We propose and numerically validate a new design concept for on-chip optical pulse shaping based on discrete space-to-time mapping in cascaded co-directional couplers. We show that under weak-coupling conditions, the amplitude and phase of the discrete complex apodization profile of the device can be directly mapped into its temporal impulse response. In this scheme, the amplitude and phase of the apodization profile can be controlled by tuning the coupling strength and relative time delay between the couplers, respectively. The proposed concept enables direct synthesis of the target temporal waveforms over a very broad range of time-resolution, from the femtosecond to the sub-nanosecond regime, using readily feasible integrated waveguide technologies. Moreover, the device offers compactness and the potential for reconfigurability.
© 2015 Optical Society of America
Optical pulse shaping techniques have been investigated for a wide variety of applications in high-speed communications, ultrafast information processing, waveform generation and control etc [1–3]. Presently, optical pulse shapers based on the well-established spatial-domain processing approach are commercially available, and they allow programmable synthesis of arbitrary waveforms with resolutions better than 100 fs [4,5]. However, the major drawback of this method is its relative complexity, requiring the use of very high-quality bulk-optics components, which also affects the device insertion losses in fiber-optics systems. To overcome some of these difficulties, devices based on similar pulse shaping principles have been implemented using on-chip arrayed diffraction gratings [6–8]. Despite their important advantages, particularly in terms of compactness, these devices still suffer from limited spectral resolution and sensitivity to fabrication errors. In the quest for more compact, lower loss and relatively simpler devices for optical pulse shaping/coding, all-fiber and integrated-waveguide grating structures have attracted considerable attention . A particularly interesting design approach for fiber or waveguide grating exploits the so-called first-order Born approximation, under which the devices’ temporal impulse response is a direct, scaled version of the grating apodization profile [9–13]. This space-to-time mapping (STM) property greatly simplifies the grating device design for temporal pulse shaping applications. However, both short-period (Bragg) and long-period grating devices have proven challenging to fabricate in integrated-waveguide configurations, particularly if they are intended to synthesize complex waveforms, e.g., with high resolutions, below the sub-picosecond regime, over long durations, above a few picoseconds [9,10]. Additionally, Bragg grating (BG) devices need to be operated in reflection and it would be also difficult to add reconfigurability in these structures, e.g., through the integration of high-resolution thermo/electro-optical controllers, due to the nanometer-scale of the grating features.
In this communication, we introduce a novel discrete co-directional coupler structure and its design approach for general time-domain optical pulse shaping. The proposed design is based on forward coupling between a main waveguide and a bus waveguide, where the coupling is controlled in a discrete fashion (point by point) through standard co-directional couplers, see scheme in Fig. 1(a).
In particular, we show that the device can be designed so that the ‘discrete’ amplitude and phase ‘apodization’ profile along the concatenated couplers, namely coupling strength and relative time delay between couplers, can be directly mapped into the output temporal response. This approach can be interpreted as a discrete version of the STM process in waveguide gratings . Similarly, the approach significantly facilitates the design of structures based on concatenated co-directional couplers for temporal pulse shaping operations, as compared with standard design approaches based on spectral-domain response synthesis, e.g., so-called lattice filters [14–18]. In particular, a lattice filter device, involving concatenated interferometer with ring resonator incorporated into its upper arm, has been recently implemented in a silicon photonics technology ; not only this device has been designed using a relatively more complex spectral-domain synthesis approach, but additionally, it incorporates a phase-shifter in each ring resonator, making it difficult to scale the device for operation bandwidths above a few GHz. In contrast, the devices achieved from our newly proposed design can be easily scaled for operation bandwidths into the THz range. Moreover, the resulting devices are notably simpler to fabricate than their waveguide-grating counterparts, while also enabling reconfigurability through well-established mechanisms. For instance, precise control of the temporal response amplitude or phase could be achieved by correspondingly tuning the coupling length, Fig. 1(b), or differential delay between couplers, Fig. 1(c), respectively.
In what follows, we first introduce the concept of discrete STM through an ideal modeling of the temporal impulse response of the proposed device. The concept is subsequently validated through numerical simulations based on a transfer-matrix method (TMM), considering practical non-idealities in the device, including losses, waveguide dispersion, tolerances in the device spatial features etc. Numerical examples are shown to confirm the possibility of scaling the concept for pulse shaping over a wide range of temporal resolutions, from the femtosecond to the sub-nanosecond regime, using readily feasible, compact designs.
2. Theoretical derivation
2.1. Temporal impulse response
Let us assume and , with , being the time-domain complex envelopes of the signals at the input of each coupler, as shown in Fig. 1(a). The proposed configuration consists of n stages, where each stage comprises a directional coupler and a delay line. These stages could be connected to each other with identical delays if a temporal impulse response with a flat phase profile is of interest. On the other hand, by changing the relative delay between different stages, it is also possible to shape the phase profile of the impulse response of the system (more details given below). In our notation, is the power coupling ratio to the cross-port of the i-th coupler. The central assumption in our design is that the device operates under weak-coupling conditions (i.e. strictly, ). In this case, the time-domain impulse response h(t) at the output port illustrated in Fig. 1(a) can be analytically calculated as follows (it is assumed that the coupling values, , are real and positive):
Hence, as predicted, under the defined weak-coupling condition, the discrete-time impulse response of the device (input: main waveguide; output: bus waveguide) is a time-scaled version of the discrete coupling profile of the concatenated couplers. Notice that the nominal time-delay difference, , depends upon the nominal length difference, , in between consecutive couplers, where , is the effective refractive index of the waveguide and is the speed of light in vacuum. Additionally, to implement a π-phase-shift at a desired point of the impulse response, the length difference ∆L before the corresponding coupler should be changed to . More generally, in order to achieve phase shifts of π/m, the change in the length of the delay line should be equal to , see Fig. 1(c). Finally, we should note that on the basis of the described performance, the nominal time-delay difference defines the device’s time resolution whereas the number of stages in the device defines the number of points (e.g., ratio between temporal duration and resolution or so-called time-bandwidth product, TBP) of the synthesized waveform.
2.2. Transfer matrix method for numerical simulations
In this paper we are specifically interested in the realization of the proposed device using silicon waveguides on a silicon-on-insulator (SOI) substrate, as this platform provides compactness and compatibility with complementary metal–oxide–semiconductor (CMOS) technology. In order to precisely model the device, waveguide dispersion, loss and also errors due to imperfections of the fabrication process have been carefully taken into account. A two-port lattice-form circuit configuration consisting of n pairs of waveguides with different path lengths and n directional couplers has been considered, in order to model the device using the TMM . The first element at each stage (period) of the device is a pair of waveguides with a delay time difference of . The upper waveguides (sections of ‘bus waveguide’) are usually considered to be straight and short while the lower waveguides (different portions of the ‘main-waveguide’) are longer and bended to realize the delay. The transfer matrix for these unit cells can be expressed as:19], i.e., the waveguides’ lengths are modeled to have each a Gaussian distribution with a mean value of (and/or ) and a variance of 5 nm. In addition, each directional coupler is modeled using the transfer matrix:17]. In particular, and in which , is a 2 × 2 matrix with the following set of elements:
In our analysis, we consider a silicon strip waveguide with a dimension of 500 nm (width) × 220 nm (height). This waveguide is fabricated on an SOI wafer. The cladding and buried oxide layers are assumed to be made of silicon dioxide with a height of 2 µm and 3 µm, respectively.
There are three important sources of error which may affect the coupling coefficient of the directional couplers: (i) changes in coupling gap due to the lateral shift of the waveguides in the coupling region; (ii) unbalanced waveguide widths in the coupling region; and (iii) wavelength dependency of the coupling ratio due to waveguide dispersion . Three-dimensional finite difference time domain (3D FDTD) numerical simulation has been used to evaluate these errors in our proposed designs.
For case (i), the coupling gap has been altered in steps of 5 nm around a typical value of 200-nm; for case (ii), we have considered two waveguides with unbalanced widths of 500 ± 4 nm at the coupling region. Finally for both cases, we have swept the coupling length (in steps of 1µm) and recorded the coupling ratio at each step, at a wavelength of 1555 nm. All directional couplers consist of 4 s-bends with the length of 4 μm and the height of 2 μm connecting the waveguides at the coupling region to cross and through ports.
The results, shown in Fig. 2, provide the relative amount of change in the coupling ratio of the coupler caused by these sources of error. According to the simulation results, we should consider a maximum of 15% and 6% variations in the coupling ratio over its nominal value due to fabrication-related fluctuations in the coupling gap or unbalanced waveguide width, respectively. Such estimates have been plugged in the TMM modeling of the couplers, Eq. (4), as follows: Two additive Gaussian distributions with mean values of and variances of and , respectively, have been introduced into the model to account for variations in the coupling ratio. We have considered these two sources of error to be uncorrelated.
Moreover, for case (iii), 3D FDTD simulations have been carried out to study the effect ofchromatic dispersion on the coupling ratio of the directional couplers. In particular has been defined to be a wavelength dependent parameter by plugging the data from the FDTD analysis into our TMM model.
3. Weak-coupling condition in cascaded co-directional couplers
The range of validity of the weak-coupling condition in the proposed device has been evaluated in deeper detail using the following strategy. We consider our output target to be a flat-top pulse, requiring the use of identical coupling ratios and delays; for a fixed number of stages, evaluations are made as the power coupling ratio at each period is increased, correspondingly increasing the peak of the device’s power spectral response (PSR). We consider an upper margin for the weak-coupling condition, where half of the transmitted input power is transferred into bus-waveguide. In other words, the PSR peak from the bus-waveguide reaches 50% of the maximum PSR from the main-waveguide (for a loss-less structure, this maximum PSR value is unity).
Typical waveguide propagation loss for the bus waveguide has been considered to be 3 dB/cm . Moreover, for the main-waveguide we add (4.n.LPB) dB to this number in order to account for the extra losses in bent waveguides (LPB being loss per bend, from Ref .). The term in parenthesis refers to the loss per bend in four (4) quarter curves, which form the delay lines. The bus- and main-waveguides have been designed to have a length difference of ( = 6.85-µm for R = 3 µm, in the case under study) at each period.
For different number of couplers (i.e. n = 7, 12, …, 27) and a fixed differential delay of 6.85-µm at each stage, we target flat-top pulses with a variety of temporal durations, depending on the number of couplers. In all cases, the input waveform is a Gaussian transform-limited pulse with a full width at half maximum (FWHM) duration of 150 fs.
Assuming that we then record the corresponding power coupling ratio and estimate the maximum relative deviation of the waveform with respect to the ideal one along the flat-top region. The recorded value of (i = 0,…, n-1) is the maximum power coupling ratio that one can achieve while still satisfying the prescribed upper margin of the weak-coupling condition.
The results shown in Fig. 3 for 7 cascaded couplers confirm that the deviation in the synthesized waveform with respect to the target one is more pronounced as the coupling strength is increased; however, a higher coupling strength leads to an increased PSR peak, which translates into a higher device energy efficiency. For the flat-top case, a reasonable waveform deviation (~6% error) is still achieved for a 50% bus-waveguide PSR peak so as anticipated above, this is chosen as the condition for optimizing the trade-off between waveform deviation and device efficiency. Results from simulation concerning the ‘optimal’ specifications (PSR peak ~50%) for different number of couplers are presented in Table 1. This table confirms that the coupling coefficient must be made weaker as the number of couplers is increased in the device, i.e., as one targets the synthesis of more complex waveforms (with a higher TBP). This is in good agreement with the assumption made in Section (2.1) for the power coupling ratio to satisfy the weak-coupling condition ().
4. Design examples for optical pulse shaping and coding
To confirm the validity of our proposed design, we have numerically simulated devices to synthesize two different waveforms, namely flat-top waveforms, and an 8-symbol optical 16-QAM signal. We target the synthesis of temporal features from the femtosecond range up to a few hundreds of picoseconds.
Our first target waveform is a ~850 fs (FWHM) flat-top pulse generated from an input ultra-short optical Gaussian pulse with a temporal duration of 150 fs (FWHM). The input optical pulse is assumed to be centered at 1529 nm. Ten couplers with identical coupling coefficients of 0.004 (corresponding to coupling gap of 200 nm and coupling length of 1 µm) are connected in series through 65.79 fs delay lines. As shown in Fig. 4(a), the upper arms are 4R long and the lower arms have a length of 2πR (R: bend radius for this specific example is equal to 3 µm). Adiabatic Bezier bends have been incorporated in the delay lines to reduce the bending loss [21,22]. The losses have been considered to be 3 dB/cm in the bus waveguide and 3dB/cm + (4.n.LPB) dB in the main waveguide. LPB for this specific case is ~0.01 dB/bend . The overall propagation loss for the light wave travelling through the main- and bus-waveguides are ~0.48 dB and ~0.06 dB, respectively. We run the simulations for 100 times, considering discrete values of error in the coupling ratio and in the phase of each delay line due to variations in waveguide lengths, following the error statistics defined above. Figure 4(d) shows the overlap of the synthesized output flat-top waveform for the 100 evaluated cases, together with their mean value. A standard deviation (STD) of ~0.02 has been estimated for the random fluctuations of the flat-top section of the waveform around its mean value.
The second target waveform is a 1.7 ps (FWHM) flat-top pulse generated from an identical structure but this time cascading 20 couplers, each with a power coupling ratio of 0.0009 (corresponding to a coupling gap of 200 nm and a coupling length of zero). The expected propagation loss for the light travelling through the main- and bus-waveguides are ~0.97 dB and ~0.13 dB, respectively. The input signal is again a 150-fs (FWHM) Gaussian transform-limited pulse. Following the same numerical simulation strategy, a similar value for the STD (~0.02) of the generated flat top has been achieved. The obtained results are shown in Fig. 4(e). Moreover, it is shown in Fig. 5, that the maximum phase variation along the duration of the pulse is ~0.25 rad.
It is worth noting that the number of sidelobes in the sync-like spectral response of the device is directly related to the TBP of the synthesized flat-top waveform (i.e., ratio of flat-top duration to the raising or decaying time of the waveform), which in turn depends on the number of concatenated couplers. Comparing Fig. 4(h) and Fig. 4(i), one can observe that the 2nd device provides a higher TBP. As predicted, a two-fold improvement in the TBP has been achieved for 20 cascaded couplers with respect to 10 cascaded couplers.
A third waveform synthesis case has been considered to illustrate the possibility of using the proposed method for synthesizing longer temporal shapes, in the sub-nanosecond regime. In particular, we have designed and numerically simulated a waveguide structure for generation of an 8-symbol 16-QAM signal, with a speed of 24 Gsymbol/s, from an input ultra-short optical Gaussian pulse with duration of 17 ps (FWHM). The designed amplitude and phase profiles for the target QAM coding operation, and the related constellation diagram, are shown in Fig. 6(b-c). Twenty four couplers were used to generate an 8-bit amplitude and phase modulated code. The length of the upper arm is equal to 415 µm and the lower arm is 1,262 µm long (see Fig. 6(a)). As anticipated, this device exhibits a far longer sampling time (differential delay of 11.78-ps and corresponding length difference of 847 µm) than for the previous flat-top pulse synthesizer designs. Consequently, the trailing copies of the input pulse will experience higher values of propagation loss. This would lead to a ramp-like response, affecting the performance of the pulse-shaping device. A solution for this problem is to use optimized designs aimed at reducing losses in the long delay lines. Multi-mode rib waveguides (MMWs) with strip widths of 3 µm, strip heights of 220 nm, slab widths of 5.1 µm, and slab heights of 90 nm have been implemented in our proposed design to reduce the propagation loss from 3 dB/cm down to less than 1 dB/cm . However, SMWs are still used in the coupling region and waveguide bends, in order to ensure single-mode operation of the device. To make sure that only the fundamental mode of the MMW is excited, 100 µm-long linear tapers are used for conversion between SMWs and MMWs . Using this design strategy the loss in the upper arm and the lower arm are 3 dB/cm and 1 dB/cm + (6.n.LPB) dB, respectively. The number 6 here refers to the number of quarter Bezier curves which have been incorporated to form the delay line. The bends are 5-µm, and we consider the use of Bezier bends with ~0.008 dB/bend loss . In this case, we estimate ~3 dB propagation loss for the light passing through the bus-waveguide and ~1.8 dB loss at the output of the main-waveguide. In practice, we expect higher values of loss for the main waveguide. The reason for that are the tapers that have been used in different sections of the device. If the tapers are not long enough, the mode mismatch loss between the SMW and MMW may become considerable. However, making the tapers longer will eventually reduce this source of loss to negligible values.
Four amplitude levels are achieved using directional couplers with power coupling ratios of κ, 9κ/16, κ/4 and κ/16. Considering κ = 0.0016, the coupling gaps are equal to 385 nm and coupling lengths are equal to 1 µm, 560 nm, 250 nm and 0, respectively.
Considering similar statistical variations in the fabrication parameters to those considered for the previously studied cases, the amplitude and phase profiles of the time-domain waveform at the output of the simulated device are shown in Fig. 6(d), demonstrating again accurate generation of the target data stream.
We have proposed a new design concept for linear optical pulse shaping based on discrete space-to-time mapping in cascaded co-directional couplers. The proposed approach bypasses the problems associated with complex, indirect designs based on synthesizing the desired response in the spectral domain, by directly mapping the target time-domain impulse response along the device’s spatial apodization profile (coupling-coefficient and phase-delay profile). The proposed design approach could be adapted for application over a wide range of temporal resolutions, from the sub-picosecond to the sub-nanosecond regime, and it should enable the synthesis of relatively complex (high TBP) temporal waveforms with energy efficiencies approaching 50%. The temporal resolution of the device depends on the differential path length between the bus- and main-waveguides so that adding a long phase shifter, as needed to tune the device spatial apodization profile, should be possible without affecting the fundamental device performance.
This research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC), and le Fonds de Recherche de Québec - Nature et Technologies (FRQNT). Hamed Pishvai Bazargani acknowledges financial support, Bourse de doctorat en recherche pour étudiants étrangers – FRQNT, from the government of Québec for his PhD studies. The authors would like to express their gratitude to Dr. Reza Ashrafi, Dr. Alexander D. Simard and Prof. Lukas Chrostowski for fruitful discussions.
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