shows spectral oscillations and a slow decrease of the absorption peak (time-constant 1/Γ) typical of the sharp extinction of the polarization decay. Around the final frequency (6THz), the spectrum shows oscillations along the delay axis at frequency Δν. They follow from coherence transfer[9]. (b) The S(τ,Ω) spectrum, derived from the Im[α(τ,ω)] spectrum of Fig. (a), is here shown for a probe duration of 400fs (FWHM). The attached movie (file size 2.2 KB) shows the similarly derived S(τ,Ω) spectra for probe durations ranging from 25 fs to 2 ps: independently of the duration, the artifacts vanish. For short probe durations, S(τ,Ω) gives access to fast evolving features at the expense of frequency resolution whereas for probe durations on the order of the dephasing time 1/Γ=1ps, the transition line is fully resolved. [Media 1]

Fig. 2.
Fig. 2.

Shift-space filter |fωt)|=exp(-|ΔωΔt|/2). f equals one along the axes (red) and zero for large ΔωΔt Fourier-product components (blue). Essentially, f rejects the components of α̂ωt) verifying |ΔωΔt|≥4. The |ΔωΔt|=4 hyperbolas (white) mark its 1/e2 limit. In shift-space, Fourier-transform limited pulses parametrized by their pulse duration Δt 0 (field rms), have a Gaussian shape : ��̂Δt0ωt)=exp(iΔωΔt/2)exp(-(Δω 2Δt02t 2t02)/4). The ellipses (black) delineating their 1/e 2 limit all fall within the |ΔωΔt|≤4 domain; the legend indicates their durations (FHWM).

Fig. 3.
Fig. 3.

Shift-space filtering procedure. The procedure is demonstrated on the frequency shift case introduced in Fig. 1. Using the filter shown in Fig. 2, the large Fourier-product components of α̂ωt) are removed. The artifacts that appear in Im[α(t,ω)] at negative delays (a) are fully eliminated in the filtered spectrum (b). The same procedure applies to noise rejection. In (c), 30% noise is added to the data of (a). The filtered spectrum (d) obtained from the data of (c) is almost the same than the noise-free spectrum shown in (b).

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

𝓟 ( t ) = 0 d Δ t 𝓡 ( t , t Δ t ) 𝓔 ( t Δ t ) .
α ( τ , ω ) = 0 d Δ t e i ω Δ t 𝓡 ( τ + Δ t , τ ) .
S ( τ , Ω ) = Im [ S ˜ ( τ , Ω ) ] = Im { d t d ω 2 π 𝓔 ( t τ ) 𝓔 ( ω Ω ) exp [ i ( ω Ω ) ( t τ ) ] α ( t , ω ) } .
α ̂ ( Δ ω , Δ t ) = d t d ω 2 π exp [ i ( t Δ ω ω Δ t ) ] α ( t , ω ) .

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