1. Introduction

Optical frequency-modulated continuous wave (FMCW) technique, thanks to its high sensitivity and high spatial resolution, is of great interest in many applications such as ranging, characterizing fiber-optic component, fiber sensing, optical coherent tomography (OCT), spectroscopy, 3-dimensional imaging, etc [1–6 ]. As the laser intrinsic frequency noise is the most important factor deteriorating the performance of optical FMCW on the aspect of range window and spatial resolution, many methods have been proposed to suppress it [7–9 ]. In these works frequency fluctuation and a static frequency-noise spectrum have been used for characterizing the laser frequency noise, however the frequency-noise characteristics of frequency-swept laser source are often not constant, since which a dynamic frequency-noise characterization is required.

Laser frequency-noise characteristics are usually evaluated in two ways: laser line shape or its linewidth, and frequency-noise spectrum, also known as power spectral density (PSD). The latter one can provide more information on the frequency domain noise distribution than the former one, which can be exactly derived from the latter one [10]. A popular method to measure linewidth is delayed self-heterodyne/homodyne (DSH) technique [11–16 ], and flicker noise's influence on broadening linewidth was also studied [17]. A digital method with coherent phase modulation detection is proposed to measure linewidth and frequency-noise PSD [18], yet adoption of a long fiber with length over many times of laser coherent length limits the measurable laser linewidth, and introduces extra noise from fiber perturbation. An unbalance Mach-Zehnder interferometer with short delay is used to discriminate the frequency noise so as to measure frequency-noise PSD [9], however the frequency range of frequency-noise PSD is limited under the reciprocal of delay time. To characterize an ultranarrow-linewidth laser, CO₂ gas is utilized as frequency-to-amplitude converter to measure frequency-noise PSD and intrinsic linewidth [19]. For dynamic linewidth measurement, a quadrature front end method is proposed [20], which still adopts a long fiber. A digital intradyne coherent receiver method is also exploited to characterize the laser dynamic linewidth [21], yet the employment of multiple reference laser increases the scheme's complexity.

In this paper, we propose a short-delayed self-heterodyne (SDSH) method to measure a laser's large-frequency-range frequency-noise spectrum over 10 Hz to 50 MHz with only 15.5 meters fiber which can reduce the impact of fiber perturbations. We verify this method by measuring a static distribution feedback (DFB) laser driven at different currents. Furthermore, we propose an averaging method to extract the frequency-swept laser's intrinsic frequency noise, then apply these two techniques into the dynamic frequency-noise characterization of a frequency-swept DFB laser driven by a sawtooth-modulated current, which enable us to obtain various information about frequency-noise spectrum variation during frequency sweeping. The dynamic frequency-noise spectrum of a servo-controlled frequency-swept DFB laser is also investigated in order to acquire a deeper insight of phase-locking process. The proposed method is capable to provide more information of laser frequency stability than conventional methods.

2. Principle

2.1 Short-delayed self-heterodyne method to restore frequency-noise spectrum

The SDSH setup is presented in Fig. 1 . The output light of laser under test is split into two branches with a polarization maintaining coupler. One of them passes through a fiber delay of ${\tau }_0$, and the other is modulated by an acoustic-optic frequency shifter (AOFS) with frequency of ${\omega }_0$ = 1 MHz to avoid the low-frequency noise of photo-detectors which is under 200 kHz. Two branches are together sent into an optical 2 × 4 90 degree hybrid, then the hybrid outputs are detected by two balanced photo-detectors (BPD). For the sake of avoiding polarization issues, all optical components and fibers here are polarization maintaining. Subsequently, the BPDs' output electrical signals are sampled with a data acquisition card (DAQ) for digital process.

 

Fig. 1 The short-delayed self-heterodyne setup. AOFS: acoustic-optic frequency shifter; BPD: balanced photo-detector; DAQ: data acquisition card.

Download Full Size | PDF

For a static laser, which refers to a non-frequency-swept laser, denote the laser's light signal as:

However, $T(f)$ has zero points at $f_k=k/{\tau }_0$,$k=1,2,\mathrm{3...}$, while $S_{\Delta \phi }(f)$ is not zero at these frequencies $f_k$ because of measurement system's noise floor, which means the actual $S_{\Delta \phi }(f)$ is like:

As known, basically the high frequency components of $S_{\Delta \nu }(f)$ are in the form of white frequency noise with level $h_a$ [23]. If the first zero point of $T(f)$, i.e., $1/{\tau }_0$ reaches this region, we can modify the transform function by adding a compensation increment $D$ onto $T(f)$. To restore the original frequency-noise spectrum, the following equation must be satisfied:

Firstly, look into the high frequency components of $S_{\Delta \nu }(f)$, and we have $S_{\Delta \phi }(f)\approx h_a$, thus:

Next, let's look into the low frequency components. We have $S_{\Delta \phi }\text{'}(f)>>N$and $T(f)>>D$(because all the zero points of $T(f)$ are in the high frequency components), thus:

The Eq. (7) remains satisfied. So in the whole Fourier frequency range, the frequency-noise spectrum can be restored using this modified transform function. According to the discussion above, if we select a ${\tau }_0$ short enough to keep $1/{\tau }_0$ reaching the white-frequency-noise region, the restored $S_{\Delta \nu }(f)$ can be simply given as:

In practice, the optimal value of $D$can be found using multiple iterations to achieve a minimum distortion of ${\tilde{S}}_{\Delta \nu }(f)$. A simulated typical frequency-noise PSD restoration process using SDSH method is illustrated in Fig. 2 . The original frequency-noise spectrum is $S_{\Delta \nu }(f)=\frac{{10}^{12}}{f}+{10}^7$(Hz²/Hz). We used two different delays to measure the SDSH phase-noise spectrum. One is 100 ns, and the other is 200 ns. The noise floor is assumed to be $N={10}^{-8}$(rad²/Hz).

 

Fig. 2 A simulated frequency-noise PSD restoration process. (a) The laser original frequency-noise spectrum $S_{\Delta \nu }(f)$; (b) The SDSH phase noise spectrum$S_{\Delta \phi }(f)$ with different delays. The purple curve is result with ${\tau }_0$ = 100 ns and the blue one is 200 ns. The dashed line is noise floor; (c) Transform function $T(f)$ plus a compensation increment $D={10}^{-15}$ with different delays. The dashed line is $D$; (d) Laser frequency-noise spectrum restored as $\frac{S_{\Delta \phi }(f)}{T(f)+D}$.

Download Full Size | PDF

One can observe from Fig. 2(b) that the longer ${\tau }_0$ is, the larger phase-noise spectrum can be obtained, which provides a higher signal-to-noise ratio. But meanwhile, in Fig. 2(d), the result of 200 ns delay shows a large distortion at 5 MHz, which is because that not all the zero points of $T(f)$ are in the white-frequency-noise range. Thus the shorter delay is, the larger frequency range we get in which frequency-noise spectrum can be restored without distortion, which means a trade-off has to be made. In our experiment system, the optimal length of delay fiber is 10~20 m.

2.2 Averaging method to extract the intrinsic laser frequency noise of a frequency-swept laser source

The frequency-swept laser source's output light can be expressed as:

Here the $\psi (t)-\psi (t-{\tau }_0)$ is a deterministic error, while $\Delta \phi (t)$ is a stationary stochastic error [24]. If ${\phi }_b(t)$is measured for many times and averaged, what remains is $\overline{{\phi }_b(t)}=2\pi \mu {\tau }_{0}t+\psi (t)-\psi (t-{\tau }_0)$(assuming the expectation of $\Delta \phi (t)$ is zero), thus the intrinsic laser phase noise is extracted as $\Delta \phi (t)={\phi }_b(t)-\overline{{\phi }_b(t)}$. Using Eq. (10), the intrinsic frequency-noise PSD is subsequently restored.

3. Experiment and result analysis

3.1 Static laser frequency-noise spectrum measurement

Firstly we are about to verify the SDSH method by measuring a static laser's frequency-noise spectrum, A commercial available DFB laser driven at different currents is taken as laser source, and a 15.5 m delay fiber is utilized in SDSH. For each driving current, 20 times SDSH measurement data for averaging spectrum is collected. In each measurement, 100 ms data was sampled by a 14-bit DAQ at sampling rate of 100 MS/s. Taking 80 mA for instance, the frequency-noise PSD restoration process is illustrated in Fig. 3 . The spikes in the results are mainly due to the receiver's noise.

 

Fig. 3 Experimental frequency-noise PSD restoration process for a DFB laser driven at 80 mA. (a) The SDSH phase noise spectrum$S_{\Delta \phi }(f)$ with${\tau }_0$ = 77.5 ns; (b) Transform function $T(f)$plus a compensation increment$D=1.19\times {10}^{-15}$; (c) The restored frequency-noise spectrum$S_{\Delta \nu }(f)$.

Download Full Size | PDF

In this process, $D$ is selected as 1.19 × 10⁻¹⁵ Hz⁻², to minimize the distortion of frequency-noise spectrum. One can see the result exhibits the laser frequency-noise PSD over 10 Hz to 50 MHz, which is actually limited by the DAQ's sampling rate and maximum sample number. In the high frequency part (>5 MHz), frequency-noise PSD presents a good similarity with Gaussian white noise distribution, and the noise level is ~10⁵ Hz²/Hz. The residual fluctuation may ascribe to the nonuniformity of noise floor.

For comparing, an experiment result was obtained using long-delayed DSH method described in [17] where we adopted a 20 km fiber delay, i.e. ~0.1 ms. Therefore, the measurement time is limited under 0.1 ms to maintain the non-coherence, which means the lowest observable Fourier frequency of this method is 10 kHz. We also calculate an approximately restored $S_{\Delta \nu }(f)$ given by $S_{\Delta \phi }(f)/4{\pi }^2{\tau }_0{}^2$ which was used in [9], and present these results in Fig. 4 . One can see the high frequency part (>100 kHz) of long-delayed DSH method's result and the low frequency part (<1 MHz) of $S_{\Delta \phi }(f)/4{\pi }^2{\tau }_0{}^2$ both agree well with corresponding part of the result of SDSH method, respectively, which verified the SDSH method is able to restore the frequency-noise spectrum in a large frequency range with little distortion.

 

Fig. 4 Comparison between different methods to measure frequency-noise PSD.

Download Full Size | PDF

By SDSH method we measured DFB laser's frequency-noise PSD under the driving currents of 80 mA, 120 mA, 160 mA, 200 mA, respectively, which are shown in Fig. 5 . From Fig. 5, it is clear that the Gaussian white noise part of frequency-noise spectrum is lowering along with the increasing current which causes an increasing laser output power, according with Schawlow-Townes-Henry theory [25]. Meanwhile due to the thermal effect of driving current, the working temperature is also going up as current increases, which produces a proportionally rising flicker noise at the low Fourier frequency.

 

Fig. 5 The static laser's frequency-noise PSD when driving current is 80 mA, 120 mA, 160 mA, 200 mA, respectively.

Download Full Size | PDF

3.2 Dynamic frequency-noise spectrum measurement of frequency-swept laser

We exploited the same DFB laser, which was injected a current sweeping like sawtooth ranging from 80 mA to 200 mA in 100 ms, as a frequency-swept laser source, whose frequency range was over 60 GHz. For eliminating the phase error introduced by frequency-swept nonlinearity, and also for averaging spectrum, 30 times SDSH measurement was made. Figure 6 shows how to extract the laser intrinsic phase noise using method described in 2.2.

 

Fig. 6 The process to extract the laser intrinsic phase noise. (a) One time SDSH phase error. (b) 30 times average of SDSH phase error. (c) Phase error in (a) minus that in (b), and what remains is intrinsic phase noise.

Download Full Size | PDF

Now we can utilize the intrinsic phase noise in Fig. 6(c) to restore the frequency-noise spectrum. But to obtain dynamic frequency-noise spectrum, we have to divide the phase noise into many equal-length time bands. In each band, the spectrum is calculated and averaged respectively. This process is illustrated in Fig. 7 . Since the time band length determines the lowest observable Fourier frequency in frequency-noise spectrum with an inverse proportional relation, the band length is set to 0.5 ms, corresponding to lower limit of 2 kHz, and therefore the band number is 200.

 

Fig. 7 The process to obtain a dynamic frequency-noise spectrum.

Download Full Size | PDF

After frequency-noise spectrums in all time bands were restored, we assembled them together into a 3-dimension form with two variables, i. e., Fourier frequency and time, which is presented in Fig. 8 . As we can see, when the driving current increases, the high frequency part of frequency-noise PSD is lowering and the low frequency part is rising, which is as same as the variation of static laser's frequency-noise PSD.

 

Fig. 8 A dynamic frequency-noise spectrum of a frequency-swept DFB laser.

Download Full Size | PDF

We compare the frequency-noise PSD of frequency-swept laser at the driving currents of 80 mA, 120 mA, 160 mA and 200 mA with that of static laser at the same driving currents, respectively (Fig. 9 ), and they agree well, which means the DFB laser's intrinsic frequency-noise PSD is barely affected by frequency sweeping. We also depict the evolution of Gaussian white frequency noise's level with respect to the laser output power during frequency sweeping in Fig. 10 , which confirms the result conforming to Schawlow–Townes–Henry theory that they are inverse proportional to each other.

 

Fig. 9 Comparison between dynamic frequency-noise PSD and static frequency-noise PSD driven at same current. (a) 80 mA; (b) 120 mA; (c) 160 mA; (d) 200 mA. The blue curve is static frequency-noise PSD, while the purple one is dynamic frequency-noise PSD.

Download Full Size | PDF

 

Fig. 10 The evolution of Gaussian white frequency noise level in dynamic frequency-noise PSD along with laser output power. The dash line is Schawlow–Townes–Henry theory fit as $\text{2}.\text{267}\times \text{1}{0}^{\text{8}}{\text{(Hz}}^2\text{/Hz}\cdot \text{W)}/\text{Power}$

Download Full Size | PDF

3.3 Dynamic frequency-noise spectrum measurement of a servo-controlled frequency-swept laser

For further application, we adopted a homemade self-locking servo loop to control the frequency-swept laser source. The servo loop setup is shown as Fig. 11 . We discriminated the frequency error using another delayed self-heterodyne interferometry and fed back the error signal into the current driver together with a sawtooth-sweeping signal, thus a phase-locked frequency-swept laser source was obtained. Next, as same process as in Fig. 7, we measured the dynamic frequency-noise spectrum of this laser during frequency sweeping, which is illustrated in Fig. 12 .

 

Fig. 11 Servo loop setup to control the frequency-swept laser.

Download Full Size | PDF

 

Fig. 12 A dynamic frequency-noise spectrum of a servo-controlled frequency-swept DFB laser.

Download Full Size | PDF

One can observe that in the beginning 7 ms of sweeping, the servo loop is losing lock, ascribing to the sudden change of frequency chirp rate. After that, the low frequency frequency noise in the loop bandwidth $B_W$ (1 MHz) has been considerably reduced, while the high frequency frequency noise out of the $B_W$ is barely changed. The comparisons between free running and servo-controlled dynamic frequency-noise PSD at the different time is presented in Fig. 13 . One can see that the protuberance at 1 MHz is rising along with time. This phenomenon can be explained as that when laser's output power is increasing, the loop gain is also rising, thus the frequency noise components under $B_W$ is lowered while the protuberance at $B_W$ is going up.

 

Fig. 13 The dynamic frequency-noise spectrum of free running and servo-controlled frequency-swept laser at specified time. (a) 7.5 ms, (b) 50 ms, (c) 100 ms.

Download Full Size | PDF

Using the power-area method [26], we can also estimate the dynamic linewidth (under the observing time of 0.5 ms) of frequency-swept laser from the dynamic frequency-noise spectrum. The results in situation of free running and servo-controlled are shown in Fig. 14 . It reveals that while free running, the laser linewidth is going up quickly from 810 kHz to 1.4 MHz because of the rising flicker noise, yet when servo-controlled (after the locking time), the flicker noise is suppressed thus laser linewidth tends to be stable around 500 kHz.

 

Fig. 14 A dynamic linewidth of frequency-swept under the situation of free running and servo-controlled.

Download Full Size | PDF

4. Conclusion

In this work, a SDSH method with simple scheme is proposed and demonstrated to restore a large-frequency-range frequency-noise spectrum noise over 10 Hz~50 MHz with little distortion. Only 15.5 m length fiber is utilized in SDSH setup which keeps the system away from the environment perturbation introduced by long fibers, and achieves the compromise between the restored frequency-noise spectrum's distortionless range and its precision. With this method, we successfully measured the frequency-noise spectrum of static DFB laser driven at different currents, and found that the white frequency noise part is lowering and flicker frequency noise part is rising when current increases.

Moreover, we proposed an averaging approach to extract intrinsic frequency noise of a frequency-swept laser. Combining these two techniques, we managed to figure out the dynamic intrinsic frequency-noise spectrum of frequency-swept DFB laser and also that with servo-loop controlled. 3-dimension measurement results with two variables i.e. Fourier frequency and time are exhibited. Compared to other results in previous works, our results are capable to reveal more information, such as the frequency-noise spectrum variation along with time on a specified Fourier frequency.

Besides the applications using frequency-swept laser, like optical FMCW, OCT, optical frequency domain reflectometer (OFDR), etc., since this method provides an additional time dimension to characterize the laser stability, it may also be of interest in the application that requires a long-time stable laser source, to name a few, optical frequency comb, photonic generated microwave, etc.