## Abstract

The first theoretical model of Fourier domain mode locking operation is presented. A specially tailored dynamic equation in a moving spectral reference frame is derived, enabling efficient numerical treatment, despite the broad laser spectrum and the extremely long cavity. The excellent agreement of the presented theory with experiment over a wide range of operation parameters enables a quantitative assessment of the relevant physical effects, such as the spectral loss modulation and gain saturation dynamics, amplified spontaneous emission, linewidth enhancement, and self-phase modulation.

©2009 Optical Society of America

## 1. Introduction

The recent introduction of Fourier Domain Mode Locking (FDML) [1] has helped to overcome physical limitations of rapidly wavelength swept laser sources. With sweep ranges of more than 150 nm at 1300 nm center wavelength [2], instantaneous linewidths of a few tens of picometers [1] and sweep repetition rates of more than 370 kHz [3], FDML lasers are the light source of choice for many biomedical imaging [1] and sensing applications [4]. In optical coherence tomography (OCT), which originally used a broadband light source [5], FDML lasers have enabled unprecedented imaging rates for densely sampled volumetric data sets, improving the visualization of tissue morphology. First *in vivo* experiments with FDML based OCT systems have already demonstrated superior performance in ophthalmology [6], gastroenterology [2] and cardiology [7]. In FDML, a narrowband optical bandpass filter is driven in resonance with the optical roundtrip time of the laser cavity. The required resonator length of several kilometers is realized by a long delay line consisting of single mode fiber (SMF). As each wavelength component circulates in the cavity such that it is transmitted through the filter at every pass, FDML represents a stationary operating regime. Lasing does not have to build up repetitively as in conventionally swept laser sources, resulting in improved noise performance, coherence length, output power and higher maximum sweep repetition rates [1]. In spite of the numerous applications of FDML lasers demonstrated so far, up to now, a theoretical description of FDML has not been available. In this paper, we derive a dynamic equation enabling us to identify the physical effects relevant for FDML, and we clarify the role of amplified spontaneous emission (ASE) for self-starting and for the steady state operation of FDML lasers. A theoretical understanding of FDML is essential for a further optimization of the laser performance. Furthermore, the derived dynamic equation sets the stage for a future development of simplified analytical FDML models.

## 2. Experimental setup

Figure 1 shows the experimental setup of the FDML laser used for this study. The entire circular part of the laser consists of polarization maintaining (PM) fiber and polarizers (POL). A semiconductor optical amplifier (SOA, Covega Corp.) with a gain maximum centered at 1320 nm is used as a gain medium. Two isolators (ISO) ensure uni-directional lasing. A tunable PM Fabry-Perot filter with a bandwidth of $\sim 0.15\text{\hspace{0.17em} nm}$ and a free spectral range of 130 nm (FFP-TF, Lambda Quest, LLC.) is used as sweep filter. The cavity design is based on a sigma-ring configuration. Light from the SOA is coupled by the polarization beam splitter (PBS) into the linear part of the cavity with a 2.0 km length of standard SMF. A Faraday rotating mirror (FRM) at the end rotates the polarization by 90° upon backreflection. The PBS transmits the 90° rotated light to the FFP-TF and then back to the SOA. The FFP-TF filter is driven sinusoidally over an 80 nm range with a sweep rate of 50.95 kHz in resonance to the optical roundtrip time of light in the cavity, or it is slightly detuned by a defined amount for comparison with the theory. The transient output power is measured with a 100 MHz photodiode (PD) at the output coupler directly after the SOA. For determining the spectral gain profile of the SOA at different levels of saturation, a second output coupler is added before the SOA. The measured transients are compared to the theoretical results for different operation parameters to investigate the accuracy of the theoretical model.

## 3. Theoretical model

Unlike for standard mode locked fiber lasers, a straightforward simulation of the FDML laser is impeded by the huge time-bandwith product of the optical output, resulting from a large spectral sweep range of 80 nm combined with extreme optical cavity lengths of $\sim 4\text{\hspace{0.17em} km}$. We solve this problem by deriving a special dynamic equation for FDML operation. The laser is modeled as an optical system consisting of an SOA, an output coupler, a length of SMF and a tunable optical bandpass filter. We start from a generalized nonlinear Schrödinger equation (NSE) based on the slowly varying amplitude approximation [8],

*t*is defined with respect to a frame moving along with the optical field, and

*z*denotes the position in the laser system along the propagation direction, suppressed in the following for a more compact notation. The coefficients ${D}_{2}$, ${D}_{3}$ and

*γ*describe the second and third order dispersion and self-phase modulation, assumed to be constant in the delay fiber and zero for the other optical components. The spectral gain of the optical amplifier and loss in the fiber system are characterized by $g(\omega )$ and $a(\omega )$, respectively. In Fourier domain, these coefficients are functions of the optical frequency $\omega =2\pi f$, corresponding to the operator $\text{i}{\partial}_{t}$ in time domain. Furthermore, coupling between the gain and the refractive index in the SOA is accounted for by the linewidth enhancement factor

*α*, and saturation effects are also included as discussed further below.

A pivotal element of the FDML laser is the sweep filter, a narrowband optical bandpass filter periodically driven at a rate ${f}_{s}$ in resonance with the optical roundtrip time *T*. This sweeping action, reflected by a time dependent bandpass center frequency ${f}_{c}+{\omega}_{0}(t)/(2\pi )$, is not yet included in Eq. (1). The huge time-bandwidth product of the FDML output ($\sim 15\text{\hspace{0.17em} THz}\times 20\text{\hspace{0.17em} \mu s}$) would require a prohibitively large number of $3\times {10}^{8}$ simulation grid points to reach sufficient temporal and spectral resolution in a direct simulation of Eq. (1). The crucial step is to introduce a swept filter reference frame moving along with the center frequency, in which the optical spectrum is limited by the narrow sweep filter bandwidth rather than being set by the sweep range, allowing a reduction of the number of grid points by a factor of $\sim 1000$. This transformation is achieved by adapting the *instantaneous* frequency of the optical carrier wave to the sweep filter position, leading to the transformed envelope

For FDML lasers, the sweep rate ${f}_{s}$ is small enough that the terms $\sim u{\partial}_{t}{\omega}_{0}$ and $\sim {\omega}_{0}{\partial}_{t}u$ can be neglected. While the contribution of third order dispersion is generally small, it is in the following considered through its leading term. Furthermore, the frequency dependent gain and loss in the first term of Eq. (3) vary only slightly over the spectral width of *u*, which is limited by the narrow sweep filter bandwidth. We then arrive at a simplified dynamic equation

## 4. Results

In the following, we investigate a typical FDML setup as shown in Fig. 1. Here, the SOA is modeled as a lumped element with an amplitude gain $G=\mathrm{exp}\left({\displaystyle \int g}\text{\hspace{0.17em} d}z\right)$, obtained by integrating over the length of the gain medium. In this context, we have to consider not only the spectral gain profile, but also the wavelength-dependent saturation behavior of the SOA. The gain of our SOA has been carefully measured as a function of both incident optical frequency and power as shown in Fig. 2(a)
, and is implemented accordingly in our simulation. The gain recovery dynamics, characterized by the carrier lifetime ${\tau}_{c}$ in the SOA [9], is fast compared to the sweep-induced optical power modulation, but not necessarily faster than optical fluctuations caused by ASE. We take this effect into account by using a quasi-instantaneous gain saturation model in our simulation, determining the saturation level of the gain in Fig. 2(a) at a given time *t* not directly from the instantaneous optical power $P(t)$, but rather based on a moving average value ${P}_{av}(t)$ with an exponential memory decay time ${\tau}_{c}$, ${P}_{av}(t)={\tau}_{c}{}^{-1}{\displaystyle {\int}_{-\infty}^{t}P(t)\mathrm{exp}\left((\tau -t)/{\tau}_{c}\right)\text{\hspace{0.17em} d}\tau}.$ For the carrier lifetime and the linewidth enhancement factor, we assume typical values of ${\tau}_{c}=380\text{\hspace{0.17em} ps}$ and $\alpha =5$ [9]. ASE is also included, allowing the simulation to self-start from noise and adding a noise floor to the laser output. This effect is here modeled by an equivalent noise source with constant spectral power density ${P}_{f}$ at the input of the ideal SOA, and numerically implemented as additive white Gaussian noise [10]. Here we use ${P}_{f}={P}_{n}/{\displaystyle \int G(f)\text{\hspace{0.17em} d}f=0.66\text{\hspace{0.17em} \mu W/THz}},$ computed from the noise power ${P}_{n}=1.7\text{\hspace{0.17em} mW}$ measured directly after the SOA for blocked laser cavity and the experimentally determined small signal gain (blue curve in Fig. 2(a)).

After the SOA and the 50/50 output coupler, the light propagates through a fiber section with total length 4.03 km, giving rise to a resonator roundtrip time *T* of about $20\text{\hspace{0.17em} \mu s}$. The fiber dispersion, absorption and self-phase modulation are implemented according to the product specifications for standard SMF. The sweep filter is modeled as a lumped element with complex Lorentzian transmission characteristic ${t}_{s}=\mathrm{exp}\left(-{\displaystyle \int {a}_{s}}\text{\hspace{0.17em} d}z\right)={T}_{\mathrm{max}}^{1/2}/\left(1-2\text{i}\omega /\Delta \right)$, using $\Delta =0.169\text{\hspace{0.17em}}{\text{ps}}^{-1}$ and ${T}_{\mathrm{max}}=0.5$. Also the overall frequency dependent loss in the experimental setup, caused by the various components in the laser cavity, has been carefully characterized (see Fig. 2(b)), and is accounted for by an absorption coefficient $a(\omega )$ in the simulation.

We present simulation results for a typical FDML setup as shown in Fig. 1, sinusoidally driven with a sweep range of 80 nm centered around 1320 nm, i.e., ${\omega}_{0}(t)=\Delta \omega /2\times \mathrm{cos}(2\pi {f}_{s}t)$ with $\Delta \omega =8.79\times {10}^{13}\text{\hspace{0.17em}}{\text{s}}^{-1}$. Equation (4) is numerically solved using the split-step Fourier method [8]. The temporal simulation window is adapted to the sweep period $1/{f}_{s}$, taking advantage of the implicit periodic boundary conditions in the algorithm. Here, 2^{23} (≈8 million) grid points are used. The simulation is self-starting from ASE noise, and typically converges after 10-1000 roundtrips, depending on the laser parameters. We have ensured convergence of the solution in the time and frequency domain, only limited by the small random fluctuations due to the ASE noise floor. We emphasize that all simulations are performed self-consistently without fitting parameters, i.e., all parameters are obtained from literature or, where not available, from experimental characterization as discussed above. The results enable us to address two crucial questions of FDML operation: (1) What is the role of ASE for self-starting and for the stationary operation of FDML? (2) What physical mechanisms are important for FDML at the various operation points?

To investigate the stationary operation, we first take into account ASE in the SOA as discussed above, and compare the result to the simulation where ASE is only used for the initial seeding. We assume exact synchronization of the sweep filter to the roundtrip time, i.e., ${f}_{s}=1/T$. Figure 3(a) shows the sweep filter frequency offset over time, and in Fig. 3(b), the experimentally measured instantaneous power at the output coupler and the corresponding numerical results with and without ASE are displayed. The agreement between the experiment and simulation is excellent, and the small deviations are mainly ascribed to the limited accuracy in the experimental characterization of the SOA, in particular the spectrally dependent saturation behavior. It should be emphasized that the simulation results with and without ASE are practically identical, apart from a small offset for the curve with ASE, which is due to the additive broadband noise floor from the SOA. A fast convergence to steady state operation after a few ten roundtrips is observed in both cases. These results show that ASE is only required for self-starting but not for stationary operation, underlining the fact that FDML operation resembles stationary lasing, rather than being dominated by noise bursts with repetitive build up from ASE.

Our numerical model makes it also possible to assess the influence and relevance of the various physical effects on the FDML dynamics in a straightforward manner. Figure 4 shows the simulated instantaneous output power as obtained by including all effects considered in Eq. (4), and by neglecting linewidth enhancement, self-phase modulation, and dispersion, respectively. In Fig. 4(a), the results are displayed for the case of no detuning, as in Fig. 3. All the curves are very close to each other, indicating that FDML operation is here only weakly affected by these effects, but is instead exclusively governed by the spectral characteristics and the gain saturation of the SOA, the cavity loss, and the sweep filter action. This is in agreement with the experimentally observed remarkable stability of FDML operation over a wide range of operation settings.

To assess the quality and robustness of the simulation results as compared to the experiment, we investigate how FDML operation is affected by a frequency detuning *δ* between the sweep filter and the roundtrip time, i.e., ${f}_{s}=1/T+\delta $. Such an effect is implemented by imposing a temporal delay ${T}^{2}\delta $ on the optical field envelope *A* before the sweep filter at each roundtrip. The dynamics become more complex if the sweep rate is slightly detuned with respect to the roundtrip frequency, i.e., $\delta \ne 0$. In Fig. 4(b), the results are shown for a detuning of 0.2% with respect to the roundtrip frequency, corresponding to $\delta =10\text{\hspace{0.17em} Hz}$. The simulation converges to steady state much more slowly than in the non-detuned case, requiring several 100 roundtrips. Moreover, this weak detuning already has a significant effect on the simulated instantaneous output power: The wavelength of the light impinging on the sweep filter is now shifted with respect to the central filter wavelength, and since the sweep filter is very narrow ($\sim 0.15\text{\hspace{0.17em} nm}$), a small shift already leads to a significant damping of the optical field. The linewidth enhancement and to a small degree also the self-phase modulation now affect the simulation result, giving rise to spectral shifting and broadening which can counteract or enhance the detuning action. Also the dispersion has some effect due to the temporal shifting of the different frequency components, but this effect is small since we operate here around the zero dispersion point of the fiber.

To assess in how far the simulations of the detuned laser in Fig. 4(b) describe real FDML operation, Fig. 5 shows the experimentally measured instantaneous output power for positive and negative detuning of the sweep filter by $\pm 10\text{\hspace{0.17em} Hz}$ (Fig. 5(a)) and $\pm 20\text{\hspace{0.17em} Hz}$ (Fig. 5(b)), along with the corresponding simulation results. Detuning affects the two sweep directions asymmetrically, see Figs. 4(b) and 5. As can be deduced from Fig. 4(b), this asymmetry is mainly caused by the wavelength shifting effect of linewidth enhancement. A sign change in detuning causes a near inversion of the instantaneous output power trace in both theory and experiment. The good agreement between experiment and theory even for the detuned case strongly indicates that Eq. (4) takes into account all physical effects relevant for FDML operation. We have also successfully tested our model for experimental setups with increased sweep rates and modified sweep filter drive waveforms like linear ramps.

## 5. Conclusions

In conclusion, we present for the first time a theoretical model for FDML operation. An equation using a sliding frequency reference frame is derived, resulting in about three orders of magnitude speed improvement and reduction in memory requirement, thus enabling a simulation of FDML lasers with their broad output spectrum and long cavity length. Comparison to the experimental results shows excellent qualitative and also quantitative agreement for various operating conditions. This model enables us to clarify the role of ASE in the FDML dynamics. The simulation results clearly demonstrate that FDML is a stationary operation mode, and ASE is only required for self-starting. Convergence to steady state is observed after 10-1000 roundtrips, depending on the detuning. We find that the FDML operation is governed by the interplay between the gain action in the optical amplifier, the cavity loss and the sweep filter dynamics. For a detuned sweep filter, additional effects like linewidth enhancement contribute to the shaping of the laser output. The introduced model for Fourier domain mode locking provides the basis for a deeper understanding of FDML operation, and is essential for a further optimization of FDML lasers.

## Acknowledgments

C. Jirauschek acknowledges support from Prof. P. Lugli at the TUM, and B. Biedermann and R. Huber would like to acknowledge support from Prof. W. Zinth at the LMU Munich. This work was supported by the German Research Foundation (DFG) within the Emmy Noether program (JI 115/1-1 and HU 1006/2-1) and under Grant No. JI 115/2-1 as well as by the European Union project FUN OCT (FP7 HEALTH, Contract No. 201880).

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