Laser linewidth characterization via self-homodyne measurement under nearly-coherent conditions

Ido Attia; Eyal Wohlgemuth; Ohad Balciano; Roi Jacob Cohen; Yaron Yoffe; Dan Sadot

doi:10.1364/OE.451758

1. Introduction

A crucial parameter in modern lightwave communications is the phase noise of the laser source, and the laser’s linewidth, in particular. It is shown that high order modulation (HOM) schemes, required for high-speed optical transceivers, are rather susceptible to phase noise [1,2]. High-end coherent optical transceivers utilize external cavity lasers (ECLs) or sampled-grating distributed Bragg reflector (SG-DBR) [3], such as the integrated tuneable laser assembly (ITLA). ITLA has an excellent optical performance in terms of high output power stability, high side-mode-suppression ratio (SMSR), ultra-narrow laser linewidth, low relative intensity noise (RIN) and high wavelength control accuracy. Process variations, and system tolerances demand individual testing and characterization of the laser source. Moreover, performance degradation towards end-of-life conditions, could be even of greater concern to transceivers’ vendors and operators [4–6]. It is therefore of utmost importance to devise a method evaluating the laser performance in its final disposition within operational product, and through self-testing mode.

Modern pluggable transceivers are densely integrated with compact optics, electronics and mechanics. A relatively new trend, is co-packaging the DSP ASIC within the pluggable module, what is termed as digital coherent optics (DCO). This approach drives an even higher level of integration, while active components such as lasers, are co-packaged with silicon photonics circuits [7]. In such systems, the laser cannot be dissemble neither externalize for testing purposes. In addition, there is an advantage of unplugging the systems and measuring it’s performance over fiber, while still hosted within a linecard or networking equipment, interrupting the service for a minimum time. The proposed method harnesses outputs of the DSP algorithms that are being used routinely for the modem normal operation, while not demanding any use of system modifications or external test equipment, thus, coherent optical communication operators may find it extremely useful.

The most common practice for accurate linewidth measurement is a delayed self-heterodyne (DSH) interferometry [8–16]. The main idea of this technique is to measure the phase fluctuations by mixing a laser with its delayed replica. In this practice, a continuous wave (CW) laser is split into two branches, where one branch is modulated with an acoustic optical modulator (AOM) and the other is delayed with a fiber. Following, the signal is re-combined and detected by a photodiode so its photocurrent fluctuations reflects the phase noise. The sinewave AOM modulation, in the range of hundreds of MHz, is used to avoid the near direct current (DC) noises and allow using electrical spectrum analyser (ESA) to analyse the phase noise and the frequency modulation (FM) noise. In addition, the OPD is usually chosen to be longer then the laser coherence length to decorrelate the signal and its replica, and to avoid interferomteric coherence pattern that distorts the Lorentzian spectral shape [8]. The laser’s linewidth is therefore provided by the half of the full-width half-maximum (FWHM) of the Lorentzian shape. However, this technique requires some non-trivial measurement equipment as an AOM, and for very narrow linewidth the required OPD can become extremely large (a few up to 10’s of kilometers). In addition, during long delay time, laser temperature drifting introduces intermediate frequency (IF), that shifts and broadens the Lorentzian shape. Therefore, some works proposed new methods of measuring linewidth in sub-coherence regime and with delayed self-homodyne interferomtry [8,12,15,16], these works still impose significant limitations on the OPD needed for measurement. While it is commonly to use an acousto-optic modulator (AOM) in order to avoid baseband noises, others offered a simple system with a short OPD, based on phase noise measurement in a self-mixing interferometer, and a 3–35 MHz modulated microphone [17].

This paper proposes a novel method for linewidth measurement via self-homodyne interferometry with a very short delay, while resolving some limitation that are associated with the conventional DSHI. It uses a common high-bandwidth coherent communication system with short fibres of approximately 0.5% of the coherent length for easy implementation of linewidth measurement. The new technique uses the nearly-coherent normalized phase noise PSD to estimate the Lorentzian line shape in a frequency regime far from its center. For that reason, although it uses self-homodyne interfermotry, the method is immune to low frequency noises, such as the 1/f noise [13,18] and DC currents that reduces the measurement system precision. In addition, the proposed technique has an inherent control parameter, the fit to the theoretical line shape, that may help to estimate the accuracy of the measurements.

In this paper, theoretical derivation and experimental proof of concept of the suggested method are presented. In Section 2, an inclusive theoretical background, is provided. In the following sections our particular experimental setup is described followed by two sets of experimental results. The first experiment followed the conventional approach of using a long OPD which is larger than the coherence length. The second experiment used the same setup including the same laser, however it used an extremely short OPD to measure the same linewidth with the new method. Quantitative comparison analysis was performed between the two methods and is elaborated as well.

2. Theoretical analysis

2.1 PSD calculation

A laser linewidth is characterized in a coherent transmission communication system that is addressed as an interferometer operating in its sub-coherence regime. An analytical analysis of the proposed approach is provided in this section. Starting from the detected currents at the output of a balanced intradyne coherent receiver, depicted in Fig. 1. The currents generated from a single polarization state are [19,20]:

(1)$$\begin{array}{lll}i_{X,I} &=& R\sqrt{P_{sig}(t) P_{LO}} \cos(\omega_{IF}t+\phi_{sig}(t)+\phi_{sn}(t)-\phi_{LOn}(t)+\phi_{LO}), \\ i_{X,Q} &=& R\sqrt{P_{sig}(t) P_{LO}} \sin(\omega_{IF}t+\phi_{sig}(t)+\phi_{sn}(t)-\phi_{LOn}(t)+\phi_{LO}) , \end{array}$$

where $R$ is the photodiodes responsivitiy, reasonably assuming equal responsitivies for all four photodiodes (two balanced photodiodes). $P_{sig}(t)$ and $\phi _{sig}(t)$ are the modulated signal power and phase, respectively, comprising the complex information baseband equivalent signal $\sqrt {P_{sig}(t)}e^{j \phi _{sig} (t)}$, with an additive phase noise $\phi _{sn} (t)$. $P_{LO}$ is the power of a continuous wave (CW) local oscillator (LO) with a phase difference from the signal $\phi _{LO}$ that distributes uniformly over $[-\pi,\pi ]$ and a phase noise denoted by $\phi _{LOn} (t)$. In addition, $\omega _{LO}$ and $\omega _{sig}$ are the carrier angular frequencies of the modulated signal and the LO, while $\omega _{IF} = \omega _{sig}-\omega _{LO} \ll \omega _{sig}$ is a random, slowly varying frequency offset. The minus sign of $\phi _{LO}$ is due to the subtraction between the phases of the signals which creates $\omega _{IF}$ as well. [19,20]

Fig. 1. Experimental setup for laser linewidth characterization via self-homodyne measurement under nearly-coherent conditions. IC-TROSA – integrated coherent transmitter-receiver optical sub-assembly; ECL – External-Cavity Laser; DP-MZM – dual parallel Mach-Zehnder modulator; PBS – Polarizing beam splitter; SSMF – standard single mode fiber; PC – polarization controller; TIA – trans-impedance amplifiers.

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During the reconstruction of the complex information signal, a carrier phase algorithm [21,22] estimates $\phi _{LO}$ and tracks the combined phase noise $\phi _{sn}(t)-\phi _{LOn}(t)$. The DSP at the receiver is depicted in Fig. 1. Using the combined phase noise, one can calculate a phase noise induced "normalized" E-field, i.e., a synthetic quantity that has only the combined phase noise and a normalized amplitude:

(2)$$E(t)= e^{j\left[\phi_{sn}(t)-\phi_{LOn}(t)+\phi_{LO}\right]} .$$

In order to obtain the linewidth, it is desired to examine the normalized E-field PSD of Eq. (2), thus we first calculate its autocorrelation function (ACF), $G_E(\tau,\tau _0)$, where $\tau$ is the ACF correlation time parameter and $\tau _0$ is the time difference between the LO and the signal. Then, Wiener-Khinchin theorem [23] can be used to obtain the PSD out of the ACF:

(3)$$G_E(\tau,\tau_0) ={<}E(t)E(t-\tau)^{*}>{=} <e^{j\left(\phi_{sn}(t)-\phi_{LOn}(t) - \phi_{sn}(t-\tau) + \phi_{LOn}(t-\tau) \right)}> ,$$

where $<f(t)>$ denotes the ensemble averaging operator i.e. $<f(t)>=\frac {1}{N}\sum _n f_n(t)$, and where $n$ is the ensemble index out of $N$ ensembles. As a result, the constant phase difference $\phi _{LO}$ will be eliminated.

As the signal and the LO are seeded from the same source their phase noise is being derived from the same realization, shifted by a delay time of $\tau _0$, namely $\phi (t) \equiv \phi _{LOn}(t)=\phi _{sn}(t-\tau _0)$. Thus, the ACF of Eq. (3) can be rewritten as:

(4)$$G_E(\tau,\tau_0) ={<}e^{ j\left( \phi(t) - \phi(t-\tau_0) - \phi(t-\tau) + \phi(t-\tau_0-\tau) \right) }> .$$

Using the well known relation for Gaussian Random variables [24], one can show that:

(5)$$<e^{j\phi}>{=} e^{-\tfrac{1}{2}<\phi^{2}>} .$$

The variance of the phase noise is therefore the dominant term of the ACF and is explicitly provided later.

The lasers we analyse are semiconductor lasers that operate above the threshold. Under this condition, the frequency noise is a white Gaussian noise with a constant PSD $\frac {\Delta f}{\pi }$ [18]. Thereby, the phase noise is a Gaussian noise with zero mean and its variance increases linearly with time [18]

(6)$$<\Delta \phi (t,\tau)^{2}>{=}<\Delta \phi (\tau)^{2}>{=}2\pi \Delta f |{\tau}|.$$

It is further shown, in Subsection 2.2, that $2 \Delta f$ is the full width at half maximum (FWHM) of the Lorentzian spectral line shape, for long delay times [8–11,13].

Expressing the ACF as $G_E(\tau,\tau _0) = e^{B(\tau,\tau _0)}$ and using Eqs. (4), (6), 5) as well as $\Delta \phi (t,\tau ) \equiv \phi (t)-\phi (t-\tau )$, one can show that the exponential argument takes the following form

(7)$$B(\tau,\tau_0) ={-}<\Delta \phi^{2} (\tau_0)>{-}<\Delta \phi^{2} (\tau)>{+}\tfrac{1}{2}<\Delta \phi^{2} (\tau+\tau_0)>{+}\tfrac{1}{2}<\Delta \phi^{2} (\tau-\tau_0)> .$$

The full derivation of Eq. (7) is provided in Appendix Subsection A.1.

Applying Wiener–Khinchin theorem:

(8)$$S(f,\tau_0)= \mathcal{F}[G_E(\tau,\tau_0)] = \mathcal{F}[e^{B(\tau,\tau_0)}] ,$$

where $\mathcal {F}$ is the Fourier transform operator. Following the transform, the PSD takes the form [9,12–14]:

(9)$$S_{s}\left(f , \tau_0\right) = \frac{\Delta f / \pi} { \Delta f^{2} +f^{2} } \left\{1-e^{- 2 \pi \Delta f |{\tau_0}| }\left[\cos \left( 2\pi f |{\tau_0}| \right)+ \frac{ \Delta f}{f} \sin \left( 2\pi f |{\tau_0}| \right)\right] \right\} +e^{ {-}2\pi f |{\tau_0}|}\delta \left(f\right)$$

The full derivation of Eq. (9) is provided in Appendix Subsection A.2.

In the next subsections, two methods for measuring the linewidth using Eq. (9) will be discussed. The first method assumes a long delay time, more than 5 times the coherence time, and was previously shown [18,25]. The second method which is introduced here for the first time, to the best of the authors knowledge, can be used for measuring the linewidth with very short delay times (delay time of approximately 0.5% of the coherence time). The fundamentals of this method are derived here in Subsection 2.3. In addition, it is further evaluated experimentally and the results are presented in this paper in the following sections.

2.2 Long delay-time interferometry

Further simplification of Eq. (9) is obtained by assuming a long optical path difference (OPD) between the LO and the signal branches. The OPD, $\ell$, is related to the delay time by $\tau _0 = \frac {\ell }{c/n}$. In addition, let us recall the definition of the coherence time, given as the temporal difference between branches that causes $e^{-1}$ degradation in the visibility ($\gamma$) of the interferometric fringes. For a Lorentzian-shaped PSD, the visibility is related to the linewidth by $\gamma (\tau _0)= e^{- 2 \pi \Delta f |\tau _0|}$ [23]. Therefore, the coherence time and coherence length are $\tau _c = \frac {1}{2 \pi \Delta f},\ell _c = \frac {c}{n}\tau _c$, respectively, where $c$ is the speed of light in vacuum and $n$ is the effective refractive index in the fiber.

It can be shown that for $\frac {|{\tau _0}|}{\tau _c} \gg 1$, i.e. the OPD is much larger then the coherence length, the exponential term, appearing twice in Eq. (9), is negligible. Due to the fact that $\delta (f)$ term has an infinitesimally narrow bandwidth, it will not be visible on the measurement curve and therefore will not be simulated on the theoretical curve, so the PSD turns into a pure Lorentzian shape.

(10)$$S_{s} \left( f, \Delta f \right) \mathop = \limits_{{\tau _0} \gg {\tau _c}} \frac{ \Delta f }{ \pi (\Delta f^{2} +f^{2}) } .$$

In this regime the linewidth measurement is often being executed by measuring the FWHM of the Lorentzian spectral shape. A shortcoming associated with this method is the long fiber needed for decorrelating the two branches, which can get to several kilometers lengths [15], as well as the IF that distorts the PSD shape. Examining Eq. (10), and assuming typical $\mathrm \mu$ ITLA laser with 100 kHz linewidth, one can observe that long optical path differences, e.g. 25 km, yield $\tau _0=125 \mathrm{\mu s} \gg \tau _c = 1.6$ $\mathrm \mu$s. Thus, the modulation of the Lorentzian shape is no longer visible neither theoretically nor in the measurements presented under Section 4. in this paper.

Another limitation to consider addresses the spectral resolution of the system. According to Nyquist’s theorem, in order to measure the interferometric modulation pattern, a system should have a resolution of at least twice the highest frequency of the measured phenomena. By examining Eq. (9), the frequency difference between consecutive peaks equals to $1/\tau _0$, i.e. the interferometric modulation pattern frequency is $\tau _0$. Therefore, in order to measure the interferometric modulation pattern, the spectral resolution of the system, $df$, should be higher than $\frac {1}{2\tau _0}$. By using the above relation between the OPD and the delay time $\tau _0$, the OPD values that result with a detectable interferometric pattern, may satisfy:

(11)$$\ell < \frac{c}{n}\frac{1}{2df} .$$

For instance, considering the testbed used in this experimental work, a 32 GBd 16-QAM signal with a total acquisition time of 4 $\mathrm \mu$s, $df = 244$ kHz and thus the largest $\ell$ value that allows a detectable interferometric pattern is 409 m, corresponding to Eq. (11).

2.3 Short delay-time interferometry

The proposed method for measuring the linewidth using short fibers (few meters long) is presented here. From Eq. (9) one can predict that the peaks of the modulation pattern imposed on a Lorentzian pedestal will correspond the peaks of $\cos (\omega \tau _0)$ and $\sin (\omega \tau _0)$. Therefore, if the peaks are detectable in terms of SNR, visibility and system’s resolution, they will appear at frequencies $f=\frac {n}{\tau _0}$ for integer $n$. For example, considering n=1.4682 at 1550 nm, we can predict that an OPD of 15 m would generate oscillations with cycles of 13.33 MHz.

In order to estimate the linewidth of the laser using our method where the data is in the nearly-coherent regime($\tau _0 \ll \tau _c$) and with most of the spectral data far beyond the linewidth ($f\gg \Delta f$), there is a need for normalization, such that $\int _{- \infty }^{\infty } S_s(f,\tau _0)df = 1$. The normalized PSD can be rewritten as:

(12)$$S_{s}\left(f ,\tau_0\right)= \frac{\Delta f}{\pi f^{2}}\left\{1-e^{- 2 \pi \Delta f |{\tau_0}| }\cos \left( 2\pi f |{\tau_0}| \right) \right\}.$$

The extraction of the linewidth from Eq. (12) can be done as follows: first find the maxima which appear when the cosine term in Eq. (12) gets the value of -1. Second, assuming $\Delta f \tau _0 \ll 1$ (nearly-coherent assumption) and thus $e^{-2\pi \Delta f |{\tau _0}|} \approx 1$,

(13)$$S_s(f)= \frac{2\Delta f}{\pi f^{2}}.$$

Next, a linear regression is used on the PSD expression in dB scale.

(14)$$10\log_{10}(S(f))= 10\log_{10}(\frac{2}{\pi})-20\log_{10}(f) +10\log_{10}(\Delta f).$$

The regression is applied over a log-log representation, i.e. $10\log _{10}(S(f))$ vs. $\log _{10}(f)$. The coefficients are expected to fit a slope of -20 [dB/dec] and the intersection point with the y axis is $10\log _{10}(\frac {2}{\pi })+10\log _{10}(\Delta f)$. The linewidth is extracted from the estimated intersection point. A successful application of this method requires identifying the appropriate frequency range. At low frequencies, the $1/f$ noise dominants, thus the assumptions used in Eq. (12) no longer holds. On the other hand, at high frequencies, a relaxation oscillation noise may dominate [13,18,25]. In both cases, the shape of the log-log spectra will deviate from an ideal, white FN-noise dominated, linear form. Furthermore, there are practical limitations imposed by the receiver’s internal electrical noise floor, therefore using low signals may lead to unreliable parameter estimation. A practical method to verify that the correct range of operation was selected, and to validate the integrity of the results, is by ensuring that the coefficient’s slope is as close as possible to -20 [$dB/dec$].

3. Experimental setup

The optical module used to demonstrate the proposed technique is an Integrated Coherent Transmitter & Receiver Optical Sub-Assembly (IC-TROSA). Such modules are highly integrated and enable small form-factor Digital Coherent Optics (DCO) transceivers in a QSFP-DD or OSFP form factors, with 400-ZR transmission format. As discussed above, the laser source of the IC-TROSA is being characterized while other subsystems in the module are used as a reference system: the coherent transmitter unit (IQ -modulator and drivers), and coherent receiver (90°hybrids), balanced photodiodes and trans-impedance amplifiers (TIAs).

In the transmitter side, offline DSP generates 16-QAM, 32 GBaud random data stream, over-sampled by a factor of 2 samples-per-symbol, and then transmitted by a 64 Gsamp/s 8-bit digital-to-analog converter (DAC). Following, the two RF streams (in-phase and quadrature) drive an IQ-MZM which is biased at a Null, Null, Quad setpoint, for both polarization states. However, the Y-polarization inputs electrodes were terminated, and no RF was injected to the Y-polarization. The IC-TROSA is configured at optical loop-back, which simply means that its transmitter is connected to its receiver via varying lengths of single-mode fiber. To adjust the polarization state of the signal and the LO, a polarization controller is inserted inline.

At the receiver side, the signal is photodetected following a mixing with the LO, and amplified with a TIA. The TIA is configured to a manual gain control (MGC) mode, so that low frequency beating (as a result of the phase noise) will not be suppressed by the high-pass filter response of the automatic gain control loop. Subsequently, the signal is digitized using 64 Gsamp/s 8-bit analog-to-digital converter (ADC). In turn, the modem algorithms, which are depict on the right-hand side of Fig. 1 are activated on the sampled signal, for bit recovery. $\phi (t)$, which relates to the E-field phase in Eq. (2), is extracted by the carrier phase estimation (CPE) with a phase agnostic algorithms such as CMA and RDE [21,22]. Following, the E-field PSD is calculated for $e^{j \phi (t)}$ using Welch’s method and Hann window. Eventually, the obtained experimental PSD is expected to behave like the PSD of Eq. (9).

The proposed method was verified by comparing the calculated linewidth with the one obtained by what is referred as the "traditional" method [18,25]. Considering that the ECL of the IC-TROSA has a linewidth of 100 kHz and corresponding coherence length of $l_c\approx$ 12.5 km, we choose OPDs of 25 km $> l_c$ to decorrelate the signal and the LO. In addition, three different OPDs were used to set near-coherent conditions: 15 m, 5 m and 60 cm $\ll l_c$. Both PSDs of the long and short OPDs cases are further analysed in the following section.

4. Results

4.1 Long delay-time results

In this subsection, measurements with delay time in the order of the coherence time, or longer, are analyzed. The resultant linewidth will be used as a reference, to be compared with the linewidth extracted by the proposed method (the short delay-time interferometry). Long delay-time interferometry using the "traditional" method, estimates the white FM noise PSD [18,25]. Note that this method is valid for long enough delay time, up to one order of magnitude below the coherent length, otherwise the interferomtric pattern appears and prevents estimating white reference noise. The direct inference of the linewidth from the FM spectrum, is the fundamental of the "traditional" methods, while it is not satisfied in short OPDs, as one can see in Fig. 2(b). In theory, the linewidth can be easily extracted using the fact that the FM-noise PSD is constant and equals to $\frac {\Delta f}{\pi }$ as long as the signal is uncorrelated enough with the LO (up to one order of magnitude below the coherence length) [18], however careful attention should be paid for choosing the right frequency band.

Fig. 2. FM noise PSD, obtained for 1.5 km, 25 km and 3 m OPDs. Dashed horizontal line indicates the estimated white reference noise, stands for the noise of the perfect Lorentzian shaped electrical field PSD. (a) Displays the PSD of the frequency noise of 25 km and 1.5 km. (b) Displays the PSD of the frequency noise of 25 km and 3 m.

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The FM noise spectrum of an ECL stems from different phenomena, where each phenomena contributes a different spectral component [25] mainly low-pass in nature , such as the 1/f noise [13,18,25]. Additionally, in the high frequency regime, the relaxation oscillation and the receiver additive white Gaussian (AWGN) noises dominate. The spontaneous emission white noise dictates the fundamental laser linewidth and was assumed as the dominate noise of the PSD examined. It was shown that low frequency noise may broaden the linewidth when measuring the FM-noise PSD at low frequencies and the relaxation oscillation noise will be more significant than the white frequency noise at the regime of few GHz [13,18]. Therefore measuring the noise floor at the range of hundreds of MHz range is chosen for most accurate linewidth estimation. As shown In Fig. 2 The measured linewidth by this method is $\Delta f = 23650 \cdot \pi = 74.3$ kHz. To conclude, according to the IC-TROSA product specifications [26] the maximal linewidth of the laser is 300 kHz, under EOL and worst-case conditions. It is assumed that the laser we have tested has fresh performances and thus introduces much lower linewidth, and thus our measurements is reasonable.

4.2 Results for short delay-time interferometry

In the described short delay-time interferometry analysis, two different parameters are extracted: delay time (related to the OPD) and linewidth. Each estimation is done separately and independently. Furthermore, we devise a criteria for the accuracy of the estimation, so one would be able to rank the reliability of each measurements. Three sets of experimental measurements are presented in this paper, each taken for a different OPD. Figures 3, 4 and 5 depict the PSD obtained for OPD of 14.5m 5m and 60cm, respectively. The OPD was verified by measuring the frequency of the interferomtric pattern, while the linewidth was fitted from the log-log envelope. The supportive theory is provided in Section. 2.3. We used two representation types to highlight different aspects of the PSD behaviour. In subfigures. 3(a), 4(a) and 5(a) the PSD is plotted over a linear frequency axis, while in subfigures. 3(b), 4(b) and 5(b) a frequency logarithmic scale is used.

Fig. 3. The measured PSD is plotted in a dotted red line, and its theoretical value, provided by Eq. (9), is plotted in a continuous blue line. The local peaks of the measured PSD, marked with blue asterisks, are then used to estimated the Loernzian envelope (yellow dashed line). The estimated OPD is 14.5 m, and the theoretical PSD has a linewidth of 66 kHz. (a) plot of PSD on a linear frequency axis. (b) plot of PSD on a log frequency axis, hence only positive frequency range is presented.

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Fig. 4. The measured PSD and the theoretical values (Eq. 9). The estimated OPD is 5 m, and the theoretical PSD has a linewidth of 66 kHz. (a) plot of PSD on a linear frequency axis. (b) plot of PSD on a log frequency axis, hence only positive frequency range is presented.

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Fig. 5. The measured PSD and the theoretical values (Eq. 9). The estimated OPD is 0.6 m, and the theoretical PSD has a linewidth of 66 kHz. (a) plot of PSD on a linear frequency axis. (b) plot of PSD on a log frequency axis, hence only positive frequency range is presented.

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Examining the measured data in Figs. 3, 4 and 5, one can observe a sinusoidal pattern with a Lorentzian envelope. Such behaviour is well predicted by Eq. (12). In these figures, continues red line denotes the measures PSD, and its peaks are calculated by basic peak search algorithm and plotted with blue asterisk markers. As the OPD gets shorter, the frequency of the interferometric patterns increases. For example, the peaks of Fig. 3 correspond $\tau _0 = 1/13.8$ MHz which sets an OPD of 14.5m, while in Fig. 5 $\tau _0 = 1/330$ MHz results with 60cm OPD. The curve that defines the envelope of the Lorentzian shape (yellow line) is a linear line drawn 3 dB below the peaks, when plotted in log-log format. Figure 3 and 4 show a good agreement between the theoretical curve to the measured one, especially within the frequency range used for the calculation of the linewidth. However, at some other frequency range, a poor agreement with the theoretical expectations is observed.

One can see that in Figs. 3, 4 and 5 the expected minima are rather far below the measured ones. This indicates that the SNR for measuring the minima is insufficient and the contrast between maxima and minima levels is unresolvable [12]. In addition, in Fig. 5 there is only a poor agreement between the theoretical curve and the measured data. An explanation for this discrepancy is the extreme short delay time used, less than 0.5% of the coherence time. When using a short OPDs, the frequency of the interferometric pattern is relatively high, for example 60 cm of OPD inflicts the appearance of the first peak at 333 Mhz. This high frequency range is more difficult to estimate as being dominated by high frequency "non-white" noise sources, as explained in the previous subsection. In the following section, it is shown that multiple short OPD measurements may be beneficially incorporated in order to yield an overall improved linewidth estimation accuracy.

Since the three measurements sets were derived from the same ECL source, information from all of them can be incorporated. The linewidth was estimated on each graph from the negative and the positive parts of the frequency axis separately. As previously mentioned, the processed frequency range was chosen manually to optimize the slope of Eq. (14). The manual choice maximizes the control parameter W which is defined in Eq. (15) Visualizing that, the peaks that were chosen for the analysis are depicted in Figs. 3, 4 and 5 with blue asterisk markers.

Weighted averaging was conducted to estimate the laser linewidth from all the measurements while using the measured slope as a weight metric. The weighting metric was defined as the mean square error (MSE in logarithmic scale) between the -20dB/dec and the actual measured slopes:

(15)$$W = \frac{1}{({-}20-S)^{2}},$$

where $W$ is the weight used in the averaging and $S$ is the measured slope.The parameters that were selected for the optimization process are summarized in Table 1, leading to a weighted averaged linewidth estimation of 66 kHz.

Table 1. The parameters used and the results for each estimation. The upper three rows results from the positive frequency axis and the lower three rows derived from the negative. The weight is proportional to the reciprocal of the square distance from the desired slope, and defined by Eq. (15)

View Table

5. Conclusions

The paper introduces a novel method for estimating the laser linewidth within a coherent communication system. No additional equipment is needed rather than a conventional coherent optical front-end and a basic coherent DSP engine. Such system is addressed as an interferometer operating in nearly-coherent regime, and the associated theoretical analysis is provided. It was demonstrated that the proposed method yields very similar linewidth estimation as of the conventional method, even though based on dramatically shorter fiber delay lines. This scheme enables linewidth measurement of operational coherent products under self-test mode.

Appendix

A.1. Developing the exponential argument of the ACF

In this part the transition from Eq. (4) to Eq. (7) will be explained. First, applying Eq. (5) on Eq. (4) will return

(16)$$G_E(\tau,\tau_0) ={<}e^{ -\tfrac{1}{2} \left( \phi(t) - \phi(t-\tau_0) - \phi(t-\tau) + \phi(t-\tau_0-\tau) \right)^{2} }>,$$

defining the exponent of Eq. (16) as $B(\tau,\tau _0)$ and simplifying it will take the form:

(17)$$\begin{aligned} B(\tau,\tau_0)=& -\tfrac{1}{2} \bigl[ \phi(t)^{2}+\phi(t-\tau_0)^{2}+ \phi(t-\tau)^{2}+\phi(t-\tau-\tau_0)^{2} -2\phi(t)\phi(t-\tau_0)\\ &-2\phi(t)\phi(t-\tau) +2\phi(t)\phi(t-\tau_0-\tau) +2\phi(t-\tau_0)\phi(t-\tau)\\ &-2\phi(t-\tau_0)\phi(t-\tau_0-\tau) -2\phi(t-\tau)\phi(t-\tau_0-\tau) \bigr] . \end{aligned}$$

Examining Eq. (6), we conclude that it is benefit to work with only complete squares of differences because there is a closed-form expression for their expectation value. Therefore completing the squares of all the cross terms by adding and subtracting: $\phi (t)^{2}, \phi (t-\tau _0)^{2},\phi (t-\tau )^{2},\phi (t-\tau -\tau _0)^{2}$ and rearranging the order will be:

(18)$$\begin{aligned} B(\tau,\tau_0)={-}\tfrac{1}{2}\{ &+\phi(t)^{2} -2\phi(t)\phi(t-\tau_0)+\phi(t-\tau_0)^{2}\\ &+\phi(t)^{2}-2\phi(t)\phi(t-\tau)+\phi(t-\tau)^{2}\\ &-\phi(t)^{2}+2\phi(t)\phi(t-\tau_0-\tau)-\phi(t-\tau_0-\tau)^{2}\\ &-\phi(t-\tau_0)^{2}+2\phi(t-\tau_0)\phi(t-\tau)-\phi(t-\tau)^{2}\\ &+\phi(t-\tau_0)^{2}-2\phi(t-\tau_0)\phi(t-\tau_0-\tau)+\phi(t-\tau_0-\tau)^{2}\\ &+\phi(t-\tau-\tau_0)^{2}-2\phi(t-\tau_0-\tau)\phi(t-\tau) +\phi(t-\tau)^{2} \} \end{aligned}$$

Using simple binomial formula will return:

(19)$${$\begin{aligned} B(\tau,\tau_0)={-}\tfrac{1}{2}\{ & +[\phi(t)-\phi(t-\tau_0)]^{2} +[\phi(t)-\phi(t-\tau)]^{2} -[\phi(t)-\phi(t-\tau_0-\tau)]^{2} \\ & -[\phi(t-\tau_0)-\phi(t-\tau)]^{2} +[\phi(t-\tau_0)-\phi(t-\tau_0-\tau)]^{2} +[\phi(t-\tau_0-\tau)-\phi(t-\tau)]^{2}\} \end{aligned}$}$$

Using Eq. (6) we will get:

(20)$$B(\tau,\tau_0) ={-}\tfrac{1}{2} \{ +\Delta\phi(\tau_0)^{2} +\Delta\phi(\tau)^{2} -\Delta\phi(\tau_0+\tau)^{2} -\Delta\phi(\tau_0-\tau)^{2} +\Delta\phi(\tau)^{2} +\Delta\phi(\tau_0)^{2} \}$$

And with basic simplifications Eq. (19) will become Eq. (7).

A.2. Derivation of the PSD

The analytic Fourier integral that transforms Eq. (8) to Eq. (9) is provided here. Once applying Eq. (6) on Eq. (7), the ACF becomes

(21)$${G_E}(\tau ,{\tau _0}) = \exp \textrm{ }\left\{ {\begin{array}{ll} { - 2\pi \mathrm{\Delta }f|\tau |}&{\textrm{for}\;|\tau | \lt {\tau _0}}\\ { - 2\pi \mathrm{\Delta }f|{\tau _0}|}&{\textrm{for}\;|\tau | \gt {\tau _0},} \end{array}} \right.$$

which can be rewritten with rectangular functions as

(22)$$G_E(\tau,\tau_0) = e^{{-}2\pi\Delta f|{\tau}|}\Pi\left(\tfrac{\tau}{\tau_0}\right)+ e^{{-}2\pi \Delta f|{\tau_0}|}\left[1-\Pi\left(\tfrac{\tau}{\tau_0}\right)\right],$$

where

(23)$$\mathrm{\Pi }(x) = \textrm{ }\left\{ {\begin{array}{ll} 1&{\textrm{for}\;|x| \lt 1}\\ 0&{\textrm{for}\;|x| \gt 1}\\ {{\textstyle{1 \over 2}}}&{\textrm{for}\;|x| = 1.} \end{array}} \right.$$

Noting that $\tau _0$ is a constant, and applying Wiener–Khinchin theorem on the ACF using the known Fourier relation [27] of $\Pi (x)$, the PSD becomes

(24)$$S_s(f,\tau_0) = \mathcal{F}\left[ e^{{-}2\pi\Delta f|{\tau}|}\Pi\left(\tfrac{\tau}{\tau_0}\right) \right] + e^{{-}2 \pi \Delta f|{\tau_0}|}\left( \delta(f) - \frac{\sin(2\pi \tau_0 f)}{\pi f} \right).$$

The second term will be calculated from the definition of the Fourier transform, and the definition of $\Pi (x)$ from Eq. (23)

(25)$$\mathcal{F}\left[ e^{{-}2\pi\Delta f|{\tau}|}\Pi\left(\tfrac{\tau}{\tau_0}\right) \right] = \int_{-\infty}^{\infty}e^{{-}2\pi \Delta f|{\tau}|}\Pi\left(\tfrac{\tau}{\tau_0}\right) e^{{-}2\pi f \tau} d\tau = \int_{-\tau_0}^{\tau_0}e^{{-}2\pi\Delta f|{\tau}|} e^{{-}2\pi f \tau} d\tau .$$

Breaking the integral into the negative and positive time axis

(26)$$\int^{\tau_0}_{0} e^{{-}2 \pi \tau (\Delta f+ jf)}d\tau + \int^{0}_{\tau_0} e^{{-}2 \pi \tau (jf- \Delta f)}d\tau = \left[ \frac{ 1-e^{{-}2\pi \tau_0(\Delta f+jf)} } { 2 \pi \left( jf + \Delta f \right) } + \frac{ e^{2\pi \tau_0(jf-\Delta f) }-1} { 2 \pi \left( jf-\Delta f \right) } \right] .$$

Rearranging the last expression yields

(27)$$\begin{aligned}&\frac{1}{2\pi} \frac{[1-e^{{-}2\pi \tau_0(\Delta f+jf)}](\Delta f-jf) + [e^{2\pi \tau_0(jf-\Delta f)}-1]({-}jf-\Delta f)}{f^{2}+\Delta f^{2}} = \\ &\left\{ \frac{2\Delta f + e^{{-}2\pi \tau_0 \Delta f} }{2\pi (f^{2}+\Delta f^{2})} \left[jf\left(e^{{-}2\pi \tau_0 jf}-e^{2\pi \tau_0 jf}\right)-\Delta f\left(e^{{-}2\pi \tau_0 jf}+e^{2\pi \tau_0 jf}\right) \right] \right\} =\\ &\frac{\Delta f}{\pi (f^{2}+\Delta f^{2})} \left\{ 1+e^{{-}2\pi \tau_0 \Delta f} \left[\frac{f}{\Delta f} \sin(2\pi f \tau_0)-\cos(2\pi f \tau_0) \right] \right\}. \end{aligned}$$

On the other hand, the second term can be rewritten as

(28)$$\frac{\sin(2\pi \tau_0 f)}{\pi f} = \frac{\Delta f}{\pi(\Delta f^{2}+f^{2})} \left(\frac{f}{\Delta f}+\frac{\Delta f}{f} \right) \sin(2\pi \tau_0 f).$$

Substituting the above into Eq. (22), results with

(29)$$\begin{aligned}S_s(f,\tau_0) & = e^{{-}2 \pi \Delta f|{\tau_0}|} \delta(f) - e^{{-}2 \pi \Delta f|{\tau_0}|} \frac{\Delta f}{\pi(\Delta f^{2}+f^{2})} \left(\frac{f}{\Delta f}+\frac{\Delta f}{f} \right) \sin(2\pi \tau_0 f)\\ & + \frac{\Delta f}{\pi (f^{2}+\Delta f^{2})} \left\{ 1+e^{{-}2\pi \tau_0 \Delta f} \left[\frac{f}{\Delta f} \sin(2\pi f \tau_0)-\cos(2\pi f \tau_0) \right] \right\}\\ & = \frac{\Delta f / \pi} { \Delta f^{2} +f^{2} } \left\{1-e^{- 2 \pi \Delta f |{\tau_0}| }\left[\cos \left( 2\pi f |{\tau_0}| \right)+ \frac{ \Delta f}{f} \sin \left( 2\pi f |{\tau_0}| \right)\right] \right\}\\ & +e^{ {-}2\pi f |{\tau_0}|}\delta \left(f\right) , \end{aligned}$$

as shown in Eq. (9).

Funding

Office of the Chief Scientist, Ministry of Economy (69284).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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OPD [m]	Meas. slope [dB/dec]	Weight	Freq. range [GHz]	Meas. LW [kHz]
14.5	-19.995	37501	0.05 to 0.74	66
5	-20.13	56	0.13 to 0.43	82
0.60	-20.22	20	1.6 to 5	39
14.5	-20.04	508	-0.6 to -0.05	71
5	-20.03	1084	-0.5 to -0.15	63
0.60	-20.2	23	-5 to -1.6	37

Laser linewidth characterization via self-homodyne measurement under nearly-coherent conditions

Abstract

1. Introduction

2. Theoretical analysis

2.1 PSD calculation

2.2 Long delay-time interferometry

2.3 Short delay-time interferometry

3. Experimental setup

4. Results

4.1 Long delay-time results

4.2 Results for short delay-time interferometry

5. Conclusions

Appendix

A.1. Developing the exponential argument of the ACF

A.2. Derivation of the PSD

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Tables (1)

Equations (29)

Optics Express