## Optical referencing technique with CW lasers as intermediate oscillators for continuous full delay range frequency comb interferometry |

Optics Express, Vol. 18, Issue 22, pp. 23358-23370 (2010)

http://dx.doi.org/10.1364/OE.18.023358

Acrobat PDF (965 KB)

### Abstract

This paper presents a significant advancement in the referencing technique applied to frequency comb spectrometry (cFTS) that we proposed and demonstrated recently. Based on intermediate laser oscillators, it becomes possible to access the full delay range set by the repetition rate of the frequency combs, overcoming the principal limitation observed in the method based on passive optical filters. With this new referencing technique, the maximum spectral resolution given by each comb tooth is achievable and continuous scanning will improve complex reflectometry measurements. We present a demonstration of such a high resolution cFTS system, providing a spectrometry measurement at 100 MHz of resolution (0.003 cm^{–1}) with a spectral signal to noise ratio of 440 for a 2 seconds measurement time. The resulting spectrum is composed of 105 · 10^{3} resolved spectral elements, each corresponding to a single pair of optical modes (one for each combs). To our knowledge, this represents the first cFTS measurement over the full spectral range of the sources in a single shot with resolved individual modes at full resolution.

© 2010 Optical Society of America

## 1. Introduction

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

5. A. Bartels, A. Thoma, C. Janke, T. Dekorsky, A. Dreyhaupt, S. Winnerl, and M. Helm, “High-resolution THz spectrometer with kHz scan rates,” Opt. Express **14**, 430–437 (2006). [CrossRef] [PubMed]

6. S. Kray, F. Spöler, M. Först, and H. Kurz, “Dual femtosecond laser multiheterodyne optical coherence tomography,” Opt. Lett. **33**, 2092–2094 (2008). [CrossRef] [PubMed]

8. G. Taurand, P. Giaccari, J.-D. Deschênes, and J. Genest, “Time Domain Optical Reflectometry Measurements Using a Frequency Comb Interferometer,” Appl. Opt. **49**, 4413–4419 (2010). [CrossRef] [PubMed]

2. 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. C. Swann, and N. R. Newbury, “Coherent linear optical sampling at 15 bits of resolution,” Opt. Lett. **34**, 2153–2155 (2009). [CrossRef] [PubMed]

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

9. B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R Holzwarth, G. Guelachvili, T. W. Hansch, and N. PicquÉ, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics **4**, 55–57 (2009). [CrossRef]

3. P. Giaccari, J.-D. Deschênes, 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]

*T*must be constant for every pulse pair. If this parameter drifts over the course of a measurement, the effect will be the same as a speed change of a moving mirror in a conventional Michelson interformeter: the delay axis of the measurement will be distorted. If this interferometer is used to produce spectroscopic measurements, this introduces smearing of the spectral features and/or noise. Two approaches are possible to eliminate this problem: one can monitor the parameter and actively adjust the frequency comb sources using feedback loops [10

_{r}10. N. R. Newbury and W. C. Swann, “Low-noise fiber-laser frequency combs (Invited),” J. Opt. Soc. Am. B **24**, 1756–1770 (2007). [CrossRef]

10. N. R. Newbury and W. C. Swann, “Low-noise fiber-laser frequency combs (Invited),” J. Opt. Soc. Am. B **24**, 1756–1770 (2007). [CrossRef]

3. P. Giaccari, J.-D. Deschênes, 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]

## 2. Post-correction algorithm

*T*

_{r}_{1}(

*k*) is the difference in pulse arrival time for each pulse pair, Δ

*φ*(

*k*) is the difference in pulse phase for each pulse pair,

*A*

_{1}and

*A*

_{2}are the slowly varying part of the bandpass filters impulse response,

*f*

_{c}_{1}and

*f*

_{c}_{2}are the center frequencies of the bandpass filters.

*A*(

_{m}*τ*) being the slowly varying part of the sample’s impulse response filtered by the probing pulse’s shape and the time-reversed shape of the other pulse shape (see appendix A).

*A*(

_{m}*τ*) is the quantity of interest and contains all the information on the sample’s linear response to any input within the bandwidth of the pulses. Removing the effect of the pulses on the measurement can be made by deconvolution with the shape found by with a calibration procedure, e.g. by measuring the inteferogram produced by the system without any sample. In the experimental demonstration presented in this paper, the sample is a gas cell with very narrow absorption lines and the shape of the pulse’s spectrum varies much more slowly than the interesting features; such a removal was not deemed necessary to demonstrate the validity of the correction algorithm.

*φ*(

*k*): This assumes that

*A*

_{1}(

*τ*) is real, ie ∠(

*A*

_{1}(

*τ*)) = 0. This is a constraint on the optical filters used. There must be no chirp in the impulse response. In practice, this constraint is met with uniform Fiber Bragg Gratings (FBG) which were used in the first referencing implementation [3

3. P. Giaccari, J.-D. Deschênes, 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]

*φ*(

*k*). Note that this first step removes all fluctuations measured at

*f*

_{c}_{1}; both the CEO and repetition rate fluctuations for this frequency are cancelled. Also,

*f*–

_{m}*f*

_{c}_{1}is much smaller than

*f*, in a ratio of about 1 : 200 in our system. The sensitivity of the carrier

_{m}*exp*[...] to jitter in Δ

*T*(

_{r}*k*) is thus accordingly reduced. The last step of the algorithm is to resample the signal on a new temporal grid such that Δ

*T*(

_{r}*k*) has constant increments from sample to sample. This grid is easily computed from the phase of the signal: It should be noted that when extracting the phase from the signal described in Eq. (5), a large span between the two chosen frequencies

*f*

_{c}_{2}and

*f*

_{c}_{1}is desirable, since this reduces the sensitivity to additive noise. The actual implementation of the algorithm differs from this description only from the fact that digital filters need to be used to generate the complex representation of the signals as well as removing some noise.

## 3. New implementation

11. C. Dorrer, D. C. Kilper, H. R. Stuart, G. Raybon, and M. G. Raymer, “Linear Optical Sampling,” *IEEE Photon. Technol. Lett.*15, 1746–1748 (2003). [CrossRef]

12. J. Reichert, R. Holzwarth, Th. Udem, and T. W. Hansch, “Measuring the frequency of light with mode-locked lasers,” Opt. Commun. **172**, 59–68 (1999). [CrossRef]

*k*is the index of the digital sample,

*P*,

_{CWx}*f*and

_{cwx}*φ*are respectively the power, the frequency and the phase deviation of each CW laser.

_{CWx}*P*is the power of the frequency comb x around wavelength

_{FCx,λy}*λ*. These signals are digitally combined to produce the two referencing signals:

_{y}*φ*

_{CW}_{1}(

*kT*

_{r}_{1}) –

*φ*

_{CW}_{1}(

*kT*

_{r}_{2}) is not present in the first equations. This term is in fact the difference in the phase of the CW laser between two successive instants. For zero optical delay between the pulses, this term is necessarily zero, while at the largest delay between these two instants, that is the period of the pulse train,

*T*, this term is maximal. This sets a constraint on the type of CW laser used in this scheme; the phase difference between the electrical field of the CW laser at time

_{r}*t*and at time

*t*+

*T*must be small. Relatively cheap narrow-linewidth lasers easily meet this criterion for our value of 10 ns for

_{r}*T*. We use PLANEX lasers from Redfern Integrated Optics which have 30 kHz of linewidth. Assuming white frequency noise, this implies a standard deviation of

_{r}*P*

_{CW}_{1}providing coherent gain. In the previous referencing scheme, the signal was produced by the interference of two filtered pulse trains, while in the new scheme, the signals are produced by the interference of one high-power CW laser (≈ 10 mW, divided in 4 paths) with each filtered pulse train. The pulse trains after the filters (ITU grid filters with 50 GHz of bandwidth) have 15

*μ*W of power.

## 4. Results

*μ*m of OPD and 1 rad of phase and are attributed to mechanical vibrations and thermal fluctuations of the fiber paths, as the referencing and measurement setup are placed on separate optical tables, linked by two 10 m cables. The post-correction IGM are presented in Fig. 2.

*μs*, as computed from one of the reference signals, which yields a linewidth before correction on the order of 100 kHz. The full-width at half-maximum (FWHM) of a sinc lineshape for a measurement time of 2 sec is 1.2/(2 s) = 0.6Hz. It is interesting to note that the correction method recovers the full coherence between the interferograms even for such a long measurement time, as seen from the width of 0.6 Hz of the individual modes in Fig. 3(D). Although performing such a long Fourier transform validates that the coherence is recovered by the referencing, in practice it is less computationally intensive to first segment the measurement into individual IGMs and to average them before performing a smaller Fourier transform. As expected, this averaging yields an IGM with an SNR equal to the SNR of a single IGM before averging times the square root of the number of IGMs in the average. The segmentation of the different IGMs is easily done after resampling as the resampled vectors are on an equidistant OPD grid. This produces the same result as evaluating the spectrum only at the discrete modes of the beating, as is shown by the red line on Figs. 3(A)–3(C).

*M*= 1.05 · 10

^{5}. The actual number of modes is a little higher than this value, but this definition eliminates any ambiguity on which modes are sufficiently higher than the noise to count as one spectral element. The spectral elements are all measured at the same time in a a single measurement. This gives a factor of merit of

14. N. R. Newbury, I. Coddington, and W. C. Swann, “Sensitivity of coherent dual-comb spectroscopy,” Opt. Express **18**, 7929–7945 (2010). [CrossRef] [PubMed]

## 5. Dynamic range reduction for SNR improvement

8. G. Taurand, P. Giaccari, J.-D. Deschênes, and J. Genest, “Time Domain Optical Reflectometry Measurements Using a Frequency Comb Interferometer,” Appl. Opt. **49**, 4413–4419 (2010). [CrossRef] [PubMed]

## 6. Comparison between stabilization and post-correction

## 7. Conclusion

*f*. One way to improve the current implementation would be to use reference lasers further apart in the optical band, to lower the effect of additive noise on the residual jitter. Future work includes implementing this referencing technique in real-time, to enable continuous measurement of a sample.

_{CEO}## A. Mathematical derivation of the referencing signals

*s*

_{1}(

*t*), at the input of the optical filter: where

*T*

_{r}_{1}is the arrival time and

*φ*

_{1}is the phase of the pulse. The impulse response of the filter is: where

*a*

_{1}(

*t*) is the baseband impulse response without the carrier frequency and

*f*

_{c}_{1}is the central frequency of the filter. The output of the filter is the convolution of the input pulse and the impulse response:

*T*

_{r}_{2}and the phase is

*φ*

_{2}:

*T*=

_{r}*T*

_{r}_{2}–

*T*

_{r}_{1}and Δ

*φ*=

*φ*

_{2}–

*φ*

_{1}. The electrical signal at the output of the photodetector is proportional to: where

*h*(

_{d}*t*) is the photodetector’s impulse response. We consider only the product term between the two fields and use the complex signal instead of only the real part. The factor of 2 is also not included in order to the keep the expressions as simple as possible. The quantity of interest will be labeled

*s̃*(

_{d}*t*):

*T*is the time over which the detector’s impulse response

_{d}*h*(

_{d}*t*) is non-zero. To evaluate this integral, we first point out that the impulse response of the optical filter

*h*

_{1}(

*t*) is much shorter than the duration of the detector’s impulse response. Hence,

*h*(

_{d}*t*) is essentially constant over the time where

*a*

_{1}(

*t*–

*T*

_{r1}–

*u*) is non-zero, which is around

*u*=

*t*–

*T*

_{r1}. Also, the limits of integration can be changed to [–∞, ∞] without affecting the result since

*a*

_{1}(

*t*) has a smaller support than the actual limits [0,

*T*]. We can then write:

_{pd}*T*and Δ

_{r}*φ*. Since there will be no interaction from pulses that are temporally separated by more than the time of the filter’s impulse response, we can simply write the electrical signal for the interference of the two pulse trains as the sum of the signal to each pulse pairs. We use the index k to indicate the k-th pulse pair. Each pulse pair will have a different Δ

*T*and Δ

_{r}*φ*and consequently we redefine them to be functions of k.

*T*

_{r}_{1}(

*k*), for each value of k. The amplitude of each pulse of this pulse train is modulated by the interference of the optical signals

*A*

_{1}(Δ

*T*(

_{r}*k*)) exp [

*j*2

*πf*

_{c}_{1}Δ

*T*(

_{r}*k*) +

*j*Δ

*φ*(

*k*)]. We now extract the amplitude of each electrical pulse by taking one sample per incident optical pulse pair, synchronized with the electrical pulse train Σ

*(*

_{k}h_{d}*t*–

*T*

_{r}_{1}(

*k*)). The discrete-time signal becomes:

*h*(0) = 1. Also, the same derivation could be done for a second optical filter with a central frequency

_{d}*f*

_{c}_{2}instead of

*f*

_{c}_{1}. This way, we obtain the following referencing signals:

*A*(

_{m}*τ*) would be the autocorrelation of the baseband part of the sample’s impulse response, filtered by the crosscorrelation of the baseband electrical field of the pulses used for probing. If only one comb probes the sample,

*A*(

_{m}*τ*) will be the sample’s baseband impulse response, filtered by the probing pulse’s shape and the time-reversed shape of the other pulse shape.

## Acknowledgments

## References and links

1. | F. Keilmann, C. Gohle, and R. Holzwarth, “Time-domain mid-infrared frequency-comb spectrometer,” Opt. Lett. |

2. | I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent Multiheterodyne Spectroscopy Using Stabilized Optical Frequency Combs,” Phys. Rev. Lett. |

3. | P. Giaccari, J.-D. Deschênes, 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 |

4. | 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. |

5. | A. Bartels, A. Thoma, C. Janke, T. Dekorsky, A. Dreyhaupt, S. Winnerl, and M. Helm, “High-resolution THz spectrometer with kHz scan rates,” Opt. Express |

6. | S. Kray, F. Spöler, M. Först, and H. Kurz, “Dual femtosecond laser multiheterodyne optical coherence tomography,” Opt. Lett. |

7. | I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent linear optical sampling at 15 bits of resolution,” Opt. Lett. |

8. | G. Taurand, P. Giaccari, J.-D. Deschênes, and J. Genest, “Time Domain Optical Reflectometry Measurements Using a Frequency Comb Interferometer,” Appl. Opt. |

9. | B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R Holzwarth, G. Guelachvili, T. W. Hansch, and N. PicquÉ, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics |

10. | N. R. Newbury and W. C. Swann, “Low-noise fiber-laser frequency combs (Invited),” J. Opt. Soc. Am. B |

11. | C. Dorrer, D. C. Kilper, H. R. Stuart, G. Raybon, and M. G. Raymer, “Linear Optical Sampling,” |

12. | J. Reichert, R. Holzwarth, Th. Udem, and T. W. Hansch, “Measuring the frequency of light with mode-locked lasers,” Opt. Commun. |

13. | C. Turcotte, “Laser a semi-conducteurs utilise comme reference metrologique dans un spectrometre par transformee de Fourier: effet du bruit,” Master’s thesis, Universite Laval, (1999). |

14. | N. R. Newbury, I. Coddington, and W. C. Swann, “Sensitivity of coherent dual-comb spectroscopy,” Opt. Express |

15. | W. C. Swann, J. J. McFerran, I. Coddington, N. R. Newbury, I. Hartl, M. E. Fermann, P. S. Westbrook, J. W. Nicholson, K. S. Feder, C. Langrock, and M. M. Fejer, “Fiber-laser frequency combs with subhertz relative linewidths,” Opt. Lett. |

**OCIS Codes**

(120.3940) Instrumentation, measurement, and metrology : Metrology

(120.4640) Instrumentation, measurement, and metrology : Optical instruments

(140.4050) Lasers and laser optics : Mode-locked lasers

(300.6300) Spectroscopy : Spectroscopy, Fourier transforms

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: September 9, 2010

Revised Manuscript: October 15, 2010

Manuscript Accepted: October 15, 2010

Published: October 20, 2010

**Citation**

Jean-Daniel Deschênes, Philippe Giaccarri, and Jérôme Genest, "Optical referencing technique with CW lasers as intermediate oscillators for
continuous full delay range frequency comb interferometry," Opt. Express **18**, 23358-23370 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-22-23358

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### References

- F. Keilmann, C. Gohle, and R. Holzwarth, "Time-domain mid-infrared frequency-comb spectrometer," Opt. Lett. 29, 1542-1544 (2004). [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]
- P. Giaccari, J.-D. Deschênes, 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]
- 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]
- A. Bartels, A. Thoma, C. Janke, T. Dekorsky, A. Dreyhaupt, S. Winnerl, and M. Helm, "High-resolution THz spectrometer with kHz scan rates," Opt. Express 14, 430-437 (2006). [CrossRef] [PubMed]
- S. Kray, F. Spöler, M. Först, and H. Kurz, "Dual femtosecond laser multiheterodyne optical coherence tomography," Opt. Lett. 33, 2092-2094 (2008). [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]
- G. Taurand, P. Giaccari, J.-D. Deschênes, and J. Genest, "Time Domain Optical Reflectometry Measurements Using a Frequency Comb Interferometer," Appl. Opt. 49, 4413-4419 (2010). [CrossRef] [PubMed]
- B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hansch, and N. Picqué, "Cavity-enhanced dual-comb spectroscopy," Nat. Photonics 4, 55-57 (2009). [CrossRef]
- N. R. Newbury, and W. C. Swann, "Low-noise fiber-laser frequency combs (Invited)," J. Opt. Soc. Am. B 24, 1756-1770 (2007). [CrossRef]
- C. Dorrer, D. C. Kilper, H. R. Stuart, G. Raybon, and M. G. Raymer, "Linear Optical Sampling," IEEE Photon. Technol. Lett. 15, 1746-1748 (2003). [CrossRef]
- J. Reichert, R. Holzwarth, Th. Udem, and T. W. Hansch, "Measuring the frequency of light with mode-locked lasers," Opt. Commun. 172, 59-68 (1999). [CrossRef]
- C. Turcotte, "Laser a semi-conducteurs utilise comme reference metrologique dans un spectrometre par transformee de Fourier: effet du bruit," Master’s thesis, Universite Laval, (1999).
- N. R. Newbury, I. Coddington, and W. C. Swann, "Sensitivity of coherent dual-comb spectroscopy," Opt. Express 18, 7929-7945 (2010). [CrossRef] [PubMed]
- W. C. Swann, J. J. McFerran, I. Coddington, N. R. Newbury, I. Hartl, M. E. Fermann, P. S. Westbrook, J. W. Nicholson, K. S. Feder, C. Langrock, and M. M. Fejer, "Fiber-laser frequency combs with subhertz relative linewidths," Opt. Lett. 31, 3046-3048 (2006). [CrossRef] [PubMed]

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