Abstract

We present a simple and quick, yet accurate method to measure the dispersion of high finesse optical cavities. By exciting the cavity with a femtosecond frequency comb and measuring the resonance condition as a function of optical frequency, the cavity’s dispersion can be determined with minimal uncertainty. Measurement results are presented from an evacuated reference cavity with low group delay dispersion as well as several differential, intra-cavity measurements of well known optical materials demonstrating the dynamic range and accuracy of this technique.

© 2009 Optical Society of America

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References

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  1. J. Ye and S. T. Cundiff, Femtosecond Optical Frequency Comb: Principle, Operation and Applications, 1st ed., (Springer, 2004).
  2. R. J. Jones and J. Ye, "Femtosecond pulse amplification by coherent addition in a passive optical cavity," Opt. Lett. 27, 1848-1850 (2002).
    [CrossRef]
  3. R. J. Jones, K. D. Moll, M. J. Thorpe, and J. Ye, "Phase-Coherent Frequency Combs in the Vacuum Ultraviolet via High-Harmonic Generation inside a Femtosecond Enhancement Cavity," Phys. Rev. Lett. 94, 193201 (2005).
    [CrossRef] [PubMed]
  4. C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hansch, "A frequency comb in the extreme ultraviolet," Nature 436, 234-237 (2005).
    [CrossRef] [PubMed]
  5. C. Gohle, B. Stein, A. Schliesser, T. Udem, and T. W. Hansch, "Frequency Comb Vernier Spectroscopy for Broadband, High-Resolution, High-Sensitivity Absorption and Dispersion Spectra," Phys. Rev. Lett. 99, 263,902-4 (2007).
    [CrossRef]
  6. M. J. Thorpe, K. D. Moll, R. J. Jones, B. Safdi, and J. Ye, "Broadband Cavity Ringdown Spectroscopy for Sensitive and Rapid Molecular Detection," Science 311, 1595-1599 (2006).
    [CrossRef] [PubMed]
  7. W. H. Knox, "Dispersion measurements for femtosecond-pulse generation and applications," Appl. Phys. B 58, 225-235 (1994).
    [CrossRef]
  8. A. Schliesser, C. Gohle, T. Udem, and T. W. Hansch, "Complete characterization of a broadband high-finesse cavity using an optical frequency comb," Opt. Express 14, 5975-5983 (2006).
    [CrossRef] [PubMed]
  9. M. Thorpe, R. Jones, K. Moll, J. Ye, and R. Lalezari, "Precise measurements of optical cavity dispersion and mirror coating properties via femtosecond combs," Opt. Express 13, 882-888 (2005).
    [CrossRef] [PubMed]
  10. J. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, Second Edition, 2nd ed. (Academic Press, 2006).
  11. R. J. Jones, "High Resolution Optical Frequency Metrology with Stabilized Femtosecond Lasers," Ph.D. thesis, University of New Mexico (2001).
  12. L. N. Trefethen, Spectral Methods in MATLAB, illustrated edition (SIAM: Society for Industrial and Applied Mathematics, 2001).
  13. F. Adler, K. Moutzouris, A. Leitenstorfer, H. Schnatz, B. Lipphardt, G. Grosche, and F. Tauser, "Phase-locked two-branch erbium-doped fiber laser system for long-term precision measurements of optical frequencies," Opt. Express 12, 5872-5880 (2004).
    [CrossRef] [PubMed]

2007 (1)

C. Gohle, B. Stein, A. Schliesser, T. Udem, and T. W. Hansch, "Frequency Comb Vernier Spectroscopy for Broadband, High-Resolution, High-Sensitivity Absorption and Dispersion Spectra," Phys. Rev. Lett. 99, 263,902-4 (2007).
[CrossRef]

2006 (2)

M. J. Thorpe, K. D. Moll, R. J. Jones, B. Safdi, and J. Ye, "Broadband Cavity Ringdown Spectroscopy for Sensitive and Rapid Molecular Detection," Science 311, 1595-1599 (2006).
[CrossRef] [PubMed]

A. Schliesser, C. Gohle, T. Udem, and T. W. Hansch, "Complete characterization of a broadband high-finesse cavity using an optical frequency comb," Opt. Express 14, 5975-5983 (2006).
[CrossRef] [PubMed]

2005 (3)

M. Thorpe, R. Jones, K. Moll, J. Ye, and R. Lalezari, "Precise measurements of optical cavity dispersion and mirror coating properties via femtosecond combs," Opt. Express 13, 882-888 (2005).
[CrossRef] [PubMed]

R. J. Jones, K. D. Moll, M. J. Thorpe, and J. Ye, "Phase-Coherent Frequency Combs in the Vacuum Ultraviolet via High-Harmonic Generation inside a Femtosecond Enhancement Cavity," Phys. Rev. Lett. 94, 193201 (2005).
[CrossRef] [PubMed]

C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hansch, "A frequency comb in the extreme ultraviolet," Nature 436, 234-237 (2005).
[CrossRef] [PubMed]

2004 (1)

2002 (1)

1994 (1)

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

Adler, F.

Gohle, C.

C. Gohle, B. Stein, A. Schliesser, T. Udem, and T. W. Hansch, "Frequency Comb Vernier Spectroscopy for Broadband, High-Resolution, High-Sensitivity Absorption and Dispersion Spectra," Phys. Rev. Lett. 99, 263,902-4 (2007).
[CrossRef]

A. Schliesser, C. Gohle, T. Udem, and T. W. Hansch, "Complete characterization of a broadband high-finesse cavity using an optical frequency comb," Opt. Express 14, 5975-5983 (2006).
[CrossRef] [PubMed]

C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hansch, "A frequency comb in the extreme ultraviolet," Nature 436, 234-237 (2005).
[CrossRef] [PubMed]

Grosche, G.

Hansch, T. W.

C. Gohle, B. Stein, A. Schliesser, T. Udem, and T. W. Hansch, "Frequency Comb Vernier Spectroscopy for Broadband, High-Resolution, High-Sensitivity Absorption and Dispersion Spectra," Phys. Rev. Lett. 99, 263,902-4 (2007).
[CrossRef]

A. Schliesser, C. Gohle, T. Udem, and T. W. Hansch, "Complete characterization of a broadband high-finesse cavity using an optical frequency comb," Opt. Express 14, 5975-5983 (2006).
[CrossRef] [PubMed]

C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hansch, "A frequency comb in the extreme ultraviolet," Nature 436, 234-237 (2005).
[CrossRef] [PubMed]

Herrmann, M.

C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hansch, "A frequency comb in the extreme ultraviolet," Nature 436, 234-237 (2005).
[CrossRef] [PubMed]

Holzwarth, R.

C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hansch, "A frequency comb in the extreme ultraviolet," Nature 436, 234-237 (2005).
[CrossRef] [PubMed]

Jones, R.

Jones, R. J.

M. J. Thorpe, K. D. Moll, R. J. Jones, B. Safdi, and J. Ye, "Broadband Cavity Ringdown Spectroscopy for Sensitive and Rapid Molecular Detection," Science 311, 1595-1599 (2006).
[CrossRef] [PubMed]

R. J. Jones, K. D. Moll, M. J. Thorpe, and J. Ye, "Phase-Coherent Frequency Combs in the Vacuum Ultraviolet via High-Harmonic Generation inside a Femtosecond Enhancement Cavity," Phys. Rev. Lett. 94, 193201 (2005).
[CrossRef] [PubMed]

R. J. Jones and J. Ye, "Femtosecond pulse amplification by coherent addition in a passive optical cavity," Opt. Lett. 27, 1848-1850 (2002).
[CrossRef]

Knox, W. H.

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

Krausz, F.

C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hansch, "A frequency comb in the extreme ultraviolet," Nature 436, 234-237 (2005).
[CrossRef] [PubMed]

Lalezari, R.

Leitenstorfer, A.

Lipphardt, B.

Moll, K.

Moll, K. D.

M. J. Thorpe, K. D. Moll, R. J. Jones, B. Safdi, and J. Ye, "Broadband Cavity Ringdown Spectroscopy for Sensitive and Rapid Molecular Detection," Science 311, 1595-1599 (2006).
[CrossRef] [PubMed]

R. J. Jones, K. D. Moll, M. J. Thorpe, and J. Ye, "Phase-Coherent Frequency Combs in the Vacuum Ultraviolet via High-Harmonic Generation inside a Femtosecond Enhancement Cavity," Phys. Rev. Lett. 94, 193201 (2005).
[CrossRef] [PubMed]

Moutzouris, K.

Rauschenberger, J.

C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hansch, "A frequency comb in the extreme ultraviolet," Nature 436, 234-237 (2005).
[CrossRef] [PubMed]

Safdi, B.

M. J. Thorpe, K. D. Moll, R. J. Jones, B. Safdi, and J. Ye, "Broadband Cavity Ringdown Spectroscopy for Sensitive and Rapid Molecular Detection," Science 311, 1595-1599 (2006).
[CrossRef] [PubMed]

Schliesser, A.

C. Gohle, B. Stein, A. Schliesser, T. Udem, and T. W. Hansch, "Frequency Comb Vernier Spectroscopy for Broadband, High-Resolution, High-Sensitivity Absorption and Dispersion Spectra," Phys. Rev. Lett. 99, 263,902-4 (2007).
[CrossRef]

A. Schliesser, C. Gohle, T. Udem, and T. W. Hansch, "Complete characterization of a broadband high-finesse cavity using an optical frequency comb," Opt. Express 14, 5975-5983 (2006).
[CrossRef] [PubMed]

Schnatz, H.

Schuessler, H. A.

C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hansch, "A frequency comb in the extreme ultraviolet," Nature 436, 234-237 (2005).
[CrossRef] [PubMed]

Stein, B.

C. Gohle, B. Stein, A. Schliesser, T. Udem, and T. W. Hansch, "Frequency Comb Vernier Spectroscopy for Broadband, High-Resolution, High-Sensitivity Absorption and Dispersion Spectra," Phys. Rev. Lett. 99, 263,902-4 (2007).
[CrossRef]

Tauser, F.

Thorpe, M.

Thorpe, M. J.

M. J. Thorpe, K. D. Moll, R. J. Jones, B. Safdi, and J. Ye, "Broadband Cavity Ringdown Spectroscopy for Sensitive and Rapid Molecular Detection," Science 311, 1595-1599 (2006).
[CrossRef] [PubMed]

R. J. Jones, K. D. Moll, M. J. Thorpe, and J. Ye, "Phase-Coherent Frequency Combs in the Vacuum Ultraviolet via High-Harmonic Generation inside a Femtosecond Enhancement Cavity," Phys. Rev. Lett. 94, 193201 (2005).
[CrossRef] [PubMed]

Udem, T.

C. Gohle, B. Stein, A. Schliesser, T. Udem, and T. W. Hansch, "Frequency Comb Vernier Spectroscopy for Broadband, High-Resolution, High-Sensitivity Absorption and Dispersion Spectra," Phys. Rev. Lett. 99, 263,902-4 (2007).
[CrossRef]

A. Schliesser, C. Gohle, T. Udem, and T. W. Hansch, "Complete characterization of a broadband high-finesse cavity using an optical frequency comb," Opt. Express 14, 5975-5983 (2006).
[CrossRef] [PubMed]

C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hansch, "A frequency comb in the extreme ultraviolet," Nature 436, 234-237 (2005).
[CrossRef] [PubMed]

Ye, J.

M. J. Thorpe, K. D. Moll, R. J. Jones, B. Safdi, and J. Ye, "Broadband Cavity Ringdown Spectroscopy for Sensitive and Rapid Molecular Detection," Science 311, 1595-1599 (2006).
[CrossRef] [PubMed]

M. Thorpe, R. Jones, K. Moll, J. Ye, and R. Lalezari, "Precise measurements of optical cavity dispersion and mirror coating properties via femtosecond combs," Opt. Express 13, 882-888 (2005).
[CrossRef] [PubMed]

R. J. Jones, K. D. Moll, M. J. Thorpe, and J. Ye, "Phase-Coherent Frequency Combs in the Vacuum Ultraviolet via High-Harmonic Generation inside a Femtosecond Enhancement Cavity," Phys. Rev. Lett. 94, 193201 (2005).
[CrossRef] [PubMed]

R. J. Jones and J. Ye, "Femtosecond pulse amplification by coherent addition in a passive optical cavity," Opt. Lett. 27, 1848-1850 (2002).
[CrossRef]

Appl. Phys. B (1)

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

Nature (1)

C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hansch, "A frequency comb in the extreme ultraviolet," Nature 436, 234-237 (2005).
[CrossRef] [PubMed]

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. Lett. (2)

R. J. Jones, K. D. Moll, M. J. Thorpe, and J. Ye, "Phase-Coherent Frequency Combs in the Vacuum Ultraviolet via High-Harmonic Generation inside a Femtosecond Enhancement Cavity," Phys. Rev. Lett. 94, 193201 (2005).
[CrossRef] [PubMed]

C. Gohle, B. Stein, A. Schliesser, T. Udem, and T. W. Hansch, "Frequency Comb Vernier Spectroscopy for Broadband, High-Resolution, High-Sensitivity Absorption and Dispersion Spectra," Phys. Rev. Lett. 99, 263,902-4 (2007).
[CrossRef]

Science (1)

M. J. Thorpe, K. D. Moll, R. J. Jones, B. Safdi, and J. Ye, "Broadband Cavity Ringdown Spectroscopy for Sensitive and Rapid Molecular Detection," Science 311, 1595-1599 (2006).
[CrossRef] [PubMed]

Other (4)

J. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, Second Edition, 2nd ed. (Academic Press, 2006).

R. J. Jones, "High Resolution Optical Frequency Metrology with Stabilized Femtosecond Lasers," Ph.D. thesis, University of New Mexico (2001).

L. N. Trefethen, Spectral Methods in MATLAB, illustrated edition (SIAM: Society for Industrial and Applied Mathematics, 2001).

J. Ye and S. T. Cundiff, Femtosecond Optical Frequency Comb: Principle, Operation and Applications, 1st ed., (Springer, 2004).

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Figures (5)

Fig. 1.
Fig. 1.

(color online) Simulation of (top) the resonance condition as a function of frequency ω and enhancement cavity length d, and (bottom) the resulting cavity reflection signal power P as a function of d. The cavity resonances are assumed to have a Lorentzian lineshape and the laser spectrum has a gaussian envelope. The overall Lorentzian lineshape of the reflection (in dotted red) as a function of d is derived in ref. [11], where a Gaussian lineshape was assumed for the cavity resonance and did not include dispersion, both of which lead to a disagreement with the simulation results. (a) The simple case where the incident FFC has zero offset frequency, ω 0 = 0, and the cavity has a null phase response, Φ(ω) = 0. At one unique enhancement cavity length d, all of the FFC’s comb elements are aligned to the enhancement cavity’s resonance peaks. In this case Δd = 0 and is labelled as the central fringe. (b) The effect of having dispersion in the enhancement cavity. Now even at the central fringe not all of the comb elements can simultaneously align to the cavity resonances at one particular cavity length d leading to a decrease in the cavity reflection amplitude. The curvature of the parabolic shape of a particular fringe number shown in the upper right pane is directly proportional to the GDD.

Fig. 2.
Fig. 2.

(color online) Experimental setup. A mode-locked Ti:Sapphire laser (with ω 0, the offset frequency, stabilized through the f - 2f interferometer) is coupled into a six mirror enhancement cavity which is under vacuum. The cavity reflection from the input coupler (IC) is separated into two spectrally resolved branches with gratings. PD1 is the reference branch photodetector; PD2 the measurement branch; IC cavity input coupler (0.25%); the small mirror attached to the PZT is used to sweep the cavity length. The scope is used to measure the (time) delay between the resonance conditions of ωref and ω as the cavity length is swept.

Fig. 3.
Fig. 3.

(color online) Group delay dispersion (GDD) measurement of an evacuated six mirror cavity. The measured delay (black points, left axes) represent the raw data collected via an oscilloscope and is converted to a path length distance through a calibration of the free spectral range. The resulting GDD (red curve, right axis) is calculated via Eq. (11) and spectral collocation methods in numerical analysis [12]. The spectral limits of the measurement are due to the finite width of the FFC. The cavity mirrors were designed for low GDD centered at 790 nm.

Fig. 4.
Fig. 4.

(color online) (a) An intra-cavity measurement of a 0.45 mm thick piece of sapphire. The GDD is calculated both by a polynomial fit to the entire data set (right axis, red) and a Chebyshev fit (right axis, green). Both show excellent agreement with the Sellmeier prediction (blue). (b) An intra-cavity measurement of a 2.2 mm thick piece of fused silica. The uncertainty in the Chebyshev fit is displayed as the gray area.

Fig. 5.
Fig. 5.

(color online) GDD of an cavity mirror with known oscillations in the GDD. The theoretical and experimentally specifications given by the manufacturer are shown as blue dashed and red dotted lines respecively. Also shown is our differential measurement with a Chebyshev fit as a solid black line

Equations (11)

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2πm=ωcL+Φ (ω) ,
GDD=d2dω2 Φ (ω) .
2πmref=ωrefcdref+Φ(ωref)
2πq=ωcd+Φ(ω).
Δd(ω)=2πc(mrefωrefqω)+c(Φ(ω)ωΦ(ωref)ωref).
ωref=ωrepmref+ω0
ω=ωrepq+ω0,
Δd=2πcωrefω[ω0(mrefq)+ωrep(mrefqmrefq)]+c(Φ(ω)ωΦ(ωref)ωref).
mrefqqmref=0.
Δd=2πcω0ωrep(1ω1ωref)+c(Φ(ω)ωΦ(ωref)ωref).
d2dω2Φ(ω)=d2dω2 (ωcΔd(ω)) .

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