Intensity Pattern Types in Broadband Fourier Domain Mode-Locked (FDML) Lasers Operating Beyond the Ultra-Stable Regime

We report on the formation of various intensity pattern types in detuned Fourier domain mode-locked (FDML) lasers and identify the corresponding operating conditions. Such patterns are a result of the complex laser dynamics and serve as an ideal tool for the study of the underlying physical processes as well as for model verification. By numerical simulation we deduce that the formation of patterns is related to the spectral position of the instantaneous laser lineshape with respect to the transmission window of the swept bandpass filter. The spectral properties of the lineshape are determined by a long-term accumulation of phase-offsets, resulting in rapid high-amplitude intensity fluctuations in the time domain due to the narrow intra-cavity bandpass filter and the fast response time of the semiconductor optical amplifier gain medium. Furthermore, we present the distribution of the duration of dips in the intensity trace by running the laser in the regime in which dominantly dips form, and give insight into their evolution over a large number of roundtrips.


Introduction
Fourier domain mode-locked (FDML) lasers produce rapidly wavelength-swept light with bandwidths of more than 100 nm at tuning rates in the range of MHz [1][2][3][4][5][6]. This is achieved by synchronizing the roundtrip time of the optical field in a ring laser setup with the sweep rate of a tunable Fabry-Pérot (FP) filter, acting as the wavelength tuning element. The record sweep speeds and the excellent coherence properties have dramatically improved imaging and sensing applications, especially optical coherence tomography (OCT) [7,8]. The overall superior compination of tuning speed, imaging depth, sensitivity and axial resolution make FDML based OCT systems an interesting alternative to high-performance OCT systems, e.g. [9][10][11][12][13][14][15][16]. Due to the large time-bandwidth product and the rapid time scales in optical systems in the order of fs, the analysis of the wavelength swept light in FDML lasers is a challenging task in general. Common experimental quantities of interest are the instantaneous frequency [17], the instantaneous lineshape [17][18][19], and the intensity trace [20][21][22][23][24][25][26]. Here, we focus on the intensity trace since it can be obtained by a relatively simple measurement with a photodiode and a real-time oscilloscope. We show that the intensity trace contains sufficient information to characterize the operation mode and the physical dynamics of the FDML laser, provided that sufficient analog bandwidth is available. Timing mismatches in FDML lasers can directly be observed in the intensity trace and are referred to as high-frequency fluctuations [23] since their time scale is much shorter than the sweep period. It has been shown that the laser can operate without high-frequency fluctuations in the intensity trace in highly dispersion compensated and highly synchronized setups over a wavelength range of more than 100 nm [23]. When timing delays, caused for example by either a residual dispersion in the fiber cavity or a detuning from the ideal sweep rate, exceed a certain amount, the intensity trace suffers from high-frequency fluctuations [23,27] which have a negative impact on the imaging quality in OCT applications. The high-frequency fluctuations can be classified as various types, such as irregular fluctuations, referred to as a modulational instability [21,28] and the Eckhaus instability [29], periodic Turing-type formations [21,28] and so-called holes [22,23]. Typically, the intensity trace of FDML lasers is recorded with measurement bandwidths of less than 10 GHz [20,[24][25][26]30,31], allowing to visualize only the low-frequency part. In this work, we characterize the intensity trace of an FDML laser at the full sweep speed using a real-time oscilloscope with a total analog bandwidth of several 10 GHz, and report on the formation of characteristic intensity patterns under different operating conditions. The relatively large measurement bandwidth is essential to fully record the high-frequency fluctuations whose temporal extensions are limited by the bandwidth of the swept bandpass filter. Our results are further validated by numerical simulations which additionally make it possible to extract the spectral features of the operating modes. Interestingly, in many cases patterns are formed instead of irregular fluctuations even under the influence of strong detunings. The advantage of the method are its simplicity and the fact that variations of the optical phase are visible in the intensity trace as amplitude fluctuations. The paper is organized as follows: First we describe the experimental setup and the underlying simulation model. Then we present the results and a comparison of experiment and simulation. In order to discuss specific properties of the highfrequency fluctuations, we demonstrate the distribution of the duration of so-called holes, i.e. dips in the intensity trace, and analyze the long-term evolution of the hole-type intensity patterns.

Intensity pattern types in non-synchronized FDML lasers
The intensity trace of an FDML laser contains rich information about the interplay of the laser components due to the strong phase-amplitude coupling in the cavity. The extremely narrow bandpass filter introduces large losses on the optical field when the instantaneous wavelength of the optical field is offset from the central position in the spectral filter transmission window over many roundtrips. This occurs in the case of strong synchronization mismatches between the roundtrip time of the opti-cal field and the sweep rate of the bandpass filter, caused by e.g. the fiber dispersion or a detuning from the ideal sweep rate. Typical values of the full width at half maximum (FWHM) bandwidth of the sweep filter are in the order of 100 pm (several tens of GHz). At a wavelength of 1300 nm (231 THz) and a filter bandwidth with a FWHM of 165 pm (30 GHz), an offset of 0.02 nm (15 GHz) from the peak transmission already causes a power loss of 50 % when the reflected power is absorbed by an isolator. Such losses are compensated by the semiconductor optical amplifier (SOA) gain medium with a fast response time in the order of several tens of ps [27]. This interplay of frequency-shift, gain and loss over a long time scale, i.e. when this process is iterated over many roundtrips, introduces highfrequency fluctuations in the intensity trace with a large amplitude. In the following, we first discuss the experimental laser setup as well as the simulation model, and present different types of highfrequency fluctuations by systematically detuning an FDML laser from the sweep rate at which the laser operates in the ultra-stable regime [20,23].

Experimental setup
We performed measurements with two similar FDML laser setups which differ in the output power, the bandwidth of the tunable bandpass filter and the mechanism to tune the sweep frequency. The first setup is described in detail in [23]. The second setup is illustrated in Fig. 1. Here the sweep frequency of the tunable bandpass filter is kept constant, except when the laser is detuned, and the cavity length is tuned by modifying the length of a free space beam path (FSBP) in order to regulate the laser in the ultra-stable regime. The other components are a single polarization SOA gain medium (Thorlabs BOA1130S), a circulator (CIRC), a fiber spool consisting of a mix of SMF-28, HI1060 and LEAF fibers, and a temperature fine-tuned chirped fiber Bragg grating (cFBG) serves as a dispersion compensating element as well as the laser output (Teraxion, custom made). Further components are a polarization controller (PC) to adjust the polarization state of the light to the maximum gain axis of the SOA, a home-made tunable bandpass filter and an isolator (ISO) to ensure unidirectional lasing. The intensity trace was recorded with a 33 GHz photodiode (Discovery semiconductors Inc. DSC20H) whose output was digitized with

Simulation model
In order to reproduce the experimental results, we performed simulations with the model presented in [27]. This model contains the fundamental interplay between the swept bandpass filter, a delay element such as a dispersive optical fiber or a detuning element and a single polarization SOA gain medium. The role of the simulation model in this work is twofold. First, by reproducing the measured intensity patterns, we can verify that the dynamics of the laser is sufficiently described by the three above mentioned processes, and we can identify the spectral features of the operating modes. Second, the simulation bandwidth is chosen to be 3.45 THz in order to show that no hidden intensity fluctuations exist which could not be captured with the limited measurement bandwidth of 33 GHz and 50 GHz, respectively. We added the effects of carrier heating (CH) and spectral hole burning (SHB) in the SOA gain medium to the model in [27]. A quasi-static approach is used, since these processes have time constants in the order of 100 fs which is much smaller than the temporal extension of the highfrequency fluctuations in FDML lasers. Within this approach the update equation for a lumped element SOA is given by where u in,out is the slowly varying complex electric field envelope in the swept filter reference frame [30] at the spatial input and output of the SOA, respectively. The retarded time is denoted by τ and α N,CH are the linewidth enhancement factors due to band-filling and carrier heating. The total gain is is given by and SHB (h SHB ). The linewidth enhancement factor due to SHB α SHB is set to zero as in [32][33][34].
A description of the related differential equations of the three parts of the total gain is given in [35]. By In Eq. (2), = CH + SHB is the combined gain compression factor where CH and SHB are the gain compression factors due to CH as well as SHB, and P in = |u in | 2 is the optical power at the spatial input of the SOA. After computing the nonlinear gain h NL with Eq. (2), the individual components can then be found by The SOA parameters are taken from [35] and are The other parameters which are different from [27] are the dispersion coefficients of the optical fiber and the cFBG which are β 2,3,4 ,β 2,3,4 = 0 for setup 1 and 2. For setup 2 the power loss factor in the fiber spool κ f is 0.33, the reflectivity of the cFGB R is 0.4, the center wavelength ω c is 2π · 230.8 THz (1300 nm), the sweep bandwidth D ω is 2π·17.8 THz (100 nm), the FP filter bandwidth ∆ ω is 2π·50.7 GHz (0.290 nm), the FP filter transmission T max is 0.075 including the losses of the FSBP as well as the PC, and the total cavity losses L are 20 dB.

Detuning from the ultra-stable regime
In Fig • In the case of a forward sweep the situation is reversed, i.e., holes occur at +100 mHz and fringes at −100 mHz, respectively.
• At strong detunings, short quasi-periodic pattern can occur at certain positions in the sweep with different shape in the backward and forward sweep, as discussed below.
These patterns are a result of the long-term interplay of the swept bandpass filter, the SOA gain medium and the frequency shift per roundtrip caused by the detuning, as will be shown in Section 3. The qualitative shape of the pattern does not depend on the temporal position in the sweep. Figure 3 presents a comparison to the same setup as above, however for strong detuning of ±10 Hz.
As can be seen in Figs. 3 (c) and (d), a pulsed output is generated rather than irregular fluctuations, which might be unexpected when looking at the full sweeps in Figs

Comparison to the simulation model
We performed numerical simulations of the first setup in order to reproduce Figs. 2 and 3. Here we neglected the dispersion in the fiber spool causing a maximum residual group delay of less than 200 fs [23], since this contribution is small compared to the delay introduced by a detuning of ±100 mHz (±10 Hz) per roundtrip which is ∼590 fs (∼5.9 ps) in the case of a linear sweep. Yet, differences in the hole density due to the unknown residual dispersion as well as residual temperature fluctuations in the experiment are to be expected, especially in the case of low detuning. As can be seen from Figs. 4 and 5, the simulation model reproduces the qualitative behavior of the experimental data in Section 2.3 extremely well and is therefore an excellent tool to study the FDML laser dynamics. Yet, differences in the exact shape exist in the case of the local fringes or the quasi-periodic pattern while holes can be reproduced almost exactly, see also [27]. This observation shows that the accuracy of the individual spectral shifts introduced by the SOA, the bandpass filter or the detuning are a bottleneck in modeling FDML lasers, which has not yet been fully addressed in literature in this detail, to the best of our knowledge. In Figs. 4 (d) and (f), the local fringes are reduced in length, and thus also the density differs by roughly a factor of three. For negative detuning, the quasiperiodic patterns have a similar shape as in experiment (see Fig. 3 (e)), but a higher periodicity in Fig. 5 (e). We found that individual shapes can indeed be reproduced by modifying the SOA parameters. In the case of Fig. 5 (e) a higher α N of around 3 qualitatively destabilizes the strong periodicity, yielding a good match of the detailed shape with the experimental result in Fig. 3 (e). The local fringes in Fig. 4 (f) and the pattern in Fig. 5 (f) agree well with experiment for τ c = 380 ps and α N = 5.0, but at the cost of a worse agreement in terms of the hole duration statistics, as will be discussed in Section 4. Generally speaking, the assumption of a static carrier lifetime τ c and linewidth enhancement factor α N limits the accuracy of the model [27,33,34], but as presented in this work it is still an ideal tool to predict and study the dynamical processes of FDML lasers from a qualitative point of view. As can be speculated from the previous discussion, an optimal set of time independent SOA parameters exists with the best match to the experimental intensity pattern in the sense of e.g. an Euclidean distance [38]. Yet, due to the nonstationary nature of the intensity patterns, the development of a suitable cost function is a challenging task and from a physical point of view this approach is questionable anyway. Therefore, such a procedure is not followed in this work. A considerable dependence of the pattern shape on the gain compression parameters α CH , CH,SHB or τ SHB could not be observed. In particular, the contribution of the last term in Eq. (2) and Eq. (3) involving τ SHB is two orders of magnitude smaller than h SHB and h CH .

Spectral signatures of the optical field related to the formation of intensity patterns
In the subsequent discussion, we address the spectral dynamics related to the formation of different patterns, such as holes or fringes in Fig. 2 and Fig. 4, with respect to the sign of the detuning from the ultra-stable regime. The formation of different patterns can be associated with different positions of the instantaneous lineshape in the spectral transmission window of the tunable bandpass filter. The instantaneous lineshape is defined and computed as in [27], and this approach has also been used to reproduce measured lineshapes [19,39]. The lineshape was averaged over 25 roundtrips to reduce spectral fluctuations. In Figs. 6 (a) and (b) the mean frequency of the instantaneous lineshape (MFL), or similarly the center of gravity, at the 40 000 th roundtrip is plotted for both sweep directions and for different signs as well as strengths in detuning. This procedure is iterated at the spatial input and output of the swept FP filter. Note that the results refer to the field in the swept filter reference frame and the linewidth is not broadened by the sweep filter movement [19]. Clearly, each operating regime is associated with either a different position of the mean frequency with respect to the center of the swept bandpass filter, here at 0 GHz, or a more pronounced spectral shift of the lineshape as in the case of the backward sweep in Fig. 6 (a) or forward sweep in Fig. 6 (b), respectively. To exemplarily illustrate the spectral shift of the lineshape by the FP filter, the transformation of the lineshape at the marked positions in Fig. 6 (a) for ±10 Hz is shown in Fig. 6 (c) and (d). The simulation also confirms that the sign of the detuning simply interchanges the position of the mean frequency in the forward and backward sweep and therefore the type of pattern which dominates the intensity trace.
with the signum function sgn. We can now compute δf for ∓100 mHz which is ∓41 MHz in the case of the forward sweep and ±41 MHz in the backward sweep case, respectively. The sign of δf explains the position of the MFL in Fig. 6 (a) and (b) where in the case of δf > 0 the mean frequency is located near the center of the filter but is greater than zero most of the time. Interestingly, in [40] an instable right hand side of the bandpass filter, which is in Figs. 6 (c) and (d) at f > 0, has been discussed in the context of SOA fiber ring lasers. Here, a combination of modulational instability and asymmetric four wave mixing between longitudinal cavity modes in the presence of the frequency dependent loss of the bandpass filter prohibits the instantaneous lineshape to stabilize at frequencies f > 0.
Furthermore, the position of the MFL scales with the strength of the detuning as well as the magnitude of the spectral shift introduced by the FP filter as presented in Fig. 6. Especially the formation of short quasi-periodic patterns is related to a large shift in the MFL compared to the case of localized fringes or hole patterns. The fact that the MFL is nearly constant in a single sweep agrees well with the observation in experiment that the type of pattern does not change over the full sweep. In summary, we have shown that our simulation model reproduces the patterns in different operating modes on a qualitative basis and we discussed that the spectral position of the instantaneous lineshape in the sweep filter is strongly correlated with the type of pattern in the intensity trace. We also computed the spectral shift in the optical field introduced by the detuning and found consistent agreement between the sign of δf and the position of the MFL with respect to the center of the filter transmission window. In addition, the symmetry between backward and forward sweep when the sign of the detuning is changed can be confirmed by an equivalent symmetry of the MFL.

Statistical evaluation of the hole duration
The patterns discussed in Section 2 are not stationary over a long time scale and the hole-type patterns in particular appear to propagate as well as modify their shape over successive roundtrips [23]. the dip. The point t 2 is found by moving forward in time: First, the algorithm moves from the dip to the intersection with the local mean and the intensity trace. Starting from this point the overshoot is tracked until the next intersection with the local mean. Based on this curve, the first local maximum is determined, and from there the algorithm moves to t 2 when the intensity has decreased by 75 % from the local maximum. The algorithm requires a smooth intensity trace since for optimal performance the analyzed curves should be de-or increasing monotonously in the time window of observation. Therefore, the simulated intensity trace was additionally smoothed with a Gaussian filter with a FWHM of 200 GHz in order to remove the impact of ASE noise. The measured intensity traces were upsampled by a factor of ten to remove a discrete timing jitter in the histogram caused by the finite sampling rate of the real-time oscilloscope of 6.25 ps which is also the width of the bins in the histograms.

Results:
We evaluated 800 consecutive backward sweeps of the second setup, corresponding to an observation time of 2 ms. The laser was detuned by −200 mHz to operate in the hole dominated regime. The center wavelength was 1300 nm and the sweep bandwidth was 100 nm. The histogram obtained from the experimental data with the above discussed algorithm is shown in Fig. 7 (b). It can be observed that the hole duration is not a constant quantity and has an asymmetric distribution around a dominating peak which is in Fig. 7 Fig. 7 (d). The loca-tion of the peak, which is in Fig. 7  , and the asymmetric broadening can be well reproduced. The variation in the hole duration can therefore be attributed to the inherent laser dynamics, particularly the frequency-shift-gain-loss interplay. The more pronounced spread around the peak interval in the experimental data can partly be attributed to the wavelength and carrier density dependency of the carrier lifetime τ c or the linewidth enhancement factor α N , which is not included in the simulation next to the time dependency of α N as mentioned in Section 3. Furthermore, the bandwidth of the tunable bandpass filter is wavelength dependent and kept constant in the simulation. The bandwidth has been measured in the second setup and it is found to vary by 20 % over most of the wavelength range. Our simulations show that the position of the peak of the histogram depends on the filter bandwidth and likely results in a broadening of the histogram when varying over the sweep. The above mentioned effects are expected to contribute to the additional broadening seen in experiment from Fig. 7 (b) and (c), yet require complex models prohibiting practical simulation times. The percentage of detected holes which are larger than 200 ps and are thus not shown in Fig. 7 (b) to (d) is maximum 0.37 % in the experimental data sets and 0.57 % in the simulation. The histogram of the hole duration presented above gives a valuable insight in the dynamical quantities of FDML lasers and can also be used to quantify simulation models and their accuracy. The key in this context is that the sign and the sweep direction determines the shape of the intensity patterns and one can control if the intensity trace is dominated by hole formation up to an upper bound when the detuning becomes too large, as shown in Fig. 3. Our results are at least valid for the presented sweep parameters in this work. In other setups with an extremely narrow bandwidth of the bandpass filter of 8.75 GHz it has been shown that also modified hole-type patterns can occur near the lasing threshold [41]. The distribution of the hole duration is also of fundamental interest since the SOA gain medium with its complex microscopic dynamics has a significant impact on the shape of the intensity patterns, as discussed above. Therefore, a qualitative agreement of experimental and simulation data is a necessary condition for the simulation model to be accurate.

Conclusion
We presented various intensity pattern types in rapidly swept FDML lasers and identified their operating conditions. By detuning the laser from the sweep rate when the laser operates in the ultrastable regime, the sign as well as the magnitude of the detuning and the sweep direction determine if the intensity trace of the laser is dominated by holes, localized fringes or short-term quasi-periodic grouping of pulsed patterns. Our experimental results are consistent with numerical simulations, showing that the formation of patterns is a result of the frequency-shift-gain-loss interplay in the laser cavity over a long time scale. By controlling the shape of the pattern, we are able to extract the distribution of the hole duration, giving insight in the dynamical quantities of the complex dynamics in FDML lasers.

Data availability statement
The datasets generated and analysed during the current study are available from the corresponding author on reasonable request.