Abstract

A spectral restoration algorithm appropriate for the asymmetric and wavelength-dependent linespread of broadband spectrographs with pixelated detectors is presented. The algorithm’s accuracy was tested on spectra of femtosecond pulse pairs with known delays from an actively stabilized interferometer. Using interleaved atomic line spectra, the spectrograph calibration and effective linespread function were retrieved with sub-pixel accuracy. The spectral restoration by Fourier pseudo-deconvolution with the effective linespread function reduced systematic artifacts and allowed recovery of the phase delay to ±2.4 as over a 2 ps range (±0.7 nm path differences over 0.6 mm). The slope delay was determined to within ±20 as and constant (intercept) phase shifts to within ±0.05 rad; these accuracies are limited by Fourier filtering of charge coupled device and interferometer imperfections.

© 2010 Optical Society of America

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2009 (1)

S. K. Debnath, M. P. Kothiyal, and S.-W. Kim, “Evaluation of spectral phase in spectrally resolved white-light interferometry: comparative study of single frame techniques,” Opt. Lasers Eng. 47, 1125–1130 (2009).
[CrossRef]

2006 (2)

2005 (1)

2004 (1)

P. Baum, S. Lochbrunner, and E. Riedle, “Generation of tunable 7-fs ultraviolet pulses: achromatic phase matching and chirp management,” Appl. Phys. B 79, 1027–1032 (2004).
[CrossRef]

2003 (3)

2002 (2)

T. P. Costello and W. B. Mikhael, “One-dimensional comparison of Wiener filtering and Richardson-Lucy methods for sectioned restoration of space-variant digital images,” IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 49, 518–522 (2002).
[CrossRef]

D. J. Jones, E. O. Potma, J. X. Cheng, B. Burfeindt, Y. Pang, J. Ye, and X. S. Xie, “Synchronization of two passively mode-locked, picosecond lasers within 20 fs for coherent anti-Stokes Raman scattering microscopy,” Rev. Sci. Instrum. 73, 2843–2848 (2002).
[CrossRef]

2001 (4)

M. Gai, D. Carollo, M. Delbo, M. G. Lattanzi, G. Massone, F. Bertinetto, G. Mana, and S. Cesare, “Location accuracy limitations for CCD cameras,” Astron. Astrophys. 367, 362–370 (2001).
[CrossRef]

A. W. Albrecht Ferro, J. D. Hybl, S. M. Gallagher Faeder, and D. M. Jonas, “‘Erratum: Experimental distinction between phase shifts and time delays: implications for femtosecond spectroscopy and coherent control of chemical reactions’ [J. Chem. Phys. 111, 10934 (1999)],” J. Chem. Phys. 115, 5691 (2001).
[CrossRef]

J. D. Hybl, A. Albrecht Ferro, and D. M. Jonas, “Two dimensional Fourier transform electronic spectroscopy,” J. Chem. Phys. 115, 6606–6622 (2001).
[CrossRef]

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]

2000 (3)

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]

C. Dorrer, N. Belabas, J. P. Likforman, and M. Joffre, “Experimental implementation of Fourier-transform spectral interferometry and its application to the study of spectrometers,” Appl. Phys. B 70, S99–S107 (2000).
[CrossRef]

J. Anderson and I. R. King, “Toward high-precision astrometry with WFPC2. I. Deriving an accurate point-spread function,” Publ. Astron. Soc. Pac. 112, 1360–1382 (2000).
[CrossRef]

1999 (3)

A. W. Albrecht, J. D. Hybl, S. M. Gallagher Faeder, and D. M. Jonas, “Experimental distinction between phase shifts and time delays: implications for femtosecond spectroscopy and coherent control of chemical reactions,” J. Chem. Phys. 111, 10934–10956 (1999).
[CrossRef]

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

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

1998 (3)

G. Bönsch and E. Potulski, “Measurement of the refractive index of air and comparison with modified Edlen's formulae,” Metrologia 35, 133–139 (1998).
[CrossRef]

J. P. De Cuyper and H. Hensberge, “Wavelength calibration at moderately high resolution,” Astron. Astrophys. Suppl. Ser. 128, 409–416 (1998).
[CrossRef]

A. W. Fountain, T. J. Vickers, and C. K. Mann, “Factors that affect the accuracy of Raman shift measurements on multichannel spectrometers,” Appl. Spectrosc. 52, 462–468 (1998).
[CrossRef]

1997 (1)

D. Meshulach, D. Yelin, and Y. Silberberg, “White light dispersion measurements by one- and two-dimensional spectral interference,” IEEE J. Quantum Electron. 33, 1969–1974 (1997).
[CrossRef]

1996 (4)

1995 (4)

1994 (1)

1992 (1)

1991 (1)

M. Beck, I. A. Walmsley, and J. D. Kafka, “Group delay measurements of optical-components near 800 nm,” IEEE J. Quantum Electron. 27, 2074–2081 (1991).
[CrossRef]

1989 (1)

1974 (1)

1973 (1)

G. Norlen, “Wavelengths and energy-levels of Ar-I and Ar-II based on new interferometric measurements in region 3400–9800 A,” Phys. Scr. 8, 249–268 (1973).
[CrossRef]

1969 (1)

1965 (1)

1950 (1)

1921 (1)

A. A. Michelson and F. G. Pease, “Measurement of the diameter of an Orionis with the interferometer,” Astrophys. J. 53, 249–259 (1921).
[CrossRef]

Adams, K. B.

Albrecht, A. W.

A. W. Albrecht, J. D. Hybl, S. M. Gallagher Faeder, and D. M. Jonas, “Experimental distinction between phase shifts and time delays: implications for femtosecond spectroscopy and coherent control of chemical reactions,” J. Chem. Phys. 111, 10934–10956 (1999).
[CrossRef]

Albrecht Ferro, A.

J. D. Hybl, A. Albrecht Ferro, and D. M. Jonas, “Two dimensional Fourier transform electronic spectroscopy,” J. Chem. Phys. 115, 6606–6622 (2001).
[CrossRef]

Albrecht Ferro, A. W.

A. W. Albrecht Ferro, J. D. Hybl, S. M. Gallagher Faeder, and D. M. Jonas, “‘Erratum: Experimental distinction between phase shifts and time delays: implications for femtosecond spectroscopy and coherent control of chemical reactions’ [J. Chem. Phys. 111, 10934 (1999)],” J. Chem. Phys. 115, 5691 (2001).
[CrossRef]

Anderson, J.

J. Anderson and I. R. King, “Toward high-precision astrometry with WFPC2. I. Deriving an accurate point-spread function,” Publ. Astron. Soc. Pac. 112, 1360–1382 (2000).
[CrossRef]

Angelow, G.

Baade, D.

M. Downing, D. Baade, P. Sinclaire, S. Deries, and F. Christen, Proc. SPIE 6276, 627609 (2006).
[CrossRef]

Baum, P.

P. Baum, S. Lochbrunner, and E. Riedle, “Generation of tunable 7-fs ultraviolet pulses: achromatic phase matching and chirp management,” Appl. Phys. B 79, 1027–1032 (2004).
[CrossRef]

Beck, M.

M. Beck, I. A. Walmsley, and J. D. Kafka, “Group delay measurements of optical-components near 800 nm,” IEEE J. Quantum Electron. 27, 2074–2081 (1991).
[CrossRef]

Belabas, N.

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]

C. Dorrer, N. Belabas, J. P. Likforman, and M. Joffre, “Experimental implementation of Fourier-transform spectral interferometry and its application to the study of spectrometers,” Appl. Phys. B 70, S99–S107 (2000).
[CrossRef]

Bell, R. J.

R. J. Bell, Introductory Fourier Transform Spectroscopy (Academic, 1972).

Bertinetto, F.

M. Gai, D. Carollo, M. Delbo, M. G. Lattanzi, G. Massone, F. Bertinetto, G. Mana, and S. Cesare, “Location accuracy limitations for CCD cameras,” Astron. Astrophys. 367, 362–370 (2001).
[CrossRef]

Bevington, P. R.

P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, 1992).

Birge, J. R.

J. Kim, J. R. Birge, V. Sharma, J. G. Fujimoto, F. X. Kartner, V. Scheuer, and G. Angelow, “Ultrabroadband beam splitter with matched group-delay dispersion,” Opt. Lett. 30, 1569–1571 (2005).
[CrossRef] [PubMed]

J. R. Birge, R. Ell, and F. X. Kartner, in Ultrafast Phenomena XV, P.Corkum, D.Jonas, R.J. D.Miller, and A.M.Weiner, eds. (Springer, 2006), pp. 160–162.

Bönsch, G.

G. Bönsch and E. Potulski, “Measurement of the refractive index of air and comparison with modified Edlen's formulae,” Metrologia 35, 133–139 (1998).
[CrossRef]

Bowie, J. L.

Burfeindt, B.

D. J. Jones, E. O. Potma, J. X. Cheng, B. Burfeindt, Y. Pang, J. Ye, and X. S. Xie, “Synchronization of two passively mode-locked, picosecond lasers within 20 fs for coherent anti-Stokes Raman scattering microscopy,” Rev. Sci. Instrum. 73, 2843–2848 (2002).
[CrossRef]

Burns, K.

Campbell, R. D.

R. D. Campbell and D. J. Thompson, in Scientific Detectors for Astronomy 2005, J.E.Beletic, J.W.Beletic, and P.Amico, eds. (Springer, 2005), pp. 507–514.

Carollo, D.

M. Gai, D. Carollo, M. Delbo, M. G. Lattanzi, G. Massone, F. Bertinetto, G. Mana, and S. Cesare, “Location accuracy limitations for CCD cameras,” Astron. Astrophys. 367, 362–370 (2001).
[CrossRef]

Cesare, S.

M. Gai, D. Carollo, M. Delbo, M. G. Lattanzi, G. Massone, F. Bertinetto, G. Mana, and S. Cesare, “Location accuracy limitations for CCD cameras,” Astron. Astrophys. 367, 362–370 (2001).
[CrossRef]

Chang, Z.

Cheng, J. X.

D. J. Jones, E. O. Potma, J. X. Cheng, B. Burfeindt, Y. Pang, J. Ye, and X. S. Xie, “Synchronization of two passively mode-locked, picosecond lasers within 20 fs for coherent anti-Stokes Raman scattering microscopy,” Rev. Sci. Instrum. 73, 2843–2848 (2002).
[CrossRef]

Chériaux, G.

Choma, M. A.

Christen, F.

M. Downing, D. Baade, P. Sinclaire, S. Deries, and F. Christen, Proc. SPIE 6276, 627609 (2006).
[CrossRef]

Corwin, K. L.

Costello, T. P.

T. P. Costello and W. B. Mikhael, “One-dimensional comparison of Wiener filtering and Richardson-Lucy methods for sectioned restoration of space-variant digital images,” IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 49, 518–522 (2002).
[CrossRef]

De Cuyper, J. P.

J. P. De Cuyper and H. Hensberge, “Wavelength calibration at moderately high resolution,” Astron. Astrophys. Suppl. Ser. 128, 409–416 (1998).
[CrossRef]

de Groot, P.

P. de Groot and L. Deck, “Surface profiling by analysis of white-light interferograms in the spatial frequency domain,” J. Mod. Opt. 42, 389–401 (1995).
[CrossRef]

Debnath, S. K.

S. K. Debnath, M. P. Kothiyal, and S.-W. Kim, “Evaluation of spectral phase in spectrally resolved white-light interferometry: comparative study of single frame techniques,” Opt. Lasers Eng. 47, 1125–1130 (2009).
[CrossRef]

Deck, L.

P. de Groot and L. Deck, “Surface profiling by analysis of white-light interferograms in the spatial frequency domain,” J. Mod. Opt. 42, 389–401 (1995).
[CrossRef]

Deckert, V.

Delbo, M.

M. Gai, D. Carollo, M. Delbo, M. G. Lattanzi, G. Massone, F. Bertinetto, G. Mana, and S. Cesare, “Location accuracy limitations for CCD cameras,” Astron. Astrophys. 367, 362–370 (2001).
[CrossRef]

DeLong, K. W.

Deries, S.

M. Downing, D. Baade, P. Sinclaire, S. Deries, and F. Christen, Proc. SPIE 6276, 627609 (2006).
[CrossRef]

Dobson, C. C.

Dorrer, C.

Downing, M.

M. Downing, D. Baade, P. Sinclaire, S. Deries, and F. Christen, Proc. SPIE 6276, 627609 (2006).
[CrossRef]

Duan, Z.

Ell, R.

J. R. Birge, R. Ell, and F. X. Kartner, in Ultrafast Phenomena XV, P.Corkum, D.Jonas, R.J. D.Miller, and A.M.Weiner, eds. (Springer, 2006), pp. 160–162.

Fish, D. A.

Fittinghoff, D. N.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vettering, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, 1986).

Fountain, A. W.

Fujihira, Y.

Fujimoto, J. G.

Gai, M.

M. Gai, D. Carollo, M. Delbo, M. G. Lattanzi, G. Massone, F. Bertinetto, G. Mana, and S. Cesare, “Location accuracy limitations for CCD cameras,” Astron. Astrophys. 367, 362–370 (2001).
[CrossRef]

Gallagher Faeder, S. M.

A. W. Albrecht Ferro, J. D. Hybl, S. M. Gallagher Faeder, and D. M. Jonas, “‘Erratum: Experimental distinction between phase shifts and time delays: implications for femtosecond spectroscopy and coherent control of chemical reactions’ [J. Chem. Phys. 111, 10934 (1999)],” J. Chem. Phys. 115, 5691 (2001).
[CrossRef]

A. W. Albrecht, J. D. Hybl, S. M. Gallagher Faeder, and D. M. Jonas, “Experimental distinction between phase shifts and time delays: implications for femtosecond spectroscopy and coherent control of chemical reactions,” J. Chem. Phys. 111, 10934–10956 (1999).
[CrossRef]

Grochmalicki, J.

Hariharan, P.

P. Hariharan and B. C. Sanders, in Progress in Optics (Elsevier Science, 1996), Vol. XXXVI, pp. 49–128.
[CrossRef]

Hecht, E.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, 1990).

Hensberge, H.

J. P. De Cuyper and H. Hensberge, “Wavelength calibration at moderately high resolution,” Astron. Astrophys. Suppl. Ser. 128, 409–416 (1998).
[CrossRef]

Homma, T.

Howell, S. B.

S. B. Howell, Handbook of CCD Astronomy, 2nd ed., Cambridge Observing Handbooks for Research Astronomers (Cambridge U. Press, 2006).

Hybl, J. D.

A. W. Albrecht Ferro, J. D. Hybl, S. M. Gallagher Faeder, and D. M. Jonas, “‘Erratum: Experimental distinction between phase shifts and time delays: implications for femtosecond spectroscopy and coherent control of chemical reactions’ [J. Chem. Phys. 111, 10934 (1999)],” J. Chem. Phys. 115, 5691 (2001).
[CrossRef]

J. D. Hybl, A. Albrecht Ferro, and D. M. Jonas, “Two dimensional Fourier transform electronic spectroscopy,” J. Chem. Phys. 115, 6606–6622 (2001).
[CrossRef]

A. W. Albrecht, J. D. Hybl, S. M. Gallagher Faeder, and D. M. Jonas, “Experimental distinction between phase shifts and time delays: implications for femtosecond spectroscopy and coherent control of chemical reactions,” J. Chem. Phys. 111, 10934–10956 (1999).
[CrossRef]

Iaconis, C.

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

Izatt, J. A.

Janesick, J. R.

J. R. Janesick, Scientific Charge-Coupled Devices (SPIE, 2001).
[CrossRef]

Jenkins, F. A.

F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 1976).

Jennings, R. T.

Joffre, M.

Jonas, D. M.

J. D. Hybl, A. Albrecht Ferro, and D. M. Jonas, “Two dimensional Fourier transform electronic spectroscopy,” J. Chem. Phys. 115, 6606–6622 (2001).
[CrossRef]

A. W. Albrecht Ferro, J. D. Hybl, S. M. Gallagher Faeder, and D. M. Jonas, “‘Erratum: Experimental distinction between phase shifts and time delays: implications for femtosecond spectroscopy and coherent control of chemical reactions’ [J. Chem. Phys. 111, 10934 (1999)],” J. Chem. Phys. 115, 5691 (2001).
[CrossRef]

A. W. Albrecht, J. D. Hybl, S. M. Gallagher Faeder, and D. M. Jonas, “Experimental distinction between phase shifts and time delays: implications for femtosecond spectroscopy and coherent control of chemical reactions,” J. Chem. Phys. 111, 10934–10956 (1999).
[CrossRef]

Jones, D. J.

D. J. Jones, E. O. Potma, J. X. Cheng, B. Burfeindt, Y. Pang, J. Ye, and X. S. Xie, “Synchronization of two passively mode-locked, picosecond lasers within 20 fs for coherent anti-Stokes Raman scattering microscopy,” Rev. Sci. Instrum. 73, 2843–2848 (2002).
[CrossRef]

Kafka, J. D.

M. Beck, I. A. Walmsley, and J. D. Kafka, “Group delay measurements of optical-components near 800 nm,” IEEE J. Quantum Electron. 27, 2074–2081 (1991).
[CrossRef]

Kakehata, M.

Kartner, F. X.

J. Kim, J. R. Birge, V. Sharma, J. G. Fujimoto, F. X. Kartner, V. Scheuer, and G. Angelow, “Ultrabroadband beam splitter with matched group-delay dispersion,” Opt. Lett. 30, 1569–1571 (2005).
[CrossRef] [PubMed]

J. R. Birge, R. Ell, and F. X. Kartner, in Ultrafast Phenomena XV, P.Corkum, D.Jonas, R.J. D.Miller, and A.M.Weiner, eds. (Springer, 2006), pp. 160–162.

Kiefer, W.

Kim, J.

Kim, S. -W.

S. K. Debnath, M. P. Kothiyal, and S.-W. Kim, “Evaluation of spectral phase in spectrally resolved white-light interferometry: comparative study of single frame techniques,” Opt. Lasers Eng. 47, 1125–1130 (2009).
[CrossRef]

King, I. R.

J. Anderson and I. R. King, “Toward high-precision astrometry with WFPC2. I. Deriving an accurate point-spread function,” Publ. Astron. Soc. Pac. 112, 1360–1382 (2000).
[CrossRef]

Kobayashi, Y.

Kothiyal, M. P.

S. K. Debnath, M. P. Kothiyal, and S.-W. Kim, “Evaluation of spectral phase in spectrally resolved white-light interferometry: comparative study of single frame techniques,” Opt. Lasers Eng. 47, 1125–1130 (2009).
[CrossRef]

Krumbügel, M. A.

Lattanzi, M. G.

M. Gai, D. Carollo, M. Delbo, M. G. Lattanzi, G. Massone, F. Bertinetto, G. Mana, and S. Cesare, “Location accuracy limitations for CCD cameras,” Astron. Astrophys. 367, 362–370 (2001).
[CrossRef]

Lepetit, L.

Li, C.

Likforman, J. P.

C. Dorrer, N. Belabas, J. P. Likforman, and M. Joffre, “Experimental implementation of Fourier-transform spectral interferometry and its application to the study of spectrometers,” Appl. Phys. B 70, S99–S107 (2000).
[CrossRef]

Likforman, J. -P.

Lochbrunner, S.

P. Baum, S. Lochbrunner, and E. Riedle, “Generation of tunable 7-fs ultraviolet pulses: achromatic phase matching and chirp management,” Appl. Phys. B 79, 1027–1032 (2004).
[CrossRef]

Longwell, J.

Malitson, I. H.

Mana, G.

M. Gai, D. Carollo, M. Delbo, M. G. Lattanzi, G. Massone, F. Bertinetto, G. Mana, and S. Cesare, “Location accuracy limitations for CCD cameras,” Astron. Astrophys. 367, 362–370 (2001).
[CrossRef]

Mann, C. K.

Massone, G.

M. Gai, D. Carollo, M. Delbo, M. G. Lattanzi, G. Massone, F. Bertinetto, G. Mana, and S. Cesare, “Location accuracy limitations for CCD cameras,” Astron. Astrophys. 367, 362–370 (2001).
[CrossRef]

Meshulach, D.

D. Meshulach, D. Yelin, and Y. Silberberg, “White light dispersion measurements by one- and two-dimensional spectral interference,” IEEE J. Quantum Electron. 33, 1969–1974 (1997).
[CrossRef]

Metcalf, M.

M. Metcalf and J. Reid, Fortran 90/95 Explained (Oxford U. Press, 1996).

Michelson, A. A.

A. A. Michelson and F. G. Pease, “Measurement of the diameter of an Orionis with the interferometer,” Astrophys. J. 53, 249–259 (1921).
[CrossRef]

Mikhael, W. B.

T. P. Costello and W. B. Mikhael, “One-dimensional comparison of Wiener filtering and Richardson-Lucy methods for sectioned restoration of space-variant digital images,” IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 49, 518–522 (2002).
[CrossRef]

Moon, E.

Morita, R.

Norlen, G.

G. Norlen, “Wavelengths and energy-levels of Ar-I and Ar-II based on new interferometric measurements in region 3400–9800 A,” Phys. Scr. 8, 249–268 (1973).
[CrossRef]

Oka, K.

Pandis, S. N.

J. H. Seinfeld and S. N. Pandis, Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, 2nd ed. (John Wiley & Sons, 2006).

Pang, Y.

D. J. Jones, E. O. Potma, J. X. Cheng, B. Burfeindt, Y. Pang, J. Ye, and X. S. Xie, “Synchronization of two passively mode-locked, picosecond lasers within 20 fs for coherent anti-Stokes Raman scattering microscopy,” Rev. Sci. Instrum. 73, 2843–2848 (2002).
[CrossRef]

Pease, F. G.

A. A. Michelson and F. G. Pease, “Measurement of the diameter of an Orionis with the interferometer,” Astrophys. J. 53, 249–259 (1921).
[CrossRef]

Perrin, H.

P. Sandoz, G. Tribillon, and H. Perrin, “High-resolution profilometry by using phase calculation algorithms for spectroscopic analysis of white-light interferograms,” J. Mod. Opt. 43, 701–708 (1996).
[CrossRef]

Pike, E. R.

Potma, E. O.

D. J. Jones, E. O. Potma, J. X. Cheng, B. Burfeindt, Y. Pang, J. Ye, and X. S. Xie, “Synchronization of two passively mode-locked, picosecond lasers within 20 fs for coherent anti-Stokes Raman scattering microscopy,” Rev. Sci. Instrum. 73, 2843–2848 (2002).
[CrossRef]

Potulski, E.

G. Bönsch and E. Potulski, “Measurement of the refractive index of air and comparison with modified Edlen's formulae,” Metrologia 35, 133–139 (1998).
[CrossRef]

Pravdo, S. H.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vettering, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, 1986).

Reader, J.

Reid, J.

M. Metcalf and J. Reid, Fortran 90/95 Explained (Oxford U. Press, 1996).

Reinheimer, A.

A. Reinheimer, e2v technologies, Tarrytown, NY (personal communication, March 11, 2008).

Riedle, E.

P. Baum, S. Lochbrunner, and E. Riedle, “Generation of tunable 7-fs ultraviolet pulses: achromatic phase matching and chirp management,” Appl. Phys. B 79, 1027–1032 (2004).
[CrossRef]

Robinson, D. K.

P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, 1992).

Sanders, B. C.

P. Hariharan and B. C. Sanders, in Progress in Optics (Elsevier Science, 1996), Vol. XXXVI, pp. 49–128.
[CrossRef]

Sandoz, P.

P. Sandoz, G. Tribillon, and H. Perrin, “High-resolution profilometry by using phase calculation algorithms for spectroscopic analysis of white-light interferograms,” J. Mod. Opt. 43, 701–708 (1996).
[CrossRef]

Sarunic, M. V.

Sawchuk, A. A.

Scheuer, V.

Schwider, J.

Seinfeld, J. H.

J. H. Seinfeld and S. N. Pandis, Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, 2nd ed. (John Wiley & Sons, 2006).

Shaklan, S.

Sharma, V.

Sharman, M. C.

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Silberberg, Y.

D. Meshulach, D. Yelin, and Y. Silberberg, “White light dispersion measurements by one- and two-dimensional spectral interference,” IEEE J. Quantum Electron. 33, 1969–1974 (1997).
[CrossRef]

Sinclaire, P.

M. Downing, D. Baade, P. Sinclaire, S. Deries, and F. Christen, Proc. SPIE 6276, 627609 (2006).
[CrossRef]

Smith, L. M.

Steel, W. H.

W. H. Steel, Interferometry, Cambridge Monographs on Physics (Cambridge U. Press, 1967).

Steinmeyer, G.

G. Steinmeyer, “Dispersion oscillations in ultrafast phase-correction devices,” IEEE J. Quantum Electron. 39, 1027–1034 (2003).
[CrossRef]

Suguro, A.

Sweetser, J. N.

Tackett, J.

Takada, H.

Takahashi, H.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vettering, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, 1986).

Thompson, D. J.

R. D. Campbell and D. J. Thompson, in Scientific Detectors for Astronomy 2005, J.E.Beletic, J.W.Beletic, and P.Amico, eds. (Springer, 2005), pp. 507–514.

Torizuka, K.

Trebino, R.

Tribillon, G.

P. Sandoz, G. Tribillon, and H. Perrin, “High-resolution profilometry by using phase calculation algorithms for spectroscopic analysis of white-light interferograms,” J. Mod. Opt. 43, 701–708 (1996).
[CrossRef]

Vettering, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vettering, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, 1986).

Vickers, T. J.

Walker, J.

J. Walker, The Analytical Theory of Light (the Cambridge University Press, 1904).

Walmsley, I. A.

Washburn, B. R.

White, H. E.

F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 1976).

Xie, X. S.

D. J. Jones, E. O. Potma, J. X. Cheng, B. Burfeindt, Y. Pang, J. Ye, and X. S. Xie, “Synchronization of two passively mode-locked, picosecond lasers within 20 fs for coherent anti-Stokes Raman scattering microscopy,” Rev. Sci. Instrum. 73, 2843–2848 (2002).
[CrossRef]

Yamane, K.

Yamashita, M.

Yang, C. H.

Ye, J.

D. J. Jones, E. O. Potma, J. X. Cheng, B. Burfeindt, Y. Pang, J. Ye, and X. S. Xie, “Synchronization of two passively mode-locked, picosecond lasers within 20 fs for coherent anti-Stokes Raman scattering microscopy,” Rev. Sci. Instrum. 73, 2843–2848 (2002).
[CrossRef]

Yelin, D.

D. Meshulach, D. Yelin, and Y. Silberberg, “White light dispersion measurements by one- and two-dimensional spectral interference,” IEEE J. Quantum Electron. 33, 1969–1974 (1997).
[CrossRef]

Zhang, Z. G.

Zhou, L.

Appl. Opt. (2)

Appl. Phys. B (2)

C. Dorrer, N. Belabas, J. P. Likforman, and M. Joffre, “Experimental implementation of Fourier-transform spectral interferometry and its application to the study of spectrometers,” Appl. Phys. B 70, S99–S107 (2000).
[CrossRef]

P. Baum, S. Lochbrunner, and E. Riedle, “Generation of tunable 7-fs ultraviolet pulses: achromatic phase matching and chirp management,” Appl. Phys. B 79, 1027–1032 (2004).
[CrossRef]

Appl. Spectrosc. (2)

Astron. Astrophys. (1)

M. Gai, D. Carollo, M. Delbo, M. G. Lattanzi, G. Massone, F. Bertinetto, G. Mana, and S. Cesare, “Location accuracy limitations for CCD cameras,” Astron. Astrophys. 367, 362–370 (2001).
[CrossRef]

Astron. Astrophys. Suppl. Ser. (1)

J. P. De Cuyper and H. Hensberge, “Wavelength calibration at moderately high resolution,” Astron. Astrophys. Suppl. Ser. 128, 409–416 (1998).
[CrossRef]

Astrophys. J. (1)

A. A. Michelson and F. G. Pease, “Measurement of the diameter of an Orionis with the interferometer,” Astrophys. J. 53, 249–259 (1921).
[CrossRef]

IEEE J. Quantum Electron. (4)

G. Steinmeyer, “Dispersion oscillations in ultrafast phase-correction devices,” IEEE J. Quantum Electron. 39, 1027–1034 (2003).
[CrossRef]

D. Meshulach, D. Yelin, and Y. Silberberg, “White light dispersion measurements by one- and two-dimensional spectral interference,” IEEE J. Quantum Electron. 33, 1969–1974 (1997).
[CrossRef]

M. Beck, I. A. Walmsley, and J. D. Kafka, “Group delay measurements of optical-components near 800 nm,” IEEE J. Quantum Electron. 27, 2074–2081 (1991).
[CrossRef]

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

IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. (1)

T. P. Costello and W. B. Mikhael, “One-dimensional comparison of Wiener filtering and Richardson-Lucy methods for sectioned restoration of space-variant digital images,” IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 49, 518–522 (2002).
[CrossRef]

J. Chem. Phys. (3)

J. D. Hybl, A. Albrecht Ferro, and D. M. Jonas, “Two dimensional Fourier transform electronic spectroscopy,” J. Chem. Phys. 115, 6606–6622 (2001).
[CrossRef]

A. W. Albrecht, J. D. Hybl, S. M. Gallagher Faeder, and D. M. Jonas, “Experimental distinction between phase shifts and time delays: implications for femtosecond spectroscopy and coherent control of chemical reactions,” J. Chem. Phys. 111, 10934–10956 (1999).
[CrossRef]

A. W. Albrecht Ferro, J. D. Hybl, S. M. Gallagher Faeder, and D. M. Jonas, “‘Erratum: Experimental distinction between phase shifts and time delays: implications for femtosecond spectroscopy and coherent control of chemical reactions’ [J. Chem. Phys. 111, 10934 (1999)],” J. Chem. Phys. 115, 5691 (2001).
[CrossRef]

J. Mod. Opt. (2)

P. de Groot and L. Deck, “Surface profiling by analysis of white-light interferograms in the spatial frequency domain,” J. Mod. Opt. 42, 389–401 (1995).
[CrossRef]

P. Sandoz, G. Tribillon, and H. Perrin, “High-resolution profilometry by using phase calculation algorithms for spectroscopic analysis of white-light interferograms,” J. Mod. Opt. 43, 701–708 (1996).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (3)

Metrologia (1)

G. Bönsch and E. Potulski, “Measurement of the refractive index of air and comparison with modified Edlen's formulae,” Metrologia 35, 133–139 (1998).
[CrossRef]

Opt. Express (2)

Opt. Lasers Eng. (1)

S. K. Debnath, M. P. Kothiyal, and S.-W. Kim, “Evaluation of spectral phase in spectrally resolved white-light interferometry: comparative study of single frame techniques,” Opt. Lasers Eng. 47, 1125–1130 (2009).
[CrossRef]

Opt. Lett. (7)

Phys. Scr. (1)

G. Norlen, “Wavelengths and energy-levels of Ar-I and Ar-II based on new interferometric measurements in region 3400–9800 A,” Phys. Scr. 8, 249–268 (1973).
[CrossRef]

Proc. SPIE (1)

M. Downing, D. Baade, P. Sinclaire, S. Deries, and F. Christen, Proc. SPIE 6276, 627609 (2006).
[CrossRef]

Publ. Astron. Soc. Pac. (1)

J. Anderson and I. R. King, “Toward high-precision astrometry with WFPC2. I. Deriving an accurate point-spread function,” Publ. Astron. Soc. Pac. 112, 1360–1382 (2000).
[CrossRef]

Rev. Sci. Instrum. (1)

D. J. Jones, E. O. Potma, J. X. Cheng, B. Burfeindt, Y. Pang, J. Ye, and X. S. Xie, “Synchronization of two passively mode-locked, picosecond lasers within 20 fs for coherent anti-Stokes Raman scattering microscopy,” Rev. Sci. Instrum. 73, 2843–2848 (2002).
[CrossRef]

Other (27)

M. Metcalf and J. Reid, Fortran 90/95 Explained (Oxford U. Press, 1996).

P. Hariharan and B. C. Sanders, in Progress in Optics (Elsevier Science, 1996), Vol. XXXVI, pp. 49–128.
[CrossRef]

The neon lines in are defined by vacuum wave numbers, which were inverted to wavelengths. The six lines used had the following vacuum wavelengths (in nanometers): 703.434 88, 717.591 20, 724.715 93, 744.094 35, 837.990 54, and 849.769 00.

R. J. Bell, Introductory Fourier Transform Spectroscopy (Academic, 1972).

The JILA loop filter was designed by J. L. Hall and T. Brown.

The standard deviation of the set of calibration constantsc1 is 2.0×10−6 nm/pixel, smaller than the 4.4×10−6 nm/pixel average standard error estimated from the fits to individual spectra. The standard deviation of the set of calibration constants c2 is 2.8×10−9 nm/pixel2, also smaller than the 1.8×10−8 nm/pixel2 average standard error of the fits to individual spectra. This means torsion or forward/backward motion of the CCD relative to the image plane is not detectable within the calibration precision and affects the calibration by less than the standard error of c1 and c2.

R.C.Weast and M.J.Astle, eds., CRC Handbook of Chemistry and Physics, 63rd ed. (CRC, 1982).

P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, 1992).

A 0.05% sinusoidal ripple with a 14 pixel period is expected on the flatfield from this procedure. This is smaller than the shot noise on the interferograms. In principle, the expected ripple could be divided out of the flatfield, but simulated interferograms show that a 0.05% sinusoidal ripple in the flatfield causes less than 0.5 mrad phase ripple, so this was not done.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, 1990).

J. R. Janesick, Scientific Charge-Coupled Devices (SPIE, 2001).
[CrossRef]

W. H. Steel, Interferometry, Cambridge Monographs on Physics (Cambridge U. Press, 1967).

A. E. Siegman, Lasers (University Science Books, 1986).

F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 1976).

R. D. Campbell and D. J. Thompson, in Scientific Detectors for Astronomy 2005, J.E.Beletic, J.W.Beletic, and P.Amico, eds. (Springer, 2005), pp. 507–514.

A. Reinheimer, e2v technologies, Tarrytown, NY (personal communication, March 11, 2008).

The argon lines used (from ) had the following vacuum wavelengths (in nanometers): 912.547 13, 867.032 50, 852.378 34, 826.679 43, 795.036 27, 763.720 78, 738.601 45, 727.494 00, and 696.735 19.

The polynomial was truncated at the quadratic term as the cubic term (predicted by the grating equation) was not well determined. Standard errors for the cubic coefficient, as estimated by least-squares fitting of single calibration spectra, were 30%–150% of its average value(5×10−11 nm/pixel3). In contrast, the quadratic term was determined to within 1%–3% of its value (4.8×10−7 nm/pixel2).

J. Walker, The Analytical Theory of Light (the Cambridge University Press, 1904).

From the standard deviation of the data in Fig. 6a of , the precision can be estimated as 0.25 fs2.

J. R. Birge, R. Ell, and F. X. Kartner, in Ultrafast Phenomena XV, P.Corkum, D.Jonas, R.J. D.Miller, and A.M.Weiner, eds. (Springer, 2006), pp. 160–162.

J. H. Seinfeld and S. N. Pandis, Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, 2nd ed. (John Wiley & Sons, 2006).

The magnitude of the largest oscillation in the phase delay error (at ∼2.35 rad/fs for 2 ps delay) gives a change in the real refractive index, Δn, on the order of 10−6. From the Kramers–Kronig relationship, the change in the imaginary portion of the refractive index, Δκ, is approximately equal to Δn. By α=2ωκ/c, the absorption coefficient associated with the Δn from the Δϕ can be found and inverted to an absorption length l=1/α=17 m, several orders of magnitude shorter than that of atmospheric O2 or water vapor (see ).

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vettering, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, 1986).

S. B. Howell, Handbook of CCD Astronomy, 2nd ed., Cambridge Observing Handbooks for Research Astronomers (Cambridge U. Press, 2006).

The wavelength axis was given by λ(p)=670 nm+(p−1)⋅0.5 nm/pixel for p=1 through N=512. The pulse spectrum was |e(λ)|2=(λ0/λ)2exp[−(λ−λ0)2/2σ2] withσ2=250 nm2 and λ0=800 nm (a FWHM of ∼75 pixels on the 512 pixel array). The simulated eLSF for each pixel was a convolution of a pixel-centered Gaussian,exp[−(P−p)2/2σ2], where σ=0.53 pixels is constant; a one-sided exponential decay, θ(p−P)exp[−(p−P)/wp]; and a 1-pixel wide pixelation function. The final eLSF was re-positioned with its maximum on the pixel center. The variation in the eLSF was created with the exponential width, wp=2−tanh[(p−(N/2))/40], producing an eLSF FWHM ranging from 2 to 3.4 pixels.

The wavelength axis was given by λ(p)=670 nm+(p−1)⋅0.25 nm/pixel for p=1 through N=1024. The eLSF was constructed in the same way described in note except that σ=0.21 pixels and wp=0.675−(0.125)tanh[(p−(N/2))/200], producing an eLSF FWHM ranging from 1.23 to 1.39 pixels, which was sufficiently undersampled on a grid of 1024 pixels.

Supplementary Material (1)

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Figures (10)

Fig. 1
Fig. 1

Simulation to illustrate pseudo-deconvolution for a spectral interferogram with 500 fs delay. Top panel: ideal interferogram I ( λ ) (gray line); blurred interferogram I p (thick black line) calculated with Eq. (2) for a wavelength dependent asymmetric eLSF, s p ; spectrum I 0 ( λ p ) as restored by pseudo-deconvolution using Eq. (5) on a 512-pixel array (diamonds); sum of pulse spectra [first two terms of Eq. (1)] (thin black line). Bottom panel: representative asymmetric pixel sensitivity functions S p (thick black lines) with their pixel centers marked at 764, 795, and 827 nm; difference between restored interferogram and ideal interferogram (crosses) and as interpolated by zero-padding (gray line). The eLSF width increases for shorter wavelengths resulting in decreased fringe depth for the blurred interferogram at shorter wavelength. The fringes in the blurred interferogram are also shifted toward shorter wavelengths with respect to the ideal interferogram; this shift is more pronounced at shorter wavelengths as the eLSF becomes increasingly asymmetric. The restored interferogram is within ±0.6% of the ideal interferogram at every pixel, with a rms error over 750–850 nm of 0.2%. In this exaggerated example, the eLSF is over two times wider than in the experiment and varies ten times more, which magnifies the errors from pseudo-deconvolution by 2 orders of magnitude.

Fig. 2
Fig. 2

Stabilized Mach–Zehnder interferometer. One arm of the interferometer has a retroreflector on a computer-controlled motorized translation stage. The other arm’s retroreflector is mounted on a PZT, which receives the feedback signal from the difference between the two interferometer outputs for the red CW laser. Stabilization was measured out of loop by monitoring the difference of the two yellow CW laser interferometer outputs. SF: spatial filter; DBS1: dichroic beam splitter (transmits Ti:sapphire; reflects red and yellow CW); C: chopper; DBS2: dichroic beam splitter (transmits yellow CW; reflects red CW); λ / 2 : half-wave plate (rotates Ti:sapphire polarization from horizontal to vertical); BS: 50–50 beam splitter, s-polarized; RR: trihedral retroreflector; P: ultrafast plate polarizer with plane of incidence parallel to polarization vector of vertically polarized light (transmits p-polarized light); DBS3: dichroic beam splitter (transmits red and yellow CW, reflects Ti:sapphire); BB: beam block; F1: longpass filter (cut-on wavelength: 650 nm); W: window; FC: fiber coupler; SMF: single-mode fiber; F2: red CW bandpass filter ( 632.8 ± 5   nm ) ; L: plano-convex lens, f = 25.4   mm ; DPD: differential photodiode (for red CW interferometer outputs); F3: yellow CW bandpass filter ( 600 ± 5   nm ) ; PD: single photodiodes (for yellow CW interferometer outputs); PLL: phase locked loop. Protected silver mirrors are unlabeled.

Fig. 3
Fig. 3

Atomic line spectra interleaved for the vicinity of λ = 764   nm (crosses). The open squares are an undersampled set of data points from a single calibration spectrum. The solid line is the smoothed cubic spline fit to the interleaved spectra which is used as the eLSF of the instrument near 764 nm.

Fig. 4
Fig. 4

eLSF in the relative wavelength domain determined at 852 nm (dotted line) and 764 nm (solid line). Each is normalized to have unit area.

Fig. 5
Fig. 5

Time domain fringe visibility ratio V ̂ / V ideal for the eLSF determined at 764 nm with the algorithm described in Subsection 3B (solid line). The time domain fringe visibility ratio is calculated from the eLSF through a Jacobian transformation from wavelength to frequency followed by a Fourier transformation to the time domain. Since V ideal 1 , the ratio is compared to experimental measurements of the fringe visibility for three sets of time delays (open squares, triangles, and diamonds). The temporal phase that results from placing the wavelength origin of the eLSF at its maximum is plotted as a dashed line.

Fig. 6
Fig. 6

Phase delay error in attoseconds for two different delays. Data are plotted both with and without spectral restoration with the eLSF. Delay time of 1915 fs: (dots) without spectral restoration and (long dashes) after spectral restoration. Delay time of 420 fs: without (dash-dot) and with (short dashes) spectral restoration. Intensity spectrum of Ti:sapphire pulse (solid line). The ×’s on the bottom horizontal axis indicate the frequencies of all unblended argon calibration lines in the range shown.

Fig. 7
Fig. 7

The top panel compares the experimental interferogram I p (black line) with the spectrally restored interferogram I 0 ( λ p ) (gray line). Fourier pseudo-deconvolution with the complex valued OTF not only improves the fringe visibility, as seen in the top panel, but it shifts the phase of the interferogram as shown in the bottom panel. The bottom panel shows intensities from the raw interferogram (filled black diamonds) and the restored interferogram (open circles). The smooth curve (gray dots) for the restored interferogram was obtained by zero-padding in quasi-time. As a result of the asymmetric eLSF, the restored spectrum is shifted toward higher pixel numbers near several maxima and minima (for example, pixels 480 and 481; pixels 491 and 492; and pixels 502 and 503).

Fig. 8
Fig. 8

Phase delay error in attoseconds for seven different delays after phase recovery algorithm on spectrally restored interferograms with the emission lamp frequency calibration described in Subsection 3C. Lines grow in darkness and thickness with increasing delay: 270, 422, 545, 871, 1261, 1636, and 1959 fs. Each delay is normalized to reference pixel 496 and the oscillatory features are more pronounced as the delay increases. This translates to a calibration error, which requires the frequency calibration refinement algorithm described in Subsection 4B.

Fig. 9
Fig. 9

Constant phase shift error versus absolute time delay. Open diamonds—raw data subjected to phase recovery algorithm; asterisks—phases recovered after Fourier deconvolution with the MTF from the central eLSF; crosses—data subjected to phase recovery algorithm after spectral restoration; open squares—phases recovered after spectral restoration and the frequency refinement algorithm described in Section 4.

Fig. 10
Fig. 10

The average phase residual (solid line) and standard deviation (dotted line) of ten interferograms ranging in delay from 199 to 1915 fs. The dashed line is the SQL for the phase uncertainty, e ϕ ( ω p ) = 1 / [ 2 M ( ω p ) ] , where M is the mean number of photons in the interferometer detectable at pixel p.

Equations (20)

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I ( ω ) = e 1 ( ω ) 2 + e 2 ( ω ) 2 + 2 e 1 ( ω ) e 2 ( ω ) cos [ Δ ϕ ( ω ) ] ,
I p = 0 d λ I ( λ ) S p ( λ p λ ) .
T ̂ p eff ( ξ ) = s p ( P ) e i ξ P d P .
I ̂ ξ = p = 0 N 1 I p e i ξ p Δ p ,
I p 0 = 1 2 π n = 0 N 1 I ̂ ξ T ̂ p eff ( ξ ) e i ξ p Δ ξ ,
I 0 ( λ p ) = ( | d λ d P | P = p ) I p 0 ,
λ ( P ) = c 0 + c 1 ( P N 2 ) + c 2 ( P N 2 ) 2 ,
e P ( P max p max ) = P max P ( λ lit ) ,
f ( ξ ) = { tanh [ ( ξ ξ 1 ) / 5 ] tanh [ ( ξ ξ 2 ) / 5 ] } / 2 ,
δ Δ ϕ ( ω ) = Δ ϕ ( ω ) Δ ϕ r ( ω ) ,
δ Δ ϕ ( ω ) = ω n ( ω ) j λ cw / ( n cw c ) = ω n ( ω ) j T / n cw ,
τ ϕ ( ω ) = δ Δ ϕ ( ω ) n cw / [ ω n ( ω ) ] ,
e ϕ ( ω ) = δ Δ ϕ ( ω ) [ n cw / n ( ω ) ] ω j T ,
e τ ϕ ( ω ) = e ϕ ( ω ) / ω .
δ Δ ϕ ( ω ) [ n cw / n ( ω ) ] = ω τ s + ϕ 0 .
e τ s = τ s j T .
Δ ϕ ( ω ) = ϕ 0 + ω t + ϕ ( ω ω ¯ ) 2 / 2 ,
e τ ϕ ( ω ) = ω τ u / [ ω e ω p ] j T ,
e τ ϕ ( ω p ) e τ ϕ ( ω o ) = τ u e ω p / ω p τ u e ω o / ω o .
Δ ϕ ( ω ) 2 π m = ( ω n ( ω ) c ) d + ( ω n FS ( ω ) c cos [ θ FS ( ω ) ] ) δ t ,

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