1. Introduction

The principle of operation of many optical devices is based on grating-assisted (GA) mode coupling within or between waveguides (see for instance Refs. [1]–[5] and references cited there). These optical devices can be divided into two broad groups depending on the direction of propagation of the coupled optical modes: contradirectional couplers, if the interacting modes propagate in opposite directions [1]; and codirectional couplers, if they propagate in the same direction [2]–[5]. Short-period Bragg gratings (BGs) written along the core of optical fibers or within integrated waveguides are particularly interesting examples of contradirectional couplers, where the coupling is induced between the forward- and backward- propagating modes in the fiber/waveguide [1]. BGs have become essential components for numerous applications in optical communications and fiber optic sensing. Preliminary studies on the time-domain behavior of BGs [6]–[10] have stimulated more recent research on the use of BGs for various optical pulse processing and shaping applications. For instance, dark solitons [11] and squared temporal waveforms [12], [13] have been generated from mode-locked optical pulses using BGs for the pulse re-shaping operation. Other time-domain applications of BGs include all-optical pulse repetition rate multiplication [13]–[15] and optical code generation and recognition [9]. Note that most of the mentioned time-domain applications are based on BGs operating within the linear regime. It is also important to mention that the BG approach is specially suited for synthesizing temporal features in the range of a few tens of picoseconds. The synthesis of faster temporal features would require a reduction of the spatial scale of the BG profiles to an extremely demanding or even unpractical level.

GA codirectional couplers (GACCs) in their different implementation platforms, namely long-period gratings (LPGs) in single mode fibers, integrated planar waveguide couplers, or double-core fiber couplers have been also widely investigated [2]–[5]. The GACC analysis (based on coupled-mode theory) has been generally limited to the spectral domain. This has similarly restricted the use of these devices to the conventional filtering – type applications (band-pass and band-stop filters, frequency equalizers etc.). Time-domain analysis of GACCs has received very little attention [16]–[19]. For instance, in [16] the idea of using weak-coupling GACCs as matched linear filters for encoding/decoding optical pulses was proposed. More recently, the temporal properties of GACCs operating in the nonlinear regime for all-optical switching applications have been also investigated [17]. In more fundamental studies, different numerical methods have been developed for calculating the temporal impulse response of GACCs [18], [19].

The present work focuses on investigating the pulse-shaping capabilities of GACCs. To the best of our knowledge, not even preliminary studies on this subject have been previously conducted. Motivated by the lack of these fundamental studies, we carry out here a detailed theoretical study of optical pulse propagation through GACCs. In particular, we calculate and analyze the temporal response of uniform GACCs to typical mode-locked optical pulses, focusing our attention in the linear regime of operation. The temporal responses associated to both interacting modes are separately analyzed and the effects of varying different physical grating parameters (e.g., grating length and coupling strength) on these temporal responses are carefully evaluated. We anticipate that the two interacting modes exhibit a very different temporal behavior and in fact, the most interesting pulse re-shaping operations occur typically for the “energy receptor” mode. Besides the intrinsic physical interest of our results, they also show the strong potential of GACCs for different optical pulse processing and shaping applications.

2. Pulse propagation dynamics in uniform GACCs

For the numerical analysis conducted in this work we assumed uniform GACCs (unapodized and unchirped) with a grating period Λ=40 µm. In our simulations, the effective index of the “energy supplier” mode (or waveguide) was set to n_s=2 and that of the “energy receptor” mode was set to n _r=1.96115. This ensures that the phase-matching condition between these two modes is exactly satisfied at a wavelength of 1554 nm. The indices were fixed to the given values targeting integrated tunable GACC designs based on EO crystals (e.g., LiNbO₃). The input pulse to the analyzed devices was assumed to be a transform-limited Gaussian pulse centered at the resonance wavelength of the GACC filter (1554 nm), with a peak intensity 100 (arbitrary units) and a full-width-half-maximum (FWHM) time width depending on the grating under analysis (in all the cases, the input pulse bandwidth was fixed to be slightly broader than the GACC spectral response in the “receptor” mode, i.e., our analysis focuses on the so-called ultrashort pulse propagation regime [9]).

The temporal responses corresponding to the “supplier” and “receptor” modes, h_s(t) and h_r(t), were calculated by taking the inverse Fourier transform (ℑ^-1) of the result of multiplying the input pulse spectrum H_in(ν) by the GACC spectral transmission response corresponding to the “supplier” mode, H_s(ν), and to the “receptor” mode H_r(ν), respectively,

We assume that only the “supplier” mode is initially excited, i.e., the input optical pulse is launched into the “supplier” mode. In this case, it is well known from coupled-mode theory that for a uniform grating, the associated spectral transmission responses are [16]

where κ is the cross-coupling coefficient, L is the grating length, β_s(ν) and β_r(ν) are the propagation constants corresponding to the “supplier” and “receptor” modes, respectively, σ is the detuning factor, σ(ν)=(β_s(ν)- β_r(ν))/2-π/Λ, Λ is the grating period, and the parameter γ=(σ²+κ²)^1/2. Note that in our following simulations we assume lossless devices. Nonetheless, it should be mentioned that we have also conducted numerical simulations assuming losses in the GACCs and have observed that the obtained temporal shapes are identical to those of the corresponding lossless device (except for the expected decreasing in the signals’ peak intensity) as long as the losses in the two interacting modes (or waveguides) are the same (this assumption is true in most practical cases).

The dynamics of evolution of an optical pulse propagating through a GACC can be evaluated by calculating the local temporal responses h_s(t) and h_r(t) along the grating length, i.e., by calculating the GACC temporal responses at different positions z, covering the region 0<z≤L (L is the grating length). It is important to note that the temporal response of a given GACC at a grating distance z is the same as that of an identical GACC (identical period and coupling strength) with a length L=z. This is associated with the fact that a GACC is a transmissive device. Thus, in order to evaluate the GACC temporal responses along the grating length (h_s(t) and h_r(t) for each z so that 0<z≤L), one only has to solve the above equations fixing L=z. Note that a much less straightforward procedure is required in order to obtain similar information (local temporal waveforms corresponding to the two interacting modes) in a reflection device (BG) [10].

Figure 1 is a movie illustrating the dynamics of optical pulse propagation through a typical uniform GACC. In particular, we assume a uniform grating with a period Λ=40 µm, a total length L=30 mm and a coupling coefficient κ so that κL=π. We also assume a Gaussian input pulse with a FWHM time-width of 0.5 ps. The temporal responses of the GACC in the “receptor” and “supplier” modes were calculated along the grating length following the numerical procedure described above. These temporal responses were then plotted and sequentially stored in order to generate the movie of Fig. 1. Note that for comparison the input pulse to the GACC is also plotted in the same movie (static representation).

 

Fig. 1. (2.1 MB) Movie showing the evolution of a Gaussian temporal pulse propagating through a GACC with a 30-mm long uniform grating and a coupling coefficient κ so that κL=π. The top (bottom) plot shows the evolution of the temporal response corresponding to the so-called “supplier” (“receptor”) mode. The input pulse to the GACC is also shown in the top plot. The label inside the figures indicates the grating distance z where the temporal responses are evaluated (z=14mm in the static figure) (14.1 MBversion).

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As a result of the continuous interaction between the “receptor” and “supplier” modes as the pulse propagates along the GACC, the corresponding temporal responses undergo a strong re-shaping process during propagation through the grating. As a first observation, the temporal responses corresponding to the two interacting modes are very different and in fact, the most interesting pulse re-shaping operations occur for the “receptor” mode. An outstanding general property is that the temporal waveform corresponding to this “receptor” mode is always temporally symmetric in any position along the grating length. Specifically, the following temporal shapes can be observed in the “receptor” mode during pulse propagation: triangular pulses (e.g., 12–14 mm), square temporal waveforms (e.g., 17–18 mm), double pulse sequences (e.g., 23–24 mm) and triple pulse sequences (e.g., 26–28 mm). A significant pulse re-shaping process is also observed for the “supplier” mode. The temporal waveform in this mode evolves from the original single Gaussian pulse into a double pulse sequence. The intensity ratio between the two pulses in the generated sequence varies during pulse propagation and in fact, these two pulses appear equalized in intensity only at a very specific location within the device (≈14.5 mm). In the final part of the grating the double pulse structure in the supplier mode develops a trailing temporal tail that becomes more significant as the pulse approaches to the output of the device.

The described temporal evolution in the “supplier” and “receptor” modes is general for any arbitrary uniform GACC, and the whole cycle (e.g. triangular – square – double pulse – triple pulse) is completed along each grating section of length L so that κL=π. This is true as long as it is ensured that the device is operated within the so-called ultrashort pulse propagation regime (where the bandwidth of the input pulse is broader than the GACC transmission bandwidth). As a general rule of thumb, we have observed that the coupling-length product κz that is required to observe a given temporal shape (among the observable temporal shapes in a uniform GACC, as described above) at a fixed location z, decreases as the distance z becomes longer (note that the same associated temporal shape is longer in time when evaluated at a longer distance z). Obviously, the temporal re-shaping process also depends on the features (shape and duration) of the pulse launched at the input of the GACC or in other words, the value of κz that is required to achieve a given temporal shape at a given location z strongly depends on the specific features of the input pulse. To give a reference, in order to reshape an ≈2-ps FWHM Gaussian input pulse into a flat-top (square) pulse within the “receptor” mode of a uniform GACC at a location z between 10 mm to 100 mm, κz must be varied approximately from 0.8π (for z=10 mm) to 0.2π (for z=100 mm). As another example, in order to reshape the same input pulse into a double pulse sequence within the “receptor” mode, κz must be varied from approximately 0.95π to 0.75π for z varying from 10 mm to 100 mm.

As mentioned above, the described reshaping processes have their origin in the continuous interaction between the “receptor” and “supplier” modes as the pulse propagates through the grating structure. Specifically, this pulse reshaping is essentially determined by the difference in group velocity between the optical radiation propagating within the “supplier” mode and that propagating within the “receptor” mode (slower and faster modes, respectively, in our example). In the first part of the device, the energy in the optical pulse is coupled from the “supplier” mode into the “receptor” mode in such a way that the radiation proceeding from the “supplier” mode arrives into the “receptor” mode with a temporal delay with respect to the radiation already propagating through the “receptor” mode. This process is essentially responsible for the slight temporal broadening observed in the “receptor” temporal pulse along the initial section of the GACC. At a specific location within the grating (z≈10 mm in our example in Fig. 1) some of the energy in the “receptor” mode starts to be coupled back into the “supplier” mode. From this location up to the point of maximum energy transfer (at the phase-matching wavelength), i.e., κz=π/2 (z=15 mm), the re-coupled energy lies far apart from the phase-matching wavelength (1554 nm). In particular, the re-coupled energy into the “supplier” mode comes from the edges of the main spectral lobe of the radiation propagating through the “receptor” mode. Due to the above mentioned velocity difference between modes, the re-coupled radiation from the “receptor” mode anticipates temporally to the radiation already in the “supplier” mode, thus giving rise to the double-pulse structure observed in the “supplier” mode along the second half of the GACC structure (after z≈10 mm). Initially, the pulse associated with radiation re-coupled from the “receptor” mode (first pulse in the sequence) is obviously less intense than the pulse associated with radiation already in the “supplier mode” (second pulse in the sequence). However, as the light continues propagating through the device, more energy is coupled back from the “receptor” mode into the “supplier” mode, thus feeding the first pulse in the “supplier” temporal sequence. Simultaneously, part of the energy in the second “supplier” pulse lies around the phase-matching wavelength and consequently, it is coupled into the “receptor” mode. The combination of these two processes leads to the observed intensity increasing [decreasing] of the first [second] pulse in the sequence. In this way, in the final section of the device, the intensity of this first pulse becomes higher that that of the second pulse.

The described energy transference dynamics is also responsible for the observed evolution of the temporal shape in the “receptor” mode. As anticipated, the re-coupled energy from the “receptor” mode into the “supplier” mode (before the point of maximum energy transfer is reached) mainly lies in the edges of the main spectral lobe of the radiation in the “receptor” mode. This also explains why the main spectral lobe in the “receptor” mode becomes narrower as it approaches to the point of maximum energy transfer. This spectral narrowing translates into the observed smoothing and duration increasing of the associated temporal waveform. This fact combined with the fact that most of the energy that it is being coupled from the “supplier” mode into the “receptor” mode feeds the central portion of the “receptor” temporal waveform leads to the observed evolution of the “receptor” waveform into a triangular temporal shape. The situation is very different after the point of maximum energy transference is reached (z=15 mm). After this point, an optical power equalization process is first observed (i.e., evolution into the observed square flat-top pulse). This is due to the fact that most of the re-coupled energy from the “receptor mode” into the “supplier” mode comes now from the optical frequencies which satisfy the phase-matching condition of the grating device (i.e. coupling frequencies) and these frequencies mainly lie in the central portion of the associated temporal waveform (around the pulse intensity peak), i.e., the energy coupled back from the “receptor” mode into the “supplier” mode mainly comes from the central portion of the “receptor” temporal waveform. The continuation of this re-coupling process causes a deployment of energy in the central portion of the “receptor” temporal waveform is such a way that this waveform evolves into the observed symmetric double-pulse temporal structure (with two clearly separated optical pulses). In the final part of the device, some energy is again coupled from the “supplier” mode into the “receptor” mode. An important part of this coupled energy has first traveled through both the “receptor” and “supplier” modes so that the average speed of this coupled energy is in between the speeds of the interacting modes. As a result, this energy, when re-coupled into the “receptor” mode, essentially contributes to the central part of the corresponding temporal waveform (in addition, this energy will lie essentially in the sidelobes of the “receptor” spectrum). This process is then responsible for the evolution of the “receptor” temporal waveform into the three-pulse sequence observed in the final part of the GACC device.

The results presented in this section indicate that by fixing properly the grating physical parameters (coupling strength κ and grating length L) one can generate different customized temporal shapes at the output of a GACC (especially in the “receptor” mode). This is a very interesting fact that can be used for optical pulse shaping applications. In the following section we show specific examples to illustrate further the optical pulse shaping capabilities of uniform GACCs. A brief comparison with the BG approach is also given.

3. Optical pulse shaping capabilities of uniform GACCs

For the numerical analysis presented in this section we assume a GACC with a short grating (14 mm) and a GACC with a long grating (60 mm). For the rest, the specifications of the interacting modes are identical to those used in our previous example.

A. Analysis of short uniform GACCs

In our first set of simulations for the short grating (14 mm long), shown in Fig. 2, we assumed an input pulse with a FWHM time-width of 450 fs, for Figs. 2(a) and (b) and Figs. 2(e) and (f) and 670 fs for Figs. 2(c) and (d). We show the GACC temporal responses (left column) and the associated spectra (right column) for different coupling strengths. In particular, each of the plots in the left column shows the average optical intensity (temporal waveform) at the output of the device for the “energy receptor” mode (red curves) and for the “energy supplier” mode (blue curves). For comparison, the average optical intensity of the input pulse to the GACC is also shown in all the figures (magenta curves). Notice that in all the cases, the optical intensities are represented in normalized units, but one has to keep in mind that the output pulses have N times lower amplitude than the input pulse, where the respective values of N are given in the plots. The corresponding spectra are shown in the column at the right (again, red curves are used for the “energy receptor” mode, blue curves for the “energy supplier” mode and magenta curves for the input pulse).

 

Fig. 2. Results corresponding to a GACC with a 14-mm long uniform grating: Temporal waveforms (left column) and spectra (right column) of the output pulses in the “receptor” mode (red curves), output pulses in the “supplier” mode (blue curves) and input pulses to the GACC (magenta curves) for different values of the grating strength (κL).

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In what follows we will focus our attention on the detailed analysis of the temporal waveforms in the “energy receptor” mode. As anticipated, very different pulse shapes can be obtained at the output of the device depending on the coupling coefficient (peak modulation amplitude) of the uniform perturbation. Specifically, for the coupling coefficient κ that provides κL=0.4π, the original Gaussian pulse is re-shaped into a nearly triangular temporal waveform with a total duration of Δt≈3 ps. Note that in this case a strong coupling from the “supplier” mode to the “receptor” mode is induced by the grating, with a peak energy transmission (at the resonance wavelength) of ≈90%. For comparison, the generation of a triangular temporal waveform with a BG would require the use of customized amplitude/phase grating spatial profiles, e.g., a weak-coupling BG (reflectivity <10%) with a triangular apodization profile along its length [12]. Moreover, in order to generate a≈3-ps triangular pulse, the BG should be around 300 µm long (assuming a grating period around 500 nm). Although achievable, the precision requirements in the fabrication process of such a grating can be extremely demanding in practice, especially as compared with those of the GACC approach (14mm long grating with a period of 40 µm). In general, our numerical results show that in addition to the differences in the temporal shapes, the optical pulses generated from a GACC exhibit a much shorter duration (generally in more than one order of magnitude) than those generated from an equivalent BG (with the same grating length). The discordance in the temporal scales that can be obtained with the BG and GACC approaches (for the same grating length) is mainly associated with the difference in their respective refractive index contrasts. The refractive index contrast is generally more than one order of magnitude larger for a BG (2n_eff) than for a GACC, (n ₁-n ₂). Note that the refractive index contrast determines the speed difference between the two coupled modes, which, in turn, fixes the temporal resolution associated with a given grating length. In this way, we can conclude that the GACC approach would result a much more attractive alternative for optical pulse shaping operations in the picosecond/subpicosecond regime.

Following with our analysis of the results in Fig. 2, other interesting temporal shapes can be obtained by simply varying the grating coupling strength. For instance, with κL=0.735π, the input Gaussian pulse is re-shaped into a low-ripple flat-top temporal pulse (i.e., nearly square pulse) with a total temporal duration of Δt≈3 ps. It should be mentioned that square temporal waveforms are highly desired for a range of non-linear optical switching and frequency conversion applications [12], [13]. In our simulations, we observe some smoothing with respect to the ideal square pulse (e.g., rise/fall times of finite duration), which is mainly associated with the limited input pulse bandwidth. In the specific case shown in Fig. 1, the grating has 55% transmission at the resonance wavelength. A square temporal waveform could be also generated with a uniform BG operated in the weak-coupling limit (peak reflectivity <10%) [12]. Alternatively, strong coupling BGs (peak reflectivity >50%) could be also used for the same purpose but in this case, much more complex amplitude/phase profiles (e.g., discrete pi-phase shifts in specific locations along the grating length) should be introduced [13] In any case, as discussed above, the GACC approach would result much more efficient for operation in the picosecond/subpicosecond regime.

Finally, when the grating strength is further increased (κL=0.89π), the input optical pulse is divided into three consecutive pulses of similar amplitude but different duration (the middle pulse is slightly shorter than the pulses at the extremes). The generation of customized optical pulse trains is important for several applications (e.g., all-optical pulse repetition rate multiplication in mode-locked lasers) and in fact, this topic has been the subject of recent intensive research [13]–[15]. In the following examples we will show that in the case of GACCs with longer uniform gratings, the individual pulses in the generated sequences can be even equalized in both amplitude and temporal duration, a factor of extreme importance for practical applications.

B. Analysis of long uniform GACCs

In our second set of simulations, we assume a GACC with a longer uniform grating of length L=6 cm and an input pulse with a FWHM time-width of 2.4 ps for Figs. 3(a) and (b), 1.2 ps for Figs. 3(c) and (d), and 1.9 ps for Fis. 3(e) and (f). Figure 3 shows the results corresponding to this second set of simulations, using the same definitions as for Fig. 2. For sufficiently low coupling coefficients, nearly square temporal waveforms can be synthesized (in the “energy receptor” mode). As an example, for κL=0.4π, a flat-top pulse with a total temporal duration of Δt≈12 ps is generated. The change in duration with respect to the optical pulses generated in our previous example is obviously associated with the increase in grating length.

As shown in Fig. 3, sequences of multiple optical pulses can be obtained from the uniform GACC (in the “receptor” mode) by simply increasing the coupling coefficient. In fact, if this coupling coefficient is properly fixed one can synthesize sequences of nearly identical optical pulses (in intensity). For instance, for κL=0.775π, the temporal response of the GACC is composed by two consecutive, identical optical pulses which are well separated by ≈ 6 ps. It can be observed that the generated individual optical pulses are similar in temporal shape but slightly broader than the optical pulse launched at the input of the GACC device. As another example of this outstanding behavior of a long uniform GACC, for κL=0.94π, the input optical pulse is re-shaped into a periodic pulse sequence comprising three nearly identical optical pulses (temporal period ≈ 3.5 ps). In this case the generated individual pulses are practically undistorted replicas of the input optical pulse (in temporal shape and duration). We emphasize that the generation of equalized multiple optical pulse sequences where the individual pulses retain the original features (e.g., temporal duration) is not a trivial task. For instance, if BGs are used for this purpose, very complex grating profiles (e.g., multiple grating structures, such as concatenated, sampled or superimposed BGs [13]–[15]) are required.

 

Fig.3. Results corresponding to a GACC with a 60-mm long uniform grating. Same definitions as for Fig. 2.

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4. Conclusions

In this paper, ultrashort pulse propagation through uniform GACCs has been studied in detail and the optical pulse shaping capabilities of these devices have been carefully evaluated. We have demonstrated that depending on the length and coupling strength of the uniform perturbation one can achieve very different temporal shapes at the output of the device, and for instance, temporal re-shaping into triangular pulses, square temporal waveforms and sequences of equalized multiple pulses can be achieved in the “receptor” mode. The evolution of the original temporal pulse into the described different temporal shapes has its origin in the continuous interaction (energy coupling) between the “supplier” and “receptor” modes and is mostly associated with the different group velocity exhibited by these two modes.

Our results are interesting from both physical and applied perspectives, showing the strong potential of GACCs for optical pulse manipulation applications and pointing at the GACC approach as an interesting alternative for implementing integrated optical pulse shapers in the picosecond/subpicosecond regime. Here we have focused on the analysis of uniform GACCs but we can anticipate that the range of achievable pulse re-shaping operations could be significantly broadened by introducing suitable non-uniformities in the grating profile (e.g. chirp or apodization profiles). In general, the grating profile in a GACC could be specifically designed to achieve customized temporal waveforms by use of well-known grating synthesis tools [20]. The results presented in this paper should stimulate future research in this direction.