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A multiheterodyne technique is presented which can accurately measure the complex spectrum and temporally reconstruct certain dynamic pulse trains. This technique is applied to periodic pulses formed in a LiNb03 Mach Zehnder modulator. The spectral amplitude and phase of 20 GHz 66% return-to-zero (RZ) pulses and 10 GHz 50% RZ pulses are measured, and compared to independent measurements from a high resolution optical spectrum analyser. The temporal pulse shape and phase is reconstructed and compared to high speed sampling oscilloscope measurements. This technique is applied to sections of a large single acquisition, allowing the reconstruction of frequency and amplitude modulated pulse trains.
© 2013 Optical Society of America
1. Introduction
The measurement of the spectral intensity and phase of optical signals is much sought after in areas such as ultrafast pulse production [1], arbitrary waveform generation [2] and the metrology of advanced modulation formats currently proposed for next generation telecommunication networks [3]. The development of a simple method to characterise the complete complex spectrum of a periodic pulse train, and therefore recreate its temporal intensity and phase shape, has been of significant interest in recent years. Techniques such as optical autocorrelation [4], frequency resolved optical gating (FROG) [5] and spectral phase interferometry for direct electric-field reconstruction (SPIDER) [6], have been developed in order to attain such a measurement of the complex spectrum. However these techniques, while informative when applied to a periodic train of pulses, can give misleading results when dealing with non periodicity, such as in slowly evolving pulse trains seen in quantum dot passively mode-locked lasers [7,8], as well as more extreme pulse-to-pulse variations [9]. Recent work has also demonstrated the possibility of sampling the electric field of a mode-locked quantum cascade laser (QCL) [10]. However this technique requires a large number of sampling events and a synchronised optical reference with a pulse length shorter than that of the sampled pulse (for example a device in the C or L telecommunication bands would require a pulse duration of approximately 2 fs with this technique). The technique presented here attempts to overcome some of the limitations present in these “large-averaging” systems, and detect changes in the pulse train that exist on timescales longer than pulse-to-pulse, but which have hitherto not been investigated.
Standard techniques to measure the phase of an optical signal involve the downconversion from optical frequencies to the RF regime of two optical modes spaced closely together in frequency [11]. Based upon this technique, previous complex spectrum analysers have utilised this heterodyne beating in order to measure the spectral intensity and phase of an optical signal by stepping a single mode local oscillator between pairs of modes of a periodic pulse train [12]. This method has significantly simplified the design considerations of complex spectrum analysers removing the requirement for external modulation of the pulse train, an external optical phase reference or a precision reference timebase, thereby ensuring that the experimental setup remains straightforward. Nonetheless, its domain of applicability is limited to periodic pulse trains.
More recently, the use of optical frequency combs (OFCs) has been suggested as a method to perform such complex spectrum measurements [13]. Optical frequency combs have seen use in fields including spectroscopy [14], waveform shaping [15], precision range finding [16] and wavelength division multiplexed systems [17], to name just a few. The basic concept in using OFCs to measure the complex spectrum of a pulse train again lies in the downconversion from all optical frequencies to a series of RF combs. Multiple discrete heterodyne beat notes are generated from the intermixing of the optical frequencies of the comb under test and the OFC. This allows for concurrent real-time sampling of individual comb lines [18]. Using largely de-tuned comb sources ensures an efficient use of the RF bandwidth, however due to an excessive number of pulses not used during the measurement, large averaging factors over the pulse train are required [19].
By replacing a local oscillator with a second optical frequency comb, it becomes possible to record the complex spectrum of the comb under test in a single real-time trace (“single shot”), while retaining the simple, low-cost setup, requiring no complex detection components, as in [12]. This paper presents such a technique which can reconstruct pulse trains over small time intervals (as low as 10 ns), thereby enabling a time-resolved single shot multiheterodyne complex spectrum measurement to be performed. A phase modulated CW source is used as a reference OFC. The algorithm employed here relies only on knowing the spectral intensities of the lines of the reference comb in order to fully characterise the comb under test. The work reports the first known results of such a technique, with successful pulse reconstructions obtained for dynamical pulse trains modulated in both amplitude and frequency.
2. Theory
The technique presented here relies on the mixing of a periodic reference optical frequency comb on a square law detector, which can be used to measure the electric field of an unknown arbitrary pulse train. This technique differs from previous complex spectrum analysers in that the measurement requires only one real time measurement of the beat tones between the two combs to completely recreate the desired pulse train intensity and phase. The following approach is similar in nature to that of [12], however the method presented here takes into account the mixing of two optical combs allowing for the characterisation of one of these combs in a single real time acquisition.
Assuming that the comb under test (CUT) is a periodic pulse train of the form
where F is the comb spacing in frequency, Pn and ϕn are the power and phase, respectively, of the nth mode, and ωcut and ϕcut(t) represent the frequency and phase noise of the highest frequency component of the optical spectrum. The negative exponent in the first term above stems from treating the comb lines in order from the highest to lowest frequency component. This comb is mixed with a similar reference comb (REF), Eref(t), which has a comb spacing f < F and whose carrier frequency is offset from the CUT comb. As seen in Fig. 1(a), the minimal inter-comb frequency offset is given by δω. This field, given by a sum over all modes of the comb, is with a comb spacing difference now given by δf = F − f, and the negative sign in the first term arising for the same reasons as eq. 1
Fig. 1 Schematic diagram of the (a) optical and (b) radio frequency spectrum of two de-tuned combs at frequencies F and f.
If the two signals are mixed on a square law photodetector, heterodyne beat tones will then be generated in the RF regime, as shown in Fig. 1. While each mode in the reference comb beats with its nearest neighbour in the CUT comb, producing RF comb A, it also beats with its second nearest neighbour thereby producing RF comb B. Considering one mode of the reference comb, say mode k, which is offset by a frequency δk from the CUT comb, beat tones are established with the nth and (n + 1)th mode of the CUT comb. These beat tones will then occur at frequencies δk and F − δk respectively, thereby contributing one line to combs A and B in Fig. 1(b). Two further signals are observed at frequencies F and f, originating from the mutual beating of the adjacent modes in the CUT and reference comb, respectively. Extracting one set of matched lines from the combs A and B (i.e at, say, frequencies δk and F − δk), as well as the signal at F, gives the following three signal terms:
where Δϕ (t) = ϕref(t) − ϕcut(t) is the carrier phase noise difference.The terms sigδk and sigF−δk can be combined in order to cancel the contribution of the reference comb phase, ϕk, and the carrier phase noise difference, Δϕ(t),
Multiplying Eq. (6) by the complex conjugate of the signal at F, ( ), yieldsThis then provides a measurement of the phase difference between the two adjecent modes in the CUT comb. The cancellation of the phase noise terms, as well as the cancellation of the reference comb phase, means that the relative phase of each mode can be measured in a single real time trace, with little restriction placed on the phase behaviour of the reference comb. The presence of the extra term ϕtot leads to an undetermined spectral phase offset, causing a temporal shift in the pulse, which will not adversely affect the measurement. The power of the nth CUT comb mode can be retrieved from the beat tone at δk, provided that the power of the reference comb modes are known. In this way, the entire complex spectrum of the CUT signal can be retrieved, and therefore the temporal intensity and phase of the field can be reconstructed.
In this work the frequency offset, δf, and the optical frequency offset, δω are chosen such that nδf + δω < F/2. This is in order to ensure that the two RF combs can fit into the available bandwidth, thereby also setting an upper limit for the number of possible comb lines that can be measured for a given δf. In the time domain, the reference signal will step across the CUT pulse, covering the entire pulse every 1/(δf) seconds, or every F/δf pulses. If the pulses are approximately periodic on this time scale then a recovery of the pulse should be possible. This then leads to the opportunity of performing a time-resolved multiheterodyne measurement. By segmenting the measured real time trace and considering each time segment as an independent measurement of a periodic pulse, measurements of amplitude and frequency variations in an pulse train are then possible.
3. Experiment
Prior to commencing measurement, it is necessary to initially characterise the reference comb. While other techniques require a complete characterisation of the reference comb or assume a flat reference comb [13], the technique proposed here simply necessitates a measurement of the spectral intensities of the individual comb lines of the reference. This spectral intensity characterisation can therefore be carried out from direct recording of a reading from an optical spectrum analyser (OSA) or by using a heterodyne mixing with a local oscillator of a known fixed power.
The experimental setup used for the single-shot multiheterodyne analysis is presented in Fig. 2. The device under test in the experiment is an intensity modulated external cavity laser (ECL), whose output is mixed with a suitably modulated optical reference comb. In this work, the reference comb was generated by phase modulation of an ECL, at an appropriate drive frequency. Optical modulators were used as the comb sources in this work. These are well-known, convenient sources of periodic pulse trains [20] but also provide easily controllable non-periodic pulse trains via RF modulation. Applying a modulation to the RF drive signal of the CUT ensured that a well known time varying pulse train could be created. The use of a phase modulated CW source as the REF comb ensured an easily tunable source. The placement of the reference comb is such that, as discussed, the associated beat signals for each mode can be unambiguously identified when mixed via a simple 50/50 coupler. One output of the coupler is captured on a high-speed photodiode connected to a 40 GSa/s real-time oscilloscope (12 GHz bandwidth). The other output of the 50/50 coupler is further connected to a 90/10 coupler with the 90 % arm connected to a 50 GHz sampling oscilloscope and the 10% arm connected to an OSA for diagnostic comparison of the results. Correct labelling and interpretation of the RF beat tones is ensured before each measurement by noting the orientation of the combs with the OSA.
The recorded real-time trace of the mixed signal is analysed off-line upon completion of the single-shot measurement. The recorded time-series is converted to the frequency domain and since each line of the reference comb beats with its nearest and second nearest neighbour, two frequency separated combs are generated (referred to as Comb A and Comb B for clarity) in the FFT signal. Figure 3 shows the resultant FFT spectrum for a CUT comb modulated at 10 GHz and a phase modulated reference comb driven at 9.8 GHz. The algorithm proposed above filters each FFT peak in both combs A and B individually, using a flat-top filter (while a 200 MHz filter was used in this work for ease of automation, it should be noted that the width of the filter is only strictly limited by the linewidth of the laser under test). Corresponding comb lines (i.e. the lines at δ and at F − δ marked in red) are then paired off, and multiplied together with the complex conjugate of the comb line at F (10 GHz in this case, marked in blue), see Fig. 3 for example. As discussed previously this allows for the calculation of the relative phase between the lines of the CUT comb. It should be noted that the line corresponding to the frequency f in Fig. 3 is significantly weaker than the frequency line at F. This is due to the fact that in this series of experiments the reference comb is a phase modulated signal, which for an ideal phase modulator would not have a frequency component at f. The residual frequency at f is not used in this technique and so does not play any role.
Fig. 3 Fourier transform of the detected multiheterodyne time trace from the 40 GSa/s real time oscilloscope. Additional tones present in the RF spectrum correspond to electrical noise signals and have no effect on the result of the algorithm.
Having successfully obtained the phase information, the complex spectrum amplitude of the CUT can then be recovered by finding the intensities of the lines of the CUT comb. This is achieved by integrating the individual FFT peaks (at δk) over the width of the filter and removing the contribution from the intensity of the corresponding mode of the reference comb, which had previously been measured. Combining the phase and intensity information, it is then possible to completely reconstruct both the spectral and temporal intensity and phase of the CUT signal. Typically previous methods [12, 13, 19] focused solely on the measurement of periodic signals, however the experimental method and algorithm presented here allows for, in addition, the measurement of dynamic signals.
3.1. Periodic pulse train reconstruction
In order to verify the single shot multiheterodyne measurements a series of experiments were carried out on an ECL, externally modulated by a LiNbO3 Mach-Zehnder modulator (MZM). The drive signal to the MZM at 10 GHz was set just below Vπ and the bias was set to quadrature in order to generate 10 GHz 50% RZ, while to generate 20 GHz 66% RZ pulses the drive signal was set to 2Vπ and bias was set to null [20]. A 3.275 μs time trace is captured on the oscilloscope incorporating 217 data points, with a 25 ps sample spacing. The reconstructed temporal and spectral complex electric fields for the two cases are plotted in red in Figs. 4 and 5. The reconstructed temporal phase response of these signals shows the expected behaviour with π phase jumps between adjacent pulses in the 66% RZ pulse train and no phase change for the case of the 50% RZ pulse train. It should be noted that in order to demonstrate the capabilities of the measurement technique, an off-null bias was applied in the case of the 66% RZ pulse train to generate 20 GHz pulses with alternating intensities.
Fig. 4 (a) Measured spectral and (b) reconstructed temporal intensity and phase of 10 Ghz 50% RZ pulses formed with a quadrature biased MZM driven at 10 GHz. Shown for comparison, in black, is in (a) the spectrum captured on a high resolution OSA and in (b) the temporal intensity measured with a fast photodiode and sampling oscilloscope.
Fig. 5 (a) Measured spectral and (b) reconstructed temporal intensity and phase, in red, of 20 Ghz 66% RZ pulses formed with a slightly off-null biased MZM driven at 10 GHz. Shown for comparison, in black, is in (a) the spectrum captured on a high resolution OSA and in (b) the temporal intensity measured with a fast photodiode and sampling oscilloscope.
In order to further validate the measurement technique, the signals were also, as mentioned previously, simultaneously measured with a high resolution optical spectrum analyser and a 50 GHz photodiode connected to a sampling oscilloscope. These independent measurements of the optical spectrum and the pulse intensity are also plotted in Figs. 4 and 5, in black for comparison, and show excellent agreement with the single-shot multiheterodyne measurements.
3.2. Dynamic pulse train reconstruction
The technique and algorithm presented previously can be adapted to allow for the reconstruction of dynamically varying pulse trains. This section focuses on amplitude and frequency modulated pulse trains. Both the AM and FM signals are applied to the CUT comb via the MZM drive signal. For the case of AM, the MZM was again biased at quadrature and a 50% depth modulation with a 100 kHz frequency was applied. A time trace of duration 255 μs was captured on the real-time oscilloscope for off-line analysis. This large time trace was split into bin sizes of 100 ns and it was assumed that over a single bin the pulse trains are approximately periodic. With this assumption, the data is then analysed as in the previous section and the temporal pulse train corresponding to each bin is recreated. The intensity of the pulse within each bin is then tracked and plotted in black in Fig. 6. In order to again verify that the single shot multiheterodyne measurements are correct, the output of a simple transfer model for amplitude modulation is overlaid in black in Fig. 6 and shows excellent agreement with the experimental data. Pulse reconstructions were carried out for modulation frequencies from 10 kHz to 500 kHz and with modulation depths ranging from 0 to 100%. The main limiting factor in the scope of this experiment was a maximum modulation frequency of 500 kHz used to vary the MZM drive signal. However, this is not a limitation in principle but rather of available equipment.
Fig. 6 Variation in the peak intensity of the pulse train over time, for a 50% amplitude modulated MZM drive signal with a 100 kHz modulation (red). The data is compared to a numerical model, computed using a simple transfer function approach with an AM drive, plotted in black. The subfigures labelled 1, 2 and 3 show the reconstructed 50% RZ pulse train at the corresponding points along the AM period.
With the MZM still biased at quadrature, the modulation type is changed from AM to a 10 kHz FM with a 10 MHz deviation. Once again a 255 μs time trace is recorded and split into bins of width 1 μs, again with the assumption that the pulses are approximately periodic over individual bins. As for each bin there is a variation in the frequency, it is necessary to measure the values of F and δk across each bin. The temporal pulse is then reconstructed for each bin and the average pulse period is plotted in Fig. 7, for both a square wave and sinusoidal frequency modulation. Again a simple FM model shows excellent agreement with the experimentally obtained results, showing the capabilities for the measurement of the intensity and phase of an arbitrary comb. Again multiple pulse reconstructions were carried out with modulations from 10 kHz to 250 kHz with frequency deviations ranging from 500 kHz to 10 MHz, which was dependent on the modulation frequency chosen.
Fig. 7 Variation in period of the pulse train over time, for a frequency modulated MZM drive signal at 10 kHz, with a 10 MHz frequency deviation, biased at quadrature. The reconstructed pulse periods are shown for both sinusoidal (blue) and square (red) modulation schemes. Note that the deviation from a perfect square modulation signal (tail and overshoot) is due to imperfections in the modulated drive signal and not from the reconstruction of the pulse train.
While the results above detail the successful pulse reconstruction of both AM and FM modulated signals, tracking both the intensity and period of a pulse train in a single bin is straight forward and the only limitation here is the equipment, as an amplitude and frequency modulated signal was not possible to achieve. Similarly, although this experiment utilises relatively narrow spectral combs, chosen so as to present a controllable dynamic pulse train for study, there should be no large obstacles to utilising this technique with wider combs.
The time resolution that can be achieved depends on the size of the bins chosen, with the set-up detailed here, bins as small as 10 ns can comfortably be chosen while still retaining the ability to reconstruct the pulse. This bin size would suggest that amplitude modulations of up to 100 MHz could be measured (assuming that 5 separate bins span a single modulation period). The measurement error in Fig. 6 suggests that measuring amplitude variations of approximately 5% is possible. In terms of frequency changes, smaller bin sizes result in a degradation of the resolution of the FFT signal thereby imposing a limit on the modulation index, h = Δν/νmod. Given the set-up proposed here, a modulation index on the order of 0.1 is attainable, again assuming that 5 bins span a single modulation period.
In the cases presented above, the measurement limitations suggested can be improved upon by taking into consideration a number of factors. Achieving a reduction in the optical detection noise (for example by using a balanced detection scheme) will lead to an improvement in the FFT signal to noise ratios, thereby increasing the accuracy of the reconstructed signal, and allowing smaller time bins to be chosen. Making more efficient use of the available RF bandwidth, for example by decreasing the mode separation, will allow more optical modes to be probed simultaneously, as would the case for semiconductor mode locked lasers. Further improvements in oscilloscope specifications will allow for significant advancements due to the vastly increased sampling rate of the real-time oscilloscope. A larger sampling rate will mean greatly improved RF bandwidth, which will enable measurements of higher repetition rate comb sources.
4. Conclusion
This work presents a novel technique to perform time-resolved reconstructions of dynamical pulse trains. Reconstructed 10 GHz 50% RZ and 20 GHz 66% RZ periodic pulse trains were shown to be in excellent agreement with concurrent measurements from a high resolution OSA and a high speed sampling oscilloscope. For the case of a dynamically varying pulse train it was shown that the technique was capable of measuring amplitude modulations of up to 500 kHz, and frequency deviations up to 500 kHz. These results are currently limited only by the available equipment, and actual limits of detection are expected to be at least one order of magnitude greater. The need for a reference comb in this technique is offset by the relaxed restrictions on its phase behaviour opening up the possibility of using various optical frequency comb sources to provide the reference depending on the application, which could vary from studies of advanced modulation formats to instabilities in mode locked lasers.
Acknowledgments
This work was conducted under the framework of the INSPIRE programme funded by the Irish Government’s Programme for Research in Third Level Institutions Cycle 5, National Development Plan 2007–2013 with the assistance of the European Regional Development Fund. The authors also gratefully acknowledge the support of Science Foundation Ireland under Contract No. 11/PI/1152. The authors would also like to express their thanks to Dr. Fatima Gunning and the Photonics Systems Group in Tyndall National Institute for their support in this work.
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