1. Introduction

The development of frequency swept laser sources in the 1300 nm wavelength range with tuning ranges of ~100 nm and sweep rates >15 kHz [1] has recently enabled a quantum leap in system performance of optical coherence tomography (OCT) systems [2]. Yet, conventional rapidly swept lasers are inherently limited in their achievable sweep rates due to the buildup time of the lasing [3]. Recently, using the technique of Fourier domain mode-locking (FDML) [4], greatly enhanced sweep rates of >5 MHz were achieved [5]. The typical instantaneous linewidth of <0.1 nm corresponds to an instantaneous coherence length of up to several millimeters. Currently, FDML lasers are the light sources of choice for OCT systems with highest imaging speed [5,6].

In FDML operation, a narrowband optical bandpass filter is tuned synchronously to the cavity roundtrip time of the laser. Therefore the laser field does not have to build up repetitively as in the standard tunable laser and the sweep rate is only limited by the mechanical response of the filter. Since FDML is a stationary laser operating regime [7] with very long cavity photon lifetimes [8], substantial linewidth narrowing compared to the width of the sweep filter can be observed [9]. Even though a connection between cavity dispersion and linewidth has been revealed, the physics behind it is not understood and a quantitative relation is currently not available. Further, a direct experimental, spectrally resolved measurement of the instantaneous FDML linewidth is not possible, especially for very narrowband emission, due to Fourier broadening [10]. Besides sweep range and rate, the instantaneous linewidth or coherence length is the most important property of an FDML laser, because it determines the maximum ranging depth in OCT imaging and it is sometimes limiting system performance. Thus, theoretical access to this parameter is of high interest.

Using the previously presented theoretical model [7], the first goal of this paper is to investigate the dependence of the FDML linewidth on cavity parameters and relevant physical effects. The gained insight might be used to find ways to increase the instantaneous coherence length from the mm range to the cm or m range in the future, enabling a whole new variety of biological and non-biological imaging and sensing applications. The second goal is to investigate if ASE and environmental instabilities ultimately limit the FDML linewidth performance, as in typical semiconductor based lasers.

2. Experimental setup

In Fig. 1(a) , the experimental setup of the FDML laser is shown. In order to study the instantaneous linewidth at different points in the cavity, three output couplers are built into a sigma ring geometry. Here, a semiconductor optical amplifier (SOA, Covega Corp., “BOA 1132”) is used as a gain medium, where the maximum of the gain lies at 1320 nm. In order to ensure unidirectional lasing, two isolators (ISO) are built in before and after the SOA. The sweeping action is performed by a tunable polarization maintaining (PM) Fabry-Perot bandpass filter (FFP-TF, Lambda Quest, LLC.), with a bandwidth of 0.15 nm.

 

Fig. 1 (a) Setup of the polarization maintaining FDML laser with outcouplers numbered 1 to 3. (b) Measurement setup for the instantaneous linewidth using a pulse generator, an electro-optical modulator and an optical spectrum analyzer. (c) Sweep filter center wavelength over time. (d) Simulated output power over time.

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The light coming from the SOA is coupled into a single mode fiber (SMF) with a length of 1.7 km by means of a polarization beam splitter (PBS). At the end of the SMF, a Faraday rotating mirror rotates the light by 90° where it is transmitted back through the fiber to the FFP-TF and the SOA. The FFP-TF filter is driven sinusoidally with a sweep frequency of 57.7 kHz in resonance to the cavity roundtrip time of T = 17.32 µs. In order to measure the linewidth, a setup as shown in Fig. 1(b) is employed, similar to the one described in [10]. The sweep range for this setup is 105 nm, see Fig. 1(c). The simulated time dependent output power is shown in Fig. 1(d).

3. Theory: propagation equation and extraction of instantaneous linewidth

The spectral sweep range for our FDML setup of 105 nm and the cavity length of 3.4 km lead to a very large time-bandwidth product of $3\times {10}^8$, inhibiting straightforward approaches to FDML simulation. This problem is overcome in [7] by the introduction of a transformation into the swept filter reference frame, reducing the number of required grid points by about two orders of magnitude. The resulting evolution equation for the envelope function u(z,t)

Amplified spontaneous emission (ASE) is modeled as an equivalent noise source at the input of the ideal SOA. It is implemented as additive white Gaussian noise [7,12] with a constant spectral power density $P_f$, computed from the noise power $P_n=3.2\text{\hspace{0.17em} mW}$ measured directly after the SOA for a blocked laser cavity together with the experimentally determined small signal gain.

The values of D ₂ and D ₃ are $-2.7603\cdot {10}^{-28}{\text{s}}^2{\text{m}}^{-1}$and $\text{1 .2183}\cdot {\text{10}}^{-41}{\text{s}}^3{\text{m}}^{-1}$, respectively, and γ is 0.00136${\text{W}}^{-1}{\text{m}}^{-1}$. Furthermore, a typical value of α = 5 is assumed [13]. The sweep filter function $-a_s(\text{i}{\partial }_t)$is implemented as a lumped element in form of a complex Lorentzian, as described in [7]. The gain and loss have been carefully measured as shown in Figs. 2(a) and 2(b), respectively, and have been implemented accordingly in our simulation.

 

Fig. 2 (a) Experimentally measured SOA power gain (linear scale) as a function of the optical frequency for different values of the incident optical power. (b) Experimentally measured overall cavity power loss (linear scale) as a function of the optical frequency. The sweep filter has been tuned to maximum transmission at each measured frequency.

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The instantaneous linewidth can be obtained from the simulation data by Fourier transforming the complex field envelope in the swept filter reference frame u(z,t), yielding the instantaneous power spectrum |u(z,f)| ², where f denotes the frequency with respect to the center frequency of the sweep filter. The duration T of one roundtrip can now be segregated into a given number of subintervals, and the instantaneous power spectrum at different times can be calculated by Fourier transforming u(z,t) for each interval. This way we can simulate the temporal evolution of the instantaneous power spectrum. Because the simulation is performed in the swept-filter reference frame, the simulation is NOT broadened by the ongoing sweeping action during a time-gate as it is in the experiment. The instantaneous power spectrum does not depend much on the position z within the laser cavity, and is here computed after the SOA. The instantaneous linewidth corresponds to the full width at half-maximum (FWHM) of the instantaneous power spectrum.

To experimentally measure the linewidth at different positions within the sweep, in the setup the time gating is performed at various times t, corresponding to different sweep filter center frequencies. The optical spectrum analyzer has a finite resolution of 20 pm, corresponding to a spectral width of 3.5 GHz at 1310 nm. Furthermore, the 1.6 ns gating window leads to a Fourier broadening of ~1 GHz according to the time-bandwidth product.

Longer gating times would suppress this effect, but lead to considerable smearing of the linewidth due to the sweep filter dynamics. For a sweep filter driven by a cosine wave of the form ${\omega }_0(t)\propto -\cos (2\pi t/T)$, the broadening has its maximum value of 3.81 GHz at 5.3 and 3.3 µs, whereas at 1.3 and 7.3 µs, it is only 1.85 GHz due to the slower sweep speed at that point. The combination of all these effects leads to a broadening of the experimentally measured spectrum by about 4-8 GHz compared to the theoretically calculated values. In the simulation, where the frequency axis moves along with the sweep filter, the gating window can be chosen sufficiently long to avoid Fourier broadening. Here, we divide the axis into 16 intervals of 1.08 µs, summing up to the total roundtrip time of 17.32 µs.

4. Results

4.1 Agreement with experiment

In Fig. 3 , the instantaneous power spectra are compared at different times t for the non-detuned case, where the sweep filter frequency matches exactly the roundtrip time. In Fig. 3(a), t = 1.3 µs, where the sweep filter center frequency varies only slowly, and in Fig. 3(b) t = 3.3 µs, where the cosine function is the steepest, thus the frequency changes fast. For this reason, the experimental spectrum in Fig. 3(b) is considerably broadened as compared to the simulated spectrum, with a full width at half-maximum (FWHM) of 12.07 GHz for the experimental result vs. 5.81 GHz for the simulation.

 

Fig. 3 Experimental (red) and simulated (blue) instantaneous power spectra after the SOA at (a) 1.3 µs and (b) 3.3 µs for no detuning.

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For further validation, the detuned case is investigated, showing that good agreement between theory and experiment is obtained under various conditions. In Figs. 4(a) , 4(b) and 4(c), the instantaneous power spectrum is displayed at t = 7.3 µs for zero, −2 Hz and + 2 Hz detuning, respectively. The detuning between the sweep period and the roundtrip time affects not only the time dependent output power [7], but also the instantaneous power spectrum. Specifically, we observe a pronounced high-frequency tail for both negative and positive detuning, see Figs. 4(b) and 4(c). This asymmetry gets reduced for a smaller amount of detuning, as can be seen by comparison with the non-detuned case shown in Fig. 4(a). The main source of this asymmetry is found to be the third order dispersion term D ₃. In Fig. 4(b), the asymmetry manifests itself as a small side peak, which is not resolved in the experiment due to the limited resolution as discussed above.

 

Fig. 4 Theoretical (blue) and experimental (red) power spectra at 7.3 µs for (a) no detuning, for (b) a detuning of −2 Hz and for (c) a detuning of + 2 Hz.

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4.2 Timing offset, linewidth enhancement factor and spectral shift

In our simulation, where the frequency axis moves along with the sweep, broadening of the instantaneous power spectrum due to the sweep filter dynamics is eliminated. Thus, we can analyze the instantaneous power spectrum averaged over the whole roundtrip time, which is not possible in the experiment. In Fig. 5(a) , the instantaneous power spectrum is plotted for zero detuning and with the laser parameters as in the experiment, where the linewidth enhancement factor is assumed to be α = 5 (red) [13] and α = 0 (blue). The sweep filter transmission function is also shown for comparison. Figure 5(b) shows the same simulation, but now with ASE only used at the start of the simulation to seed lasing. From Fig. 5(a) we can extract that the frequency shift of the power spectrum is due to the linewidth enhancement in the SOA, as also observed for conventional swept laser sources [13].

 

Fig. 5 (a) Simulated instantaneous power spectrum for α = 5 (red) and α = 0 (blue), the sweep filter transmission function is drawn in black. (b) The instantaneous power spectrum for α = 5 (red) and α = 0 (blue) but without ASE.

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The cause of this frequency shift can be understood if we separately investigate the gain term containing α in Eq. (1), ${\partial }_{z}u=g({\omega }_0)\left(1-\text{i}\alpha \right)u,$ in the frequency domain. We found that for α>0 and an asymmetric gain function g(ω₀) which falls off more rapidly on its high-frequency side (compare Fig. 2(a)), the power spectral peak of a pulse u gets shifted to lower frequencies after propagation through the gain medium.

In our case, we additionally observe a significant broadening of the linewidth from 7.25 and 7.41 GHz to 10.08 and 9.99 GHz, respectively, indicating that SOAs with low or optimized values for α might be preferred. This finding might also indicate that for light sources with very narrow instantaneous linewidth, post amplification as presented in [3] might lead to decreased coherence properties, depending on α.

The qualitative and quantitative agreement of experimental and simulation results presented above show the validity of our model. Furthermore, these results clearly indicate that the linewidth is NOT dominated by external noise sources, such as fluctuations of the pump current, frequency or amplitude instabilities of the filter drive waveform or acoustic vibrations, since such effects are not contained in the simulation. Our model can now be used to identify the physical effects governing the instantaneous linewidth by successively switching on and off the effects in the simulation, which is not possible in experiment. The central question is if the linewidth is dominated by ASE, as is usually the case for semiconductor and other lasers in the absence of external noise sources [11]. In Fig. 5(b), the linewidth is displayed as obtained with ASE used only for initial seeding. Comparison with Fig. 5(a) shows that the power spectrum is virtually unchanged without ASE. Rather, the instantaneous lineshape is governed directly by the FDML dynamics due to the sweep filter and gain action, dispersion and self-phase modulation.

5. Conclusions

In conclusion, the instantaneous power spectrum of an FDML laser is theoretically and experimentally investigated. The linewidth enhancement factor results in a frequency shift relative to the sweep filter center frequency as well as a broadening, and third order dispersion leads to an asymmetry of the instantaneous power spectrum. Good agreement between simulation and measurement is obtained for both the non-detuned and the detuned case, confirming the validity of our theoretical model. The simulations reveal that the instantaneous linewidth is not governed by external noise sources or ASE, but results directly from the FDML dynamics due to the sweep filter and gain action, dispersion and self-phase modulation. Such a theoretical understanding of the effects governing the instantaneous power spectrum is important for a further optimization of the linewidth and thus the coherence properties of FDML lasers.