Referenced passive spectroscopy using dual frequency combs

Sylvain Boudreau; Jérôme Genest

doi:10.1364/OE.20.007375

1. Introduction

In the past few years, frequency combs have been used to perform infrared spectroscopy. Most applications in the literature show active spectroscopy measurements, where the combs are used to probe a sample and the returning light is combined with another comb to obtain spectral and impulse response information about the sample [1–3]. However, passive spectroscopy is also possible using frequency combs. In this case, the combs can be seen as taking samples of the source’s field at different lag values, which can be used to estimate the autocorrelation of the source.

In this paper, the method is first described and it is shown mathematically that the autocorrelation of the source can indeed be obtained from the samples taken from the combs, even when the pulses are chirped enough that they cannot be seen as sampling the source at a single point in time. Then, a noise analysis taking thermal and shot noises into account is performed. The results of this analysis are used to compare to the expected performance of a classical Fourier transform spectrometer (FTS). Finally, we present experimental results of the sampling of both a stable single-mode laser and a wideband incoherent source to corroborate the theoretical results.

2. Description of the method

The measurement setup is shown in Fig. 1 . It is very similar to the setup used in [4, 5], which demonstrated the method presented here, but with stabilized combs instead of a referencing scheme. Laser 1 and Laser 2 represent the two mode-locked lasers used to sample the source to be measured, called Source here. The blocks labeled BPD represent balanced photodiodes.

Fig. 1 Dual comb passive spectroscopy setup.

Download Full Size | PDF

The block called Ref represents the referencing setup, where two fiber Bragg gratings are used to measure and correct for the sampling delay and phase offset perturbations, as fully described in [6].

The pulses from each laser are mixed with the source using a fiber coupler and then photodetected. The beating between a pulse and the source measured at a photodiode ( $M_{1}$ and $M_{2}$ ) essentially corresponds to a sample of the source’s field at the sampling time. Both pulsed lasers have a slightly different repetition rate, resulting in the source being sampled twice each repetition period with a delay that increases approximately linearly with each pair of pulses. Throughout this paper, optical path difference (OPD) and sampling delay will be used interchangeably to highlight the similarity with a classical interferometer. The autocorrelation of a source is defined as the expected value of the product of the source and a delayed version of itself. This means that, assuming ideal Dirac delta sampling, the product of the obtained samples on $M_{1}$ and $M_{2}$ is an estimator of the source autocorrelation at the lag corresponding to the delay between the sampling pulses. Since that estimator has a variance on the order of its expected value, multiple samples for each lag need to be averaged to get acceptable signal-to-noise ratio (SNR). The autocorrelation of a random process and its power spectrum being related by a Fourier transform, it is easy to get the spectral signature of the source once enough samples are averaged to get good noise performance.

In the above description, ideal sampling of the field was assumed for simplicity. It will now be shown that the sampling lasers don’t need to be ideal Dirac deltas for the sampling scheme to work as described above. The incoming fields on $M_{1}$ and $M_{2}$ are the sum of the sampling laser and source fields. Since the detectors square the incoming electrical fields, the photodetected signals are composed of two constant terms due to self beating and a cross beating term. This is the interesting term and the one that will be considered here. The photodetected signals of interest can be written as

\begin{array}{l} M_{1} = \int_{- \infty}^{\infty} E_{1} (t_{1}) \exp (j θ_{1}) s^{*} (t_{1}) d t_{1} \\ M_{2} = \int_{- \infty}^{\infty} E_{2} (t_{2} + τ) \exp (j 2 π f_{c} τ) \exp (j θ_{2}) s^{*} (t_{2}) d t_{2}, \end{array}

where

E_{1} (t)

and

E_{2} (t)

are the field envelopes of a single pulse from each sampling comb,

s (t)

is the source complex field envelope and f_c is the optical carrier frequency, common to both sampling pulses and the sampled source. In Eq. (1),

τ

is the temporal offset between the sampling pulses. It is the sweeping of this variable that allows the autocorrelation of the source to be computed at different lags. The constant phase factors (θ) take the carrier envelope offset (CEO) into account, namely that each pulse in the comb is only identical to the others up to a constant phase.

M_{1}

and

M_{2}

are proportional to the modulated energy detected when the comb fields (E₁ or E₂) sample the source. This signal is contained in the amplitude of the photodetector impulse response corresponding to each pulse pair. The peak values of the detected pulses thus contain the information on

M_{1}

and

M_{2}

. In Eq. (1), the integrals should formally be a convolution integral containing the impulse responses of the detectors, but it is assumed here that said impulse responses are both much longer and slowly varying than the sampling pulse duration and shorter than the repetition period of the sampling laser. In this case, the value of the peak of the convolution integral can be reduced to the given equations.

For a given $τ$ , the product $M_{1}^{*} M_{2}$ is calculated in post-processing and is given by

\begin{array}{l} M_{1}^{*} M_{2} = \int_{- \infty}^{\infty} E_{1}^{*} (t_{1}) \exp (- j θ_{1}) s (t_{1}) d t_{1} \int_{- \infty}^{\infty} E_{2} (t_{2} + τ) \exp (j 2 π f_{c} τ) \exp (j θ_{2}) s^{*} (t_{2}) d t_{2} \\ = \exp (j Δ θ) \exp (j 2 π f_{c} τ) \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E_{1}^{*} (t_{1}) E_{2} (t_{2} + τ) s^{*} (t_{2}) s (t_{1}) d t_{1} d t_{2}, \end{array}

where

Δ θ = θ_{2} - θ_{1}

. By making the change of variables

t_{2} = t_{1} + α

, and taking the expected value (averaging many sample pairs), we then get

\begin{array}{l} E {M_{1}^{*} M_{2}} = E {\exp (j Δ θ)} \exp (j 2 π f_{c} τ) \\ \times \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E_{1}^{*} (t_{1}) E_{2} (t_{1} + α + τ) E {s^{*} (t_{1} + α) s (t_{1})} d t_{1} d α \\ = E {\exp (j Δ θ)} \exp (j 2 π f_{c} τ) \int_{- \infty}^{\infty} R_{s s} (α) \int_{- \infty}^{\infty} E_{1}^{*} (t_{1}) E_{2} (t_{1} + α + τ) d t_{1} d α \\ = E {\exp (j Δ θ)} \exp (j 2 π f_{c} τ) \int_{- \infty}^{\infty} R_{s s} (α) R_{12}^{*} (α + τ) d α, \end{array}

where

R_{s s}

and

R_{12}

denote respectively the autocorrelation of the source and cross-correlation between the two sampling pulses. It was assumed that

Δ θ

is independent from s(t), which is reasonable since it only requires that the incoherent source and the combs be statistically independent. Equation (3) gives a lot of insight as to what needs to be done to make the measurement successful. The term

\int_{- \infty}^{\infty} R_{s s} (α) R_{12} (α + τ) d α

is the convolution of the combs’ cross-correlation with the auto-correlation of the source. In the spectral domain, this means that the measurement spectral width is limited and weighted by the cross-spectrum of the combs, which is not surprising. Another thing this implies, which might not be as obvious, is that the pulses from the combs need not be shorter than the coherence length of the source to sample it appropriately. The only effect of chirp in the pulses is that a differential chirp between the two sampling lasers will cause the measured source auto-correlation to be chirped as well. This may be undesirable as it might force a longer measurement OPD span than what would be necessary for the desired spectral resolution.

Since the sampling lasers are not perfectly stable in repetition rate and carrier-envelope offset (CEO), the measurements need to be corrected. The block called Ref in Fig. 1 represents the referencing system that is used to correct the measurements. The referencing algorithm used in this paper is the one presented in [6]. The purpose of this algorithm is to track and correct for the two parameters that need to be known in Eq. (3): $τ$ and $Δ θ$ . For a digital Fourier transform to give accurate spectral information, the samples need to be taken on an evenly spaced $τ$ grid. In a dual comb setup, this corresponds to the combs having a perfectly stable repetition rate. For unstable combs, the referencing setup measures the actual $τ$ grid, which makes it possible to resample the acquired signals on an even grid. The other parameter that needs to be tracked, $Δ θ$ , is the phase difference between both sampling pulses. Again, in a perfectly stable comb setup, $Δ θ$ is a ramp and does not distort the spectrum. For unstable combs, this phase needs to be tracked and removed for each sample pair before resampling and averaging. The referencing scheme makes that possible.

Due to the ambiguity of absolute phase when unwrapping, the referencing algorithm does not yield absolute values of lag and phase difference between the sampling lasers. It only gives those parameters up to an arbitrary constant. For this reason, the corrected source autocorrelation estimates obtained from different sweeps need to be phase and delay aligned before the averaging process. Since each autocorrelation estimate does not have a good signal-to-noise ratio, it cannot be used for alignment. This is the role of the signal labeled BP in Fig. 1. It is the beating between the combs, which has a good single shot signal-to-noise ratio. This beating is corrected using the same method used for the autocorrelation estimation. Using the fact that the same lag and phase offsets of the referencing algorithm apply to the comb beating and the autocorrelation estimation, these offsets can be found from the former signal and used to align the latter.

Since each beating signal in the setup is generated from the coherent interaction between the different optical sources, the system needs inteferometric stability over the full measurement time. It is especially important for the phase offset alignment operation: If the effective length of any of the arms after the sources are split drifts by a considerable fraction of a wavelength, the assumption that the phase offsets for both signals are the same does not hold anymore. The autocorrelation estimates then cannot be averaged in phase. For this reason, great care must be taken to be sure that thermal drifts are limited as much as possible.

3. Noise analysis

This section discusses the noise performance of the passive sampling technique and explains the fundamental differences between optical sampling and a traditional spectrometer when it comes to noise.

One interesting characteristic of the sampling technique is the coherent gain provided by the sampling laser. When sampling with a comb, the random fluctuations of the optical field are amplified by the multiplication with the large deterministic field of the optical pulse. This makes it possible to bring the level of those fluctuations above the thermal noise more easily without having to rely on amplification.

Another major difference is the fact that a conventional spectrometer continuously averages the product of the source with itself, whereas the sampling method takes one sample each comb repetition period. This is the most important limit on achievable signal-to-noise ratio: If samples are not taken at each source coherence time, maximum noise performance is not attained. Since it is obviously impossible to meet this condition for a reasonably wideband source, because of electronics speed reasons among others, the signal-to-noise ratio obtained using comb sampling is always less than ideal.

An optical field, being a random process, continually fluctuates, both in amplitude and in phase. Its spectrum is therefore only defined in a statistical manner, even before considering the usual thermal and shot noises. The product of the source field with a delayed version of itself is also a random variable and thus fluctuates. Those fluctuations need to be averaged out to get an accurate estimate of the desired autocorrelation. This source of noise, arising from the random nature of the field is always present in optical measurements, but often neglected because it is usually below the shot noise level. Derickson [7] breaks down those fluctuations into the usual two categories: intensity and phase noises. In the context of this paper, the combs sample the random field fluctuations. We rely on this noise to get a statistical estimate of the field. We will therefore call this the estimation noise. In the high optical power limit, this estimation noise is the dominant noise source. The SNR obtained in this case depends on the statistical nature of the source. For example, the annex shows that the single shot SNR obtained for a single mode laser source is $\sqrt{2}$ _, when it is modeled as a white frequency noise source.

Assuming that the condition where estimation noise is the dominant noise source is met, it is possible to get an SNR value as a function of measurement time and number of spectral bins to be able to directly compare it to the conventional Fourier-transform spectrometer. It is assumed that no sample is wasted, that is, the lasers sweep OPD in a fashion that does not scan the full OPD range, but only the range we’re interested in.

To get N independent spectral bins, 2N autocorrelation bins need to be taken. This is due to the fact that the Fourier transform operation is spectrally redundant for real signals. The time for one full OPD sweep is given by $2 N / f_{r}$ , where $f_{r}$ is the repetition rate of the lasers. The time for K sweeps is then

T = 2 K N / f_{r} .

The SNR for the average of K sweeps is thus given by

S N R_{t} = S N R_{single} \sqrt{\frac{T f_{r}}{2 N}},

where

S N R_{single}

is the signal to noise ratio of a single measurement.

For a broadband source, the frequency domain SNR is simply $\frac{1}{\sqrt{N}}$ times the time domain SNR. This is easily explained by the fact that, while the characteristics of white noise are not changed by a discrete Fourier transform, a very short temporal signal is spread across the N spectral bins. The signal amplitude is therefore reduced by a factor of $\sqrt{N}$ . The frequency domain SNR is then given by

S N R_{f} = \frac{S N R_{single}}{N} \sqrt{\frac{T f_{r}}{2}} .

We see that the SNR grows with $f_{r}$ . In fact, sampling with a low $f_{r}$ can be seen as using a low measurement duty cycle. While a FTS is continuously averaging, a comb sampler wastes time after each pair of pulses waiting for the next one.

Table 1 shows the SNR values for the optical sampling method and a conventional Fourier-transform spectrometer. The values for the conventional FTS can be retrieved from the SNR analysis performed in [8]. For both methods, the same dependencies in 1/N and in $\sqrt{T}$ are observed, which is to be expected. Comparing with the results for an FTS limited by aconstant noise floor, the term P/(2 NEP) for the classical FTS is replaced by $S N R_{single} \sqrt{f_{r}}$ in the sampling approach. To give some numbers, let us suppose a conventional FTS is measuring a 10µW source with a 20 pW/Hz^0.5 NEP, a typical photodetector noise value. To perform equally, the baseband optical sampling approach should sample at around 60 GHz, a quite high figure. These values assume a single shot SNR of unity, which is very close to reality, as will be shown in the results section. From a measurement time perspective, in those conditions, it would take 600 times as long as the FTS to obtain the same SNR with the optical sampling method with a 100 MHz sampling rate.

Table 1. SNR for FTS and Optical Sampling Spectroscopy

View Table | View all tables in this article

Of course, it is not possible to get arbitrarily high SNR by choosing an arbitrarily high repetition rate. In the previous analysis, it was assumed that each sample is independent from the others. If not, a new sample does not give new information and cannot be used to improve the estimation noise SNR. The repetition rate that would yield the maximum SNR is thus given by the correlation time of the source, which is the inverse of its bandwidth. The maximum achievable SNR is thus

S N R_{f (\max)} = \frac{\sqrt{T B W}}{2 N},

where BW is the source’s optical bandwidth.

Of course, achieving this SNR poses technical problems. The first obvious limitation is detection electronics: the repetition rate needed to maximize SNR is a lot higher than the electrical bandwidth of photodetectors. Since the optical pulses need to be resolved for the technique to work, the maximum measurement SNR is limited by the speed of the complete electronics chain. Additionally, if one has a sampler fast enough to prevent aliasing of the source spectral width, it is possible to directly sample the signal with one comb rather than using two sliding combs to measure its autocorrelation.

The second limitation is computational. Comb-based optical sampling is more computationally expensive than other spectroscopy techniques because the averaging needs to be done digitally instead of being done in the photodetector.

It was assumed in the development of Eq. (6) that estimation noise is the dominant source of noise. The conditions that make that assumption true will now be examined. The beating amplitude (estimation noise), shot noise from the combs, shot noise from the source and thermal noise all need to be considered.

Starting with the beating amplitude, we consider a rectangular pulse of peak power $P_{L p e a k}$ and duration $1 / B W_{L}$ and a source of average power $P_{S}$ and coherence length of more than $1 / B W_{L}$ beating together. The beating amplitude, which is to be understood as the standard deviation of the integrated power on the photodetector, is given by

B = \frac{\sqrt{P_{L p e a k} P_{S}}}{B W_{L}} = \sqrt{\frac{P_{L} P_{S}}{B W_{L} f_{r}}},

where

P_{L}

is the comb’s average power. The shot noise amplitude is given by

N_{s h} = \sqrt{E_{p h} E_{s i g n a l}} = \sqrt{E_{p h} P_{s i g n a l} T},

where

E_{p h}

,

P_{s i g n a l}

and

T

are the photon energy, power of integrated signal and integration time, respectively. Applied to the source and the comb, which both contribute to shot noise, we get

N_{s h S} = \sqrt{\frac{E_{p h} P_{S}}{B W_{P D}}}

and

N_{s h L} = \sqrt{\frac{E_{p h} P_{L}}{f_{r}}} .

Here, $N_{s h S}$ and $N_{s h L}$ are the shot noise from the source and the laser pulse respectively, while $B W_{P D} = 1 / T$ is the bandwidth of the photodiode. The last noise term to consider is thermal noise. It is expressed as

N_{t h} = \frac{N E P}{\sqrt{B W_{P D}}},

where NEP is the noise equivalent power of the photodetector in W/ Hz^0.5.

From Eq. (9) to (12) and choosing conditions where the beating signal is stronger than every noise term, the following conditions for the validity of Eq. (6) can be deduced:

P_{S} > E_{p h} B W_{L}

P_{L} > \frac{E_{p h} B W_{L} f_{r}}{B W_{P D}}

P_{L} P_{S} > \frac{N E P^{2} B W_{L} f_{r}}{B W_{P D}^{}} .

Using values corresponding to the setup at Université Laval, with 20 mW, 100 nm wide combs around 1550 nm, a 100 MHz repetition rate, a photodiode with a 350 MHz bandwidth and a 7.5 pW/ Hz^0.5 NEP, the power conditions are then $P_{S} > 2$ µW, $P_{L} > 450$ nW and $P_{S} P_{L} > 2 \times 10^{- 10}$ W. With 20 mW of comb power, Eq. (13) is more restrictive on source power than Eq. (15). This means that the source power must be at least in the microwatts range to get optimal sampling performance so that Eq. (6) can be used to evaluate SNR. It should be noted that Eq. (13) simply states that, in average, at least one photon from the source must be measured during the sampling period of a pulse. This ensures that the shot noise signal to noise ratio is greater than 1.

4. Measurement results

This section presents the results of the autocorrelation measurement of both a stable laser source and a broadband source filtered by an HCN cell. The spectrums of both sources are computed and compared to a measurement performed with an optical spectrum analyzer. The SNR of the measurements are calculated and confirmed to match expectations.

Both sources were sampled with the lasers locked at a repetition rate difference of 100 Hz. The feedback loop used for this lock is very slow (four updates per second). Its main purpose is to keep the beating from drifting too much during the measurement, preventing aliased copies of the spectrum from overlapping and rendering the signals unusable. Any fluctuation between the sampling lasers is removed by the post-correction algorithm described in [6]. The referencing system used for the measurements consist of two fiber Bragg gratings (FBG) selecting two known optical frequencies. Because the gratings have a spectral width corresponding to close to 100 modes of the lasers, the signals from the referencing system decay with lag, which makes them unusable for correcting the autocorrelation at more than around 100 ps of lag. This means that the measurement spectral resolution is limited to 5 GHz, which corresponds to roughly 0.04 nm at 1550 nm. The wavelengths of the gratings used for the measurements are 1548.94 nm and 1567.35 nm, which provide reasonable separation while still having good SNR for each reference. This resolution limit can be removed by using the referencing technique presented in [3].

4.1 Stable laser source

The first measured source is a stable laser. The measured laser is a PLANEX laser from Redfern Integrated Optics. Its center wavelength is 1562.236 nm and it has a 30 kHz line width. This means that our system cannot resolve the line and that there should be no measurable decay in the autocorrelation over the measurement span.

The laser source measurement is the average of 1000 traces. Each of those traces contains 20000 sample pairs. The full measurement is thus composed of 100 million samples. In the ideal case where the lasers do not sweep the full OPD span, this corresponds to a measurement time of one second. If the lasers do sweep the full OPD span, it corresponds to a measurement time of 50 seconds. Since the referencing algorithm is applied in post-processing, each trace is saved to the hard drive of the acquisition PC, which takes a lot more time, the actual measurement took around 15 minutes, but no care was taken to optimize that duration.

Figure 2 shows the measured autocorrelation of the laser source. The first thing to notice is the decay in envelope near the edge of the autocorrelation. Since the laser has a much longer coherence time than the measurement span, this decay in correlation has to come from the system. It can be explained by a decreased SNR in the referencing signals near the edge. Degradation of the referencing signals results in more sampling and phase jitter, which in turn, after averaging multiple traces, results in attenuation in the sine wave. This is a very interesting form of self-apodization.

Fig. 2 Autocorrelation of the laser source.

Download Full Size | PDF

To validate the SNR values, it will be assumed that the laser field is a white frequency noise process. It is shown in the annex that such a process has a single shot peak autocorrelation SNR of $\sqrt{2}$ around ZPD, making the SNR for $k$ samples $\sqrt{2 k}$ . This means that, for 100 samples, the expected SNR is around 14. The measured SNR for 100 samples was obtained in the following way: The 1000 samples autocorrelation trace was used as a reference which was subtracted from a 100 samples trace. The result of this subtraction wasused as a noise trace whose standard deviation was calculated. The SNR obtained with this method is almost 11, which is very close to the expected 14.

As for frequency accuracy and resolution, the zoomed line, as seen on Fig. 3 , has both the expected center wavelength of 1562.24 nm and a width of 0.08 nm, which corresponds exactly to the resolution calculated from the lag span of the measurement.

Fig. 3 Zoom on the laser line.

Download Full Size | PDF

4.2 Broadband source filtered by HCN cell

The next measurement is the more interesting case of a gas absorption spectrum. The source consists of an EXFO broadband light-emitting diode filtered by an HCN gas cell. It is then amplified by an erbium-doped fiber amplifier to bring its power at a level where the measurement is estimation noise limited. Since it is a broadband source, more traces need to be averaged than in the single mode laser case to get an equivalent spectral signal-to-noise ratio. The reason for this is twofold: First, a wideband signal consists of a strong correlation peak at ZPD with weak ripples elsewhere which account for the absorption lines. There is therefore less total modulated energy in the wideband signal than in a narrowband one. Moreover, the available energy is spread on a wide spectral band, which reduces further the spectral signal-to-noise ratio.

The results presented here were obtained by averaging 22300 autocorrelation traces. This corresponds to 446 million pulse pairs at the resolution given by the reference signals. This corresponds to a full OPD sweep measurement time of 223 seconds, which could be optimized to 4.5 seconds by sweeping a reduced OPD span. As with the stable laser measurement, the actual measurement time was much longer than that, but it is not indicative of the measurement time needed for a properly optimized setup.

Figure 4 shows the measured autocorrelation of the HCN filtered source. The central peak, which corresponds to the spectral baseline, has been clipped to better show the periodic peaks characteristic of the HCN signature. The trace was filtered to reject spectral content where the noise dominates and thus has much better SNR than the raw averaged autocorrelations.

Fig. 4 Autocorrelation of the HCN filtered source. The clipped peak has a unitary amplitude

Download Full Size | PDF

To validate the results in terms of noise, the SNR as a function of the number of averaged traces was computed. The noise amplitude on the autocorrelation traces was calculated by taking the standard deviation of the trace far away from ZPD, where noise dominates. The correct ZPD amplitude was obtained by filtering the signal so that the signal dominates spectral noise everywhere and by removing the spectral phase. This results in a non-chirped IGM that is minimally affected by the rectification of the spectral noise. Figure 5 shows the progression of the measured squared SNR as the number of averaged traces grows. It can beseen that the measured curve is approximately linear, which shows that the referencing, alignment and averaging steps are working properly. The slope of approximately $1 / 2$ suggests a single shot SNR of $1 / \sqrt{2}$ . Figure 6 shows the Fourier transform of the measured autocorrelation trace, which corresponds to the spectrum of the source. The baseline shape, given by the sampling combs’ and the broadband source’s spectrums, was removed to better show the absorption lines and highlight the HCN transmittance.

Fig. 5 Squared SNR of the autocorrelation estimate as a function of the number of averaged traces.

Download Full Size | PDF

Fig. 6 Transmittance of the HCN filtered source. The red dotted lines represent the reference HCN line frequencies.

Download Full Size | PDF

As can be seen on Table 2 , the measured line positions are accurate to the nearest GHz. The average deviation is 66 MHz and the standard deviation is 600 MHz. These results are most likely limited by the signal to noise ratio. The reference wavelengths were taken from the NIST SRM 2519a certificate of analysis.

Table 2. - Reference and Measured Frequencies of HCN Absorption Lines

View Table | View all tables in this article

7. Conclusion

In this paper, it was shown that it is possible to use dual frequency combs and a referencing scheme to perform passive spectroscopy. Experimental results agreed very well with theoretical values given by the noise analysis. However, said noise analysis showed that the sampling rate needs to be quite high to match a traditional Fourier transform spectrometer in typical conditions. Given the significant constraints highlighted in the paper, it seems very difficult to imagine a practical application where asynchronous sampling of an external source with two combs would provide better measurement performance than classical spectrometers. Nevertheless, passive sampling spectroscopy has the advantage of coherent gain from the sampling lasers, which can be used to get better performance in thermal noise limited scenarios without the need for amplification.

Appendix: Single shot SNR for a white frequency noise source

This section aims to compute the single shot sampling SNR for stable single mode laser source, which is often described by a white frequency noise process.

A white frequency noise process is a random process whose amplitude is constant and whose phase is the sum of a ramp and a Brownian motion process. The slope of the ramp gives the center frequency of the source. The phase deviation from that ramp at a given delay is simply a normally distributed variable with a variance proportional to said delay. The field of such a laser source can thus be mathematically described as

s (t) = \cos (2 π f t + ϕ (t)),

where $f$ is the frequency of the source and $ϕ$ is the Brownian motion phase term, whose variance is equal to $| t | σ_{0}^{2}$ . The product of two delayed samples from the combs is given by

S = \cos (θ) \cos (θ + ϕ (τ) + 2 π f τ) .

Here, the first sample is, without loss of generality, assumed to have been taken at $t = 0$ . $θ$ is a uniformly distributed variable which accounts for the fact that the first sample has a random phase. The autocorrelation of $s (t)$ is the expected value of $S$ and can be shown [9] to be equal to

\begin{matrix} R_{s s} (τ) & = & \frac{1}{2} e^{- \frac{| τ | σ_{_{0}}^{2}}{2}} \cos (2 π f τ) \end{matrix} .

This result is the familiar double sided decaying exponential function, whose Fourier transform is a Lorentzian function.

Using a similar method, the variance of the estimate can be obtained. First, its second moment is found:

\begin{matrix} E {S^{2}} & = & \int_{- \infty}^{\infty} \int_{0}^{2 π} \cos^{2} (θ) \cos^{2} (θ + ϕ + 2 π f τ) \frac{1}{2 π} \frac{1}{\sqrt{2 π | τ | σ_{_{0}}^{2}}} e^{- \frac{ϕ^{2}}{2 | τ | σ_{_{0}}^{2}}} d θ d ϕ \\ = & \frac{1}{4} [1 + \frac{1}{2} e^{- 2 | τ | σ_{_{0}}^{2}} \cos (4 π f τ)] \end{matrix} .

The variance of the estimate can now be obtained:

\begin{matrix} σ_{S}^{2} & = & E {S^{2}} - R_{S S}^{2} \\ = & \frac{1}{4} [1 + \frac{1}{2} e^{- 2 | τ | σ_{_{0}}^{2}} \cos (4 π f τ) - e^{- | τ | σ_{_{0}}^{2}} \cos^{2} (2 π f τ)] \\ = & \frac{1}{4} [1 - \frac{1}{2} e^{- | τ | σ_{_{0}}^{2}} + \frac{1}{2} \cos (4 π f τ) (e^{- 2 | τ | σ_{_{0}}^{2}} - e^{- | τ | σ_{_{0}}^{2}})] \end{matrix} .

The variance is thus composed of a slowly varying term and a fast modulated term. If the frequency domain is the domain of interest, it is possible to neglect the modulated term without changing the SNR value. The reason for that is simple: a Fourier transformed non-stationary white noise is a spectrally flat noise whose variance is the same as the one from a stationary white noise with the same average variance. As far as spectral noise is concerned the following low-pass filtered noise variance is equivalent:

{\bar{σ}}_{S}^{2} = \frac{1}{4} [1 - \frac{1}{2} e^{- | τ | σ_{_{0}}^{2}}] .

Around ZPD, the peak SNR is $\sqrt{2}$ . As autocorrelation decays, the variance of the estimate increases which further degrades SNR.

Acknowledgments

This work was supported by Defense Research and Development Canada (DRDC, Valcartier) under contract W7701-094432/A awarded to ABB inc. by Public Works and Government Services Canada. The content presented herein is however the sole responsibility of the authors.

References and links

1. I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs,” Phys. Rev. 100, 13902 (2008).

2. J. Mandon, G. Guelachvili, and N. Picqué, “Fourier transform spectroscopy with a laser frequency comb,” Nat. Photonics 3(2), 99–102 (2009). [CrossRef]

3. J.-D. Deschênes, P. Giaccarri, and J. Genest, “Optical referencing technique with CW lasers as intermediate oscillators for continuous full delay range frequency comb interferometry,” Opt. Express 18(22), 23358–23370 (2010). [CrossRef] [PubMed]

4. F. R. Giorgetta, I. Coddington, E. Baumann, W. C. Swann, and N. R. Newbury, “Dual frequency comb sampling of a quasi-thermal incoherent light source,” in Conference on Lasers and Electro-Optics (2010).

5. I. R. Coddington, W. C. Swann, and N. R. Newbury, “Frequency comb spectroscopy,” in Fourier Transform Spectroscopy (2009).

6. 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(6), 4347–4365 (2008). [CrossRef] [PubMed]

7. D. Derickson, “Noise sources in optical measurement” in Fiber Optic Test and Measurement (Prentice Hall PTR, 1998), pp. 597–613.

8. J. W. Brault, “Fourier transform spectrometry,” in Saas-Fee Advanced Course 15: High Resolution in Astronomy (1985), Vol. 1, pp. 1–61.

9. K. Petermann, Laser Diode Modulation and Noise (Springer, 1991).

METHOD	SNR
FTS (shot noise limited)	$\sqrt{\frac{P}{2 E_{ph}}} \frac{1}{N} \sqrt{\frac{T}{2}}$
FTS (detector noise limited)	$\frac{P}{2 N E P} \frac{1}{N} \sqrt{\frac{T}{2}}$
Optical sampling	$S N R_{single} \sqrt{f_{r}} \frac{1}{N} \sqrt{\frac{T_{}}{2}}$
P is the optical power,
E_ph is the energy of a single photon,
N is the number of measured points,
T is the total measurement duration
NEP is the noise equivalent power of the detection chain
SNR_single is the single shot SNR of the estimation noise
f_r, is the repetition rate of the sampling combs

Reference frequencies (THz)	Measured frequencies (THz)
192.086	192.086
192.1976	192.1966
192.3080	192.3079
192.4173	192.4167
192.5255	192.5258
192.6326	192.6327
192.7385	192.7380
192.8433	192.8434
192.9470	192.9481

METHOD	SNR
FTS (shot noise limited)	$\sqrt{\frac{P}{2 E_{ph}}} \frac{1}{N} \sqrt{\frac{T}{2}}$
FTS (detector noise limited)	$\frac{P}{2 N E P} \frac{1}{N} \sqrt{\frac{T}{2}}$
Optical sampling	$S N R_{single} \sqrt{f_{r}} \frac{1}{N} \sqrt{\frac{T_{}}{2}}$
P is the optical power,
E_ph is the energy of a single photon,
N is the number of measured points,
T is the total measurement duration
NEP is the noise equivalent power of the detection chain
SNR_single is the single shot SNR of the estimation noise
f_r, is the repetition rate of the sampling combs

Reference frequencies (THz)	Measured frequencies (THz)
192.086	192.086
192.1976	192.1966
192.3080	192.3079
192.4173	192.4167
192.5255	192.5258
192.6326	192.6327
192.7385	192.7380
192.8433	192.8434
192.9470	192.9481

Referenced passive spectroscopy using dual frequency combs

Abstract

1. Introduction

2. Description of the method

3. Noise analysis

4. Measurement results

4.1 Stable laser source

4.2 Broadband source filtered by HCN cell

7. Conclusion

Appendix: Single shot SNR for a white frequency noise source

Acknowledgments

References and links

Cited By

Figures (6)

Tables (2)

Equations (21)

Optics Express