Abstract

Optical cavities provide high sensitivity to dispersion since their resonance frequencies depend on the index of refraction. We present a direct, broadband, and accurate measurement of the modes of a high finesse cavity using an optical frequency comb and a mechanical Fourier transform spectrometer with a kHz-level resolution. We characterize 16000 longitudinal cavity modes spanning 16 THz of bandwidth in terms of center frequency, linewidth, and amplitude. Using the center frequencies we retrieve the group delay dispersion of the cavity mirror coatings and pure N2 with 0.1 fs2 precision and 1 fs2 accuracy, as well as the refractivity of the 3ν1 + ν3 absorption band of CO2 with 5 × 10−12 precision. This opens up for broadband refractive index metrology and calibration-free spectroscopy of entire molecular bands.

© 2017 Optical Society of America

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Corrections

Lucile Rutkowski, Alexandra C. Johansson, Gang Zhao, Thomas Hausmaninger, Amir Khodabakhsh, Ove Axner, and Aleksandra Foltynowicz, "Sensitive and broadband measurement of dispersion in a cavity using a Fourier transform spectrometer with kHz resolution: erratum," Opt. Express 28, 13290-13291 (2020)
https://opg.optica.org/oe/abstract.cfm?uri=oe-28-9-13290
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References

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  5. A. Cygan, P. Wcisło, S. Wójtewicz, P. Masłowski, J. T. Hodges, R. Ciuryło, and D. Lisak, “One-dimensional frequency-based spectroscopy,” Opt. Express 23(11), 14472–14486 (2015).
    [Crossref] [PubMed]
  6. S. Tan, P. Berceau, S. Saraf, and J. A. Lipa, “Measuring finesse and gas absorption with Lorentzian recovery spectroscopy,” Opt. Express 25(7), 7645–7656 (2017).
    [Crossref] [PubMed]
  7. M. J. Thorpe, D. Balslev-Clausen, M. S. Kirchner, and J. Ye, “Cavity-enhanced optical frequency comb spectroscopy: application to human breath analysis,” Opt. Express 16(4), 2387–2397 (2008).
    [Crossref] [PubMed]
  8. A. Foltynowicz, T. Ban, P. Masłowski, F. Adler, and J. Ye, “Quantum-noise-limited optical frequency comb spectroscopy,” Phys. Rev. Lett. 107(23), 233002 (2011).
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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2017 (1)

2016 (4)

LIGO Scientific Collaboration and Virgo Collaboration, “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116(6), 061102 (2016).
[Crossref] [PubMed]

P. F. Egan, J. A. Stone, J. E. Ricker, and J. H. Hendricks, “Comparison measurements of low-pressure between a laser refractometer and ultrasonic manometer,” Rev. Sci. Instrum. 87(5), 053113 (2016).
[Crossref] [PubMed]

B. Spaun, P. B. Changala, D. Patterson, B. J. Bjork, O. H. Heckl, J. M. Doyle, and J. Ye, “Continuous probing of cold complex molecules with infrared frequency comb spectroscopy,” Nature 533(7604), 517–520 (2016).
[Crossref] [PubMed]

P. Masłowski, K. F. Lee, A. C. Johansson, A. Khodabakhsh, G. Kowzan, L. Rutkowski, A. A. Mills, C. Mohr, J. Jiang, M. E. Fermann, and A. Foltynowicz, “Surpassing the path-limited resolution of Fourier-transform spectrometry with frequency combs,” Phys. Rev. A 93(2), 021802 (2016).
[Crossref]

2015 (3)

2014 (2)

2013 (1)

G. W. Truong, K. O. Douglass, S. E. Maxwell, R. D. van Zee, D. F. Plusquellic, J. T. Hodges, and D. A. Long, “Frequency-agile, rapid scanning spectroscopy,” Nat. Photonics 7(7), 532–534 (2013).
[Crossref]

2011 (1)

A. Foltynowicz, T. Ban, P. Masłowski, F. Adler, and J. Ye, “Quantum-noise-limited optical frequency comb spectroscopy,” Phys. Rev. Lett. 107(23), 233002 (2011).
[Crossref] [PubMed]

2010 (1)

2009 (1)

2008 (1)

2006 (1)

2005 (1)

2002 (1)

1998 (1)

1995 (1)

1966 (1)

1871 (1)

W. Sellmeier, “Zur Erklärung der abnormen Farbenfolge im Spectrum einiger Substanzen,” Ann. Phys. (Berlin) 219(6), 272–282 (1871).
[Crossref]

Abd Alrahman, C.

Adler, F.

A. Foltynowicz, T. Ban, P. Masłowski, F. Adler, and J. Ye, “Quantum-noise-limited optical frequency comb spectroscopy,” Phys. Rev. Lett. 107(23), 233002 (2011).
[Crossref] [PubMed]

Axner, O.

Balslev-Clausen, D.

Ban, T.

A. Foltynowicz, T. Ban, P. Masłowski, F. Adler, and J. Ye, “Quantum-noise-limited optical frequency comb spectroscopy,” Phys. Rev. Lett. 107(23), 233002 (2011).
[Crossref] [PubMed]

Berceau, P.

Bjork, B. J.

B. Spaun, P. B. Changala, D. Patterson, B. J. Bjork, O. H. Heckl, J. M. Doyle, and J. Ye, “Continuous probing of cold complex molecules with infrared frequency comb spectroscopy,” Nature 533(7604), 517–520 (2016).
[Crossref] [PubMed]

Bor, Z.

Changala, P. B.

B. Spaun, P. B. Changala, D. Patterson, B. J. Bjork, O. H. Heckl, J. M. Doyle, and J. Ye, “Continuous probing of cold complex molecules with infrared frequency comb spectroscopy,” Nature 533(7604), 517–520 (2016).
[Crossref] [PubMed]

Ciurylo, R.

Cygan, A.

Di Domenico, G.

Douglass, K. O.

G. W. Truong, K. O. Douglass, S. E. Maxwell, R. D. van Zee, D. F. Plusquellic, J. T. Hodges, and D. A. Long, “Frequency-agile, rapid scanning spectroscopy,” Nat. Photonics 7(7), 532–534 (2013).
[Crossref]

Doyle, J. M.

B. Spaun, P. B. Changala, D. Patterson, B. J. Bjork, O. H. Heckl, J. M. Doyle, and J. Ye, “Continuous probing of cold complex molecules with infrared frequency comb spectroscopy,” Nature 533(7604), 517–520 (2016).
[Crossref] [PubMed]

Egan, P. F.

P. F. Egan, J. A. Stone, J. E. Ricker, and J. H. Hendricks, “Comparison measurements of low-pressure between a laser refractometer and ultrasonic manometer,” Rev. Sci. Instrum. 87(5), 053113 (2016).
[Crossref] [PubMed]

Ehlers, P.

Fermann, M. E.

P. Masłowski, K. F. Lee, A. C. Johansson, A. Khodabakhsh, G. Kowzan, L. Rutkowski, A. A. Mills, C. Mohr, J. Jiang, M. E. Fermann, and A. Foltynowicz, “Surpassing the path-limited resolution of Fourier-transform spectrometry with frequency combs,” Phys. Rev. A 93(2), 021802 (2016).
[Crossref]

Foltynowicz, A.

P. Masłowski, K. F. Lee, A. C. Johansson, A. Khodabakhsh, G. Kowzan, L. Rutkowski, A. A. Mills, C. Mohr, J. Jiang, M. E. Fermann, and A. Foltynowicz, “Surpassing the path-limited resolution of Fourier-transform spectrometry with frequency combs,” Phys. Rev. A 93(2), 021802 (2016).
[Crossref]

C. Abd Alrahman, A. Khodabakhsh, F. M. Schmidt, Z. Qu, and A. Foltynowicz, “Cavity-enhanced optical frequency comb spectroscopy of high-temperature H2O in a flame,” Opt. Express 22(11), 13889–13895 (2014).
[Crossref] [PubMed]

A. Foltynowicz, T. Ban, P. Masłowski, F. Adler, and J. Ye, “Quantum-noise-limited optical frequency comb spectroscopy,” Phys. Rev. Lett. 107(23), 233002 (2011).
[Crossref] [PubMed]

Gohle, C.

Hall, J. L.

Hammond, T. J.

Hänsch, T. W.

Hashimoto, M.

Hausmaninger, T.

Hebling, J.

Heckl, O. H.

B. Spaun, P. B. Changala, D. Patterson, B. J. Bjork, O. H. Heckl, J. M. Doyle, and J. Ye, “Continuous probing of cold complex molecules with infrared frequency comb spectroscopy,” Nature 533(7604), 517–520 (2016).
[Crossref] [PubMed]

Hendricks, J. H.

P. F. Egan, J. A. Stone, J. E. Ricker, and J. H. Hendricks, “Comparison measurements of low-pressure between a laser refractometer and ultrasonic manometer,” Rev. Sci. Instrum. 87(5), 053113 (2016).
[Crossref] [PubMed]

Hodges, J. T.

A. Cygan, P. Wcisło, S. Wójtewicz, P. Masłowski, J. T. Hodges, R. Ciuryło, and D. Lisak, “One-dimensional frequency-based spectroscopy,” Opt. Express 23(11), 14472–14486 (2015).
[Crossref] [PubMed]

G. W. Truong, K. O. Douglass, S. E. Maxwell, R. D. van Zee, D. F. Plusquellic, J. T. Hodges, and D. A. Long, “Frequency-agile, rapid scanning spectroscopy,” Nat. Photonics 7(7), 532–534 (2013).
[Crossref]

Hsieh, Y.-D.

Inaba, H.

Jiang, J.

P. Masłowski, K. F. Lee, A. C. Johansson, A. Khodabakhsh, G. Kowzan, L. Rutkowski, A. A. Mills, C. Mohr, J. Jiang, M. E. Fermann, and A. Foltynowicz, “Surpassing the path-limited resolution of Fourier-transform spectrometry with frequency combs,” Phys. Rev. A 93(2), 021802 (2016).
[Crossref]

Johansson, A. C.

P. Masłowski, K. F. Lee, A. C. Johansson, A. Khodabakhsh, G. Kowzan, L. Rutkowski, A. A. Mills, C. Mohr, J. Jiang, M. E. Fermann, and A. Foltynowicz, “Surpassing the path-limited resolution of Fourier-transform spectrometry with frequency combs,” Phys. Rev. A 93(2), 021802 (2016).
[Crossref]

Jones, D. J.

Jones, R.

Khanna, B. N.

Khodabakhsh, A.

P. Masłowski, K. F. Lee, A. C. Johansson, A. Khodabakhsh, G. Kowzan, L. Rutkowski, A. A. Mills, C. Mohr, J. Jiang, M. E. Fermann, and A. Foltynowicz, “Surpassing the path-limited resolution of Fourier-transform spectrometry with frequency combs,” Phys. Rev. A 93(2), 021802 (2016).
[Crossref]

C. Abd Alrahman, A. Khodabakhsh, F. M. Schmidt, Z. Qu, and A. Foltynowicz, “Cavity-enhanced optical frequency comb spectroscopy of high-temperature H2O in a flame,” Opt. Express 22(11), 13889–13895 (2014).
[Crossref] [PubMed]

Kirchner, M. S.

Kovacs, A. P.

Kowzan, G.

P. Masłowski, K. F. Lee, A. C. Johansson, A. Khodabakhsh, G. Kowzan, L. Rutkowski, A. A. Mills, C. Mohr, J. Jiang, M. E. Fermann, and A. Foltynowicz, “Surpassing the path-limited resolution of Fourier-transform spectrometry with frequency combs,” Phys. Rev. A 93(2), 021802 (2016).
[Crossref]

Kurdi, G.

Lalezari, R.

Lee, K. F.

P. Masłowski, K. F. Lee, A. C. Johansson, A. Khodabakhsh, G. Kowzan, L. Rutkowski, A. A. Mills, C. Mohr, J. Jiang, M. E. Fermann, and A. Foltynowicz, “Surpassing the path-limited resolution of Fourier-transform spectrometry with frequency combs,” Phys. Rev. A 93(2), 021802 (2016).
[Crossref]

Lipa, J. A.

Lisak, D.

Long, D. A.

G. W. Truong, K. O. Douglass, S. E. Maxwell, R. D. van Zee, D. F. Plusquellic, J. T. Hodges, and D. A. Long, “Frequency-agile, rapid scanning spectroscopy,” Nat. Photonics 7(7), 532–534 (2013).
[Crossref]

Ma, L.-S.

Ma, W.

Maslowski, P.

P. Masłowski, K. F. Lee, A. C. Johansson, A. Khodabakhsh, G. Kowzan, L. Rutkowski, A. A. Mills, C. Mohr, J. Jiang, M. E. Fermann, and A. Foltynowicz, “Surpassing the path-limited resolution of Fourier-transform spectrometry with frequency combs,” Phys. Rev. A 93(2), 021802 (2016).
[Crossref]

A. Cygan, P. Wcisło, S. Wójtewicz, P. Masłowski, J. T. Hodges, R. Ciuryło, and D. Lisak, “One-dimensional frequency-based spectroscopy,” Opt. Express 23(11), 14472–14486 (2015).
[Crossref] [PubMed]

A. Foltynowicz, T. Ban, P. Masłowski, F. Adler, and J. Ye, “Quantum-noise-limited optical frequency comb spectroscopy,” Phys. Rev. Lett. 107(23), 233002 (2011).
[Crossref] [PubMed]

Maxwell, S. E.

G. W. Truong, K. O. Douglass, S. E. Maxwell, R. D. van Zee, D. F. Plusquellic, J. T. Hodges, and D. A. Long, “Frequency-agile, rapid scanning spectroscopy,” Nat. Photonics 7(7), 532–534 (2013).
[Crossref]

Mills, A. A.

P. Masłowski, K. F. Lee, A. C. Johansson, A. Khodabakhsh, G. Kowzan, L. Rutkowski, A. A. Mills, C. Mohr, J. Jiang, M. E. Fermann, and A. Foltynowicz, “Surpassing the path-limited resolution of Fourier-transform spectrometry with frequency combs,” Phys. Rev. A 93(2), 021802 (2016).
[Crossref]

Mills, A. K.

Mohr, C.

P. Masłowski, K. F. Lee, A. C. Johansson, A. Khodabakhsh, G. Kowzan, L. Rutkowski, A. A. Mills, C. Mohr, J. Jiang, M. E. Fermann, and A. Foltynowicz, “Surpassing the path-limited resolution of Fourier-transform spectrometry with frequency combs,” Phys. Rev. A 93(2), 021802 (2016).
[Crossref]

Moll, K.

Morville, J.

Okubo, S.

Onae, A.

Osvay, K.

Patterson, D.

B. Spaun, P. B. Changala, D. Patterson, B. J. Bjork, O. H. Heckl, J. M. Doyle, and J. Ye, “Continuous probing of cold complex molecules with infrared frequency comb spectroscopy,” Nature 533(7604), 517–520 (2016).
[Crossref] [PubMed]

Peck, E. R.

Plusquellic, D. F.

G. W. Truong, K. O. Douglass, S. E. Maxwell, R. D. van Zee, D. F. Plusquellic, J. T. Hodges, and D. A. Long, “Frequency-agile, rapid scanning spectroscopy,” Nat. Photonics 7(7), 532–534 (2013).
[Crossref]

Qu, Z.

Ricker, J. E.

P. F. Egan, J. A. Stone, J. E. Ricker, and J. H. Hendricks, “Comparison measurements of low-pressure between a laser refractometer and ultrasonic manometer,” Rev. Sci. Instrum. 87(5), 053113 (2016).
[Crossref] [PubMed]

Rutkowski, L.

P. Masłowski, K. F. Lee, A. C. Johansson, A. Khodabakhsh, G. Kowzan, L. Rutkowski, A. A. Mills, C. Mohr, J. Jiang, M. E. Fermann, and A. Foltynowicz, “Surpassing the path-limited resolution of Fourier-transform spectrometry with frequency combs,” Phys. Rev. A 93(2), 021802 (2016).
[Crossref]

L. Rutkowski and J. Morville, “Broadband cavity-enhanced molecular spectra from Vernier filtering of a complete frequency comb,” Opt. Lett. 39(23), 6664–6667 (2014).
[Crossref] [PubMed]

Saraf, S.

Schiller, S.

Schilt, S.

Schliesser, A.

Schmidt, F. M.

Sellmeier, W.

W. Sellmeier, “Zur Erklärung der abnormen Farbenfolge im Spectrum einiger Substanzen,” Ann. Phys. (Berlin) 219(6), 272–282 (1871).
[Crossref]

Silander, I.

Spaun, B.

B. Spaun, P. B. Changala, D. Patterson, B. J. Bjork, O. H. Heckl, J. M. Doyle, and J. Ye, “Continuous probing of cold complex molecules with infrared frequency comb spectroscopy,” Nature 533(7604), 517–520 (2016).
[Crossref] [PubMed]

Stone, J. A.

P. F. Egan, J. A. Stone, J. E. Ricker, and J. H. Hendricks, “Comparison measurements of low-pressure between a laser refractometer and ultrasonic manometer,” Rev. Sci. Instrum. 87(5), 053113 (2016).
[Crossref] [PubMed]

Szipocs, R.

Tan, S.

Thomann, P.

Thorpe, M.

Thorpe, M. J.

Truong, G. W.

G. W. Truong, K. O. Douglass, S. E. Maxwell, R. D. van Zee, D. F. Plusquellic, J. T. Hodges, and D. A. Long, “Frequency-agile, rapid scanning spectroscopy,” Nat. Photonics 7(7), 532–534 (2013).
[Crossref]

Udem, T.

van Zee, R. D.

G. W. Truong, K. O. Douglass, S. E. Maxwell, R. D. van Zee, D. F. Plusquellic, J. T. Hodges, and D. A. Long, “Frequency-agile, rapid scanning spectroscopy,” Nat. Photonics 7(7), 532–534 (2013).
[Crossref]

Wcislo, P.

Wójtewicz, S.

Yasui, T.

Ye, J.

Ann. Phys. (Berlin) (1)

W. Sellmeier, “Zur Erklärung der abnormen Farbenfolge im Spectrum einiger Substanzen,” Ann. Phys. (Berlin) 219(6), 272–282 (1871).
[Crossref]

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

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

Nat. Photonics (1)

G. W. Truong, K. O. Douglass, S. E. Maxwell, R. D. van Zee, D. F. Plusquellic, J. T. Hodges, and D. A. Long, “Frequency-agile, rapid scanning spectroscopy,” Nat. Photonics 7(7), 532–534 (2013).
[Crossref]

Nature (1)

B. Spaun, P. B. Changala, D. Patterson, B. J. Bjork, O. H. Heckl, J. M. Doyle, and J. Ye, “Continuous probing of cold complex molecules with infrared frequency comb spectroscopy,” Nature 533(7604), 517–520 (2016).
[Crossref] [PubMed]

Opt. Express (8)

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(3), 882–888 (2005).
[Crossref] [PubMed]

A. Schliesser, C. Gohle, T. Udem, and T. W. Hänsch, “Complete characterization of a broadband high-finesse cavity using an optical frequency comb,” Opt. Express 14(13), 5975–5983 (2006).
[Crossref] [PubMed]

T. J. Hammond, A. K. Mills, and D. J. Jones, “Simple method to determine dispersion of high-finesse optical cavities,” Opt. Express 17(11), 8998–9005 (2009).
[Crossref] [PubMed]

A. Cygan, P. Wcisło, S. Wójtewicz, P. Masłowski, J. T. Hodges, R. Ciuryło, and D. Lisak, “One-dimensional frequency-based spectroscopy,” Opt. Express 23(11), 14472–14486 (2015).
[Crossref] [PubMed]

S. Tan, P. Berceau, S. Saraf, and J. A. Lipa, “Measuring finesse and gas absorption with Lorentzian recovery spectroscopy,” Opt. Express 25(7), 7645–7656 (2017).
[Crossref] [PubMed]

M. J. Thorpe, D. Balslev-Clausen, M. S. Kirchner, and J. Ye, “Cavity-enhanced optical frequency comb spectroscopy: application to human breath analysis,” Opt. Express 16(4), 2387–2397 (2008).
[Crossref] [PubMed]

C. Abd Alrahman, A. Khodabakhsh, F. M. Schmidt, Z. Qu, and A. Foltynowicz, “Cavity-enhanced optical frequency comb spectroscopy of high-temperature H2O in a flame,” Opt. Express 22(11), 13889–13895 (2014).
[Crossref] [PubMed]

S. Okubo, Y.-D. Hsieh, H. Inaba, A. Onae, M. Hashimoto, and T. Yasui, “Near-infrared broadband dual-frequency-comb spectroscopy with a resolution beyond the Fourier limit determined by the observation time window,” Opt. Express 23(26), 33184–33193 (2015).
[Crossref] [PubMed]

Opt. Lett. (4)

Phys. Rev. A (1)

P. Masłowski, K. F. Lee, A. C. Johansson, A. Khodabakhsh, G. Kowzan, L. Rutkowski, A. A. Mills, C. Mohr, J. Jiang, M. E. Fermann, and A. Foltynowicz, “Surpassing the path-limited resolution of Fourier-transform spectrometry with frequency combs,” Phys. Rev. A 93(2), 021802 (2016).
[Crossref]

Phys. Rev. Lett. (2)

A. Foltynowicz, T. Ban, P. Masłowski, F. Adler, and J. Ye, “Quantum-noise-limited optical frequency comb spectroscopy,” Phys. Rev. Lett. 107(23), 233002 (2011).
[Crossref] [PubMed]

LIGO Scientific Collaboration and Virgo Collaboration, “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116(6), 061102 (2016).
[Crossref] [PubMed]

Rev. Sci. Instrum. (1)

P. F. Egan, J. A. Stone, J. E. Ricker, and J. H. Hendricks, “Comparison measurements of low-pressure between a laser refractometer and ultrasonic manometer,” Rev. Sci. Instrum. 87(5), 053113 (2016).
[Crossref] [PubMed]

Other (3)

P. R. Griffiths and J. A. de Haseth, Fourier Transform Infrared Spectrometry (John Wiley & Sons, Inc., 2006).

L. Rutkowski, P. Masłowski, A. C. Johansson, A. Khodabakhsh, and A. Foltynowicz, “Optical frequency comb Fourier transform spectroscopy with sub-nominal resolution - principles and implementation,” arXiv:1612.04808v2 (2017).

“HITRAN database,” http://www.hitran.org .

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Figures (5)
Fig. 1
Fig. 1 (a) Experimental setup: f-2f - f-2f interferometer beat note; DBM - double-balanced mixer; DDS - direct digital synthesizer; FC - fiber collimator; λ/2 - half-waveplate; PBS - polarizing beam splitter; λ/4 - quarter-waveplate; PZT - piezoelectric transducer; FTS - Fourier transform spectrometer; EOM - electro-optic modulator; G - diffraction grating; (b) Matching of the comb lines (red) and the cw laser (blue) to the cavity modes (black). The reflected comb lines are shown with the red dashed lines.

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Fig. 2
Fig. 2 (a) Spectrum of the empty cavity transmission spanning 16 THz consisting of 16000 discrete modes. Inset: enlargement of 3 modes separated by 3FSR. (b) A zoom of a single mode at 1600 nm (black markers) together with a Lorentzian fit (red line) and the residuum (lower panel).

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Fig. 3
Fig. 3 (a) Shift of the cavity mode frequencies Δν measured when the cavity is empty (black curve, left y-axis) and when the cavity is filled with pure N2 at 750 Torr (red curve, right y-axis) plotted together with a calculated shift based on the Sellmeier equation for N2 (blue markers). Note the three orders of magnitude difference between the two y-axis scales. (b) Residuum of a polynomial fit to the shift of the empty cavity modes. (c) Residuum of a polynomial fit to the mode shift of the cavity filled with N2.

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Fig. 4
Fig. 4 Group delay dispersion (GDD) of the empty cavity (black solid curve), and the cavity filled with pure N2 at 750 Torr (red solid curve). The dashed curve is the sum of the GDD of N2 calculated using the Sellmeier equation and the experimentally determined empty cavity GDD (dashed black curve).

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Fig. 5
Fig. 5 (a) Refractivity of the 3ν1 + ν3 absorption band of 1% of CO2 in N2 at 750 Torr (black markers) together with a fit (red curve). (b) Residual of the fit.

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Equations (6)

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Φ( ν q )= ϕ 0 ​( ν q )+ ϕ n ​( ν q )=2πq,
ν q 0 =  qFSR ref 0 +f 0 ,
Δν =  ν q − ν q 0  = FSR ref 2π [ 2πq−Φ( ν q 0 ) ],
GDD( ν q 0 )≡ 1 4 π 2 ∂ 2 Φ ∂  ν 2 =− 1 2π  FSR ref ∂ 2 Δν ∂  ν 2 .
Δ ν n =− ν q 0 [ n​( ν q 0 )−1 ].
n abs ​( ν q 0 )−1=− c 4 πν q 0 ρ ∑ i S i Im[ χ i ​( ν q 0 ) ]  ,

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