1. Introduction

The generation of arbitrary optical waveforms is the ultimate limit of optical signal manipulation [1], and progress toward this has helped to revolutionise the fields of telecommunication [2], quantum control [3], and high energy laser based methods such as laser wakefield accelerators [4]. Arbitrary radio frequency (RF) waveforms can be created because their lower frequency allows direct synthesis through generation of arbitrary time varying voltages. At optical frequencies, the electric field oscillates far too rapidly for a similar direct synthesis technique, however equivalent effects can be achieved by lower frequency modulation of the high frequency carrier signal, a technique used ubiquitously in RF signal processing for decades.

In the optical domain, the low frequency modulation can be induced by two general methods: either direct modulation of the amplitude or phase of the carrier itself [5,6], or addition of other optical frequency components with known amplitude and phase relationships to the carrier [7]. While both methods are ultimately equivalent, amplitude and phase modulation of the carrier is induced by elements such as acousto-optic and electro-optic modulators (AOMs and EOMs), while addition of new frequency components is typically achieved through generation of optical pulses in mode locked lasers. Both methods create optical sidebands with a specific frequency offset from the carrier, rather than continuous spectra, and where the number of sidebands is large, the resulting field is referred to as a comb.

Where simultaneous arbitrary amplitude and phase modulation of the time domain optical signal is possible, any desired frequency domain signal can be achieved. The required amplitude and phase modulation is obtained simply by inverse Fourier transforming the frequency domain signal. While simultaneous amplitude and phase modulation is possible, it adds significant complexity to the experimental setup, and can require non-standard components [8].

Phase modulating EOMs however are ubiquitous in optics labs, and are often already incorporated into experimental setups. While sinusoidal modulation is sufficient for many applications, where more complex spectra are desired, other waveforms must be imprinted. The concatenation of serrodyne signals can be used to generate more complex spectra [9], but this technique offers poor control over both amplitude and phase, and requires relatively high bandwidth EOMs and electronics compared to the desired frequency shifts.

More precise control of the frequency domain amplitude and phase may also aid in the development of optical based quantum computing and sensing. Electro-optical modulators can be viewed as beamsplitters in frequency space and can be useful in this regard in the construction of quantum sensor networks. In such situations an EOM functions to entangle quantum-correlated light in different frequency modes. Recent work has pointed to how such modulators, driven at low modulation index, can be used to construct optical cluster states for measurement-based quantum computation experiments [10]. While in this example a small modulation index and nearest-neighbor connections via one pair of sidebands is required, the use of EOMs to generate strong mixing of a number of frequency modes, and with different phase relations can be envisioned to create a frequency-space analogy to the quantum sensing experiments performed in Ref. [11]. In this case several nearly-equally-weighted sidebands would be desired for the directly analogous experiment, and the phase relationship could be varied to enable sensing of specific amplitude-phase patterns.

In this paper, we present a method of using only an electro-optic modulator to directly generate optical combs where teeth have individually customisable amplitude and phase, without the need for any further spectral shaping apparatus. Thus, pseudo-arbitrary sidebands can be introduced into a setup simply by adjusting the input waveform to the EOM. The required time domain phase to imprint is calculated using an iterative algorithm based on a generalisation to the Gerchberg-Saxton algorithm. The algorithm progresses by numerically transforming the signal back and forth between the time and frequency domains, and applying constraints at each step until the desired frequency domain signal is achieved.

Using phase-only modulation to arbitrarily tailor some sidebands necessitates the redistribution of optical power into other sidebands in order to keep the total field amplitude constant. We find that despite the strong constraint of phase-only modulation in the time domain, significant tailoring of the frequency domain spectrum can be achieved with high fidelity, and an experimental validation of the method is presented.

2. Phase retrieval algorithm

The classic Gerchberg-Saxton algorithm was developed to determine the phase in the object and diffraction planes of a forward propagating two-dimensional monochromatic optical field, where the intensity of the field was recorded over both planes [12]. Where the diffraction plane is in the far-field, the field at the two planes is related simply by a Fourier transform. The phase is recovered by numerically propagating the complex field back and forth between the two planes, replacing the amplitude with the measured value after each propagation, but allowing the phase to freely settle on a solution. A plethora of variations of this technique have since been investigated, showing the general result that given some combination of constraints of amplitude and phase over both planes, it is often possible to find a self-consistent solution for the amplitude and phase of the unconstrained regions of both planes [13].

The time and frequency domains of an optical signal are also related simply by a Fourier transform, so iterative phase retrieval algorithms like those just described can also be employed to control the optical waveform [14–19]. While many variations of phase and amplitude modulation are routinely used to control the time domain envelope of an ultrafast optical pulse using pulse shapers [20], here we use just a single EOM to modify the time domain phase in order to tailor the amplitude and phase of the spectral comb teeth in the frequency domain.

The real, time-varying electric field of an optical signal at a given position can be represented by:

The power spectral density (PSD) of the optical field is proportional to $|\breve {E}(\nu _{\rm {opt}})|^2$. Writing $\nu \equiv \nu _{\rm {opt}}-\nu _0$ and assuming that $\mathcal {E}(t)$ varies slowly, for optical frequencies close to the reference frequency the PSD can be written as:

It may be noted that the factored complex exponentials in Eq. (1) have the opposite sign to the usual quantum mechanical convention. This was done so that positive frequencies of $\breve {\mathcal {E}}(\nu )$ correspond to frequencies above the carrier frequency, and negative frequencies to those less than the carrier. The opposite convention could also have been chosen, but this can make the relationship between $\breve {\mathcal {E}}(\nu )$ and $\breve {E}(\nu _{\rm {opt}})$ less obvious.

The goal of the phase retrieval process is to determine the time domain phase, $\phi (t)$ (defined in Fig. 1) to imprint on the optical carrier, such that the resulting field has a spectrum with sidebands of amplitude $|\breve {\mathcal {E}}(\nu )|$ and phase $\theta (\nu )$. The time domain phase will repeat with a period $T$: $\phi (t)=\phi (t+T)$, so the Fourier spectrum, $\breve {\mathcal {E}}(\nu )$ can have nonzero amplitude only at integer multiples, $n$, of the sideband frequency: $\nu _{\rm {SB}} = \frac {n}{T}$.

 

Fig. 1. Iterative solution finding algorithm. The unprimed functions represent the experimentally realisable values for amplitude and phase in the time and frequency domains, because the time domain constraints must be the final step in the algorithm. This is due to the experimental fact that the phase modulation leaves the time domain amplitude unchanged.

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For a given set of desired sideband amplitudes and phases, the algorithm progresses as in Fig. 1. The field is transformed back and forth between the time and frequency domains, and some constraints are placed on the amplitude and phase in each domain at each iteration. The key to the algorithm is to determine what sort of constraints to apply in order to achieve an optical spectrum that is as close as possible to the desired spectrum.

2.1 Constraints

The most stringent constraint is that the time domain amplitude must remain constant: $|\mathcal {E}(t)|=\mathcal {E}_0$. This follows directly from the fact that EOMs can only modulate time domain phase, not amplitude. With that rigid constraint in place, a simple demonstration of the algorithm’s ability to find a solution was performed. It is known that if a sinusoidal time domain phase modulation is applied to an optical carrier, then the resulting spectral amplitudes are given by a set of Bessel functions [21]. For the demonstration, these known frequency domain amplitudes were set as the constraint, and the algorithm was allowed to progress with no constraints on the phases of either the time or frequency domain. The result can be seen in Fig. 2, where the algorithm perfectly recovers the desired spectrum using a solution for the time domain phase that matches a sine wave. The same test was done for a serrodyne signal [22], where the desired Fourier amplitude is to shift all the optical power into the first sideband, which is achieved with a time domain sawtooth phase modulation. Again, the recovered spectrum perfectly matched the desired one, with the time domain phase solution perfectly matching the known sawtooth solution. It should be remembered that only the marked points in the power spectra represent the actual power of the corresponding sideband, and that the power is zero at the frequencies between sidebands. The line joining the points is given only as a visual aid, and does not indicate the optical power at intermediate frequencies.

 

Fig. 2. Validating the iterative phase retrieval process. The frequency domain amplitude constraints shown in a) were set to functions with analytically known time domain phases. The iterative algorithm was then run, and the recovered time domain phases compared to the exact solutions shown in b).

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The above example is contrived because all the Fourier amplitudes have been set to a value which is known to result from a given modulation. That is, it was known in advance that a solution actually existed for $\phi (t)$, and all the algorithm had to do was find it. In the general case, if all the frequency domain amplitudes were exactly set, a solution would not exist because the time domain amplitude is also rigidly set by necessity, and so the algorithm would fail. Therefore, to find solutions for generating a desired spectrum, some freedom must be introduced to the values of at least some frequency domain amplitudes (and phases).

The success of the algorithm is not particularly sensitive to precisely what constraints are used, however as a general rule of thumb, the more loosely the solution is constrained overall, the more precisely it will match those constraints that are tightly set. The challenge in developing a useful algorithm is therefore to establish a way of constraining the time and frequency domain amplitudes and phases in a way that results in a final frequency domain solution which is as close as possible to the one desired, and which also satisfies experimental limitations to phase modulation bandwidth and maximum phase retardance.

2.2 Frequency domain constraints

In this algorithm, constraints in the frequency domain are set in three ways, with all three methods being applied at every iteration. Firstly, sideband amplitudes and phases with a precisely required value are set to exactly that value. Secondly, all sidebands amplitudes and phases that are not exactly defined are given upper and/or lower bounds. The phase and amplitude of each of these sidebands is completely free to take any value within its bounds, but if it exceeds these values it is set back to the nearest bounding limit. Finally, all sideband amplitudes that are not exactly set, are subject to multiplication by a decay coefficient at each iteration which is close to, but less than unity. Slightly decaying all undesired sideband amplitudes creates a cost to maintaining them, so that only sidebands which are actually required to satisfy the exactly defined constraints are included. Without this decay step, the algorithm would have no way to favour solutions with minimal power in additional sidebands, reducing the efficiency with which optical power would be transferred from the carrier into the desired sidebands. The amplitude decay can be frequency dependent, and we use a Gaussian function centred around zero frequency so that sidebands further from the carrier are more strongly penalised. It is desirable to decay sidebands further from the carrier because these are more difficult to generate experimentally due to the higher bandwidth requirements of the phase modulator. However this function could be tailored to preferentially decay sidebands around any desired frequency region if the application required it.

The decay step is not applied to the phases of the sidebands, since it is usually the case that no particular phase is preferred for sidebands that are not directly constrained, whereas the amplitude for these sidebands would preferably be zero.

The effect of the different frequency domain constraints can be seen in Fig. 3. In this simple example, the goal was to generate just a single positive sideband with amplitude $-25\,\rm {dB}$ relative to the carrier, and to suppress the corresponding negative sideband to $-60\,\rm {dB}$. Generating this configuration of sidebands using only phase modulation requires that some power be transferred to additional sidebands, but it can be seen that the ancillary constraints have a significant effect as to how this power gets distributed. Without either limiting or decaying them, the additional sidebands close to the carrier grow in amplitude to be significantly larger than the desired single positive sideband, with the power of higher order sidebands fluctuation around the $-40\,\rm {dB}$ level across the whole frequency range. The effect of these high frequency sidebands can clearly be seen in Fig. 3(b), where the required time domain phase also has significant high frequency components.

 

Fig. 3. Frequency domain constraints. The effect that different frequency domain constraints have on the optical power spectrum can be seen in a). The power of the exactly set sidebands are shown as red dots. The grey horizontal line indicates the maximum allowed power for unset sidebands in the "limit on" trace. b) shows the calculated time domain phases which synthesise the corresponding power spectra. The constant offset in phase is only used as a visual aid, and has no effect on the resulting optical spectra.

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By applying the gaussian multiplicative decay function (with standard deviation width of $1000\,\rm {MHz}$), the sideband power rapidly decreases as sideband frequency increases, however the sidebands in the immediate vicinity of those that are exactly defined actually increase, since these are now being used preferentially in the synthesis of the exactly confined components.

Switching on the limiting constraint yields the final spectrum. This spectrum is a compromise solution, with reduced sideband amplitude in the vicinity of the exactly defined sidebands, at the expense of increasing the amplitude of those sidebands further away.

2.3 Time domain constraints

A strict constraint in the time domain is that the amplitude of the optical electric field be completely unaltered by the EOM. For a regular continuous wave laser beam this means that the amplitude of $\mathcal {E}(t)$ is set to a constant value, $\mathcal {E}_0$, though the iterative calculation method generalises to beams with a deterministic amplitude modulation.

The more interesting constraint in the time domain is that of the phase, $\phi (t)$, which must conform to something that is realisable experimentally, taking into account all practical constraints of the EOM and voltage synthesis electronics.

In our algorithm, we first limit the phase to the desired range, where all values above and below the allowed range are set to the nearest limit. This clipping is necessary because phase modulators have a maximum retardation which they can induce, so the algorithm must limit the requested phase range accordingly.

The other constraint in the time domain is on the frequency spectrum of the time domain phase. The time domain phase is a real signal directly induced by the EOM, so the spectral power of the phase must be constrained by the bandwidth capabilities of the EOM itself, and all driving electronics. While the spectral amplitude of the time domain phase will be related to the final optical sideband amplitude, they are not precisely the same, and so constraints on the spectral power of the time domain phase must be applied in the time domain.

The effect of the constraints on the time domain phase can be seen in Fig. 4, which shows an example of a synthesised flat optical frequency comb with 100 teeth centered about the carrier, where all other sidebands are suppressed to less than $-10\,\rm {dB}$. The calculated optical spectrum (Fig. 4(a)) successfully conforms to the desired frequency comb whether or not the phase filtering and clipping are imposed. However the form of the calculated time domain phase (Fig. 4(b)), and its corresponding power spectrum (Fig. 4(c)), are quite different.

 

Fig. 4. Time domain constraints. a) The optical power spectra obtained using different combinations of time domain constraints. b) The calculated time domain phase to be imprinted on the carrier. c) The power spectrum of the phase to be imprinted. The exactly set constraints are shown by large red dots, which overlap so appear as a line.

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The inset in Fig. 4(b)(i) illustrates that without frequency filtering, $\phi (t)$ can develop a $n\times 2\pi$ discontinuity because within a complex exponential, all values modulo $2\pi$ are equivalent. These discontinuities are only revealed once $\phi (t)$ is unwrapped, where the step function necessarily introduces high frequency components which can be seen in the corresponding power spectrum.

Phase discontinuities are avoided by adding a filter, which at each iteration smooths out any discontinuity that happens to be introduced, so the algorithm naturally settles on a phase that is continuous once unwrapped. This filtering also allows tailoring of the phase to the specific limitations of the electro-optic and driving systems, and in this example a 6th order Butterworth filter with a frequency cutoff of $200\,\rm {MHz}$ was used to match the experimental apparatus presented later.

The result of running the algorithm with the filter enabled is shown in Figs. 4(ii), where no discontinuity can be seen in $\phi (t)$, and the power spectrum of $\phi (t)$ shows negligible power above $200\,\rm {MHz}$ as required. The optical power spectrum does show slightly increased power of the unconstrained sidebands, as is expected given that the system is more tightly constrained overall.

Enabling the phase clipping results in the final experimentally generatable phase, $\phi (t)$, which in shown in Figs. 4(b)(iii). In this example the phase was clipped to a range of $2\pi$, which reduced the power of the low frequency components of $\phi (t)$ (Fig. 4(c)(iii)), but resulted in slightly higher amplitude of the unconstrained optical sidebands (Fig. 4(a)(iii)).

An important numerical consideration during the filtering step is to not use a frequency filter that introduces any dispersion. Standard digital filters introduce a frequency dependent phase delay which can significantly distort the waveform when it is applied, reducing the ability of the algorithm to converge on a stable solution for $\phi (t)$. The dispersion can be cancelled out by running the signal through the digital filter twice - once forward, once backwards, which leaves the signal filtered, but otherwise undistorted (this is known as “zero-phase filtering”). To enforce continuity between the first and last elements in $\phi (t)$, three duplicate arrays can be concatenated, filtered, and then the central segment extracted. The filtering can also be performed directly by Fourier transforming $\phi (t)$, multiplying it by the real frequency dependent gain coefficients of the desired filter, and then inverse Fourier transforming. With either filtering method, care should be taken to ensure the cutoff frequency is far enough below the Nyquist frequency such that $n\times 2\pi$ discontinuities are avoided.

At the start of the algorithm iteration, an initial guess must be made for the time domain phase. The initial guesses used for $\phi (t)$ throughout this paper were uniform random samples in the range $[0, \frac {2\pi }{100}]$. A fairly small range was used so the time domain constraining processes of clipping and filtering didn’t immediately flatten out the time domain phase. In practice, the precise range of the random samples didn’t seem to have a large impact of the ability of the algorithm to find a solution.

3. Constraining sideband phase

It is possible to constrain the frequency domain phase as well as amplitude. Setting the phase of only one frequency component is usually meaningless, since it amounts to a redefinition of $t=0$. Setting the relative phase of two or more components however does have a real effect, and the well defined relative phases between sidebands of an optical waveform are exploited in many techniques of the optical sciences, such as frequency stabilisation of lasers or locking of optical cavities.

A simple example synthesised spectrum with constrained phases is shown in Fig. 5, where the amplitude of the $\pm 1$ sidebands have been set to $-10\,\rm {dB}$. The phase of the central and lower sidebands were set to $0$, while the upper sideband was varied to demonstrate the affect that doing so would have on the generated optical spectrum (Fig. 5(a)), and required time domain phase (Fig. 5(c)). It can be seen in Fig. 5(b) that the desired phases are generated successfully, and that the required power in the higher order sidebands can vary significantly depending on the combination of phases. The case of $\theta _{+1}=\pi$ corresponds to a set of phases generated when a carrier is phase modulated with a sine wave in the time domain. Similar to sine wave modulation, the amplitude of higher order sidebands drops off steeply with frequency, though the form is not exactly the Bessel function generated with sine modulation. Requiring $\theta _{+1}=0$ corresponds to a set of phases generated when a carrier is amplitude modulated with a sine wave. As such, generating this combination of phases with phase-only modulation required significantly more power in the higher order sidebands. The case of $\theta _{+1}=\pi /2$ requires only slightly less power in the higher order sidebands than does $\theta _{+1}=0$, but represents an arrangement of phases that is not easily achievable by more conventional modulation means.

 

Fig. 5. Setting sideband phase. In addition to controlling the optical power spectrum shown in a), the phase of the sidebands can also be selected as illustrated in b). Three examples are shown with different combinations of phase for the central three frequency components, and the effect on the required time domain phase can be seen in c). The red dots show the exactly set constraints. The small dots in b) show the achieved sideband phase.

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4. Limitations of phase-only spectral tailoring

As noted previously, there are limitations to how a spectrum of sidebands can be shaped using phase-only modulation. While a limited set of sidebands can be tailored arbitrarily, additional sidebands must also be generated to cancel out any amplitude modulation which would otherwise result. The maximum power of these additional sidebands can also be constrained, but this comes at the expense of increasing the amplitude of other additional sidebands. An example of the limitations due to phase-only modulation can be seen in Fig. 6(a), where a simple arrangement of two unbalanced sidebands and a carrier have been exactly set, and the maximum power of any additional sidebands has been varied to the indicated levels. Ultimately, while the additional power can be shifted to wherever it will be least problematic for a given application, constraining a system as little as possible will generally yield the best results, as even small increases in how tightly the spectrum is constrained may significantly increase the amplitude of additional sidebands. The effect that limiting the power of the additional sidebands has on the required time domain phase can be seen in Fig. 6(b).

 

Fig. 6. Limitations of phase-only spectral tailoring. Setting the optical power of some sidebands exactly (red dots in a) usually introduces additional sidebands which are required to ensure the optical field isn’t amplitude modulated as a whole. Limiting the maximum amplitude of these additional sidebands (limiting power indicated by number) simply increases the amplitude of other sidebands further from the carrier. This typically also increases the frequency of oscillations in the required time domain phase, as shown in b).

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5. Convergence and uniqueness

For a given set of constraints there is no guarantee that the algorithm will converge on a solution, or if it does that the solution will be unique, or even that it will satisfy the constraints particularly well, especially if heavily constrained. In practise however, if the desired spectral shaping is relatively simple, after a sufficiently large number of iterations the calculated time domain phase can often converge to the same form, despite being initialised with a different random sample. Where the time domain phase does converge to the same form, it may still have trivial ambiguities such as a global phase shift, a ‘conjugate inversion’, and ‘spatial shifts’ (see Ref. [13] for a more comprehensive discussion of uniqueness in iterative phase retrieval methods). Not all these trivial phase ambiguities apply if the frequency domain phase is also constrained.

If the desired spectrum is more complicated, it becomes much more likely that the algorithm will settle on a different solution every time the algorithm is run, but these solutions will usually satisfy the constraints to a similar extent. There is nothing particularly special about obtaining an identical solution for many different runs with the same constraints (but with different starting conditions), except that it indicates the solution is more likely to be close the the best achievable.

Figure 7 gives an indication of how many iterations it takes to converge on a time domain phase which generates a simple spectrum. The desired spectrum was the same as that shown in Fig. 3(a) (with all constraints applied), and the algorithm was run ten times for several million iterations, with each run starting with an initial guess for the time domain phase of uniformly disturbed random numbers. In each run, both the optical spectrum and time domain phase converged, and appeared practically identical to those shown in Fig. 3. To get an indication of how quickly the spectrum converged to the desired solution, the sum of the residuals of optical power was calculated as the algorithm progressed, and is plotted in Fig. 7(a). The sum of residuals in optical power for the exactly set teeth decrease over time, before stabilising to some minimum value after about one million iterations. Conversely, the sum of residuals for all other teeth increases as the algorithm progresses, before also stabilising after around one million iterations. The residuals for these teeth increase as their power would ideally be zero, but has only been constrained to be less than some maximum value. Although the residuals of the exactly set teeth in Fig. 7 take around a million iterations to stabilise, the solution has mostly been formed after a few hundred iterations, with only very minor improvements in the match between the desired and achieved spectra after this time. A similar behaviour is seen in the remaining teeth (max. limited teeth), where they have mostly obtained their total aggregate power after a few hundred iterations, with very minor changes afterwards.

 

Fig. 7. Convergence and uniqueness. a) shows the sum of residuals of optical power in spectral teeth as a fraction of total exactly set power, for ten different runs of the iterative algorithm. The sum of residuals for teeth that have been set exactly, and those that have only been limited to a maximum value, have been plotted separately. b) shows the similarity between a final solution for the time domain phase, and the time domain phase for each run as the algorithm proceeds.

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Observing the progression of the residuals indicates if the algorithm has converged, but not if the solution found in each instance is the same. To check for uniqueness, the final time domain phase calculated in one instance was selected as the reference phase function, and phases calculated in other instances were compared to this reference phase as the algorithm progressed. To get a measure of similarity of these functions, a normalised cross correlation was performed between the test phase and the reference phase, and the maximum value obtained was plotted, shown in Fig. 7(b). The normalisation factor was taken as the peak value of the cross correlation of the reference function with itself, such that a similarity of unity indicates the function is probably identical.

For the simple constraints tested, all instances of the calculated time domain phase converge to the same function between $10^5$ and $10^6$ iterations. It should be noted that if a given time domain phase is inverted and time reversed, it yields the same optical spectrum as if it was imprinted in its original form. The algorithm converges to these mirrored and inverted phase functions about half the time, which are equivalent, but which the cross correlation metric would indicate are dissimilar. These instances of calculated phase were manually inverted and time reversed before the cross correlation was performed so they could be meaningfully compared.

While calculation of the time domain phase for more complex spectral constraints may take more iterations to converge, or may not be unique, the example shown indicates that the algorithm is easily implementable with the computational power of modern personal computers. The example shown consists of 512 samples, and took around 10 minutes to compute a million iterations with a standard 2015 model laptop computer.

The use cases for the algorithm are constrained by the fact that it is not guaranteed to find a good solution for a given set of constraints, or if it does find a good solution, it may take many iterations to do so. Therefore, the algorithm is not applicable to any application that requires dynamically generating a new spectrum in response to a some input in a latency sensitive manner.

6. Experimental validation

The arbitrary spectrum generation technique was validated experimentally using an optical setup shown in Fig. 8. To generate the tailored spectrum, a $200\,\rm {\mu W}$ laser beam at $795\,\rm {nm}$ was injected into a fibre coupled EOM with $10\,\rm {GHz}$ bandwidth, which was driven by an arbitrary waveform generator (AWG) with $200\,\rm {MHz}$ bandwidth producing $500\,\rm {MSa/s}$. To measure the spectrum and phase of the tailored beam, a heterodyne measurement was performed by splitting off a fraction of the initial beam to act as a local oscillator, downshifting it by around $80\,\rm {MHz}$, and then recombining it with the tailored beam on a non-polarising beam splitter cube. The beam at one of the output ports was directed onto an amplified silicon photodiode with bandwidth $380\,\rm {MHz}$, and its output was recorded on a $1\,\rm {GHz}$, $10\,\rm {GSa/s}$, $12$bit digital oscilloscope. A series of waveplates (not shown in figure) were used to control power splitting and polarisation before and after the active optical elements.

 

Fig. 8. Experimental setup. The optical spectra were tailored using only the EOM with an attached AWG. A heterodyne measurement was performed using a local oscillator which was downshifted relative to the signal beam using an AOM.

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The need to measure optical phase required that the local oscillator and the signal beam be phase locked over the timescale of the measurement. To achieve this, a dual output arbitrary waveform generator with common output sample clock was used to produce the signal for both the AOM and EOM. A common $10\,\rm {MHz}$ reference clock was shared between the waveform generator and the oscilloscope so their relative sample rates did not drift, simplifying analysis. The signal at the photodiode was recorded for only around $2\,\rm {ms}$, so drifts in phase between local oscillator and signal beam due to slow mechanical shift did not affect the measurement, and so active phase stabilisation was not required.

For the demonstration spectrum, the time domain phase was chosen to consist of 512 samples, which is enough to produce very complex frequency domain spectra. At the maximum sample rate of the wavefrom generator, this gave a time domain envelope period of $1.024\,\rm {\mu s}$, and a corresponding sideband frequency of just less than $1\,\rm {MHz}$, which is the same as is used in all examples throughout this paper. The spectral amplitude and phase constraints were set as shown in Fig. 9, which were chosen to cover a wide range of powers, phases, and to be asymmetric and obviously synthetic, but were otherwise fairly arbitrary.

 

Fig. 9. Experimentally achieved arbitrarily tailored frequency comb. Both the amplitude and phase of the individual teeth were set, with the experimentally measured values (orange dots) closely matching the desired value (red dots). For this spectrum, the algorithm managed to achieve an almost exact numerical match to the desired constraints, so the red dots represent both the exactly set input constraints, and the output achieved by the algorithm. The small deviation between the desired values and experimental results comes from experimental factors discussed in the main text.

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The magnitude $|\breve {\mathcal {E}}(\nu )|$ and phase $\theta (\nu )$ of the frequency components of the signal beam can be extracted from the Fourier transform of the heterodyne beatnote intensity, $\breve {I}_{\rm beat}(\nu )$ [23]:

It can be seen from Fig. 9 that the experimentally realised spectrum closely matches the desired spectrum both in magnitude and phase. The cause of the small variation that does exist in some of the teeth was not investigated, but was possibly due to the finite fidelity of arbitrary signal generation, and some nonlinearity between applied voltage and phase retardance in the EOM. EOM phase retardance as a function of input voltage may drift over time, so regular spectrum measurement and voltage calibration may be necessary for applications where the precision of the produced spectrum is important.

7. Conclusion

We have presented an extremely simple method of arbitrarily tailoring the optical amplitude and phase of teeth in an optical frequency comb generated with an EOM, which requires only an electro-optic modulator and arbitrary waveform generator. To achieve the desired spectrum an iterative phase retrieval algorithm is used, with constraints that optimise the result to be as close as possible to the desired spectrum, with as little power in additional comb teeth as possible. The spectrum tailoring technique was experimentally validated using heterodyne measurement, and the resulting spectrum closely matched the desired one.

The accuracy limit in generating a desired spectrum ultimately comes from the strict constraint that time domain amplitude cannot be altered using only an EOM. So while a limited set of sidebands can be tailored arbitrarily, the phase-only modulation necessitates that additional sidebands must also be generated to cancel out any amplitude modulation which would otherwise result. Whether or not these additional sidebands are detrimental would depend completely on the application.

This technique could find wide use in quantum, optical, and atomic physics experiments where precise control over optical waveforms is needed, and may aid in the development of cluster state quantum computing and sensing. In future the algorithm could be generalised to the case where the incoming optical signal has a periodic, rather than constant amplitude. Example code implementing the algorithm herein described is provided in Code File 1 (Ref. [24]).