1. Introduction

Generation, control, and processing of microwave and millimeter-wave arbitrary signals is a major challenge due to potential applications in ultra-wideband (UWB) fiber-wireless communication systems, pulsed radar, radio frequency communications, and sensor networks, among others [1, 2]. Electronic devices used to generate high-frequency complex waveforms are seriously limited by the bottleneck of digital-to-analog technology. In practice, current electromagnetic arbitrary waveform generators (AWGs) are restricted to signals with a bandwidth below 2 GHz. Photonically-assisted AWGs far exceed this limit [3–16]. These devices are designed to achieve a user-defined optical sequence which is converted to the electrical domain by means of a high-speed photodiode or a photoconductive antenna. Commercially available photodiodes have a bandwidth approaching the 60 GHz regime and the photoconductive antennas exceed the 1 THz, which make them useful for sub-millimeter applications.

High-resolution photonic AWGs based on an array of fiber Bragg gratings have been proposed and experimentally verified. Here, the user-defined output optical waveform is obtained by the interference of the time-delayed signals emerging from each grating [5, 6]. The use of a nonlinear birrefringent fiber has also been considered [7]. Optical pulse shapers have also been exploited to extend microwave AWG at higher frequencies. These AWGS are certainly the most appreciated as they enable adaptive waveform synthesis with unprecedented agility. Direct space-to-time pulse shapers using a virtually imaged phased-array have demonstrated the generation of 10–50 GHz arbitrary microwave waveforms with time apertures as large as 1.0 ns and peak-to-peak amplitudes as high as 400 mV [8–10]. On the other hand, electromagnetic AWG based on Fourier transform pulse shapers rely on spectral shaping by means of a pixelated spatial light modulator and subsequent electrical conversion [11, 12]. Here, the time aperture constraint has been circumvented by wavelength-to-time mapping through a dispersive medium such as an optical fiber. Furthermore, coherent control of the field at all stages through heterodyne optical-to-electrical conversion has been demonstrated to be of relevance for UWB electromagnetics [13]. The need for high-quality bulk optical elements and limited integration with waveguide devices motivate research on alternative solutions. Recently, an integrated solution to generate UWB pulse signals where the filtering stage is performed in the time domain has been experimentally demonstrated [14]. It is based on phase-modulation of a continuous-wave (CW) which is distributed over a single mode fiber (SMF) link. Phase modulation provides both a higher energetic efficiency and a larger variety of output waveforms than those based on intensity modulation [15].

In this paper, we propose a new method for the generation of millimetre-wave and microwave signals via low-frequency phase-modulation of a time-stretched optical pulse and subsequent signal dispersion through a group delay dispersion (GDD) circuit. In the first stage, a time-stretched linearly chirped pulse is generated from a transform-limited ultrashort pulse launched a distance z through a SMF with group velocity dispersion coefficient β ₂. Note that both the temporal aperture and the chirp imparted over the input pulse are fully controlled by the product β ₂ z. In the second stage, the dispersed optical pulse is phase-modulated within an electrooptic phase modulator (EOPM) driven by a low-frequency, up to 2 GHz, periodic microwave waveform V(t). Finally, the modulated signal is launched through a linearly chirped fiber Bragg grating (LCFG) with reversed dispersion, total dispersion Φ₂, which allows for achieving the output electrical sequence after optical-to-electrical conversion in a fast photodiode.

We show that, although arbitrary waveform generation capabilities are limited, a wide variety of user-defined high-frequency microwave signals can be achieved by appropriate choice of the input dispersion β ₂ z, the output dispersion Φ₂, and the modulating signal V(t). Particular emphasis is paid on the shaping problem. We find that it is best formulated when the LCFG behaves as a transversal filter with N taps. For this situation, the dispersed pulse shape is expressed as a weighted sum of contributions from a finite number of temporal points within the input phase function. Specifically, we derive the amount of output dispersion required to reach the above goal. The above dispersion is connected with a fraction of the temporal Talbot dispersion of the periodic signal V(t) through a simple expression. Furthermore, we demonstrate that the scaling problem at the output plane, which in turn determines the output bandwidth, is fully determined by the ratio Φ₂/β ₂ z whereas the temporal aperture effect is dominated by β ₂ z. As a result, the proposed device allows for independent tuning of the shape and the bandwidth of the output signal, which provides a high flexibility. This is the first time to our knowledge that a phase-only modulator is proposed as the driving signal for nearly arbitrary waveform generation basing on an N-fractional Talbot configuration.

The proposed device offers high integrability, low cost, reconfigurability, high speed of operation, high stability, and, due to the fact that the modulation process is phase-only, essentially there is no loss of energy at this stage. Moreover, it would also allow for the feedback of the generated electrical signal into the EOPM [16], which could substantially increase the operational stability. Finally, we present some simulations concerning square-wave-type waveforms within the range 40–60 GHz with time apertures as long as 0.5 ns from an EOPM driven at 2 GHz. This work is structured as follows. First we give a schematic preview of the setup under study. In the third section we provide the analytical expressions which allow optical waveform synthesis in an easy way. In the fourth section we discuss the bandwidth tuning capabilities of our proposal. Some simulations including the temporal aperture effect are presented in section five. Finally, we highlight the main results of the work.

2. Theoretical analysis

A schematic view of the proposed device is shown in Fig. 1. A transform-limited optical pulse emerging from a modelocked laser is launched through a SMF. We assume an input Gaussian pulse with amplitude U _o, carrier frequency ω _o, and intensity root-mean-square (rms) width σ _o. It is well-known that, up to first-order, the complex field envelope after linear propagation a distance z along the SMF, $U_{\mathit{\text{in}}}^{-}$ (t), is given by

The effect of the chromatic dispersion of the fiber on the input pulse is twofold. On the one hand, the pulse broadens to a rms width σ ²(z)=${\sigma }_o^2$ [1+(β ₂ z/2${\sigma }_o^2$ )²]. Furthermore, a linear chirp is impressed onto the pulse upon propagation with chirp parameter C(z)=(4${\sigma }_0^4$+(β ₂ z)2)/${\beta }_{2}z$ . Note that both σ(z) and C(z) scale linearly with the propagation distance z when β ₂ z/2${\sigma }_o^2$ ≫1. This assumption holds for the practical situations considered through this paper. Thus, we will assume σ(z)≅β ₂ z/2σ _o and C(z)≅β ₂ z.

 

Fig. 1. Experimental setup under study

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In the second stage, the phase of the dispersed optical pulse is temporally modulated by a periodic microwave signal V(t) by use of an EOPM. The resulting optical field, $U_{\mathit{\text{in}}}^{+}$ (t), is given by $U_{\mathit{\text{in}}}^{+}$ (t) $U_{\mathit{\text{in}}}^{-}$ (t)exp[jV(t)]. Thus, $U_{\mathit{\text{in}}}^{+}$ (t) can be expressed as

We mention that the bandwidth of V(t) is around 2 GHz, which is the upper limit for electronics-based AWGs. It is important to recognize that the temporal aperture of the system (the temporal window over which the phase modulation operates) is determined by σ(z).

We proceed to propagate the phase-modulated signal through a subsequent GDD circuit, which is represented in Fig. 1 by means of a LCFG operating in reflection. The dispersive medium acts as a spectral phase-only filter with a transfer function given by

Here the coefficients Φ₁ and Φ₂ are, respectively, the first and the second derivative of the phase spectral transfer function evaluated at ω_o . Aside from some irrelevant constant factors, the output complex field envelope is found from the convolution integral

Here, τ denotes the so-called proper time; namely, τ=t-Φ₁. Finally, the output intensity I_out (τ,Φ₂)=U_out (τ, Φ₂)|² is mapped from the optical to the electrical domain with a high-speed photodetector.

3. Waveform synthesis

In this section we deal with the synthesis problem; namely, given the user-specified optical intensity waveform at the output of the LCFG I_out (τ, Φ₂) we seek the values of the modulation function V(t) that will give rise to the above intensity. The theoretical analysis will be performed by assuming that the signal impinging onto the EOPM $U_{\mathit{\text{in}}}^{-}$ (t) has an infinite width. This is equivalent to assume an ideal periodic phase modulation V(t). In mathematical terms, we require σ(z)≫T, where T is the temporal period of V(t). However, in practice, the width of the signal $U_{\mathit{\text{in}}}^{-}$ (t) has a finite value. The effect of the time window of the phase modulation will be included through numerical simulations in Section 5.

Next, we will take advantage of the periodic nature of the driving signal. The analysis of the synthesis problem is best performed when the field outgoing the EOPM is written as

In the above equation the symbol * stands for the convolution operation, δ() denotes the Dirac’s delta function, and

The complex field envelope after dispersion through the LCFG, U_out (τ, Φ₂), is found by substitution of Eq. (5) into Eq. (4). In this way we obtain

where

A relevant expression that connects I_out (τ, Φ₂) to V(t) can be obtained for an output GDD Φ₂ satisfying the relationship

or, equivalently,

In the above equation P and N are coprime integer numbers, N≥2, and Φ_2T=T ²/π is the Talbot dispersion parameter of the periodic signal [17]. For the above dispersion, we can rewrite Eq. (8) as

Finally, inserting Eq. (11) into Eq. (7) and after a long but straightforward calculation, we find the output optical field is given, aside from an irrelevant quadratic phase factor, by

where

From Eq. (12) we can easily find the output intensity waveform once the driving signal V (t) is given. In fact, Eq. (12) indicates that the output complex envelope depends only on the input field values at a finite number of regularly spaced time intervals. In this way, the output GDD circuit behaves as a nonrecursive filter with N taps. Note that the weight of each tap is given by Eq. (13) and, thus, only can be modified by the choice of the parameter P. It should be emphasized that the above result only holds for the set of the output GDD coefficients given by Eq. (10). Thus, the capabilities to generate an arbitrary optical waveform are limited and the inverse problem, i.e., to obtain a specified waveform from a periodic V(t) function is only analytic in some special cases. This subject will be addressed from a general point of view elsewhere. At this point it is worth mentioning that a close spatial analogue of the above formula was derived in Ref. [18], in the context of diffractive optics, to describe the properties of the irradiance distribution corresponding to the Fresnel diffraction patterns of a one-dimensional phase grating. However, Ref. [18] only deals with the plane wave-front case, which corresponds to an unchirped input signal in the temporal domain.

In this paper, we restrict our analysis to the cases where the summation in Eq. (12) can be performed to obtain an analytical expression for the output light intensity. First, we consider the case N=4 and P=1, namely, Φ₂=β ₂ ZΦ_{2 T} /(4β ₂ Z-Φ_2T). From Eq. (12), we find

The above formula was derived for an unchirped input signal both in the temporal [19] and in the spatial domain [20]. Alternatively, for N=3 and P=1, Φ₂=β ₂ ZΦ_2T/(3β ₂ Z-Φ_2T), we get

The spatial analogue of this formula, for plane wave-front illumination, was derived in [21].

4. Bandwidth tuning

Once we have obtained an analytical expression for the shape of I_out (τ, Φ₂), now we highlight the bandwidth tuning capabilities of the optical device in Fig. 1. With this aim, the above results are particularized for an unchirped input phase signal; namely, when z→∞. In this way from Eq. (11) and Eq. (12) we obtain,

where the weight of the taps is given again by Eq. (13). Several conclusions can be drawn from the above result. First, we infer that U_out (Mτ, Φ₂=β ₂ zΦ_2T P/N/(β ₂ z-Φ_2T P/N)) is a scaled replica of U_out , _z→∞(τ, Φ₂=Φ2TP/N). From Eq. (12) we learn that the scaling factor of the waveform generator (compression or magnification) is given by M=β ₂ z/(β ₂ z-Φ_2T P/N). Even more, from the above expression, we recognize that when sign(β ₂ zΦ_2T P/N)=-1, then |M|<1. In other words, the system must be configured with two dispersive lines with opposite sign dispersions to achieve frequency upshifting. Note that, within the framework of the space-time duality, the linearly chirped optical source can be understood as the temporal counterpart of spherical-wave illumination [22, 23].

From a practical point of view, for a fixed driving period T, once an output shape profile has been chosen, which in turn determines the factor Φ_2T P/N, the temporal scale of the output can be tuned through the dispersion of the SMF β _2z. It is assumed that the optical-to-electrical conversion has a bandwidth large enough so that a direct mapping from the optical to the electrical domain is performed. In this way, there is no distinction between the optical intensity shape and the electrical one. Finally, we note that the amount of GDD dispersion required to achieve the user-specified output profile is β _2z-dependent as indicated in Eq. (10). Furthermore, the choice for β ₂ z also determines the temporal window over which the phase modulation operates.

5. Microwave and millimeter-wave signal generation

To illustrate further the capabilities of the proposed device, we conduct a series of numerical simulations of Eq. (4) to obtain a variety of useful waveforms approaching the 50 GHz bandwidth range. The temporal aperture effect has been included into the calculations by taking into account a finite width for the signal $U_{\mathit{\text{in}}}^{+}$ (t), which is given by Eq. (2). Results of the simulations demonstrate that the intensity output is mainly affected by a temporal window which is in essence a Gaussian function. In this way the output electrical signal has a finite duration. In all cases, we consider an initial transform-limited Gaussian pulse obtained from a modelocked laser with a temporal duration of σ _o =1ps and carrier wavelength λ _o =1.55 µm. Note that at this wavelength and for that bandwidth (~8 nm), Erbium-doped fiber amplifiers present a nearly flat gain curve. Thus, they can be included in any stage of the setup depicted in Fig. 1, if necessary, to increase the energy of the waveform without any essential distortion of the shape. A SMF with a GVD coefficient of β ₂=-21.6 ps²/km is assumed. Concerning the EOPM, we assume a realistic repetition rate of f=2GHz for the driving function, V(t), which provides a temporal period of T=500ps. Its corresponding Talbot dispersion parameter is Φ_2T=79578ps².

The first example deals with a sinusoidal driving signal V(t)=Δθsin(2πft). Here, Δθ is the so-called modulation index that has been fixed to Δθ=π/4rad. The device parameters have been set so that the LCFG behaves as a transversal filter with 4 taps and P=1. This choice fixes the shape of the output pulse which corresponds to a periodic signal with a nearly flat-top unit cell, as can been devised from Eq. (14). Furthermore, a 20 GHz output repetition rate is demanded. From Eq. (10) and Eq. (11), we infer that a fiber length of 102 km and an output dispersion of Φ₂=1989ps² are required. In Fig. 2 we can see the results of the computer simulations where a burst of nearly square signals at 20 GHz is clearly appreciated. It is inferred that the bandwidth of a single pulse within the periodic sequence is ~40 GHz, since it occupies half of a period. We find a time aperture around 0.5 ns.

 

Fig. 2. In solid line, results of computer simulation for square-wave-type burst generation with nanosecond duration. For comparison, the case when aperture effects are neglected is also plotted in the same figure by means of a dashed line.

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Fig. 3. Results of computer simulation for 40 GHz microwave tone burst generation from triangular phase modulation at 2GHz.

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In the second example we demand an output intensity profile I_out (τ, Φ₂) that corresponds to a microwave tone burst at 40 GHz. Again, this signal is obtained from an LCFG that behaves as a 4 taps transversal filter with P=1. As devised from Eq. (14), now, a triangular driving signal with a phase jump of π/2 is needed. In mathematical terms,

From Eq. (10) and Eq. (11), we infer that a fiber length of 48 km and an output dispersion of Φ₂=994ps² are required. The corresponding time aperture is 0.2 ns. In Fig. 2 we can see the results of the computer simulations. Again, for comparison, the case when aperture effects are neglected is also plotted in the same figure by means of a dashed line.

Finally, we demonstrate square-wave-type train pulse generation with adjustable duty cycle and time slot. First, we consider the generation of a pulse waveform with a duty cycle of 50%. Such waveform has proved to be a useful signal for switching ultrafast optoelectronic devices. This signal can also be obtained from a transversal filter with 4 taps and P=1. Now, we require a driving signal in the form of a serrodyne-like function with π jump phase. In mathematical terms,

For an output period of 50 ps, which corresponds to a bandwidth of 40 GHz for a single pulse within the periodic sequence, the required fiber length and output GDD are 102 km and Φ₂=1989ps², respectively. Results of computer simulations are shown in Fig. 4(a). The duty cycle is varied if we consider a LCFG design corresponding to a 3 taps transversal filter and P=1. From Eq. (15), we infer to obtain square-wave-type pulse generation the EOPM should be driven with a binary modulation format with a jump phase of -2π/3. In mathematical terms,

In this way, a square-wave signal with a duty cycle of 33% is achieved. For a repetition rate of 20 GHz, which corresponds to a bandwidth of 60 GHz for a single square pulse, the required fiber length and output GDD are 136 km and Φ₂=2652ps ². Results of computer simulation are plotted in Fig. 4(b).

 

Fig. 4. Square-wave-type pulse train generation with variable duty cycle: a) the EOPM is driven with a serrodyne phase function and the LCFG behaves as a transversal filter with 4 taps; b) the EOPM is driven with a binary signal and the LCFG behaves as a 3 taps transversal filter.

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5. Conclusions

In short, we have proposed and numerically illustrated a new configuration for waveform generation of electrical signals approaching the 50 GHz bandwidth with time apertures as large as a few nanoseconds, by low-frequency, up to 2 GHz, electro-optic phase modulation of time-stretched optical pulses. The feasibility of the device has been tested by means of numerical simulations. Different useful waveforms in the microwave and millimeter-wave range have been demonstrated. We have performed an analytical treatment of the problem based on the space-time analogy by recalling the counterpart phenomenon of Fresnel array illuminators under spherical wave illumination. The effect of finite time aperture has also been numerically considered.