Coherence properties of short cavity swept lasers

: It has been shown theoretically and experimentally that short cavity swept lasers are passively mode locked. We develop a mathematical model of these lasers and the light ﬁeld solutions are used to predict the coherence length and coherence revival behavior. The calculations compare favorably with data from a 990-1100 nm laser swept at 100 kHz suitable for optical coherence tomography applications.


Introduction
Coherence is the most important issue for a swept laser used for optical coherence tomography. In the case of short cavity swept lasers, the coherence revival properties [1,2] are of interest as well as the dynamic coherence length itself. The maximum imaging depth is limited by the coherence length. Coherence revival peaks can lead to artifacts in images, if imaging systems are not designed carefully. We show in this paper that both coherence length and coherence revival can be accurately calculated from numerical models. The first numerical model of a similar laser was by Bilenca, et al.
[3] that showed theoretically the importance of self-phase modulation [4] in the assistance of blue-to-red tuning [5]. Subsequently, similar models have been developed for a variety of swept laser configurations [6-9]. The model described here is for a linear, rather than a ring cavity. The method of calculating coherence lengths described in [7] is extended to coherence revival phenomena.
The short-cavity swept laser described in [10,11] is passively mode locked. The emitted pulses modulate the gain medium in such a way as to promote short-to-long wavelength tuning. Instead of new wavelengths being built up from spontaneous emission, each pulse hops to a longer wavelength by nonlinear means, tracking the tunable Fabry-Perot filter. This allows very high speed tuning in the blue to red direction. In addition, the pulsation is stable, so these lasers have low relative intensity noise (RIN) and have demonstrated shot-noise-limited optical coherence tomography (OCT) system performance [11,12].
This paper starts with a description of the laser and a brief physical description of how the mode-locked behavior aids rapid wavelength tuning in the short-to-long wavelength direction. A more complete analysis of the laser, based on the mode locking theories of Haus [13,14], is presented along with numerical results from the theory. These calculations are compared with experimental data. The "interference spectrogram" measurement provides strong evidence supporting the model. The model allows accurate calculation of coherence length, an important parameter for OCT, and provides a quantitative treatment of coherence revival [1].

Laser operation
In general, a swept laser consists of a gain medium, a tunable wavelength selection filter, and a laser cavity that supports lasing over the desired wavelength range. The laser architecture is shown in Fig. 1 and contains a reflective MEMS tunable Fabry-Perot filter, a broadband 1060nm gain chip, and a fiber reflector that forms the other end of the laser cavity and serves as the output coupler [10,11]. The filter exhibits a tunable reflection peak due to the filter optical axis tilt with respect to the optical beam [15] and tuning is accomplished by changing the drive voltage on the MEMS filter. The MEMS filter can be tuned from DC up to several hundred kilohertz depending on the desired repetition rate. The fiber extension brings the equivalent air length of the cavity to 104 mm such that there are a handful of laser cavity modes within the filter passband at all times. The laser is constructed in a hermetically sealed butterfly package with a fiber-optic feed-through, and exhibits stable polarization due to the strong TE/TM gain asymmetry of the semiconductor optical amplifier (SOA) chip.
At some speed, a rapidly swept short cavity laser tunes too quickly for lasing to build up anew from spontaneous emission at each new wavelength [16]. A nonlinear optical mechanism is required to shift the wavelength of light circulating within the laser cavity to match the wavelength of the filter on successive round trips. In the case of the laser of Fig. 1, a Doppler shift from the moving MEMS filter mirror does part of the job, although it is small compared to  the wavelength shift required. Most of the shift comes from self-phase modulation induced by depletion of the gain as the mode-locked pulse travels through the semiconductor gain medium. Gain depletion is accompanied by a rise in refractive index. The coupling between the index and the power gain can be described using the linewidth enhancement factor [17], α , as: The mode locking process is illustrated in Fig. 2. The SOA becomes optically longer as the pulse travels through, red shifting the light field. The laser does not tune continuously, but rather hops discretely to the next wavelength on each new pulse. The frequency hop for a SOA of length L is: The pulse energy and width determine the magnitude of the frequency hop. The laser operates in this manner because the lowest threshold is obtained when the pulse frequency hops to follow the filter tuning. A feedback mechanism built into the laser dynamics naturally ensures the pulse frequency hops to follow the filter. The filter linewidth and sweep rate play a critical role in ensuring proper mode locking behavior throughout the sweep. The MEMS filter sweep is linearized to provide a nearly constant sweep rate during the tuning cycle. Stable passive mode locking behavior can be maintained over the 100 nm data collection range of the 1060 nm laser. This is essential for obtaining low relative intensity noise (RIN) and maintaining clean k-clocks for the optical engine.

Mathematical model of laser
The mode-locking theories of Haus [13] are the starting point for the mathematical simulation of the laser. A detailed introduction to Haus' methods is provided in reference [14], where a number of mode locking problems are attacked in a similar way. We follow his basic method, but add the effects of the imaginary part of the gain and add a tunable intracavity filter. Instead of developing a master equation and solving it analytically, we move directly to numerical solution.
The pseudo-ring cavity of Fig. 3 is used to model the linear cavity of the physical device. The diagram and Table 1 define the variables used in the analysis. Two oppositely directed gain blocks with a common carrier density account for the gain of the forward and backward directions. It is assumed that any gain gratings excited through interference between forward and backward waves are negligible. It is expected that a gain grating will dissipate in less than 1 ps through ambipolar diffusion [18]. In any case, the success of the model as constructed confirms that these effects are in fact negligible.
The analysis starts with a rate equation for the dynamics of the semiconductor optical amplifier (SOA). A F (t) is the electric field delivered to the SOA from the swept optical filter; g is the field gain integrated over the length of the SOA (the power gain is 2g); p is the pump; τ the carrier lifetime; E the SOA saturation energy. The pump, p, is normalized so that p = 1 at threshold for a zero tuning rate (R = 0).
The tunable filter is swept mathematically by mixing the light to baseband, low pass filtering, and then mixing back up to lightwave frequencies as shown in Fig. 3. It is done this way to separate the linear and nonlinear parts of the problem of sweeping the filter. This method naturally accounts for the Doppler shift produced by the Fabry-Perot mirror movement. This happens because the low pass filter has a group delay that spaces the up mixing from the down mixing, realizing the Doppler shift. The optical tuning rate R is defined in Fig. 3. The exact optical carrier frequency does not matter in the simulation. Frequencies lower than lightwave are actually used (0-0.63 THz in this paper) to prevent aliasing for numerical integration steps of 0.25 ps. The impulse response of the MEMS tunable filter, translated to DC, is a step exponential as described in Eq. (4) below, where u(t) is the unit step function and B is the FWHM bandwidth of this Lorentzian filter.
The computer code implements this as an infinite impulse response (IIR) filter [19] for efficiency. The simulation variablesĀ i and A i are defined in Fig. 3.
For an integration time step ∆t, the IIR filter coefficients are: Finally, the fields out of the SOA are linked to the fields at the inputs, where l O and l F are the fixed cavity losses and α is the linewidth enhancement factor.
This model has mode-locked solutions in the sense that regular pulsation at harmonics of the round trip frequency are possible. The model does not assume uniform pulsation. It is a general laser model and other (chaotically pulsed) solutions are possible.

Pulsed solutions to the laser model
A numerical solution to the laser model showing mode locked behavior is shown in Fig. 4 for the parameters listed in Table 1, which correspond to a 1060 nm laser swept 110 nm at 100 kHz with a 45% duty cycle. The model is seeded with noise and then run for 10's of nanoseconds at which point it achieves a dynamic steady state. Figure 4 plots the last 2000 ps of the calculation. In this simulation, the laser pulses twice per round trip, strongly modulating the gain. Each pulse "hops" to a new optical frequency, but there is also considerable chirp to the pulses. The computer model fills in the details of the process outlined in Fig. 2.
The computer model is very efficient. A typical 100 ns MATLAB simulation with 0.25 ps integration steps runs in 0.9 seconds of CPU time. This means that searching the space of solutions with varying laser parameters is possible in a reasonable amount of time. An example of this is shown in Fig. 5. Each pixel in the diagram is a unique simulation run. The plot is color coded to show number of pulses in the cavity. The plot clearly shows the pulsation mechanism causes red shifting since stable, pulsed solutions are found only for negative tuning rates (shortto-long wavelength tuning). The results in the unstable regions are legitimate solutions since no assumption about pulsation is built into the model. Those solutions are chaotically pulsed, not driven by noise, since no spontaneous emission effects are built into the model. The number of pulses in the cavity generally goes up with the pump level. Higher pulse energies modulate the gain medium more, increasing the hop. If the hop becomes too great for stability, the laser will shift to a larger number of pulses, because distributing the energy between more pulses will decrease the hop. Figure 5 plots out the laser behavior for the design parameters in Table 1. Other designs and higher speeds are possible. Single pulse per round trip designs are helpful for stability. Higher speeds pressure the laser designer to shrink the cavity length and broaden the tunable filter bandwidth. The bandwidth needs to be broadened to shorten the pulses to fit inside the smaller cavity. These constraints reduce coherence length and bring coherence revival peaks to lower depths. Both of these trends are undesirable, but may be acceptable depending on the application. Modeling these designs before construction is very helpful. Sweeps faster than 200 kHz have  been demonstrated, but at some point designs will become untenable.
Pulsation with 1, 2 and 3 pulses per round trip has been observed in these lasers. Photodiode traces for 1 and 2 pulses are shown in Fig. 6. These were taken with a high speed photodiode and 2.5 GHz bandwidth oscilloscope sampling at 10 GS/s. The points between samples were filled in using band-limited interpolation. The SOA bias current was 100 mA for the N=1 example, and 220 mA for N=2. This observation qualitatively confirms the prediction of Fig. 5.
Another way of looking at the pulsation data is by plotting the spectrogram of the photodiode signal. This shows the pulsation behavior in relation to the sweep over wavelength and shows that pulsation is stable across the wavelength band. The cavity mode spacing corresponding to Fig. 7 is 1.4 GHz so beats are observed at harmonics of 1.4 GHz for N=1 pulsation. For N=2, pulsation is seen at 2.8 GHz, twice the mode spacing. Harmonics of 2.8 GHz are outside the bandwidth of the oscilloscope. Many laser parameters, including the saturation energy, carrier lifetime, linewidth enhancement factor, filter linewidth, cavity loss, etc. are varying over the wide tuning range. The fact that pulsation is stable over the whole wavelength range in the face of these variations is an important demonstration. The spectrogram is an important diagnostic tool since unstable lasers do not maintain clean pulsation, which shows up as a distinctive disruption in the spectrogram. Maintaining uniform pulsation is essential to obtain low relative intensity noise (RIN). The spectrogram in Fig. 7d shows a low noise floor between DC and 200 MHz. These are the frequencies used for OCT detection and low RIN in this frequency range is essential.
All of the results in this paper are for passive mode-locking induced by a rapid sweep of the tunable Fabry-Perot filter. As an aside, these lasers have also been actively mode locked, while swept, by modulating the SOA injection current in synchronism with harmonics of the cavity round trip frequency [20]. This has a stabilizing effect, especially on the pulse repetition rate which varies slightly over the sweep in the case of pure passive mode locking. Aside from this, the active locking achieved did not change the behavior of the laser significantly from its passively locked state.

Coherence properties
Output from the model is the light field's magnitude and phase, a complete description of the laser emission. Any experiment that can be performed in the laboratory can be simulated with these results. For example, Fig. 8a shows the results of a coherence calculation using the output from the simulation that produced Fig. 4. Experimental values for a 1060 nm linear cavity laser modified with the filter output port shown by dashed lines in Fig. 1 are 13 mm through the normal output coupler and 26 mm through the filter. Theoretically, a coherence length of 13 mm is found for light emitted through the output coupler (A O ) and 29 mm at the MEMS filter output (A F ) for the laser parameters of Table 1, but with p = 3.4. Changing the pump level, p, is done for a better fit. The experimental data in this paper come from three different, but similarly constructed, lasers. They would all have parameters similar to, but not exactly the same as, those  Table 1, but with p = 3.4.
in Table 1. All the calculations in this paper use the parameters of Table 1, unless otherwise noted.
The dynamic coherence calculation proceeds the way an experiment would. First simulated pulses are obtained. They are interfered over a range of time delays and the RF beat on a photodiode between DC and 900 MHz is calculated. This is wide enough that the simulated measurement produces the actual laser-limited coherence length, not being affected by a virtual instrumentation bandwidth limit.
In a coherence measurement, a multimode laser source will exhibit periodic peaks in interference fringe amplitude at multiples of the cavity length. This is sometimes referred to as coherence revival in the literature [2]. In the case of a multimode laser with organized pulsation coherence revival can be thought of as interference between neighboring mode-locked pulses. The situation, which is more complicated than that simple explanation, can be mapped out by the interference spectrogram shown in Fig. 9a. A variable path interferometer is set up and monitored by a high-speed photodiode. The RF spectrum of the output vs. interferometer depth is plotted in a spectrogram style. All of the interference components that lead to coherence revival peaks can be traced out to their origin. A simple swept light source producing a sinusoidal interference waveform in time would generate the diagonal line f = 2DR/c in the interference spectrogram, where R = the sweep rate, and 2D = the interferometer path mismatch. The far more complex interference spectrogram of the short cavity laser is due to its mode locked behavior. The interference waveform can be thought of as a sampled sine wave due to the short pulses. The sampling theorem [21] says that the spectrum of a time-sampled signal is duplicated at multiples of the sampling rate. Tracing diagonal lines through the interference spectrogram shows that aliased frequencies can stray into the OCT system's receiver bandwidth to create artifacts. The artifact map (Fig. 9c) shows the OCT system engineer where to avoid placing reflecting optics to avoid image artifacts.
The raw interference spectrogram in GHz shown in Fig. 9a, can be resampled into the depth interference spectrogram of Fig. 9b. An OCT receiver with a 4 mm imaging window will see the display peak magnitude seen in Fig. 9c. The display peak is the maximum signal between 0.4 and 4 mm seen in Fig. 9b.
The output from the model plotted in Fig. 4 can be used to compute an interference spectrogram (Fig. 10a) and artifact map (Fig. 10b). Comparison of the experimental results in Fig. 9 to the  theoretical results of Fig. 10 shows remarkable agreement.
The results in Figs. 9 and 10 are for a two pulse per round trip laser. There are two interleaved, semi-independent pulse trains in the cavity. Adjacent pulses are not amplified copies of one another. Pulses in a one pulse per round trip laser are all related through direct amplification: father, son, grandson. . . Pulses in the two pulse per round trip laser might be described as: father, brother, son, nephew. . . This means that interference between pulse trains an odd number apart is different than those an even number apart. In particular, odd interference is noisier because the pulses are only loosely related, rather than directly amplified descendants. This is why there are diamond-shaped voids in the interference spectrogram (Fig. 9a) centered at y = 1.4 GHz with x = 104, 208, 312 mm . . .

Summary
Theory and experiment both show that short-cavity swept lasers are passively mode locked. The emitted pulses modulate the gain medium in such a way as to promote short-to-long tuning. Instead of new wavelengths being built up from spontaneous emission, each pulse hops to a longer wavelength by nonlinear means, tracking the MEMS tunable Fabry-Perot filter. This mechanism produces a low RIN laser with speed and coherence lengths suitable for a wide range of optical coherence tomography applications.
The theoretical model can make many detailed predictions that match the observed laser behavior, such as calculations of relative intensity noise and chaotic behavior. Most importantly, the model successfully predicts the coherence behavior of these lasers. It describes the constraints on the coherence length due to the pulse width and enables detailed calculation of the pulse chirp, which further restricts the coherence length. The model successfully predicts coherence revival behavior and makes the physical origin understandable as interference between neighboring mode-locked pulses.
Two experimental techniques, the spectrogram and the interference spectrogram, were applied to track evolution of the laser operation during the wavelength sweep. These methods are potentially useful in the characterization of other swept sources.
The theoretical understanding of this laser is important in allowing further commercial development of this technology and in enabling exploration of new designs, leading to performance improvements and more cost effective production.