1. Introduction

The ability of photonic crystals (PhCs) to control the propagation of light in a small area makes them one of the main candidates for the development of reduced-size photonic devices. By using a raw PhC, the propagation of light can be inhibited for the frequencies inside its photonic bandgap (PBG) [1]. If a linear defect is introduced in the otherwise perfect PhC, an ideally lossless waveguide can be obtained [2]. This PhC waveguide is very compact (its typical width is around λ/2) and allows sharp bends without losses [3]. In theory, this full control of light can be only achieved by using three-dimensional (3D) PhCs [4, 5], but the introduction of linear defects in such 3D structures is not easy. A real alternative to 3D PhCs in order to achieve a full control of light are two-dimensional (2D) planar PhCs, also called PhC slabs [6]. They consist of a 2D PhC of finite height, where light is confined in the vertical direction by total internal reflection (TIR). The interest in PhC slabs has grown quickly in the last years due to the easier fabrication process (current semiconductors fabrication processes can be used for this purpose) [7]. A waveguide can be created in a PhC slab by introducing a linear defect in the in-plane 2D periodic structure. Nevertheless, waveguides in simple 2D PhCs (vertical confinement is not considered) will be used throughout the text. The effect studied in this text is based on the gap-guided condition of the modes for the PhC plane, so a similar behavior will occur for 3D modes if they are properly confined in the vertical direction.

Usually, when analyzing the transmission properties of a PhC waveguide the analysis is carried out only for a monochromatic signal and its transmission spectrum is obtained, i.e., an infinitely long sinusoidal signal of a fixed frequency is sent into a waveguide and the amplitude of the output signal is then measured (the output signal vanishes for frequencies inside the PBG) [8, 9]. However, the study of the transmission of non-monochromatic signals (i.e., pulses) has a great interest for many practical cases such as data transmission, switching functions, etc. When a pulse is used in an optical system, the information is usually encoded in the amplitude of the pulse, but other parameters such as temporal width, time delay or temporal symmetry are also important. The effect of the propagation over each frequency of the input signal can be different, so the shape of the output signal may be distorted. As PhCs are expected to be part of these systems it becomes necessary to study the behavior of a pulse in a PhC-based structure.

In this work, the influence of the finite length of a PhC waveguide on the transmitted pulse has been studied. In a real case, any PhC waveguide will have a certain finite length and it will be accessed at its ends with other type of structures (e.g., dielectric waveguides, PhC waveguides with other characteristics or even raw PhCs). If the coupling at the input and output of the PhC waveguide is not perfect, reflections take place at the interfaces and a Fabry-Perot (FP) cavity is generated, leading to the appearance of an infinite number of pulses at the end of the waveguide. Each pulse experiments its own distortion due to the different length of the path traveled and the reflections suffered. If the PhC waveguide is short enough, all these pulses overlap themselves to create the total output pulse. It will be shown how, if the length of the PhC waveguide is changed, the parameters that define the total pulse (amplitude, temporal width, time delay and group velocity have been studied) will vary significantly, showing an oscillatory behavior. As the cavity is generated because of the non-perfect coupling at the interfaces of the waveguide, the amplitude of these oscillations will be reduced if the coupling between waveguides is improved [10].

The method used to obtain the output pulse at the end of the PhC waveguide is based on theoretically propagate each cavity pulse separately. Both the propagation constant in the PhC waveguide and the transmission and reflection coefficients at the interfaces are calculated by means of the eigenmode expansion (EME) method using the numerical tool CAMFR [11]. These parameters are used to obtain each cavity contribution to the output pulse separately. Even though in the EME simulations the parameters of a structure of infinite length are obtained, the approximation is proved to be quite accurate. If we compare the time and resources requirements of this method with FDTD simulations, a great saving is achieved. Other numerical methods have been proposed to calculate the response of finite-size periodic structures [12, 13], but only transmission and reflection spectra have been calculated and the propagation of non-monochromatic signals through them has not been considered.

2. Pulse propagation model

The theoretical model used to describe the pulse propagation is schematically depicted in Fig. 1. Media 1 and 3 represent the input and output semi-infinite waveguides that couple light in and out of the PhC waveguide (medium 2). The PhC waveguide has a certain length L. As the coupling between the PhC waveguide and the input and output media is not perfect, a transmitted and a reflected signal is generated at each interface. Therefore, each interface is characterized by a transmission and a reflection coefficient for a direction of incidence (r_ij, t_ij for incidence from the medium i to the j). These frequency-dependent coefficients have been calculated from the EME results with a semi-analytic method similar to that proposed in [14]. For this reason, the PhC waveguide behaves as a FP cavity where an infinite number of output signals are generated (see Fig. 1). If a signal with spectrum E₀ enters the PhC waveguide from medium 1, the generic expression for the spectrum of these output signals can be written as:

being β the propagation constant (i.e. the wavevector k calculated by means of the EME method) of a guided mode in the PhC waveguide and n indicates each of the pulses obtained at the waveguide output (n ranges from 1 to infinite). In the calculations, the dependence of the input signal spectrum, transmission and reflection coefficients and propagation constant with frequency has to be taken into account. Combining all the contributions given by Eq. 1, the spectrum of the total output signal when E₀ enters the cavity is obtained:

Once the spectrum of the signal at the end of the waveguide is obtained (Eqs. 1 and 2), we only have to apply the inverse Fourier transform in order to calculate its corresponding temporal signal. Although Eq. 2 seems to be a more compact expression than Eq. 1, Eq. 1 will be used to analyze the pulse propagation in the PhC waveguide in order to achieve a better understanding of the variations in the shape of the output pulse.

Therefore, the method used to simulate the propagation of a pulse in a PhC waveguide can be summarized as follows. First, the propagation constant and the transmission and reflection coefficients are theoretically obtained from EME simulations. Next, we multiply the spectrum of the selected input pulse with the frequency response of the waveguide, which is given in Eq. (1) (for the isolated pulses) or in Eq. 2 (for the total output pulse). As said before, we will calculate the isolated pulses generated in the cavity and combine them to obtain the total output pulse of the waveguide. Only a limited number of cavity pulses has been taken into account (this number will depend on the reflection coefficient value), since their amplitudes get smaller and smaller after each reflection at the interfaces. Finally, the temporal signal at the output of the waveguide is achieved by applying the inverse Fourier transform to the previously obtained spectrum.

The main advantage of this method of calculation is the huge saving of time and resources achieved, compared with other methods such as FDTD (used in this work to validate the results). In this case, the calculation of the parameters that characterize the PhC waveguide has to be made only once (it only takes a few minutes), and the results can be used to simulate any propagated pulse inside this waveguide (it only takes a few seconds). Moreover, contrary to what occurs in FDTD, if the length of the PhC waveguide changes, the calculation time remains the same. Thus, it is easier to make a complete study of the structure when any of its parameters changes. In addition, the results obtained with this method are in good agreement with FDTD results.

 

Fig. 1. Schematic of the theoretical model used to calculate the pulse propagation. The three media depicted represent: 1) input waveguide, 2) PhC waveguide of length L, and 3) output waveguide. The transmission and reflection coefficients at each interface are also depicted (t_ij, r_ij). The expressions of each signal getting out the cavity are shown, as well as the combination of all of them.

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3. Results and discussion

3.1. Validation of the proposed method

The structure that has been analyzed is shown in Fig. 2. It consists of a single-line-defect waveguide (SLWG) created by removing a row of holes in a PhC of air holes in a background of Silicon (n = 3.4). The radius of the air holes is r = 0.3a and the length of the waveguide is L. This structure is accessed at both ends by Silicon waveguides of width √3a. The transmission and reflection coefficients are schematically depicted at the interfaces between waveguides.

 

Fig. 2. PhC waveguide created by removing a row of air holes. The structure is accessed at its ends by means of two Silicon waveguides of width √3a.

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As a first step, the dispersion diagram for TE-polarized modes (modes with electric field in the plane) of the infinite PhC waveguide is obtained by the EME method. A square grid has been used to discretize the basic unit cell, using a resolution of N = 32 points per lattice constant. Optimum results are obtained for this resolution with a moderate computational charge. When N is decreased, the frequencies of the calculated band significantly vary as N changes. On the other hand, for N values over 32 the band remains almost unchanged while the computational and time resources hugely increase. The obtained dispersion diagram is depicted in Fig. 3. It can be seen that two guided modes appear inside the PBG. We have worked around the lower edge of the even band, in the range around 0.2165 (c/a) to 0.23 (c/a) (the exact lower band edge is determined to be 0.21748 (c/a)). For these frequencies, the PhC waveguide is strictly single-mode and the even mode is the only that will be excited inside the structure. If frequencies where both even and odd modes exist were selected, the behaviour would not vary if a source with even symmetry is selected (e.g., an optical fiber). This effect is because an even-shaped source can only excite an even-shaped mode, so the odd mode remains unexcited and the PhC can still being considered as single-mode. The problem arises for real physical structures, where discontinuities, roughness, etc. may cause the coupling of light from the even to the odd mode.

 

Fig. 3. Dispersion diagram for TE-polarized modes of the selected single-line-defect waveguide.

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The next step is to obtain the transmission and reflection coefficients at each interface. These coefficients will depend on the termination of the PhC waveguide at the interfaces, as explained in [15]. The case shown in Fig. 2, where the interface doesn’t cross any hole, has been analyzed. Although this is only one of the infinite possible cases, the analysis would be similar for the rest of terminations. The transmission and reflection coefficients have been calculated for the frequencies of the lower edge of the even guided band and the obtained results are shown in the Fig. 4. The reflection coefficient depicted is the one for the incidence from the PhC to the dielectric waveguide (the only needed to obtain the output pulse (r ₂₃, r ₂₁), see Eqs. (1) and (2)). Regarding the transmission coefficient, it is the same for the incidence from the PhC to the dielectric waveguide and vice versa due to reciprocity properties (t_ij = t_ji) [14], so only a generic coefficient has been depicted. It can be seen that an almost perfect coupling is achieved for frequencies around the centre of the guided band. Nevertheless, the reflection coefficient increases as frequency gets closer to the band edge, leading to the generation of the cavity pulses that deteriorate the PhC waveguide response. We are specially interested in observing the performance of the PhC waveguide for frequencies around the band edge due to the special properties that these regions exhibit (mainly the extremely low group velocity [16] and the extremely high group velocity dispersion (GVD) [17]); therefore, pulses whose central frequencies are closer to the band edge will be used in the simulations. Care must be taken when selecting the central frequency of the transmitted pulse: if it is selected extremely close to the band edge some of the pulse bandwidth can be allocated out of the band and these frequencies will be filtered, dramatically distorting the pulse.

 

Fig. 4. Transmission and reflection coefficients for the interface between the Silicon and the PhC waveguides.

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In order to analyze the propagation inside the PhC waveguide, a Gaussian pulse with central frequency 0.21823 (c/a), 7.5×10^-4 (c/a) away from the band edge, and a temporal full width half maximum (FWHM) of 2138 (a/c) is sent through it. In order to serve as an example of real physical parameters, for a lattice constant of 340 nm, a working wavelength of 1558 nm and a temporal FWHM of 2.42 ps are obtained. The propagation results obtained when the pulse travels through a waveguide of length L = 15a are shown in Fig. 5. All the pulses used throughout the text are defined in power. The input pulse can be seen in Fig. 5(a). The isolated output pulses that have been generated in the FP cavity are shown in Fig. 5(b). As mentioned before, the amplitude of these pulses gets smaller and smaller (each pulse has been scaled by the depicted factor), so only a few contributions are necessary to obtain the total output pulse. The combination of the contributions generated in the cavity (i.e., the total output pulse) is shown with solid line in Fig. 5(c). The output pulse directly obtained by means of the Eq. (2) when the infinite cavity pulses are considered is also shown in Fig. 5(c) with dashed line, but it cannot be seen because it perfectly matches with the solid curve.

In order to check the validity of these results, an equivalent structure has been simulated with FDTD. First, the transmission spectrum is calculated for the structure shown in Fig. 2 with a PhC waveguide long enough (L = 21a is selected) to obtain the band edge for the FDTD simulations. Again, a grid with resolution N = 32 has been used to discretize the structure (the suitability of this N has also been checked). Input signal is sent at the access Si waveguide 15.5a away from the entrance of the PhC waveguide. Output signal is measured at the access Si waveguide 6.5a away from the output of the PhC waveguide. The obtained power spectrum is plotted in Fig. 6 with solid line. The theoretical power transmission spectrum for EME calculations obtained from Eq. 2 with L = 21a has also been depicted with dashed line, showing a good agreement with the FDTD result, with only a slight frequency shift and an additional ripple at the edge of the band for the EME calculations. This ripple is determined by the great variation of the angular phase when working close to the band edge of a PhC waveguide. The frequency separation between these ripple peaks is very small and it doesn’t appear in FDTD simulations because frequency resolution is not enough to resolve them (an extremely long simulation would be needed to obtain such great frequency resolution). Concerning the frequency shift, it can be seen that the band edge cannot be accurately determined for the FDTD simulation due to the smooth variation of the plotted magnitude, but a possible edge frequency has been determined by projecting the band slope (see dotted line). Doing this, the band edge for FDTD simulations is selected to be 0.21947 (c/a) (vs. 0.21748 (c/a) for EME calculations). The determination of the new band edge allows us to overcome the “intrinsic deviations” between both calculation methods, deviations that have a great influence when working near band edges. Since the structure analyzed with both methods is the same, we consider that both spectra are equivalent with only a slight frequency shift between them. So, a proper comparison between both methods can be carried out if the carrier frequency of the pulse is determined with the difference to the band edge instead of with an absolute frequency.

 

Fig. 5. Propagation for a PhC waveguide of length L = 15a. a) Input pulse. b) Output pulses generated in the cavity. Each subsequent pulse has smaller amplitude, so each one has been scaled by an appropriate factor in order to achieve a better representation. c) Total output pulse obtained by combining the contributions shown in b) (solid line). The output pulse obtained by directly applying the Fabry-Perot formula of Eq. 2 is also shown (dashed line), but it perfectly matches the solid curve. The output pulse for the FDTD simulation is also plotted with dotted line in c).

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Fig. 6. Transmission spectrum of a PhC waveguide with length L = 21a for a FDTD simulation (solid line). The beginning of the band is determined by the projection of the edge slope (dotted line) as 0.21947 (c/a). The theoretical transmission spectrum for the EME calculations obtained from Eq. 2 for L = 21a has also been depicted with dashed line.

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Once the band edge for FDTD simulations has been determined, the pulse propagation for a PhC waveguide of L = 15a is calculated in order to check the previous result obtained by using our method. The central frequency of the pulse is 0.22022 (c/a), 7.5×10^-4 (c/a) away from the band edge. The FDTD result is depicted in Fig. 5(c) with dotted line. The additional time delay introduced by the access dielectric waveguides has been taken away when plotting the results. It can be seen as this result is very close to the obtained with our method. The parameters that characterise the output pulse when L = 15a for both methods are shown in Table I. These parameters are: 1) the amplitude of the pulse (the maximum output power measured), 2) the temporal FWHM (temporal pulse width when amplitude is a half of the maximum), 3) the time delay (difference between the maxima positions of the input and output pulses), and 4) the group velocity, v_g (it has been calculated as the ratio between the PhC waveguide length and the time delay, as it would be defined in experimental results). It should be denoted that output pulse is no longer Gaussian, since its shape is distorted by dispersion and the cavity reflections.

Table 1. Amplitude, temporal width and time delay of the output pulse for different PhC waveguide lengths. Two values are shown for each length, the first corresponding to the FDTD simulations and the second corresponding to the theoretical calculations carried out by means of Eq. 1. The input pulse has normalized amplitude and a temporal FWHM of 2138 (c/a).

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The same case has been simulated for shorter (L = 12a) and longer (L = 18a) PhC waveguides. The obtained results are shown in Fig. 7 and Fig. 8 respectively and the parameters of the output pulses are again shown in Table I. From Figs. 5, 7 and 8 it can be seen that the pulses generated at the cavity (shown in subplot b) are almost equal for the three cases but with different temporal positions. As PhC waveguide length changes these contributions arrive to the waveguide end at different times and with different angular phases, what makes the total output pulse to be different for each case. This effect can be more clearly seen in Fig. 9, where the evolution of the pulse shape after considering a new contribution of the cavity pulses is depicted along each row (only the first three contributions have been considered). Each row depicts the evolution of the output pulse shape for the three PhC waveguide lengths analyzed: L = 12a for the first row, L = 15a for the second and L = 18a for the third. The first contribution is almost the same for the three lengths, but it can be seen that the interaction with the second contribution is constructive for L = 12a and L = 18a (the first and the third rows) and destructive for L = 15a (the second row). As shown in Fig. 9, a constructive interaction gives a higher and wider pulse with its maximum shifted to a later temporal position (group velocity decreases), whilst the effect when a destructive interaction occurs is completely the opposite. In Fig. 9 can also be seen that the influence of the cavity pulses over the total output is almost negligible from the third contribution for this example. Therefore, the consideration of taking only the first five contributions to do the calculations is completely correct, as it was shown in Fig. 5(c) where the results from Eqs. (1) and (2) were compared. A quantitative criterion to calculate the required number of cavity pulses in order to obtain proper results will be provided later. These effects of constructive and destructive interaction are reflected in the parameters of the output pulse shown in Table I. As said before, it can be seen that regardless of whether we increase (L = 18a) or reduce (L = 12a) the PhC waveguide length, a constructive interaction between pulses occurs, leading to an increase on the amplitude and the width of the pulse and a decrease on the group velocity. This effect is not such clear for the FDTD simulations as for the theoretical calculations (it doesn’t happen when shortening the waveguide).

 

Fig. 7. (a) Input pulse, (b) output pulses generated in the cavity and c) total output pulse when a length of the waveguide L = 12a is selected (solid - theoretical propagation, dotted - FDTD simulation).

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Fig. 8. (a) Input pulse, (b) output pulses generated in the cavity and c) total output pulse when a length of the waveguide L = 18a is selected (solid - theoretical propagation, dotted - FDTD simulation).

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In order to better understand the influence of the PhC length over the total response and to check why the constructive/destructive effect was not the same for both simulation methods, the evolution of the four parameters of the pulse (amplitude, temporal width, time delay and group velocity) with the waveguide length has been studied. This parameters evolution is shown in Fig. 10. Solid line depicts the results obtained with Eq. (1). It can be seen that a significant ripple appears for these parameters for short waveguide lengths (approx. until L = 40a), this ripple is more significative for the width parameter (Fig. 10(b)). As said before, this ripple is due to the overlapping of the cavity pulses at the output of the PhC waveguide; the phase difference between contributions changes as the length increases and hence the constructive/destructive interaction is responsible for this oscillatory behaviour. If the PhC waveguide is long enough, the pulses will not overlap and the several repetitions will appear separate at the output of the waveguide, being the main output pulse defined only by the first contribution and no ripple in the parameters is observed. The theoretical parameters for the output pulse when no reflection at the waveguide interfaces is considered in the calculations (no FP cavity is generated) are also depicted in Fig. 10 with dashed line as a reference. This is the desirable optimum case (with a transmission efficiency around 0.46 instead of 1 due to the non-perfect transmission), so efforts in improving the coupling efficiency between the different sections are necessary in order to avoid the ripple originated by the pulse repetitions at the ends of the PhC structure.

 

Fig. 9. Evolution of the pulse shape after each contribution is taken into account. Each row depicts the evolution for a PhC waveguide length: a) L = 12a, b) L = 15a, c) L = 18a. Each column depicts the pulse shape after each subsequent contribution is taken into account, i.e., column 1 depicts the pulse shape when only the first contribution is considered, column 2 depicts the pulse shape when the second contribution is also considered, and column 3 depicts the pulse shape when the third contribution is also considered. The pulse shapes corresponding to the previous column are also depicted with dashed line in order to compare their evolution (obviously, this is only made for columns 2 and 3).

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Fig. 10. (a) Pulse amplitude, (b) temporal width, (c) time delay and (d) group velocity of the output pulse versus the PhC waveguide length. Solid lines depict the results obtained when the pulses are theoretically propagated. Dashed lines depict the theoretical evolution of the parameters when no reflections are considered at the waveguide interfaces. Dotted lines depict the results obtained with FDTD for PhC waveguide lengths between a and 21a. The dashed-dotted line in plot d) depicts the “instantaneous” group velocity obtained by derivating the time delay results of the FDTD simulations.

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FDTD simulations have been carried out for PhC waveguide lengths between L = a and L = 21aand results are also depicted in Fig. 10 with dotted line (around 1 day was needed to calculate each point). Several conclusions can be deduced from the obtained results. First, a similar ripple to that of theoretical curves is observed for the amplitude and width evolutions: the ripple amplitude is in the same range and only the mean value of the pulse amplitude varies around 0.03 between both methods. Concerning the position of the maxima and minima of the ripple and the separation between them, it can be seen that they are almost the same for both methods for PhC waveguides longer than ~ 7a. The difference for lengths shorter than ~ 7a is due to the non-PBG behaviour of the structure for these lengths, a fact that is not taken into account for the theoretical calculations. Because of this, the parameters ripple appears from the beginning for the solid line (theoretical calculation), being an unreal behaviour and leading to the slight difference between maxima and minima positions.

This effect for short PhC waveguide lengths is also shown for the dependence of the time delay of the pulse (Fig. 10(c)). On the one hand, it can be seen that the slope of the curve is constant for the theoretical calculations (solid line). On the other hand, low delays are obtained for short PhC waveguides in FDTD calculations (dotted line) as the PBG effect doesn’t appear yet and higher group velocities are experimented by the pulse. When PhC length is over 7a the slope of the curve increases reaching the same as for the theoretical calculations, appearing the low group velocity that characterise the modes of a PhC waveguide. It can be seen in Fig. 10(c) how the lack of the PBG effect for short lengths is responsible of the shift between both results. A ripple appears for these plots in Fig. 10(c) too, but due to the high slope of them it cannot be clearly seen. This ripple can be more clearly seen in the group velocity of the propagated mode, shown in Fig. 10(d). This ripple can be seen for the theoretical results (its amplitude is not very high), but not for the FDTD calculations (dotted line) where a very different behaviour is observed. Because the group velocity is calculated as the ratio between the PhC waveguide length and the time delay, the results for FDTD calculations are influenced by the change in the slope for short waveguides (where group velocities are higher than 0.1c) and the offset delay caused by this. Nevertheless, it can be seen that as the PhC waveguide length increases, the group velocity for FDTD tends to the theoretical value of 0.02c. In order to see more clearly the change in the group velocity for FDTD calculations reducing the influence of the short-lengths section as much as possible, an “instantaneous” group velocity has been calculated by derivating the time delay results. The results are shown in Fig. 10(d) with dashed-dotted line. Now, the ripple in the group velocity can be seen and we can also see how group velocities around 0.02c are obtained for lengths from 7a.

3.2. Number of cavity pulses required in the calculations

A criterion to determine the minimum number of cavity pulses needed when doing the calculations has been obtained. This criterion is based on the previously commented ripple of the output pulse parameters. The influence of each contribution over the ripple of the pulse amplitude is analyzed, considering that cavity pulses are only affected by the transmission and reflection coefficients at the interfaces and they constructively interact at the same temporal position (they are not considered to be affected by the propagation constant of the PhC waveguide mode). An upper limit of the ripple is obtained by using these considerations. So, the pulse amplitude (in power) at the output of the PhC waveguide when only a limited number of contributions n are considered would be given by the expression:

where n can range from 1 to infinite. In this approximation transmission and reflection coefficients are considered to be the ones for the central frequency of the input signal. The main ripple of the amplitude parameter is determined by the first cavity pulse (E₂ from Eq. (1)) and the amplitude of this ripple can be approximated by O ₂ - O ₁ (On given in Eq. (3)). It will be considered that the influence of a cavity pulse over the total response is negligible when the ripple that it causes is much lower than the ripple caused by the first cavity pulse previously commented. This criterion has been expressed as:

When this value Qn is below a certain threshold (e.g. 1% or 0.1%) we can consider that the influence of this contribution over the total response is negligible. Finally, Eq. (4) can be expressed using the transmission and reflection coefficients at the interfaces of the PhC waveguide by using Eq. (3):

Equation (5) shows that the number of contributions needed to obtain the final response is only determined by the reflection coefficients at the interfaces of the cavity.

Now, this criterion is used to check if the number of contributions previously selected to do the calculations (5 contributions) is correct. In our case, the reflection coefficient has a value of 0.1759 for the selected working frequency. If a threshold value of 1% is selected (a tight threshold), only 3 contributions are needed to do the calculations, as Q ₃ = 3.142 (> 1) and Q ₄ = 0.097 (< 1). So, the 5 contributions taken to do the calculations are enough to obtain correct results. For higher values of the reflection coefficient r, more contributions will be needed to obtain the total response, e.g., 5 contributions are needed when r = 0.5.

3.3. Other calculations

Once the validity of the proposed scheme to explain the behaviour of finite length PhC waveguides has been checked, we will use it to also analyze other cases of pulses propagation. First, the effect of the carrier frequency of the input pulse over the output response has been analyzed. A worse coupling is achieved as the central frequency gets closer to the band edge, leading to a higher ripple and even to the filtering of some frequencies of the input pulse if it is moved too close to the edge. The effect of the carrier frequency over the pulse propagation can be seen in Fig. 11, where pulses with central frequencies f = 0.2176, 0.218, 0.21823 and 0.2185 (c/a) have been theoretically propagated through the structure. Their respective spectra are also shown in Fig. 12. For the case of f = 0.2176 (c/a), the interface between PhC and access waveguides presents a reflection coefficient of 0.5501 for this central frequency, so 6 cavity pulses are needed to do the calculations when the criterion mentioned above is applied. It can be seen how the ripple amplitude diminishes as the pulse frequency gets far from the band edge for all the parameters. Very high ripple amplitude is observed for the pulse with frequency 0.2176 (c/a), which has some of its frequencies out of the guided band. In Fig. 11 can also be seen the reduction of the group velocity (see Fig. 11(d) and the increment of the time delay in Fig. 11(c)) and the increment of the GVD (see the broadening of the pulse in Fig. 11(b) and the reduction of the pulse amplitude in Fig. 11(a)) as the central frequency gets close to the band edge.

 

Fig. 11. (a) Pulse amplitude, (b) temporal width, (c) time delay and (d) group velocity of the output pulse versus the PhC waveguide length for different working frequencies. f = 0.2176 (c/a) - solid line; f = 0.218 (c/a) - dashed line; f = 0.21823 (c/a) - dotted line; f = 0.2185 (c/a) -dash-dotted line.

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Fig. 12. Spectra of the four input pulses simulated. f = 0.2176 (c/a) - solid line; f = 0.218 (c/a) - dashed line; f = 0.21823 (c/a) - dotted line; f = 0.2185 (c/a) - dash-dotted line. The transmission and reflection coefficients have been also depicted.

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The effect of the pulse width has also been studied. Fig. 13 shows the evolution of the four parameters when three input pulses with central frequency 0.218 (c/a) and temporal FWHM widths T = 2138 (a/c) (solid line), 2T (dashed line) and 0.5T (dotted line) are sent into the PhC waveguide. Apart from the known effects related with the GVD (faster broadening and amplitude reduction for narrower pulses that have wider spectral width), it can be seen that the ripple is higher and it remains for longer waveguides when a wider pulse is used. This is because the overlapping between generated repetitions is avoided for low travel times in the cavity (low delays between repetitions) when narrow pulses are propagated. However, if a wider pulse is selected, its cavity pulses will remain overlapping for these travel times. This can be more clearly seen in Fig. 14 where the propagation of the pulses of width 0.5T (dotted line) and 2T (dashed line) over a 20a long PhC waveguide has been calculated. It can be seen how the cavity pulses appears almost in the same positions in both cases (differences in the position are due to GVD and frequency filtering), but the narrower width of the second case prevents them from significantly overlap.

 

Fig. 13. (a) Pulse amplitude, (b) temporal broadening, (c) time delay and (d) group velocity of the output pulse versus the PhC waveguide length for a central frequency 0.218 (c/a) and pulse widths T (T = 2138 (a/c), solid line), 2T (dashed line) and 0.5T (dotted line).

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Fig. 14. Propagation over a PhC waveguide of length 20a for two pulses with central frequency 0.218 (c/a) and different widths: 0.5T and 2T (where T =2138 (a/c)). (a) Input pulses for both widths (dashed - 2T, dotted - 0.5T), (b) output pulses generated in the cavity for the 2T case and (c) output pulses generated in the cavity for the 0.5T case.

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Finally, Fig. 15 shows the evolution of the output pulse as the several cavity contributions are taken into account for the previous case with central frequency 0.218 (c/a) and temporal widths 2T. The propagation for 3 PhC waveguide lengths is depicted: L = 11a, 14a and 18a. Figure 15 represents the same as in Fig. 9, but now the effect can be more clearly seen due to the wider pulses and the higher reflection coefficient.

 

Fig. 15. Evolution of the pulse shape after each contribution is taken into account for a case where a higher reflection coefficient exists (f = 0.218 (c/a), 2T) . Each row depicts the evolution for a PhC waveguide length: a) L = 11a, b) L = 14a, c) L = 18a. Each column depicts the pulse shape after each subsequent contribution is taken into account. The pulse shapes corresponding to the previous column are also depicted with dashed line.

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4. Conclusions

An extensive study in the temporal domain of the influence of finite length over propagation through PhC waveguides has been presented. The study has been carried out for a SLWG in a PhC of air holes in Silicon accessed by two Silicon waveguides, but it can be extended to any other PhC or access structures that can be analyzed with the model in [14]. A theoretical method to easily analyze the propagation has been proposed. This method uses the numerically calculated propagation constant and frequency-dependent transmission and reflection coefficients to propagate the desired signal through the cavity formed by the PhC waveguide. This method has been proved to give accurate results and allows us hugely reduce the computation time required for the calculations (~ 1 day for FDTD vs. few seconds for this method for each calculation).

The obtained results show how the non-prefect coupling between waveguides originates several reflections inside the PhC waveguide that behaves as a FP cavity. These reflections overlap at the output when the PhC is short enough and distort its shape depending on the phase difference between the contributions. This phase difference varies with length, so an oscillatory behaviour in the different pulse parameters (amplitude, width, delay and group velocity) appears. This ripple can be minimised if an optimum coupling method between structures is used.