Abstract

A new model for the determination of group delay (GD) and GD dispersion of dispersive mirrors is presented. The algorithm based on this model enables one to process interferometric data pro vided by a white-light interferometer and to obtain GD wavelength dependence over a broad spectral range.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed. (Academic, 2006).
  2. R. Szipöcs, K. Ferencz, C. Spielmann, and F. Krausz, “Chirped multilayer coatings for broadband dispersion control in femtosecond lasers,” Opt. Lett. 19, 201-203 (1994).
    [CrossRef] [PubMed]
  3. V. Pervak, A. V. Tikhonravov, M. K. Trubetskov, S. Naumov, F. Krausz, and A. Apolonski, “1.5-octave chirped mirror for pulse compression down to sub-3 fs,” Appl. Phys. B 87, 5-12 (2007).
    [CrossRef]
  4. A. V. Tikhonravov, M. K. Trubetskov, V. Pervak, F. Krausz, and A. Apolonski, “Design, fabrication and reverse engineering of broad band chirped mirrors,” in Proceedings, Optical Interference Coatings on CD-ROM, Presentation WB4, Tucson, Arizona (OSA, 2007).
  5. V. Pervak, C. Tiesset, A. Sugita, S. Naumov, F. Krausz, and A. Apolonski, “High-dispersive mirrors for femtosecond lasers,” Opt. Express 16, 10220-10233 (2008).
    [CrossRef] [PubMed]
  6. V. Pervak, S. Naumov, F. Krausz, and A. Apolonski, “Chirped mirrors with low dispersion ripple,” Opt. Express 15, 13768-13772 (2007).
    [CrossRef] [PubMed]
  7. V. Pervak, F. Krausz, and A. Apolonski, “Dispersion control over the UV-VIS-NIR spectral range with HfO2/SiO2 chirped dielectric multilayers,” Opt. Lett. 32, 1183-1185 (2007).
    [CrossRef] [PubMed]
  8. A. Ozawa, J. Rauschenberger, C. Gohle, M. Herrmann, D. R. Walker, V. Pervak, A. Fernandez, R. Graf, A. Apolonski, R. Holzwarth, F. Krausz, T. W. Haensch, and T. Udem, “High harmonic frequency comb for high resolution spectroscopy,” Phys. Rev. Lett. 100, 253901 (2008).
    [CrossRef] [PubMed]
  9. W. H. Knox, N. M. Pearson, K. D. Li, and C. A. Hirlimann, “Interferometric measurements of femtosecond group delay in optical components,” Opt. Lett. 13, 574-576 (1988).
    [CrossRef] [PubMed]
  10. W. H. Knox, “Dispersion measurements for femtosecond-pulse generation and applications,” Appl. Phys. B 58, 225-235 (1994).
    [CrossRef]
  11. A. Gosteva, M. Haiml, R. Paschotta, and U. Keller, “Noise-related resolution limit of dispersion measurements with white-light interferometers,” J. Opt. Soc. Am. B 22, 1868-1874 (2005).
    [CrossRef]
  12. M. Beck and I. A. Walmsley, “Measurement of group delay with high temporal and spectral resolution,” Opt. Lett. 15, 492-494 (1990).
    [CrossRef] [PubMed]
  13. K. Naganuma, K. Mogi, and H. Yamada, “Group-delay measurement using the Fourier transform of an interferometric cross correlation generated by white light,” Opt. Lett. 15, 393-395 (1990).
    [CrossRef] [PubMed]
  14. S. Daddams and J.-C. Diels, “Dispersion measurements with white-light interferometry,” J. Opt. Soc. Am. B 13, 1120-1129 (1996).
    [CrossRef]
  15. V. N. Kumar and D. N. Rao, “Using interference in the frequency domain for precise determination of thickness and refractive indices of normal dispersive materials,” J. Opt. Soc. Am. B 12, 1559-1563 (1995).
    [CrossRef]
  16. A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, 1977).
  17. A. P. Kovács, K. Osvay, Z. Bor, and R. Szipöcs, “Group-delay measurement on laser mirrors by spectrally resolved white-light interferometry,” Opt. Lett. 20, 788-790 (1995).
    [CrossRef] [PubMed]
  18. S. A. Akhomanov and S. Y. Nikitin, Physical Optics (Oxford University, 1997).
  19. C. H. Reinsch, “Smoothing by spline function,” Numer. Math. 10, 177-183 (1967).
    [CrossRef]

2008

A. Ozawa, J. Rauschenberger, C. Gohle, M. Herrmann, D. R. Walker, V. Pervak, A. Fernandez, R. Graf, A. Apolonski, R. Holzwarth, F. Krausz, T. W. Haensch, and T. Udem, “High harmonic frequency comb for high resolution spectroscopy,” Phys. Rev. Lett. 100, 253901 (2008).
[CrossRef] [PubMed]

V. Pervak, C. Tiesset, A. Sugita, S. Naumov, F. Krausz, and A. Apolonski, “High-dispersive mirrors for femtosecond lasers,” Opt. Express 16, 10220-10233 (2008).
[CrossRef] [PubMed]

2007

V. Pervak, A. V. Tikhonravov, M. K. Trubetskov, S. Naumov, F. Krausz, and A. Apolonski, “1.5-octave chirped mirror for pulse compression down to sub-3 fs,” Appl. Phys. B 87, 5-12 (2007).
[CrossRef]

A. V. Tikhonravov, M. K. Trubetskov, V. Pervak, F. Krausz, and A. Apolonski, “Design, fabrication and reverse engineering of broad band chirped mirrors,” in Proceedings, Optical Interference Coatings on CD-ROM, Presentation WB4, Tucson, Arizona (OSA, 2007).

V. Pervak, F. Krausz, and A. Apolonski, “Dispersion control over the UV-VIS-NIR spectral range with HfO2/SiO2 chirped dielectric multilayers,” Opt. Lett. 32, 1183-1185 (2007).
[CrossRef] [PubMed]

V. Pervak, S. Naumov, F. Krausz, and A. Apolonski, “Chirped mirrors with low dispersion ripple,” Opt. Express 15, 13768-13772 (2007).
[CrossRef] [PubMed]

2006

J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed. (Academic, 2006).

2005

1997

S. A. Akhomanov and S. Y. Nikitin, Physical Optics (Oxford University, 1997).

1996

1995

1994

1990

1988

1977

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, 1977).

1967

C. H. Reinsch, “Smoothing by spline function,” Numer. Math. 10, 177-183 (1967).
[CrossRef]

Akhomanov, S. A.

S. A. Akhomanov and S. Y. Nikitin, Physical Optics (Oxford University, 1997).

Apolonski, A.

V. Pervak, C. Tiesset, A. Sugita, S. Naumov, F. Krausz, and A. Apolonski, “High-dispersive mirrors for femtosecond lasers,” Opt. Express 16, 10220-10233 (2008).
[CrossRef] [PubMed]

A. Ozawa, J. Rauschenberger, C. Gohle, M. Herrmann, D. R. Walker, V. Pervak, A. Fernandez, R. Graf, A. Apolonski, R. Holzwarth, F. Krausz, T. W. Haensch, and T. Udem, “High harmonic frequency comb for high resolution spectroscopy,” Phys. Rev. Lett. 100, 253901 (2008).
[CrossRef] [PubMed]

V. Pervak, S. Naumov, F. Krausz, and A. Apolonski, “Chirped mirrors with low dispersion ripple,” Opt. Express 15, 13768-13772 (2007).
[CrossRef] [PubMed]

V. Pervak, A. V. Tikhonravov, M. K. Trubetskov, S. Naumov, F. Krausz, and A. Apolonski, “1.5-octave chirped mirror for pulse compression down to sub-3 fs,” Appl. Phys. B 87, 5-12 (2007).
[CrossRef]

V. Pervak, F. Krausz, and A. Apolonski, “Dispersion control over the UV-VIS-NIR spectral range with HfO2/SiO2 chirped dielectric multilayers,” Opt. Lett. 32, 1183-1185 (2007).
[CrossRef] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, V. Pervak, F. Krausz, and A. Apolonski, “Design, fabrication and reverse engineering of broad band chirped mirrors,” in Proceedings, Optical Interference Coatings on CD-ROM, Presentation WB4, Tucson, Arizona (OSA, 2007).

Arsenin, V. Y.

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, 1977).

Beck, M.

Bor, Z.

Daddams, S.

Diels, J.-C.

Ferencz, K.

Fernandez, A.

A. Ozawa, J. Rauschenberger, C. Gohle, M. Herrmann, D. R. Walker, V. Pervak, A. Fernandez, R. Graf, A. Apolonski, R. Holzwarth, F. Krausz, T. W. Haensch, and T. Udem, “High harmonic frequency comb for high resolution spectroscopy,” Phys. Rev. Lett. 100, 253901 (2008).
[CrossRef] [PubMed]

Gohle, C.

A. Ozawa, J. Rauschenberger, C. Gohle, M. Herrmann, D. R. Walker, V. Pervak, A. Fernandez, R. Graf, A. Apolonski, R. Holzwarth, F. Krausz, T. W. Haensch, and T. Udem, “High harmonic frequency comb for high resolution spectroscopy,” Phys. Rev. Lett. 100, 253901 (2008).
[CrossRef] [PubMed]

Gosteva, A.

Graf, R.

A. Ozawa, J. Rauschenberger, C. Gohle, M. Herrmann, D. R. Walker, V. Pervak, A. Fernandez, R. Graf, A. Apolonski, R. Holzwarth, F. Krausz, T. W. Haensch, and T. Udem, “High harmonic frequency comb for high resolution spectroscopy,” Phys. Rev. Lett. 100, 253901 (2008).
[CrossRef] [PubMed]

Haensch, T. W.

A. Ozawa, J. Rauschenberger, C. Gohle, M. Herrmann, D. R. Walker, V. Pervak, A. Fernandez, R. Graf, A. Apolonski, R. Holzwarth, F. Krausz, T. W. Haensch, and T. Udem, “High harmonic frequency comb for high resolution spectroscopy,” Phys. Rev. Lett. 100, 253901 (2008).
[CrossRef] [PubMed]

Haiml, M.

Herrmann, M.

A. Ozawa, J. Rauschenberger, C. Gohle, M. Herrmann, D. R. Walker, V. Pervak, A. Fernandez, R. Graf, A. Apolonski, R. Holzwarth, F. Krausz, T. W. Haensch, and T. Udem, “High harmonic frequency comb for high resolution spectroscopy,” Phys. Rev. Lett. 100, 253901 (2008).
[CrossRef] [PubMed]

Hirlimann, C. A.

Holzwarth, R.

A. Ozawa, J. Rauschenberger, C. Gohle, M. Herrmann, D. R. Walker, V. Pervak, A. Fernandez, R. Graf, A. Apolonski, R. Holzwarth, F. Krausz, T. W. Haensch, and T. Udem, “High harmonic frequency comb for high resolution spectroscopy,” Phys. Rev. Lett. 100, 253901 (2008).
[CrossRef] [PubMed]

Keller, U.

Knox, W. H.

Kovács, A. P.

Krausz, F.

A. Ozawa, J. Rauschenberger, C. Gohle, M. Herrmann, D. R. Walker, V. Pervak, A. Fernandez, R. Graf, A. Apolonski, R. Holzwarth, F. Krausz, T. W. Haensch, and T. Udem, “High harmonic frequency comb for high resolution spectroscopy,” Phys. Rev. Lett. 100, 253901 (2008).
[CrossRef] [PubMed]

V. Pervak, C. Tiesset, A. Sugita, S. Naumov, F. Krausz, and A. Apolonski, “High-dispersive mirrors for femtosecond lasers,” Opt. Express 16, 10220-10233 (2008).
[CrossRef] [PubMed]

V. Pervak, F. Krausz, and A. Apolonski, “Dispersion control over the UV-VIS-NIR spectral range with HfO2/SiO2 chirped dielectric multilayers,” Opt. Lett. 32, 1183-1185 (2007).
[CrossRef] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, V. Pervak, F. Krausz, and A. Apolonski, “Design, fabrication and reverse engineering of broad band chirped mirrors,” in Proceedings, Optical Interference Coatings on CD-ROM, Presentation WB4, Tucson, Arizona (OSA, 2007).

V. Pervak, A. V. Tikhonravov, M. K. Trubetskov, S. Naumov, F. Krausz, and A. Apolonski, “1.5-octave chirped mirror for pulse compression down to sub-3 fs,” Appl. Phys. B 87, 5-12 (2007).
[CrossRef]

V. Pervak, S. Naumov, F. Krausz, and A. Apolonski, “Chirped mirrors with low dispersion ripple,” Opt. Express 15, 13768-13772 (2007).
[CrossRef] [PubMed]

R. Szipöcs, K. Ferencz, C. Spielmann, and F. Krausz, “Chirped multilayer coatings for broadband dispersion control in femtosecond lasers,” Opt. Lett. 19, 201-203 (1994).
[CrossRef] [PubMed]

Kumar, V. N.

Li, K. D.

Mogi, K.

Naganuma, K.

Naumov, S.

Nikitin, S. Y.

S. A. Akhomanov and S. Y. Nikitin, Physical Optics (Oxford University, 1997).

Osvay, K.

Ozawa, A.

A. Ozawa, J. Rauschenberger, C. Gohle, M. Herrmann, D. R. Walker, V. Pervak, A. Fernandez, R. Graf, A. Apolonski, R. Holzwarth, F. Krausz, T. W. Haensch, and T. Udem, “High harmonic frequency comb for high resolution spectroscopy,” Phys. Rev. Lett. 100, 253901 (2008).
[CrossRef] [PubMed]

Paschotta, R.

Pearson, N. M.

Pervak, V.

V. Pervak, C. Tiesset, A. Sugita, S. Naumov, F. Krausz, and A. Apolonski, “High-dispersive mirrors for femtosecond lasers,” Opt. Express 16, 10220-10233 (2008).
[CrossRef] [PubMed]

A. Ozawa, J. Rauschenberger, C. Gohle, M. Herrmann, D. R. Walker, V. Pervak, A. Fernandez, R. Graf, A. Apolonski, R. Holzwarth, F. Krausz, T. W. Haensch, and T. Udem, “High harmonic frequency comb for high resolution spectroscopy,” Phys. Rev. Lett. 100, 253901 (2008).
[CrossRef] [PubMed]

V. Pervak, A. V. Tikhonravov, M. K. Trubetskov, S. Naumov, F. Krausz, and A. Apolonski, “1.5-octave chirped mirror for pulse compression down to sub-3 fs,” Appl. Phys. B 87, 5-12 (2007).
[CrossRef]

V. Pervak, S. Naumov, F. Krausz, and A. Apolonski, “Chirped mirrors with low dispersion ripple,” Opt. Express 15, 13768-13772 (2007).
[CrossRef] [PubMed]

V. Pervak, F. Krausz, and A. Apolonski, “Dispersion control over the UV-VIS-NIR spectral range with HfO2/SiO2 chirped dielectric multilayers,” Opt. Lett. 32, 1183-1185 (2007).
[CrossRef] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, V. Pervak, F. Krausz, and A. Apolonski, “Design, fabrication and reverse engineering of broad band chirped mirrors,” in Proceedings, Optical Interference Coatings on CD-ROM, Presentation WB4, Tucson, Arizona (OSA, 2007).

Rao, D. N.

Rauschenberger, J.

A. Ozawa, J. Rauschenberger, C. Gohle, M. Herrmann, D. R. Walker, V. Pervak, A. Fernandez, R. Graf, A. Apolonski, R. Holzwarth, F. Krausz, T. W. Haensch, and T. Udem, “High harmonic frequency comb for high resolution spectroscopy,” Phys. Rev. Lett. 100, 253901 (2008).
[CrossRef] [PubMed]

Reinsch, C. H.

C. H. Reinsch, “Smoothing by spline function,” Numer. Math. 10, 177-183 (1967).
[CrossRef]

Rudolph, W.

J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed. (Academic, 2006).

Spielmann, C.

Sugita, A.

Szipöcs, R.

Tiesset, C.

Tikhonov, A. N.

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, 1977).

Tikhonravov, A. V.

V. Pervak, A. V. Tikhonravov, M. K. Trubetskov, S. Naumov, F. Krausz, and A. Apolonski, “1.5-octave chirped mirror for pulse compression down to sub-3 fs,” Appl. Phys. B 87, 5-12 (2007).
[CrossRef]

A. V. Tikhonravov, M. K. Trubetskov, V. Pervak, F. Krausz, and A. Apolonski, “Design, fabrication and reverse engineering of broad band chirped mirrors,” in Proceedings, Optical Interference Coatings on CD-ROM, Presentation WB4, Tucson, Arizona (OSA, 2007).

Trubetskov, M. K.

A. V. Tikhonravov, M. K. Trubetskov, V. Pervak, F. Krausz, and A. Apolonski, “Design, fabrication and reverse engineering of broad band chirped mirrors,” in Proceedings, Optical Interference Coatings on CD-ROM, Presentation WB4, Tucson, Arizona (OSA, 2007).

V. Pervak, A. V. Tikhonravov, M. K. Trubetskov, S. Naumov, F. Krausz, and A. Apolonski, “1.5-octave chirped mirror for pulse compression down to sub-3 fs,” Appl. Phys. B 87, 5-12 (2007).
[CrossRef]

Udem, T.

A. Ozawa, J. Rauschenberger, C. Gohle, M. Herrmann, D. R. Walker, V. Pervak, A. Fernandez, R. Graf, A. Apolonski, R. Holzwarth, F. Krausz, T. W. Haensch, and T. Udem, “High harmonic frequency comb for high resolution spectroscopy,” Phys. Rev. Lett. 100, 253901 (2008).
[CrossRef] [PubMed]

Walker, D. R.

A. Ozawa, J. Rauschenberger, C. Gohle, M. Herrmann, D. R. Walker, V. Pervak, A. Fernandez, R. Graf, A. Apolonski, R. Holzwarth, F. Krausz, T. W. Haensch, and T. Udem, “High harmonic frequency comb for high resolution spectroscopy,” Phys. Rev. Lett. 100, 253901 (2008).
[CrossRef] [PubMed]

Walmsley, I. A.

Yamada, H.

Appl. Phys. B

V. Pervak, A. V. Tikhonravov, M. K. Trubetskov, S. Naumov, F. Krausz, and A. Apolonski, “1.5-octave chirped mirror for pulse compression down to sub-3 fs,” Appl. Phys. B 87, 5-12 (2007).
[CrossRef]

W. H. Knox, “Dispersion measurements for femtosecond-pulse generation and applications,” Appl. Phys. B 58, 225-235 (1994).
[CrossRef]

J. Opt. Soc. Am. B

Numer. Math.

C. H. Reinsch, “Smoothing by spline function,” Numer. Math. 10, 177-183 (1967).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

A. Ozawa, J. Rauschenberger, C. Gohle, M. Herrmann, D. R. Walker, V. Pervak, A. Fernandez, R. Graf, A. Apolonski, R. Holzwarth, F. Krausz, T. W. Haensch, and T. Udem, “High harmonic frequency comb for high resolution spectroscopy,” Phys. Rev. Lett. 100, 253901 (2008).
[CrossRef] [PubMed]

Other

J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed. (Academic, 2006).

A. V. Tikhonravov, M. K. Trubetskov, V. Pervak, F. Krausz, and A. Apolonski, “Design, fabrication and reverse engineering of broad band chirped mirrors,” in Proceedings, Optical Interference Coatings on CD-ROM, Presentation WB4, Tucson, Arizona (OSA, 2007).

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, 1977).

S. A. Akhomanov and S. Y. Nikitin, Physical Optics (Oxford University, 1997).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Schematic of the WLI. BS, beam splitter; Ag, silver mirror; M, dispersive mirror under investigation.

Fig. 2
Fig. 2

Typical spectral scan with removed background spectrum.

Fig. 3
Fig. 3

Interferograms corresponding to three different wavelength values. Coordinates τ 1 , τ 2 , and τ 3 of the interferograms maxima are indicated by dashed lines. Relative shifts of these coordinates are specified by GDs at respective wavelength values.

Fig. 4
Fig. 4

Introduction of local minima and local maxima of an interferogram.

Fig. 5
Fig. 5

Typical fitting of local extrema of the measured inter ferogram (gray circles) by the model envelope function defined by Eq. (5) (solid black curve).

Fig. 6
Fig. 6

Schematic illustrating the procedure of calculating time increments.

Fig. 7
Fig. 7

Time increments caused by changing the motor position. Black narrow curve, increments corresponding to the wavelengths of 750 nm ; gray medium-width curve, 830 nm ; light gray wide curve, 910 nm .

Fig. 8
Fig. 8

Comparison of analytical GD and GDD wavelength dependencies (dashed gray curves) and calculated GD and GDD wavelength dependencies (black curves).

Fig. 9
Fig. 9

GDD obtained from the experimental data by two different methods: Fourier transform technique (dashed curve) and new algorithm described in this paper (solid curve). Theoretical GDD dependence of this mirror is shown by dotted curve for comparison.

Tables (1)

Tables Icon

Table 1 Effect of Error Level in Simulated Data on the Accuracy of Group Delay and Group Delay Dispersion Calculation

Equations (12)

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

t i = k = 1 i 1 Δ t k , Δ t k = 2 Δ s k / c ,
I ( t ) = 1 2 I γ ( t ) cos ω t , ω = 2 π c λ ,
γ ( t ) = sin ( Δ ω t / 2 ) Δ ω t / 2 , Δ ω = 4 π Δ λ λ 2 Δ λ 2 .
I ( t ) = 1 2 I γ ( t τ ( λ ) ) cos ( ω t + α ( τ ) ) , ω = 2 π c λ .
I env ( t ) = 1 2 I sin ( Δ ω ( t τ ) / 2 ) Δ ω ( t τ ) / 2 , Δ ω = 4 π Δ λ λ 2 Δ λ 2 .
DF = k = 1 N max ( | I env ( t max , k ) | I max , k ) 2 + k = 1 N min ( | I env ( t min , k ) | + I min , k ) 2 .
GDD ( λ ) = λ 2 2 π c · d τ ( λ ) d λ .
δ p , 1 = I p , 1 δ p I p , 2 I p , 1 , δ p , 2 = I p , k δ p I p , k I p , k 1 .
ξ r ξ l = ( K 3 ) δ p + δ p , 1 + δ p , 2 .
δ p = 1 2 T [ K + I p , 2 I p , 2 I p , 1 + I p , k I p , k I p , k 1 ] 1 .
I i = I ( t i ) ( 1 + δ i ) , i = 1 , , N ,
σ GD = 1 M j = 1 M ( τ ( λ j ) τ S ( λ j ) ) 2 , σ GDD = 1 M j = 1 M ( GDD ( λ j ) GDD S ( λ j ) ) 2 ,

Metrics