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Abstract
We present a theoretical overview and a proposed methodology which demonstrates SLASOPS (single laser asynchronous optical sampling) as a single-laser alternative to the conventional two-laser ASOPS technique. We propose the optical and electronic setup in which SLASOPS may be achieved experimentally with a single 2-section mode-locked laser diode as the pulsed-laser source and simulate how asynchronous optical sampling is generated and detected theoretically. We highlight the technique’s ability to provide customizable scan ranges, scan rates and scan resolutions through variation of the imbalance in the interferometer arms and by tuning the repetition rate of the pulsed-laser source, which we present as optical cross-correlations between pulse pairs. We incorporate jitter into the system mathematically to assess the limitations on resolving both intensity and interferometric cross-correlation traces and to investigate the effects of averaging such traces in real-time. Analysis is then carried out on cross-correlation trace amplitude, width, and temporal positioning in order to discuss the technique’s ability for deployment in typical optical sampling applications. In particular we note SLASOPS’ ability to conduct asynchronous optical sampling using only a single laser, halving both the expense and technical requirements, doing so at megahertz scan rates, and within a spatial precision of just a few microns.
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1. Introduction
1.1 Optical sampling
Applications in time-resolved spectroscopy range across the sciences from fluorescence life-time microscopy in life-sciences [1] to metrology [2], such as distance measurement [3–5] and LIDAR (Light Detection And Ranging) [6]. Optical sampling (OS) techniques and their development are therefore driven by a variety of technical demands.
The choice of technique (and pulsed laser source) will influence not only the cost and footprint of the system, but the scanning parameters such as temporal resolution (step-size), maximum scan range and, of course, scan rate. As an example, pump-probe spectroscopy relies on optical sampling, whereby a train of higher-energy pump pulses excite a sample of interest, whilst a train of lower-energy probe pulses investigate the target’s excitation response over a delay time, often achieved by splitting pulses in a Michelson interferometer and delaying the probe pulses via a delay stage. Among the state-of-the-art, we shall discuss the two which have most influenced the development of SLASOPS (Single Laser Asynchronous Optical Sampling). First of these, we consider ASOPS [7–9] – Asynchronous Optical Sampling – which uses two lasers with a slight difference in their repetition rate to act as pump and probe pulse trains. When directed towards a target of interest, this offset leads to an increasing delay between pulse pairs, in such a manner that one scan may be completed with a scan range equal to the roundtrip period of one of the lasers, and at a scan rate equivalent to the offset in repetition rates. Compared to the delay stage method, ASOPS requires no moving parts, which has given rise to fast scan rates up to 100 kHz to date [10], and has been deployed in a wide variety of applications such as terahertz time-delay spectroscopy [11], multiheterodyne spectroscopy [12], and absolute distance measurement [13,14]. However, ASOPS is often demonstrated using expensive solid-state systems such as titanium-sapphire lasers [15], the major drawbacks then being the requirement of two such lasers, alongside a great deal of technical intricacy surrounding the electronic timing and locking of the repetition rates. Additionally, it is not possible to customize the scan range, which is dictated entirely by the roundtrip period (the time between successive pulses), meaning unwanted dead-time can occur in the scan. Similarly, the step-size (scan resolution) is dictated by the fixed repetition rate parameters of the two lasers, highlighting ASOPS as a difficult-to-optimise technique. The issue of tunable scan range was investigated by ECOPS [16,17] – Electronically Controlled Optical Sampling – where one of the repetition rates was modulated via square wave in order to narrow the scanning window, however this introduced the need for mechanical parts (tuning the intra-cavity laser length using a piezo) which are a limiting factor in speed and stability.
Secondly, OSCAT [18] – Optical Sampling by Cavity Tuning – makes use of the Michelson interferometer used in conventional optical sampling, but instead of two equal-length arms where one is variable, one of the arms in OSCAT is significantly longer than the other, and is entirely passive, i.e., a passive delay line (PDL) of length l and refractive index n. Scanning then occurs instead by tuning the intra-cavity length of the laser (using a piezo stack), thus tuning the repetition rate f by some tunability Δf. A scan range of Δτ is then walked through such that
Achieved by a hybridisation of both of these techniques, SLASOPS (first theorised in 2016 [19]), aims to achieve ASOPS-type sampling using only a single laser, halving the expense and technical requirements of the system whilst still maintaining the high-speed advantages of ASOPS. This, along with the added bonus of having the full customizability of OSCAT-type sampling, places SLASOPS in the unique position of being able to conduct two-laser asynchronous optical sampling using a single laser: a mode-locked laser diode (MLLD). These edge-emitter devices are entirely electronically pumped, meaning there is no pump laser required to enable lasing as in the case of most solid-state systems, which are optically pumped. Indeed, a single semiconductor-based MIXSEL (Mode-locked Integrated External-cavity Surface Emitting Laser) has previously been adopted for ASOPS, where two repetition rates were simultaneously generated in one device [20]. However, as a ‘surface-emitter’ laser, the MIXCEL must be optically pumped, often with an additional electronically pumped edge-emitter diode – an expense and complication which can be eliminated when using edge-emitting MLLDs such as those recommended for SLASOPS. PHIRE (Parallel Heterodyne Interferometry ia Rep-rate Exchange [21]), a conceptually similar technique, was demonstrated two years later for use in dual-comb interferometry. The crucial difference between this and SLASOPS is that our proposed technique takes advantage of entirely electronically driven scanning using MLLDs, requiring no mechanical parts whatsoever, and therefore is simulated to access MHz scan rates even with relatively high levels of pulse-to-pulse jitter, unlike the kHz scan rates achieved by PHIRE. Furthermore, we delve into the analysis of the effects of jitter on systems like SLASOPS which require a highly imbalanced interferometer, quantitatively highlighting the limitations in precision and optical sampling resolution.
1.2 Mode-locked laser diodes
As discussed, optical sampling of this nature requires a pulsed photonic source. Conventionally, techniques such as ASOPS and OSCAT have been demonstrated using commercial lasers such as solid state titanium sapphire [22] or doped fibre systems [23]. Such lasers can be prohibitively expensive and cumbersome, as well as complicated to run and maintain. In this work we propose MLLDs as low-cost, highly compact, and versatile alternatives which (in 2-section form) are fabricated with two isolated sections which may be independently electronically biased. Whilst one section (the gain section) is forward biased, a saturable absorber section is simultaneously reverse biased until passive mode-locking gives rise to a steady repetition rate which is proportional to the full length of the device, see Fig. 1.
Fig. 1. Diagram of a typical MLLD with a reverse biased saturable absorber section which is electrically isolated from a forward biased gain section.
Since SLASOPS requires the repetition rate of the source laser to be varied, MLLDs make an excellent candidate due to the fact their repetition rate can be tuned entirely electronically by varying the biasing levels to each section (and furthermore by temperature control) [24–26]. Numerous explanations are discussed in the literature as to why this tuning occurs, for example the Pockels’ effect, which may lead to variations in the semiconductor material refractive index and therefore roundtrip period [25] – a similar explanation considers the plasma effect [27,28], though the change here in refractive index arises due to a change in carrier density [29]. The Quantum Confined Stark Effect [30,31] also describes the effect of an external electric field upon the light absorption or emission spectrum of quantum wells, wires or dots.
Crucially, this means we can vary the repetition rate and conduct optical sampling without physical tuning of the cavity length (which is the conventional approach in OSCAT, for example). Additionally, quantum-dot MLLDs exhibit lower jitter than their quantum-well counterparts [32] due to the reduced amplified spontaneous emission as a result of greater carrier confinement [33]. Indeed, our recent work [34] demonstrated Optical Sampling By Repetition-rate Tuning (OSBERT) which used a MLLD in an OSCAT-type arrangement, where the repetition rate tuning was driven entirely electronically. Furthermore, we have shown that these 2-section devices can be driven at MHz scan rates and beyond by electronic modulation of the absorber section, without the need for any additional electronic stabilisation components [35]. To that end, this work combines our understanding of MLLDs, their electronic repetition rate tunability, and optical sampling to simulate SLASOPS as a single-laser alternative to ASOPS.
1.3 SLASOPS concept
We begin our simulation assuming the setup of a SLASOPS system to be identical to that of OSCAT. With the initial pulse train of some tuneable pulsed laser being split into two and one of the trains being passed through a Passive Delay Line (PDL); as in the case of many OSCAT setups, an optical fibre here allows for a large imbalance with much smaller footprint. Despite the OSCAT-type arrangement, the scanning proceeds in the same manner as ASOPS, which fundamentally requires two different repetition rates to be present at the detector at the same time, see Fig. 2.
Fig. 2. A) Simplified ASOPS setup featuring two lasers which deliver two repetition rates to the detector, and b) SLASOPS setup delivering ASOPS-type scanning with a single laser
As can be seen, when combined with a passive delay line, the ability to tune the repetition rate from one value, f, to another, f+Δf, allows for the situation to arise where two different repetition rates are present at the detector at the same time. This is the same key premise achieved using ASOPS, with the fundamental difference that SLASOPS is using, by definition, only one laser.
For our purposes, we switch between repetition rates using a square wave similarly to that of ECOPS. Depending on the preference of the user, the period of the square ${T_{sq}}$ wave can be selected to match the real-time period ${T_{PDL}}$ associated with the period of the PDL, such that
where this provides the special case that the two square waves are exactly out of phase with respect to each other due to the exact length of the PDL. It is important to note that it is not essential to meet this special case; scanning will still occur within any window where two repetition rates are present, however they may be optimised by the user’s preference of scan rate or scan range, both of which are determined by the length of PDL, and which will be a consideration for pulse characteristics such as dispersion. We firstly prove this concept in Fig. 3, which shows how the scan propagates. For the ‘pulse trains’, optical pulses are generated with a temporal spacing proportionate to the ‘laser repetition rate’ which exists in each arm in real time; for our purposes, we have selected a square wave period which meets the criteria of expression (2). The difference in repetition rates means the pulses walk around each other asynchronously, giving rise to an increasing ‘time delay’ between successive pulse pairs, until the moment the square wave swaps the repetition rate of the laser, which (due to the PDL) leads to different repetition rates being present in each arm. From here the scan repeats in reverse order, simply backtracking the time delay points which were previously acquired. We consider that since two traces will be acquired in one full scan (one forward and one reverse) that for practical purposes it would be appropriate for the user to monitor one preferred direction for their measurements, though both may be compared since each direction should be identical in their acquisition. Finally, an intensity cross-correlation is built in real time whose data points are generated directly from the cross-correlation of pulse pairs, where a cross-correlation is simply an autocorrelation which occurs between pulses which were not copies of each other.Fig. 3. Diagram showing the operating principles of SLASOPS. Dashed vertical line highlighting the peak of a cross-correlation occurs when two pulses are temporally overlapped. All units are arbitrary
We theorise mathematically and confirm by investigation, of these and other sample traces, that the step-size (scan resolution) $\delta t$ of the cross-correlation trace is given by the same expression which governs the step-size $\delta \tau $ of ASOPS, such that
where ${\varDelta }f = {f_{min}} - {f_{max}}$. Similarly, if the PDL in a given OSCAT system has a pulse index a according to expression (1), it then follows that there are a scan points, and so the scan range ${\varDelta }\tau $ of SLASOPS is the product of the pulse index and step size,As confirmation of the unification between both OSCAT and ASOPS, we resolve that since ${\varDelta }f = {f_{max}} - {f_{min}}$ then expression (4) is simply a rearrangement of expression (1). This is significant point, as in ASOPS the scan range is fixed by the repetition rates of the laser, whereas in OSCAT the range can be altered by changing the PDL length (and thus the pulse index a). Therefore, it is shown that it is possible to achieve asynchronous optical sampling while still giving the end user the ability to customize the parameters of the scan using a single laser.
In SLASOPS, the real time ${T_{SLASOPS}}$ which elapses over one scan is equal the number of roundtrip periods ${T_{min}}$ in each scan, and so it follows that the scan rate ${r_{SLASOPS}}$ is its inverse, such that
where factor $a - 1$ represents the fact that the starting time of the scan has been defined as when the first scan point occurs, i.e. when $a = 1$, $t = 0\; s$. In the case where expression (2) is satisfied the scan rate can be more simply expressed as the inverse of PDL period.With the concept demonstrated clearly, we move on to simulate realistic cross-correlations of pulses originating from lasers which exhibit realistic repetition rate linewidths, as well as pulse-to-pulse jitter built into the system in order to assess the potential and limitations of SLASOPS.
1.4 Simulation setup
In the simulation the electric field of each pulse is given by,
where ${\omega _0} = 2\pi f$ is the frequency of the pulse. $I(t )$ is the gaussian intensity envelope of the pulse in the form, where ${t_0}$ is the pulse centre, A is an arbitrary amplitude set to unity, and it’s width w is related to the FWHM byCross-correlation traces were demonstrated in a system similar to that shown in Fig. 3 above, where the detection was assumed to be capable of resolving both an intensity and a second order interferometric trace. We assume the cross-correlation may be described using pump and probe pulses which are functionally identical and described in real time t with respect to delay time $\tau $ using $E(t )$ and $E({t - \tau } )$. Following from this we may write the equations which represent both measurements, the intensity and second order interferometric cross-correlations are given below using the same approach outlined in [36]:
andPractically speaking, this could be implemented as shown in Fig. 4, where the switch in repetition rate is brough on by square wave modulation of the absorber section of a MLLD whose output is sent to a highly imbalanced interferometer. Upon recombination of the beams after beam-splitting, the asynchronous signal is sent to a detector which is then resolved at an oscilloscope.
Fig. 4. Proposed practical setup of a mode-locked laser diode (MLLD) within a SLASOPS setup, featuring a highly imbalanced interferometer.
2. Simulation results
2.1 Scan customizability
A significant advantage SLASOPS holds over ASOPS other than cost is the ability to modify the scan, therefore in the following simulations we explore this niche where the PDL length and the magnitude of repetition rate tunability gives rise to varied outputs in terms of scan range, scan rate and scan resolution. Here, and for all simulated results, we assume an optical pulse of 1 ps duration, 5 GHz repetition rate and 1040 nm center wavelength, as well as the ideal case of zero-jitter. These output characteristics resemble those fairly typical of mode-locked laser diodes [32], which would be ideal candidate laser sources for SLASOPS in practice, owed to their ability to electronically vary their repetition rate [37].
In Fig. 5, top, we simulate two SLASOPS scans with the same step size $\delta \tau = \; $60 fs (given by a repetition rate tunability Δf of 1.5 MHz according to expression (3)); the first is simulated assuming a PDL length of 6 m and the second with 100 m. Increasing the PDL length increases the scan range from 8.76 ps to 146 ps according to expression (4) due to the increase in pulse pairs contained therein. It should be noted that intrinsically an increase in PDL length will mean a long time to complete one acquisition, in this case the 6 m PDL scan has a scan rate of 34.2 MHz whilst the 100 m PDL has a scan rate of 2.1 MHz. We then demonstrate that a lower value of 17.5 kHz of repetition rate tunability will decrease the step-size to 0.7 fs, Fig. 5, bottom, which will inherently lower the scan range. We can mitigate this as necessary by then electing to use a longer PDL length, in this case 370 m to provide a scan range of 6.3 ps, to resolve an entire cross correlation as in the previous scan. This lowers the scan rate to 555 kHz but allows for a significantly higher resolution scan.
Fig. 5. Customizable optical sampling: Top left trace acquired with a 6 m PDL at 34.2 MHz and top right with a 100 m PDL at 2.1 MHz, both with a repetition rate tunability of 1.5 MHz. Bottom trace shows a high-resolution scan with 0.7 fs step size acquired with a PDL of 370 m and a repetition rate tunability of 17.5 kHz scanning at 555 kHz.
2.2 Introducing jitter to SLASOPS scans
The simulation considers two sources of noise within the cross-correlation traces: the laser’s RF linewidth (${\varDelta }v$) and the pulse-to-pulse jitter (${\sigma _{pp}}$) inherent in the system due to the imbalance in the two arms. For the RF linewidth, instead of using a fixed value for repetition rate, we impose a normal distribution of values as an approximation of the assumed linewidth, where the greater the linewidth correlates with greater instability in the timing of the pulses. The pulse-to-pulse timing jitter is simulated according to the Kefelian expression [38] which we have previously used to describe the jitter in our optical sampling methods [34]. This can be expressed as a function of the RF linewidth ${\varDelta }v$, along with the number of pulses in the PDL N, and the repetition rate ${f_{rep}}$.
Therefore, it can be seen that as either the PDL length or the linewidth increases, the pulse-to-pulse jitter will increase. Figure 6 shows diagrammatically how we apply the effects of RF linewidth and pulse to pulse jitter to the ideal fixed repetition rate.
Fig. 6. Visualisation of the application of RF linewidth ${\varDelta }v$ and pulse-to-pulse timing jitter ${\sigma _{pp}}$ to the expected timing of the pulses given by the repetition rate ${f_{rep}}$.
Whilst the temporal spacing was calculated using the repetition rate and a normally distributed random value with a standard deviation of ${\varDelta }v$, the jitter offset was generated in a similar way to the linewidth and applied after the temporal spacing of the pulse was calculated, so the temporal spacing is defined by ${({f_{rep}} \pm {\varDelta }v)^{ - 1}}$ and the pulses are experience jitter around that point according to ${\sigma _{pp}}$. Ultimately, increasing jitter levels will cause temporal displacements in the expected data point positions, which will therefore have a greater impact on the interferometric traces than the intensity traces.
2.3 Signal processing with trace-averaging
In optical sampling, it is standard practice to present traces which have been time-averaged as a means to suppress the deleterious effects of jitter and to improve the signal-to-noise ratio of the scan. We simulate multiple SLASOPS scans which we then time-average in order to assess the effects of jitter on both intensity and interferometric traces.
Firstly, we qualitatively illustrate the effect of jitter on the interferometric scan by applying a jitter of 10 fs, 100 fs and 200 fs to the same 0.7 fs step size trace, see Fig. 7.
Fig. 7. SLASOPS interferometric scans with 0.7 fs step-size scanning at 555 kHz with different levels of jitter
Illustratively, we found that although up to 200 fs levels of jitter have affected the second-order interferometric traces, they remained very recognizable. This was also true for traces whose jitter approached picosecond levels, though as may be predicted, the trace broadens and eventually becomes too noisy to distinguish any features. Further research is planned to bring to light a quantitative understanding of signal processing measures which would allow noisy traces to be retrievable.
For intensity traces, we quantitatively explore the effects of averaging on SLASOPS traces exhibiting jitter levels as high as 500 fs, which were then averaged over a set of 5, 15 and 30 traces, see Fig. 8, which shows a 60 fs step size trace resolved using a 1.5 MHz tunability and a 6 m PDL. These parameters give rise to a scan rate of 34.2 MHz which even after averaging 30 times maintains a MHz scan range.
Fig. 8. SLASOPS intensity traces simulated with 500 fs of jitter which are time-averaged over 5, 15 and 30 traces.
We then compared the goodness of fit to SLASOPS intensity traces with jitter values ranging between 100 fs and 500 fs, for the same trace parameters and the effects of increasing the number of averages in the final traces, quantifying the effects on the amplitude and the FWHM, see Fig. 9.
Fig. 9. Deviation from pure (red dashed lines) trace amplitude (left) and FWHM (right) for traces exhibiting jitter of up to 500 fs, with time-averaging up to 30 traces.
The resulting effects on the traces are as one would predict; the amplitude drops for all values of jitter when the number of averages is increased compared to the ideal case, because the centre peak of the intensity is different in each trace due to the effects of jitter. Whilst this is only a few % in lower-jitter cases, around a 15% drop was found for 500 fs of jitter. A similar trend is found for the same reason when studying the FWHM of the traces which increase slightly for increasing the number averages. As seen in Fig. 8 above the widening of the FWHM and reduction in amplitude still leave the trace easily retrievable even at 500 fs of jitter. At fewer averages, we see that for some jitter values the FWHM narrows, which occurs due to the FWHM being retrieved by fitting a gaussian to the trace, the quality of which reduces for particularly few averages and for particularly high jitter.
We next investigate how the temporal centre of the trace was affected by jitter and averaging, see Fig. 10.
As can be seen, even for 500 fs levels of jitter, the temporal centre of the SLASOPS traces converge towards a 0-offset after approximately 30 averages. In the case of 100 fs of jitter, this corresponds to a spatial error of just 0.5 µm, and 4.9 µm for the upper limit of 500 fs of jitter. This is crucial, since for accurate distance measurements to be made, for example in metrology applications, the position of the traces centre must also remain consistent for different levels of jitter.
To assess the repeatability and reliability of this accuracy, the simulation was repeated 20 times for 0 to 30 trace averages. The maximum standard deviation (${\sigma _{dev}})$ measured was 16.3 fs at 500 fs of jitter, see Fig. 11.
We found our results were well described by a straight-line fit which was given by ${\sigma _{dev}} = 0.033{\sigma _{pp}}$ and exhibited a goodness-of-fit value of R2 = 0.96. This gives first indications that SLASOPS-type scanning for metrology and other applications are reproducible to the extent of being characterised (under these conditions) by a standard deviation in central position which is approximately 3.3% that of the jitter value – an extremely useful metric which would allow a user to estimate the accuracy of their SLASOPS system for any given value of jitter which arises from either the RF-spectrum linewidth, the PDL length or a combination of both.
3. Conclusions
SLASOPS offers ASOPS-type scanning at half the complexity and cost by utilising a 2-section mode-locked laser diode as the pulsed-laser source, which has the advantage of being purely electronically pumped and supporting fast repetition rate switching We illustrate the potential for competitive precision at fast, megahertz scan rates, alongside the ability to user-customize the scan range, scan resolution and scan rate, where the effects of jitter as temporal noise can be mitigated through averaging. For systems with sufficiently low jitter, SLASOPS can resolve interferometric traces very well, whilst even especially high-jitter systems will be able to resolve intensity traces. In particular we explored the potential for distance measurements with precision of a few microns in a system exhibiting 500 fs of jitter at MHz scan rates, even after averaging. The results from the simulation are extremely promising and show that SLASOPS has the potential to disrupt the status quo in optical sampling techniques with a fast, customizable, single-laser alternative to ASOPS.
Funding
Science and Technology Facilities Council (ST/T000651/1, ST/T003242/1).
Acknowledgments
D. Bajek would like to acknowledge the support given by S. Pollard, Open University.
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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