Abstract

Coherent dual comb spectroscopy can provide high-resolution, high-accuracy measurements of a sample response in both magnitude and phase. We discuss the achievable signal-to-noise ratio (SNR) due to both additive white noise and multiplicative noise, and the corresponding sensitivity limit for trace gas detection. We show that sequential acquisition of the overall spectrum through a tunable filter, or parallel acquisition of the overall spectrum through a detector array, can significantly improve the SNR under some circumstances. We identify a useful figure of merit as the quality factor, equal to the product of the SNR, normalized by the square root of the acquisition time, and the number of resolved frequency elements. For a single detector and fiber-laser based system, this quality factor is 106 – 107 Hz1/2.

© 2010 Optical Society of America

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References

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  1. F. Keilmann, C. Gohle, and R. Holzwarth, "Time-domain mid-infrared frequency-comb spectrometer," Opt. Lett. 29, 1542-1544 (2004).
    [CrossRef] [PubMed]
  2. A. Schliesser, M. Brehm, F. Keilmann, and D. van der Weide, "Frequency-comb infrared spectrometer for rapid, remote chemical sensing," Opt. Express 13, 9029-9038 (2005).
    [CrossRef] [PubMed]
  3. T. Yasui, Y. Kabetani, E. Saneyoshi, S. Yokoyama, and T. Araki, "Terahertz frequency comb by multifrequency heterodyning photoconductive detection for high-accuracy, high-resolution terahertz spectroscopy," Appl. Phys. Lett. 88, 241104 (2006).
    [CrossRef]
  4. P. Giaccari, J. D. Deschenes, P. Saucier, J. Genest, and P. Tremblay, "Active fourier-transform spectroscopy combining the direct RF beating of two fiber-based mode-locked lasers with a novel referencing method," Opt. Express 16, 4347-4365 (2008).
    [CrossRef] [PubMed]
  5. B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hansch, and N. Picque, "Cavity-enhanced dual-comb spectroscopy," Nat. Photon. 4, 55-57 (2009).
    [CrossRef]
  6. I. Coddington, W. C. Swann, and N. R. Newbury, "Coherent multiheterodyne spectroscopy using stabilized optical frequency combs," Phys. Rev. Lett. 100, 013902 (2008).
    [CrossRef] [PubMed]
  7. I. Coddington, W. Swann, and N. Newbury, "Time-domain spectroscopy of molecular free-induction decay in the infrared," Opt. Lett., accepted (2010).
    [CrossRef] [PubMed]
  8. I. Coddington, W. Swann, and N. Newbury, "Coherent dual-comb spectroscopy at high signal to noise," http://arxiv.org/abs/1001.3865 (2010).
  9. R. J. Bell, Introductory Fourier transform spectroscopy (Academic Press, 1972).
  10. J. Chamberlain, The Principles of Interferometric Spectroscopy (John Wiley and Sons, Inc, 1979).
  11. J. R. Birch, "Dispersive Fourier-transform spectroscopy," Mikrochimica Acta 3, 105-122 (1987).
  12. N. Almoayed and M. Afsar, "High-resolution absorption coefficient and refractive index spectra of carbon monoxide gas at millimeter and submillimeter wave-lengths," IEEE T. Instrum. Meas. 55, 1033-1037 (2006).
    [CrossRef]
  13. S. Schiller, "Spectrometry with frequency combs," Opt. Lett. 27, 766-768 (2002).
    [CrossRef]
  14. J. W. Brault, High Resolution in Astronomy (Geneva Observatory, 1985), Fourier transform spectrometry, pp. 1-65.
  15. L. A. Sromovsky, "Radiometric errors in complex Fourier transform spectrometry," Appl. Opt. 42, 1779-1787 (2003).
    [CrossRef] [PubMed]
  16. S. P. Davis, M. C. Abrams, and J. W. Brault, Fourier Transform Spectrometry (Academic Press, 2001).
  17. V. V. Protopopov, Laser Heterodyning (Springer Berlin / Heidelberg, 2009).
    [CrossRef]
  18. W. Demtroder, Laser Spectroscopy (Springer, 1996), 2nd ed.
  19. I. Coddington, W. C. Swann, and N. R. Newbury, "Coherent linear optical sampling at 15 bits of resolution," Opt. Lett. 34, 2153-2155 (2009).
    [CrossRef] [PubMed]
  20. N. R. Newbury and W. C. Swann, "Low-noise fiber-laser frequency combs (invited), " J. Opt. Soc. Am. B 24, 1756-1770 (2007).
    [CrossRef]
  21. J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006).
    [CrossRef]
  22. N. R. Newbury, B. R. Washburn, K. L. Corwin, and R. S. Windeler, "Noise amplification during supercontinuum generation in microstructure fiber," Opt. Lett. 28, 944-946 (2002).
    [CrossRef]
  23. K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, "Fundamental noise limitations to supercontinuum generation in microstructure fiber," Phys. Rev. Lett. 90, 113904 (2003).
    [CrossRef] [PubMed]
  24. T. W. Hänsch, "Nobel lecture: Passion for precision," Rev. Mod. Phys. 78, 1297-1309 (2006).
    [CrossRef]
  25. J. L. Hall, "Nobel lecture: Defining and measuring optical frequencies," Rev. Mod. Phys. 78, 1279-1295 (2006).
    [CrossRef]

2010 (1)

I. Coddington, W. Swann, and N. Newbury, "Time-domain spectroscopy of molecular free-induction decay in the infrared," Opt. Lett., accepted (2010).
[CrossRef] [PubMed]

2009 (2)

B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hansch, and N. Picque, "Cavity-enhanced dual-comb spectroscopy," Nat. Photon. 4, 55-57 (2009).
[CrossRef]

I. Coddington, W. C. Swann, and N. R. Newbury, "Coherent linear optical sampling at 15 bits of resolution," Opt. Lett. 34, 2153-2155 (2009).
[CrossRef] [PubMed]

2008 (2)

2007 (1)

2006 (5)

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

N. Almoayed and M. Afsar, "High-resolution absorption coefficient and refractive index spectra of carbon monoxide gas at millimeter and submillimeter wave-lengths," IEEE T. Instrum. Meas. 55, 1033-1037 (2006).
[CrossRef]

T. Yasui, Y. Kabetani, E. Saneyoshi, S. Yokoyama, and T. Araki, "Terahertz frequency comb by multifrequency heterodyning photoconductive detection for high-accuracy, high-resolution terahertz spectroscopy," Appl. Phys. Lett. 88, 241104 (2006).
[CrossRef]

T. W. Hänsch, "Nobel lecture: Passion for precision," Rev. Mod. Phys. 78, 1297-1309 (2006).
[CrossRef]

J. L. Hall, "Nobel lecture: Defining and measuring optical frequencies," Rev. Mod. Phys. 78, 1279-1295 (2006).
[CrossRef]

2005 (1)

2004 (1)

2003 (2)

L. A. Sromovsky, "Radiometric errors in complex Fourier transform spectrometry," Appl. Opt. 42, 1779-1787 (2003).
[CrossRef] [PubMed]

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, "Fundamental noise limitations to supercontinuum generation in microstructure fiber," Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

2002 (2)

1987 (1)

J. R. Birch, "Dispersive Fourier-transform spectroscopy," Mikrochimica Acta 3, 105-122 (1987).

Afsar, M.

N. Almoayed and M. Afsar, "High-resolution absorption coefficient and refractive index spectra of carbon monoxide gas at millimeter and submillimeter wave-lengths," IEEE T. Instrum. Meas. 55, 1033-1037 (2006).
[CrossRef]

Almoayed, N.

N. Almoayed and M. Afsar, "High-resolution absorption coefficient and refractive index spectra of carbon monoxide gas at millimeter and submillimeter wave-lengths," IEEE T. Instrum. Meas. 55, 1033-1037 (2006).
[CrossRef]

Araki, T.

T. Yasui, Y. Kabetani, E. Saneyoshi, S. Yokoyama, and T. Araki, "Terahertz frequency comb by multifrequency heterodyning photoconductive detection for high-accuracy, high-resolution terahertz spectroscopy," Appl. Phys. Lett. 88, 241104 (2006).
[CrossRef]

Bernhardt, B.

B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hansch, and N. Picque, "Cavity-enhanced dual-comb spectroscopy," Nat. Photon. 4, 55-57 (2009).
[CrossRef]

Birch, J. R.

J. R. Birch, "Dispersive Fourier-transform spectroscopy," Mikrochimica Acta 3, 105-122 (1987).

Brehm, M.

Coddington, I.

I. Coddington, W. Swann, and N. Newbury, "Time-domain spectroscopy of molecular free-induction decay in the infrared," Opt. Lett., accepted (2010).
[CrossRef] [PubMed]

I. Coddington, W. C. Swann, and N. R. Newbury, "Coherent linear optical sampling at 15 bits of resolution," Opt. Lett. 34, 2153-2155 (2009).
[CrossRef] [PubMed]

I. Coddington, W. C. Swann, and N. R. Newbury, "Coherent multiheterodyne spectroscopy using stabilized optical frequency combs," Phys. Rev. Lett. 100, 013902 (2008).
[CrossRef] [PubMed]

Coen, S.

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, "Fundamental noise limitations to supercontinuum generation in microstructure fiber," Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

Corwin, K. L.

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, "Fundamental noise limitations to supercontinuum generation in microstructure fiber," Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

N. R. Newbury, B. R. Washburn, K. L. Corwin, and R. S. Windeler, "Noise amplification during supercontinuum generation in microstructure fiber," Opt. Lett. 28, 944-946 (2002).
[CrossRef]

Deschenes, J. D.

Diddams, S. A.

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, "Fundamental noise limitations to supercontinuum generation in microstructure fiber," Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

Dudley, J. M.

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, "Fundamental noise limitations to supercontinuum generation in microstructure fiber," Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

Genest, J.

Genty, G.

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

Giaccari, P.

Gohle, C.

Guelachvili, G.

B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hansch, and N. Picque, "Cavity-enhanced dual-comb spectroscopy," Nat. Photon. 4, 55-57 (2009).
[CrossRef]

Hall, J. L.

J. L. Hall, "Nobel lecture: Defining and measuring optical frequencies," Rev. Mod. Phys. 78, 1279-1295 (2006).
[CrossRef]

Hansch, T. W.

B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hansch, and N. Picque, "Cavity-enhanced dual-comb spectroscopy," Nat. Photon. 4, 55-57 (2009).
[CrossRef]

Hänsch, T. W.

T. W. Hänsch, "Nobel lecture: Passion for precision," Rev. Mod. Phys. 78, 1297-1309 (2006).
[CrossRef]

Holzwarth, R.

B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hansch, and N. Picque, "Cavity-enhanced dual-comb spectroscopy," Nat. Photon. 4, 55-57 (2009).
[CrossRef]

F. Keilmann, C. Gohle, and R. Holzwarth, "Time-domain mid-infrared frequency-comb spectrometer," Opt. Lett. 29, 1542-1544 (2004).
[CrossRef] [PubMed]

Jacquet, P.

B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hansch, and N. Picque, "Cavity-enhanced dual-comb spectroscopy," Nat. Photon. 4, 55-57 (2009).
[CrossRef]

Jacquey, M.

B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hansch, and N. Picque, "Cavity-enhanced dual-comb spectroscopy," Nat. Photon. 4, 55-57 (2009).
[CrossRef]

Kabetani, Y.

T. Yasui, Y. Kabetani, E. Saneyoshi, S. Yokoyama, and T. Araki, "Terahertz frequency comb by multifrequency heterodyning photoconductive detection for high-accuracy, high-resolution terahertz spectroscopy," Appl. Phys. Lett. 88, 241104 (2006).
[CrossRef]

Keilmann, F.

Kobayashi, Y.

B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hansch, and N. Picque, "Cavity-enhanced dual-comb spectroscopy," Nat. Photon. 4, 55-57 (2009).
[CrossRef]

Newbury, N.

I. Coddington, W. Swann, and N. Newbury, "Time-domain spectroscopy of molecular free-induction decay in the infrared," Opt. Lett., accepted (2010).
[CrossRef] [PubMed]

Newbury, N. R.

I. Coddington, W. C. Swann, and N. R. Newbury, "Coherent linear optical sampling at 15 bits of resolution," Opt. Lett. 34, 2153-2155 (2009).
[CrossRef] [PubMed]

I. Coddington, W. C. Swann, and N. R. Newbury, "Coherent multiheterodyne spectroscopy using stabilized optical frequency combs," Phys. Rev. Lett. 100, 013902 (2008).
[CrossRef] [PubMed]

N. R. Newbury and W. C. Swann, "Low-noise fiber-laser frequency combs (invited), " J. Opt. Soc. Am. B 24, 1756-1770 (2007).
[CrossRef]

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, "Fundamental noise limitations to supercontinuum generation in microstructure fiber," Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

N. R. Newbury, B. R. Washburn, K. L. Corwin, and R. S. Windeler, "Noise amplification during supercontinuum generation in microstructure fiber," Opt. Lett. 28, 944-946 (2002).
[CrossRef]

Ozawa, A.

B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hansch, and N. Picque, "Cavity-enhanced dual-comb spectroscopy," Nat. Photon. 4, 55-57 (2009).
[CrossRef]

Picque, N.

B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hansch, and N. Picque, "Cavity-enhanced dual-comb spectroscopy," Nat. Photon. 4, 55-57 (2009).
[CrossRef]

Saneyoshi, E.

T. Yasui, Y. Kabetani, E. Saneyoshi, S. Yokoyama, and T. Araki, "Terahertz frequency comb by multifrequency heterodyning photoconductive detection for high-accuracy, high-resolution terahertz spectroscopy," Appl. Phys. Lett. 88, 241104 (2006).
[CrossRef]

Saucier, P.

Schiller, S.

Schliesser, A.

Sromovsky, L. A.

Swann, W.

I. Coddington, W. Swann, and N. Newbury, "Time-domain spectroscopy of molecular free-induction decay in the infrared," Opt. Lett., accepted (2010).
[CrossRef] [PubMed]

Swann, W. C.

Tremblay, P.

Udem, T.

B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hansch, and N. Picque, "Cavity-enhanced dual-comb spectroscopy," Nat. Photon. 4, 55-57 (2009).
[CrossRef]

van der Weide, D.

Washburn, B. R.

Weber, K.

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, "Fundamental noise limitations to supercontinuum generation in microstructure fiber," Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

Windeler, R. S.

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, "Fundamental noise limitations to supercontinuum generation in microstructure fiber," Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

N. R. Newbury, B. R. Washburn, K. L. Corwin, and R. S. Windeler, "Noise amplification during supercontinuum generation in microstructure fiber," Opt. Lett. 28, 944-946 (2002).
[CrossRef]

Yasui, T.

T. Yasui, Y. Kabetani, E. Saneyoshi, S. Yokoyama, and T. Araki, "Terahertz frequency comb by multifrequency heterodyning photoconductive detection for high-accuracy, high-resolution terahertz spectroscopy," Appl. Phys. Lett. 88, 241104 (2006).
[CrossRef]

Yokoyama, S.

T. Yasui, Y. Kabetani, E. Saneyoshi, S. Yokoyama, and T. Araki, "Terahertz frequency comb by multifrequency heterodyning photoconductive detection for high-accuracy, high-resolution terahertz spectroscopy," Appl. Phys. Lett. 88, 241104 (2006).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

T. Yasui, Y. Kabetani, E. Saneyoshi, S. Yokoyama, and T. Araki, "Terahertz frequency comb by multifrequency heterodyning photoconductive detection for high-accuracy, high-resolution terahertz spectroscopy," Appl. Phys. Lett. 88, 241104 (2006).
[CrossRef]

IEEE T. Instrum. Meas. (1)

N. Almoayed and M. Afsar, "High-resolution absorption coefficient and refractive index spectra of carbon monoxide gas at millimeter and submillimeter wave-lengths," IEEE T. Instrum. Meas. 55, 1033-1037 (2006).
[CrossRef]

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

Mikrochimica Acta (1)

J. R. Birch, "Dispersive Fourier-transform spectroscopy," Mikrochimica Acta 3, 105-122 (1987).

Nat. Photon. (1)

B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hansch, and N. Picque, "Cavity-enhanced dual-comb spectroscopy," Nat. Photon. 4, 55-57 (2009).
[CrossRef]

Opt. Express (2)

Opt. Lett. (5)

Phys. Rev. Lett. (2)

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, "Fundamental noise limitations to supercontinuum generation in microstructure fiber," Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

I. Coddington, W. C. Swann, and N. R. Newbury, "Coherent multiheterodyne spectroscopy using stabilized optical frequency combs," Phys. Rev. Lett. 100, 013902 (2008).
[CrossRef] [PubMed]

Rev. Mod. Phys. (3)

T. W. Hänsch, "Nobel lecture: Passion for precision," Rev. Mod. Phys. 78, 1297-1309 (2006).
[CrossRef]

J. L. Hall, "Nobel lecture: Defining and measuring optical frequencies," Rev. Mod. Phys. 78, 1279-1295 (2006).
[CrossRef]

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

Other (7)

J. W. Brault, High Resolution in Astronomy (Geneva Observatory, 1985), Fourier transform spectrometry, pp. 1-65.

S. P. Davis, M. C. Abrams, and J. W. Brault, Fourier Transform Spectrometry (Academic Press, 2001).

V. V. Protopopov, Laser Heterodyning (Springer Berlin / Heidelberg, 2009).
[CrossRef]

W. Demtroder, Laser Spectroscopy (Springer, 1996), 2nd ed.

I. Coddington, W. Swann, and N. Newbury, "Coherent dual-comb spectroscopy at high signal to noise," http://arxiv.org/abs/1001.3865 (2010).

R. J. Bell, Introductory Fourier transform spectroscopy (Academic Press, 1972).

J. Chamberlain, The Principles of Interferometric Spectroscopy (John Wiley and Sons, Inc, 1979).

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Figures (4)

Fig. 1.
Fig. 1.

(a) Schematic of dual-comb layout including the possibility of sequential or parallel data acquisition. In the figure, the signal and reference are time-multiplexed [7, 8], but the reference could also be acquired in a separate interferometer [6] or sequentially with an empty cell. (b) Simulated time-domain signal versus the effective down-converted time (and not laboratory time). (c) Corresponding frequency-domain sample response over a 10 THz window at 100 MHz resolution for M = 100,000 resolved elements. For illustrative purposes, we assume a sample with 20 equally spaced, Lorentzian transitions 2 GHz wide and a spectral domain SNR of σ -1 H = 100. In the time domain, the corresponding signal is a one-sided interferogram with two components: a centerburst and a trailing free induction decay signal. In the frequency domain, each transition is highly resolved in magnitude and phase.

Fig. 2.
Fig. 2.

SNR and quality factor for a dual comb spectrometer versus the total comb power, Pc, for a single detector with no optical filtering (F = Nd = 1) from Eq. (2), normalized to a 1 sec total acquisition time. The quality factor is the product of the number of resolved spectral elements and the SNR for either amplitude or phase. The SNR for M =100,000 resolved elements is also shown. The SNR (or quality factor) (solid blue line) is limited by detector noise (long dashed brown line) at low comb powers, by shot noise (short dashed grey line) at medium comb powers, and by either laser RIN (dash-dot light green line) or detector dynamic range (dash-dotted dark green line) at high comb powers. With our assumption of a uniform spectrum and Gaussian filter shape, the average power per comb tooth is Ptooth = 0.8Pcfr ν. Values used: NEP = 2 pW/Hz1/2, D = 7000, RIN = -145 dBc/Hz, η = 0.9, b = 1, ε=l, γ = cγ = cγ2 =1, Δν= 10 THz, fr = νres = 100 MHz.

Fig. 3.
Fig. 3.

Quality factor and SNR at M = 100,000 resolved elements versus the total comb power, Pc, for different values of sequential acquisition at F = 10, 20, 50 and Nd = 1 (black lines) and parallel data acquisition at Nd = 10, 20, 50 detectors and F = 2 (red lines) under the otherwise identical conditions to Figure 2. The additional lines are carried over from Figure 2 for reference. The use of sequential acquisition comes at a cost for low powers, but at high comb powers can remove the limitations otherwise imposed by laser RIN or detector dynamic range. The effects of multiplicative noise (see Section 2.4) are not included. (The curves for Nd > 1 use F = 2 since an interleaved approach would likely be used.)

Fig. 4.
Fig. 4.

Schematic of the different optical frequencies involved in the calculation. The light blue (dark blue) lines are the source (LO) comb lines. Their repetition-rate difference is Δfr. The entire measured bandwidth is Δν, but for any given interferogram, the bandwidth is filtered to ΔνA either sequentially by tuning the filter across the spectrum or in parallel with a fixed filter bank to multiple detectors (or a combination of the two.) Note the “dead zones” where the rf beat notes are at either zero or Nyquist frequency; at these filter locations an overall frequency shift is applied to one source [7, 8].

Tables (1)

Tables Icon

Table 1. Definition of variables

Equations (32)

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H ˜ ( ν ) = V ˜ ( ν ) V ˜ ref ( ν ) = H ˜ 0 ( ν ) + σ H
σ H = M ε 0.8 T { a NEP F P c 2 + a shot 1 N d P c + ( a RIN + a range ) 1 N d 2 F } 1 / 2 ,
M σ H = 0.8 N d T ε F 2 RIN + 8 D 2 f r 1
( α 0 L ) min effective = 2 σ H × 4 ν res π j Δ ν Lor , j ( α j / α 0 ) 2 ,
σ H , mult ~ 3 Δ ν T N d f r σ φ , fast ,
K T S Δ T = T L Δ T 1 .
K f rL Δ f r = f rS Δ f r 1 .
E s ( t ) = e i φ S e i 2 π ν 0 t n A S ( t n T S ) , E L ( t ) = e i φ S + Δ φ e i 2 π ν 0 t m A L ( t m T L ) ,
V ( t ) = aR ( t ) E L * ( t ) E S ( t ) ,
V k = V ( t = k T L ) = a e i Δ φ n , m R ( τ ) A L * ( ( k m ) T L τ ) A S ( k T L n T S τ ) ′.
V ( t eff ) = e i Δ φ r = 0 r = T / T frame V ( t eff r T S ) ,
V ( t ) = S ( t ) A S ( t t ) dt S ( t ) A s ( t ) ,
S ( t ) = a Δ T 1 R [ ( K + 1 ) t ] A L * [ ( K + 1 K ) t ] .
E S ( t ) = e i φ S e i 2 π ν 0 t q A ˜ S ( q f rS ) e i 2 πq f rS t
E L ( t ) = e i φ S + Δ φ e i 2 π ν 0 t q A ˜ L ( p f rL ) e i 2 πp f rL t ,
V ( t ) = e i Δ φ a w , q R ˜ ( q Δ f r w f rL ) A ˜ L * ( q f rL + w f rL ) A ˜ S ( q f rS ) e i 2 πq Δ f r t e i 2 πw f rL t .
V ( t ) = e i Δ φ a q = q min q = q max R ˜ ( q Δ f r ) A ˜ L * ( q f rL ) A ˜ S ( q f rS ) e i 2 πq Δ f r t ,
V ( f ) = e i Δ φ a q = q min q = q max R ˜ ( q Δ f r ) A ˜ L * ( q f rL ) A ˜ S ( q f rS ) δ ( f q Δ f r ) ,
V ˜ ( ν ) = e i Δ φ S ˜ ( ν ) A ˜ S ( ν ) ,
S ˜ ( ν ) = a K + 1 R ˜ ( ν K + 1 ) A ˜ L * ( K K + 1 ν ) ,
V ( t ) = S ( t ) E S H ( t ) + n ( t ) T A Δ f r ,
( S / N ) t = T A Δ f r V ( 0 ) σ t .
V ˜ ( ν ) = S ˜ ( ν ) E ˜ s ( ν ) H ˜ ( ν ) + n ˜ ( ν ) T A Δ f r ,
( S / N ) ν = T A Δ f r V ˜ ( ν ) σ ν ,
σ H = 2 ( S / N ) ν ,
σ t = σ d 2 + η n S + η n LO + ( RIN S ) ( f r / 2 ) η 2 n L 2 + ( RIN L ) ( f r / 2 ) η 2 n S 2 .
( S / N ) t = T A Δ f r 2 η n S γ 1 σ d 2 + 2 c γ η n S + c γ 2 b ( RIN ) f r η 2 n S 2 + 4 D 2 η 2 n S 2 ,
( S / N ) ν = 0.8 T A M A 2 P cA γ 1 NE P 2 + 4 c γ 2 η 1 P cA + 2 b c γ 2 ( RIN ) P cA 2 + 8 D 2 f r 1 P cA 2 ,
σ H = M T { 1 0.8 ( NEP P c ) 2 γ 1 F + c γ ( 4 η P c ) 1 N d + 2 b c γ 2 RIN F N d 2 + 8 D 2 f r F N d 2 } .
( S / N ) ν = 2 T A P tooth NE P 2 + 4 η 1 P cA + 2 b P cA 2 RIN .
σ H , mult 2 ( ν , ν ) = ( 4 T A Δ f r ) n ˜ ( ν ) n * ( ν ) V ˜ ( ν ) V ˜ * ( ν )
σ H , mult ( ν , ν ) ( 2 π ln ( 2 ) ) 1 / 4 Δ ν A T A f r 2 σ φ , fast e 4 ln ( 2 ) ( ν ̄ ν A ) 2 / Δ ν A 2 ,

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