Ultrafast lasers have a crucial function in many fields of science; however, up to now, high-energy pulses directly from compact, efficient and low-power semiconductor lasers are not available. Therefore, we introduce a new approach based on temporal compression of the continuous-wave, wavelength-swept output of Fourier domain mode-locked lasers, where a narrowband optical filter is tuned synchronously to the round-trip time of light in a kilometre-long laser cavity. So far, these rapidly swept lasers enabled orders-of-magnitude speed increase in optical coherence tomography. Here we report on the generation of ~60–70 ps pulses at 390 kHz repetition rate. As energy is stored optically in the long-fibre delay line and not as population inversion in the laser-gain medium, high-energy pulses can now be generated directly from a low-power, compact semiconductor-based oscillator. Our theory predicts subpicosecond pulses with this new technique in the future.
Compact and reliable ultrashort pulse, high-energy laser sources1 are of great importance for many fields in science and engineering, such as material processing, biomedical optics or optical communication. Compared with other gain media, the application of fully semiconductor-based laser sources can have many significant advantages owing to optimal compactness, robustness, cost-effectiveness, very-high wall-plug efficiency, simple current pumping, broadband gain bandwidth or spectral versatility. Since their first demonstration2, there has been great effort to improve the performance of ultrashort pulse generation with semiconductor lasers, reducing the pulse duration and increasing the output power3,4. Active-, passive- or colliding-pulse mode-locking techniques5,6 were the prerequisite for the generation of few picosecond and subpicosecond pulses, considerably impeded due to non-linear phase changes associated with gain saturation4. So far, the shortest pulse durations from semiconductor lasers, reaching several tens of femtoseconds, have been demonstrated with optically pumped, passively mode-locked, vertical external-cavity surface emitting lasers (VECSELs)7 based on wideband semiconductor saturable absorber mirrors8.
However, a drawback of conventional mode-locked, q-switched or gain-switched semiconductor laser systems are the comparably low pulse energies as a direct consequence of the short carrier-relaxation time (~100 ps), high gain and low saturation energy in the semiconductor gain medium4,5,9. A major problem is that short carrier-relaxation times in the semiconductor gain media impede the realization of low repetition rate mode-locked systems. Thus, semiconductor lasers with higher pulse energies have very high levels of average output power, making them impracticable for many applications where compact, low-power light sources without extensive infrastructure are desired. In general, pulse energy can be increased using semiconductor-based master oscillator power amplifier systems. Whereas in case of single-spatial-mode semiconductor optical amplifiers (SOAs) the achievable pulse energies typically range from a few pJ to few tens of pJ3,9, higher energies become feasible using multispatial-mode semiconductor-based amplifiers like inverse bowtie SOAs10 or tapered amplifiers11,12,13,14. Increased pulse energies can be achieved using the semiconductor-based eXtreme chirped pulse amplification concept9, where pulses are stretched temporally before amplification and the output energy of the recompressed pulse then exceeds the fundamental energy-storage limit of the semiconductor gain medium. Besides edge-emitting diodes, ultrashort pulses can be generated with optically pumped, passively mode-locked VECSELs15 exhibiting increased saturation power and, thus, allowing for comparably high average output powers and pulse energies16. However, disadvantages of optical pumping can be, especially at high average power, a lower wall-plug efficiency and the need for a more sophisticated thermal management. In this work we introduce and investigate a fundamentally new concept of short-pulse generation based on Fourier domain mode-locked (FDML) lasers17,18, a recently introduced novel type of rapidly wavelength-swept lasers. Whereas picosecond or subpicosecond pulses with energies exceeding several nJ are hardly achievable with state-of-the-art semiconductor-based lasers and amplifiers, the FDML approach provides the potential for pulse energies reaching 100 nJ at repetition rates of several 100 kHz directly from the electrically pumped semiconductor-based oscillator.
Fast wavelength-swept lasers with tuning rates of 0.1–1 MHz and sweep bandwidths of ~100 nm in the near infrared are the light sources of choice for today’s fastest optical coherence tomography (OCT) systems. OCT is a novel technique for high-resolution biomedical imaging, which finds widespread use from research set-ups to commercial systems already in clinical practice19,20,21.
In standard wavelength-swept lasers, a narrowband optical filter is tuned over the gain spectrum of the laser medium. Lasing has to continuously build up from amplified spontaneous emission (ASE), causing a fundamental limit to the achievable sweep speed22. This limit can successfully be overcome using FDML lasers17,18, allowing for an up to 100-fold increase in sweep speed and, therefore, ultra-high OCT imaging speeds23. A main element of FDML lasers is the fibre delay line of ~1 km in the laser cavity, enabling the synchronization of the filter sweep frequency and the round-trip time of light in the cavity. This concept is similar to the one presented by Telle and Tang24. The ideal FDML laser is a semiconductor-based swept laser in a stationary operation mode, emitting periodic continuous-wave (cw) wavelength sweeps equivalent to highly chirped laser pulses. In this work we report for the first time on the optical compression of these sweeps, which is accomplished using a 15 km long, highly dispersive optical fibre. In this way we introduce a new approach of ultrashort pulse generation, with a very unique set of performance parameters regarding repetition rate and output pulse energy. The combination of low repetition rate, high pulse energy and high plug-wall efficiency might open access to many new applications.
Standard mode-locked semiconductor-based lasers suffer from comparably low pulse energies because of the energy-storage limit in the semiconductor gain medium. However, FDML picosecond or subpicosecond pulses would have a unique advantage, as here the entire wavelength sweep with energies of up to 1 μJ is optically stored within the long delay fibre in the laser cavity and not as polulation inversion in the gain medium. Similar to eXtreme chirped pulse amplification concepts9, FDML lasers therefore have the potential to overcome this limit. Apart from high pulse energies, another advantage is that FDML lasers in principle enable direct access to the phase, that is, the chirp of the sweeps/pulses via control of the tunable bandpass filter and direct access to the amplitude by the current of the semiconductor gain medium. This means an FDML laser has a pulse shaper inherently built in. It should be emphasized that this method of ultrashort pulse generation is fundamentally different from conventional mode locking, such as the well-known soliton mode locking and stretched-pulse (dispersion-managed) solutions to the Haus master equation25, as well as the recently discovered all-normal-dispersion26, similariton27 and soliton-similariton28 regimes. Besides the fact that this concept might be an attractive candidate for future high-energy, low repetition rate semiconductor pulse lasers, our findings, that is, the measured compression factor, also provide valuable insight into the coherence of the FDML output.
In this work we demonstrate that pulse duration very critically depends on the coherence properties of the FDML laser, currently allowing for temporal compression down to 60–70 ps. The experimental results are compared with numerical simulations based on a theoretical model of the FDML laser29. Simulations also enable us to give an outlook how pulse duration might significantly be improved in the future.
Temporal compression of the FDML sweeps
The FDML resonator in Fig. 1a uses a 1,550±50 nm SOA as gain medium and a bulk Fabry–Pérot tunable filter (BFP-TF)23 with a finesse of ~400 for very-high sweep speeds of νsweep~250 nm μs−1 at 2 × 390 kHz repetition rate. The 524 m fibre delay line is dispersion compensated to improve the coherence of the laser30,31. Light from the FDML output passes a second, single polarization (16 dB polarization dependence) booster SOA, which ensures a stable polarization state and acts as an optical switch, only transmitting and amplifying a part of each forward (short to long wavelengths) sweep. This determines the temporal width of the sweep τsweep, the sweep bandwidth Δλsweep and the centre wavelength λc of the sweep used for temporal compression. Our optical compressor is a 15 km long dispersion compensation fibre (DCF), which is passed either once (‘1 × pass’) or four times (‘4 × pass’). To ensure optimum compression, the filter-drive parameters have to be chosen very accurately (see Methods). An additional chirp, which might be introduced by the rather slow ~18 ns switching of the SOA gain, should be very small and can, if necessary, be compensated by fine tuning of the filter-drive parameters (see Methods). For all measurements, a good polarization contrast is observed after the optical compressor. Fig. 1b shows the wavelength tuning of the bandpass filter for 1 × pass of the DCF (amplified part red). The pulses are analysed with a fast-sampling oscilloscope and a second-harmonic intensity autocorrelator.
Internal FDML parameters and their impact on pulse duration
Figure 2a shows the ~127±3 ps apparatus function of the sampling oscilloscope and photodiode, measured with pulses from a titanium sapphire laser (see Methods). Figure 2a also shows the signal from the shortest pulse (full width at half maximum (FWHM) ~144±3 ps). By numerical deconvolution, a duration of ~68±12 ps (assuming a Gaussian shape) can be estimated. The uncertainty of the pulse duration caused by electronic measurement and deconvolution was calculated from the fluctuations of the pulse duration on the sampling oscilloscope via error propagation methods (here ±12 ps). In this measurement, light passed the DCF once and the bandwidth Δλsweep is set to ~6 nm (τsweep~24 ns) to reduce effects of higher-order dispersion in the DCF. Figure 2b show the pulse durations for frequency detunings ranging from −1 to 1 Hz. It is obvious that already a slight detuning of the FDML filter-drive frequency on the order of Δf~0.1 Hz (that is, a relative frequency change of ~3 × 10−7) results in considerable temporal broadening of the pulses (increase of pulse duration by a factor of ~1.5). These values of detuning are far too small to significantly change the amount of dispersion required for compression. This shows that internal laser dynamics governs the output of the FDML sweep rather than only specifications of the individual components, for example, the sweep filter. As shown in Fig. 2c, the experimental values found with our set-up are in good agreement with numerical simulations using the theoretical model described in 29 and 32, even though the discrepancy between theory and experiment reaches a factor of up to ~2. The deviation of the minimum value of the achievable pulse duration measured in the experiment to the minimum value predicted by theory is ~30%. However, the observed asymmetry for detuning is much less pronounced in the theory; only the trend is predicted qualitatively. To mimic experimental conditions (sampling oscilloscope) and suppress fluctuations, the theoretical pulse durations have been achieved by averaging the simulated intensity profiles of over 20 non-sequential pulses, evaluating every 100th round trip. In this way a variation of the pulse duration of up to ~20% is observed, depending on the chosen trigger condition for averaging, giving an idea of the uncertainty of the presented theoretical data.
For a more systematic investigation, Fig. 2d shows the minimum achievable pulse duration τpulse in case of different spectral widths ΔλFilter of the BFP-TF for 1 × pass (Δλsweep~6 nm, τsweep~24 ns, νsweep~250 nm μs−1) and 4 × pass (Δλsweep~1.5 nm, τsweep~24 ns, νsweep~62.5 nm μs−1) of the DCF fibre compressor. With a fixed finesse of ~400 (mainly given by the coatings) the filter width ΔλFilter can be adjusted by adapting the free spectral range (FSR) of the BFP-TF. Depending on ΔλFilter, the FSR therefore ranges from ~40 to ~170 nm. The measurements indicate that τpulse correlates to a certain extent with ΔλFilter (smaller ΔλFilter yields smaller τpulse), but also with νsweep (larger νsweep gives shorter τpulse for same ΔλFilter). Currently, there are two different limits to the filter width ΔλFilter accessible in the experiment: (1) For the case of 1 × pass of the DCF fibre compressor, ΔλFilter was not reduced below ~315 pm, as the necessary further reduction of the FSR requires an increase in mechanical excursion of the BFP-TF, which is not possible owing to mechanical restrictions. (2) For the case of 4 × pass of the DCF fibre compressor, mechanical restrictions are currently not limiting. The 4 × pass generates 4 × more dispersion; hence, it can compensate a sweep/pulse with 4 × more chirp. A 4 × higher chirp of the sweep translates to a 4 × slower sweep speed of the FDML laser, reducing the mechanical requirements for the filter. Hence, in the case of 4 × pass we were not able to reduce ΔλFilter below ~110 pm, because the centre wavelength λc of the sweep does not coincide with the gain maximum and, therefore, smaller values of FSR induce lasing at wrong wavelengths. In the future, the application of a BFP-TF with higher finesse can enable considerably smaller filter widths ΔλFilter. Currently, a minimum pulse duration τpulse of ~68±12 ps has been achieved in both cases, that is, in case of 1 × pass and of 4 × pass.
Intensity autocorrelation and continuum generation
To confirm the electronic measurements with autocorrelation data, we applied erbium-doped fibre amplification (EDFA) to raise the power levels (1 × pass of DCF). To minimize non-linear effects and prevent pulse break-up, we limited our EDFA gain to ~16 dB, resulting in a pulse width τpulse~59±12 ps, measured with the sampling oscilloscope. Figure 3a shows the corresponding autocorrelation signal, which exhibits two features. A narrow coherence spike (consistent with the bandwidth limit of ~600 fs) is centred on top of a broad pedestal (FWHM, autocorrelation time τAC~89 ps; contrast ratio ~1:1). This implies a pulse width of τpulse~63 ps consistent with the results acquired with the sampling oscilloscope, but also indicates the existence of an intensity substructure33,34 caused by internal FDML dynamics (see Discussion)33,34. Note that the uncertainty of the autocorrelation measurement itself can be considered negligible. However, the error caused by the unknown pulse shape can be estimated to be ~10 ps (Gaussian: 63 ps; Sinc2: 67 ps; Sech2: 58 ps). The average power and the instantaneous power after the DCF are ~2.2 mW and ~96 W, respectively. This corresponds to a pulse energy of ~5.6 nJ at a pulse repetition rate of 390 kHz. Figure 3b (top) shows the spectrum after the booster SOA, whereas Fig. 3b (bottom) demonstrates the potential of continuum generation, realized by propagation of the pulses (pulse energy ~11 nJ) through 1.7 km of dispersion shifted fibre. We observe remarkable spectral flatness over ~100 nm even on a linear scale. Although continua can even be generated with cw-sources, the extremely low average power of our source might be attractive in several configurations.
From a laser physics point of view, the observed pulse duration τpulse gives a direct indication of the degree of coherence in the FDML sweep. In case of a fully incoherent wavelength sweep, the shortest achievable pulse length τinc for the compression is determined by the instantaneous linewidth Δλinst (refs 32,35) and the sweep speed νsweep (see Methods), as sketched in Fig. 4a (dark green curve, 1 × pass; dark blue curve, 4 × pass). Temporal compression of a per se incoherent wavelength-swept ASE36,37 with adjustable instantaneous linewidth36 (see Methods) demonstrates very good agreement with this model (see crosses in Fig. 4a). Assuming equal filter width, the minimum pulse lengths achieved with incoherent light are ~20 × longer than with FDML (compare achievable pulse durations for incoherent wavelength-swept ASE (Fig. 4a) and for FDML operation (Fig. 2b)). In case of a fully coherent wavelength sweep with Δλsweep~6 nm (1 × pass) or Δλsweep~1.5 nm (4 × pass), the minimum achievable pulse durations τcoh are ~600 fs or ~2.4 ps (see Methods), respectively; both are considerably smaller than the ~68 ps achieved with FDML. This indicates that the FDML sweep has non-coherent contributions in its amplitude and phase evolution. However, a purely incoherent sweep can never generate a τpulse of ~68 ps, as illustrated in Fig. 4b (only 4 × pass), as all values of the dark blue curve (expected pulse duration for incoherent sweep, see Methods) are at least ~5 × larger. On the basis of the measured instantaneous linewidth31,35 (not filter width) of ~10 pm (see Methods), the red oval indicates the approximate operation range of the FDML laser, clearly located within the parameter range of partial coherence (yellow area).
Hence, the entire FDML sweep is at least partially or section-wise coherent. The question whether the electric field of the FDML sweep consists of many sections with internally high phase and amplitude stability, but random phases between each other, or if random phase and amplitude fluctuations have some increased probability near the optimum phase and amplitude evolution cannot be judged at this point with certainty. However, as we operate our FDML laser in the ultra-stable regime as observed by Kraetschmer et al.38, it can be speculated that the FDML sweeps show a section-wise very good coherence, interrupted by wavelength/phase discontinuities. This assumption is supported by the observation of interference fringe signals from a Mach–Zehnder interferometer, where sections of very smooth and clean fringe signals can be observed, separated by more unstable regions of phase discontinuities. After compression, the weakly defined phase between the different sections causes strong amplitude fluctuations, which have also been observed in the numerical simulations.
To conclude, in this work we demonstrate for the first time the temporal compression of the wavelength-swept cw output of an FDML laser. Approximately 60–70 ps pulses at a repetition rate of 390 kHz have been achieved. Because of uncompensated higher-order chirp, only 6 nm sweep range has been used, which currently is the reason for limited pulse energies of ~140 pJ (without EDFA) and ~5.6 nJ (with EDFA). Concerning the achievable pulse energy, these results cannot compete with master oscillator fibre power amplifier systems39, but are comparable to semiconductor-based, optically pumped VECSELs15,16 or semiconductor-based master oscillator power amplifier systems12,14. Nevertheless, it should be emphasized that in spite of current experimental restrictions, already the ~140 pJ is higher than typical pulse energies achievable with conventionally mode-locked SOA-based semiconductor lasers9, even in the case of using an eXtreme chirped pulse oscillator approach40. In the future, the additional application of chirped fibre Bragg gratings30 for arbitrary dispersion compensation might make >100 nm sweep ranges usable, which would already push the pulse energy of the ~60–70 ps long pulses directly from this laser oscillator without an amplification well into the nJ range, even including compressor losses30. Moreover, the additional use of chirped fibre Bragg gratings could minimize non-linear effects in the optical compressor and, therefore, reduce or prevent the effect of pulse break-up limiting pulse energy. For example, if the final compression from nanoseconds to picoseconds is performed by a chirped fibre Bragg grating at the very end of the fibre set-up, the propagation length of the pulse at maximum peak power can be minimized. Alternatively, a final bulk optics compressor may be used.
Using an EDFA, a pulse energy of more than 11 nJ is feasible tolerating degradation of pulse duration. This is sufficient for the generation of a spectrally flat continuum. The ~60–70 ps pulse duration is far shorter than the result expected from purely incoherent superposition, indicating, at least partially coherent superposition of the different spectral components of the FDML sweep during the compression process. The experiments are in good agreement with numerical simulations, which predict that <10 ps pulses can be achieved with the current set-up by simply using a more narrowband sweep filter (higher finesse). Remarkably, an even further reduction in pulse duration is predicted by theory, reaching almost bandwidth-limited pulses if, in addition to a more narrowband filter, the cavity dispersion is further reduced. These findings indicate that, in the future, modifications in the experiment will enable the generation of considerably shorter pulses well into the lower picosecond range.
However, it should be emphasized that even the 60–70 ps pulse durations of our set-up presented here are highly attractive for many applications. Usually, picosecond pulses have a very narrow spectrum due to the time-bandwidth product; this makes them prone to parasitic, coherent, non-linear effects like Brillouin scattering41, which limits the maximum achievable pulse energy, especially in fibre-based delivery systems. Moreover, all applications that are susceptible to interference caused by spurious reflections greatly benefit from the wide spectral bandwidth. In imaging applications the wide spectrum reduces problems with speckle noise, whereas in sensing applications it reduces spectral modulations caused by back reflections.
Considering the described potential for improvement, FDML lasers might become an interesting option as compact, all fibre-based and highly robust source for ultrashort pulses. Even though pulse energies from the FDML laser without EDFA currently do not reach the performance of state-of-the-art semiconductor-based pulse laser systems9,15,16, the low repetition rate of FDML lasers, the fact that they are constructed of all fibre-based low-cost telecom components and the fact that they are electrically pumped may make them highly attractive for applications that demand compactness, low power consumption and robust design.
FDML laser resonator and booster SOA
The optical delay line in the FDML resonator consists of a 246 m standard, single-mode fibre (Corning SMF 28) and a 16 m DCF (OFS-HFDK) used to minimize residual dispersion in the resonator in the wavelength range around 1,560 nm to less than 0.25 ps nm−1. A σ-ring configuration is used with an optical circulator and a Faraday rotation mirror; hence, light passes the delay line twice during one resonator round trip and birefringence effects in the delay line are cancelled. The circulator and the optical isolator ensure unidirectional lasing. Fifty per cent of the optical power is coupled out via a fused fibre coupler. The gain of the SOA in the cavity is polarization-independent (Covega SOA1117), whereas the gain of the booster SOA is polarization-dependent (Covega BOA1004). Polarization controllers enable adjustment of the polarization state in the fibre. A two-channel, phase-locked voltage control is used. Channel 1 (DC+AC voltage) generates a sinusoidal wavelength tuning of the BFP-TF. The DC voltage is actively feedback-controlled to stabilize the centre wavelength λc, using an optical spectrum analyser. Channel 2 yields a synchronous switching of the gain of the booster SOA via a fast current modulator (Wieserlabs WL-LDC10D). Light is either amplified from ~10 mW average power before the booster SOA to ~47 mW instantaneous power (SOA on) after the booster SOA or suppressed by >10 dB (SOA off).
Temporal compression and filter-drive parameters
The optical compressor is an ~15-km long, highly dispersive DCF fibre (OFS type HFDK). Light passes the fibre once (‘1 × pass’) or four times (‘4 × pass’). For all experiments, a centre wavelength of λc=1,560 nm is chosen, as this is a good trade-off between large amount of negative dispersion (−4 ns nm−1) of the DCF, minimal loss (approximately −8.5 dB) and good dispersion compensation in the FDML resonator. The ~211 nm total-filter sweep range of the BFP-TF is set such that the filter sweep speed νsweep(λc) equals the inverse of the magnitude of the dispersion in the DCF compressor. To guarantee minimal loss, a very high sweep speed of the BFP-TF is crucial, as it minimizes the required DCF fibre length. However, some measurements are performed with light passing ~15 km of DCF fibre four times (two directions and two orthogonal polarizations) using an optical circulator and a polarization beam splitter. In this case, dispersion and loss are increased and the required total filter sweep range (~53 nm) is reduced by a factor of 4. Because of the dispersion slope of the DCF and the sinusoidal drive of the optical filter, there exists a limit for temporal compression τcom(Δλsweep) caused by remaining higher-order dispersion, depending on the chosen bandwidth Δλsweep. Besides accurate adjustment of the total filter sweep range, a reasonable choice of the phase difference ΔΦ between filter-drive and booster SOA modulation signal is important, so the curvature of the drive can partly compensate the dispersion slope in the DCF. Simulations using the measured dispersion42 of the DCF predict τcom <10 ps for an optical sweep bandwidth Δλsweep~6 nm (1 × pass) and Δλsweep~1.5 nm (4 × pass).
Erbium-doped fibre amplification
The EDFA used in this experiment consists of ~6 m of doped fibre (MetroGain M12-980-125) and two wavelength division multiplexers. Pump light from a pump diode (JDSU-S34) at 1,457 nm is used in forward direction. Benefiting from a small duty cycle (~1%), an amplification of the truncated sweep (τsweep~24 ns, Δλ~6 nm) by at least 16 dB is easily possible. By far higher amplification factors can be achieved, but the spectral shape and the pulse duration degrade because of non-linear processes in the compression fibre. However, for our experiment the 16 dB amplification is sufficient to measure a second-harmonic intensity autocorrelation.
Sampling oscilloscope and autocorrelator
For measurement and characterization of the pulse duration, two methods are used. First, a direct electronic measurement using an 8-GHz photo receiver (Kyosemi Corporation KPDX10GV3) and an analogue-sampling oscilloscope (Tectronix 7603). Second, a home-built second-harmonic intensity autocorrelator. For the electronic measurement, 1% of the compressed light is adjusted in power by a bulk optic variable attenuator. To guarantee proper triggering of the sampling oscilloscope (~25 ps rise time), the train of pulses is split by a 50/50 fibre coupler; one part is directly used to trigger the oscilloscope (2 GHz photodiode (Wieserlabs WL-PD2GA)), the other part is optically delayed by a length of ~100 m single-mode fibre and measured with the fast photodiode. To determine the apparatus function of photodiode and oscilloscope, short pulses (several picoseconds) from the amplified output of a 74 MHz titanium sapphire oscillator (Femtolasers Integral, 975 nm part, post-amplified by Yb fibre amplifier—set-up described in 43) were measured in the same manner yielding an FWHM of ~127±3 ps. The home-built second-harmonic intensity autocorrelator for measurements up to 500 ps pulse length uses a BIBO crystal in a non-collinear geometry for frequency doubling, generating a background-free signal. The frequency-doubled signal is detected with an integrating silicon photodiode (Wieserlabs WL IPD4A) adjusted to 250 μs integration time and a read-out rate of ~900 Hz. Each data point is a 100 times average.
Theoretical model of pulse duration after compression
To gain a comprehensive understanding of pulse generation based on temporal compression of wavelength-swept waveforms, it is important to theoretically analyse the minimum achievable pulse duration for both limiting cases, assuming fully coherent and fully incoherent sweeps. This is the aim of the following description.
For a fully coherent sweep, which is 100% identical to a highly chirped laser pulse, the minimum achievable pulse duration τcoh is given by the total sweep bandwidth and the time-bandwidth product (0.44 for a Gaussian envelope):
To estimate the shortest pulse duration that can be achieved by compressing a fully incoherent wavelength-swept waveform, we assume a tunable optical bandpass filter with a spectral width ΔλFilter where the filter passband is swept over a spectral range Δλsweep in a time τsweep. The tuning speed is assumed to be constant:
Here the ratio between filter bandwidth ΔλFilter and tuning range Δλsweep defines the number of resolvable wavelengths n=Δλsweep/ΔλFilter within the sweep. Note that in the case of tunable lasers with mode competition, the output can exhibit a more narrowband instantaneous linewidth Δλinst than the width of the filter ΔλFilter. Therefore, in general, the number of resolvable wavelengths has to be defined as n=Δλsweep/Δλinst. The minimum pulse duration achievable by compressing fully incoherent wavelength sweeps is determined by two different limits (limit 1 and limit 2), which are discussed separately in the following. Depending on the sweep speed and the spectral filter width, either limit 1 or limit 2 is the dominant contribution.
Limit 1, which arises as a direct consequence of wavelength sweeping, is the dominant contribution in the case of a sufficiently slow sweeping operation or a sufficiently wide filter. In this case, the resolvable spectral sections in the wavelength-swept waveform are equivalent to resolvable temporal sections. Consequently, we can define a switching time of τswitch=τsweep/n, where τswitch is the time the filter needs to sweep over its bandwidth. As amplitude and phase of the optical waveform fluctuate randomly during each time interval τswitch, this, under given assumptions, is the shortest pulse duration that can be expected by compressing a wavelength sweep generated from an incoherent light source. Hence, the minimum achievable incoherent pulse width τinc=τswitch is determined by the filter width (or the instantaneous linewidth) and the sweep speed:
More narrowband filters and higher sweep speed will result in shorter pulses.
However, there is an additional condition determining the minimum pulse duration. The time-bandwidth limit (limit 2) sets a lower boundary to the shortest pulse duration, and in incoherent swept sources the filter width ΔλFilter or the instantaneous linewidth Δλinst determines this limit (in fully coherent sources it is the sweep width Δλsweep). Limit 2 is the dominant contribution assuming a sufficiently high tuning speed (or sufficiently narrowband filter widths). In this case, we can estimate the minimum achievable pulse duration for a Gaussian shape at a centre wavelength of λc and the speed of light c:
In general, the minimum achievable pulse duration τinc of fully incoherent swept sources can be obtained by convoluting both contributions (limit 1 and limit 2). Assuming Gaussian-shaped temporal windows (τinc is FWHM), the minimum pulse duration reads:
Experimentally, fully incoherent swept sources can be realized by filtering temporally incoherent light, such as a thermal source or, in our case, spontaneous emission from an SOA. Such a source has been described in 36 and 37. In such sources, the instantaneous linewidth Δλinst is always identical to the filter width.
Instantaneous linewidth of the FDML laser
Measuring the instantaneous linewidth Δλinst of a rapidly wavelength-swept light source is not trivial, especially as, at very high sweep speed, the values approach the time-bandwidth limit35. One approach, that is very common in swept source OCT, but yielding only an average value for the instantaneous spectrum (average over one sweep), is the following: the interference signal from a Michelson interferometer is acquired and a combined analysis of phase jitter and loss in fringe contrast using a Fourier transform is used to determine the coherence length, and with it the instantaneous spectrum. As the fringe frequency increases beyond our measurement bandwidth (1 GHz—Tektronix real-time oscilloscope type DPO 7104), for a very narrow instantaneous linewidth Δλinst, we cannot measure down to the −6 dB coherence roll-off, but only to ~−0.5 dB. Extrapolation of our data allows for a rough estimation of Δλinst to ~10 pm, which corresponds to ~1.2 GHz at 1,560 nm. This value is used in the diagram in Fig. 4b to estimate the level of coherence in the FDML sweep.
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We acknowledge support from Professor W. Zinth at the Ludwig–Maximilians University Munich. This research was sponsored by the Emmy Noether programme of the German Research Foundation (DFG—HU 1006/2-1), the project OCT-Labs (DFG: HU1006/3-1), the European Union project FUN OCT (FP7 HEALTH, contract number 201880) and FDML-Raman (ERC, contract number 259158). S.T. and C.J. acknowledge support from Professor P. Lugli at the TUM and by the German Research Foundation (DFG) within the Emmy Noether programme (JI 115/1-1) and under DFG grant number JI 115/2-1.
The authors declare no competing financial interests.
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Eigenwillig, C., Wieser, W., Todor, S. et al. Picosecond pulses from wavelength-swept continuous-wave Fourier domain mode-locked lasers. Nat Commun 4, 1848 (2013). https://doi.org/10.1038/ncomms2870
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