1. Introduction

Photonic devices performing required temporal and spatial transformations of optical signals are of great interest for a wide range of applications including all-optical information processing and analog optical computing. Among the important operations of analog processing of optical signals are the operations of temporal and spatial differentiation and integration. Various types of resonant structures performing these operations were previously proposed, such as Bragg gratings [1–8], resonant diffraction gratings [9–11], and micro- and nanoresonators [12–14]. The use of resonant structures for the implementation of these operations is possible due to the fact that the Fano profile describing the reflection or transmission coefficient in the vicinity of the resonance can approximate the transfer function of a differentiating or integrating filter [9].

For spectral filtering and temporal shaping of optical pulses, phase-shifted Bragg gratings (PSBG) are widely used. PSBG consist of two symmetrical Bragg gratings separated by a defect layer and provide zero reflectance (and, hence, unity transmittance) at a required frequency or incidence angle lying in the spectral or angular band gap of the Bragg gratings, respectively [3, 6]. This effect is caused by the excitation of the eigenmodes localized in the defect layer. It is this spectral shape that enables the use of PSBG as spectral filters and elements for pulse transformations, in particular, for temporal differentiation of the pulse envelope in reflection and integration of the pulse envelope in transmission [3–5]. To perform these transformations, the structure has to be designed so that the reflectance vanishes at the central frequency of the pulse.

The use of the PSBG for spatial differentiation and integration of the profile of an optical beam was for the first time considered in our previous works [6–8]. Similarly to temporal transformations, in the case of spatial transformations the angle of incidence of the beam has to coincide with the zero-reflectance angle.

The development and investigation of PSBG analogues for surface electromagnetic waves propagating in metal-dielectric and all-dielectric structures of different kind are of great interest. In particular, this interest is due to the prospects of application of surface waves in on-chip all-optical information processing systems [15].

The most thoroughly studied among surface waves are surface plasmon-polaritons (SPP) and plasmonic modes propagating in various metal-dielectric structures [16]. A common drawback of plasmonic structures are high absorption losses in the metal, which are directly proportional to the mode energy localization. Furthermore, zero reflectance necessary to perform the operations of temporal and spatial differentiation is possible only in resonant structures without absorption and parasitic scattering losses [17]. Therefore, despite the fact that several configurations of Bragg gratings were proposed for SPP [18–20], the implementation of the operations of temporal and spatial differentiation of plasmonic pulses and beams using resonant structures is hardly possible.

At the same time, there exist surface waves that are supported by all-dielectric structures. Such waves are called Bloch surface waves (BSW) or surface states of photonic crystals [21, 22] and propagate along the interfaces between a homogeneous medium and a photonic crystal or between two different photonic crystals (PhC). For all-dielectric structures, the absorption losses upon the surface wave propagation are negligibly small. Transmission of BSW through dielectric ridges deposited on the surface of a photonic crystal was experimentally studied in [23], and the excitation and propagation of Bloch surface waves bounded in two dimensions in such ridges and ridge arrays was experimentally analyzed in [24–27]. The presence of band gaps of Bloch surface waves in dielectric gratings located on the surface of a 1D photonic crystal was also demonstrated in [28, 29].

In our previous paper, phase modulation of BSW propagating along the boundary between one-dimensional PhC and free space was theoretically and numerically studied [30]. The modulation was carried out by changing the geometrical parameters of the microrelief located at the PhC surface. In the present work, this modulation mechanism is applied for the design of the PSBG for Bloch surface waves. The results of rigorous numerical simulations of the BSW diffraction on the proposed PSBG are compared with the plane wave diffraction at a conventional PSBG consisting of a periodic set of plane-parallel layers. The obtained results demonstrate the possibility of using the proposed PSBG for temporal differentiation of the envelope of picosecond BSW pulses.

2. Dispersion relation of Bloch surface waves

For the sake of completeness, let us first present a brief derivation of the dispersion relation of surface electromagnetic waves propagating along the interface between a one-dimensional semi-infinite photonic crystal and a uniform medium (Bloch surface waves) shown in Fig. 1(a).

 

Fig. 1 (a) Geometry of the structure. (b) Effective refractive indices of BSW of two adjacent orders vs. $h_c$ (ТЕ-polarization).

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The electromagnetic field of such surface wave decays exponentially away from the interface. In the uniform medium over the PhC, the field is represented by an evanescent plane wave. On the other side of the interface, the field corresponds to a decaying eigenmode of the PhC. Thus, to derive the BSW dispersion relation, it is necessary to obtain a general representation of the PhC eigenmode.

In what follows, we consider the TE-polarization case, when $E=\left(0,E_y,0\right)$ and $H=\left(H_x,0,H_z\right)$. As shown in Fig. 1(a), the PhC period consists of two homogeneous plane-parallel layers with thicknesses $h_1$ and $h_2$ ($d=h_1+h_2$) and dielectric permittivities ${\epsilon }_1$ and ${\epsilon }_2$. Electric field in each of the crystal layers at $-d\le z<0$ can be represented as a superposition of two plane waves:

Now that we have obtained the field representation and dispersion relation of the PhC eigenmodes, let us write the required dispersion relation of a BSW propagating along the interface between the considered PhC and a uniform semi-infinite medium with dielectric permittivity ${\epsilon }_d$. The BSW propagation constant $k_{x,0}$ depends on the thickness of the first (upper) layer of the photonic crystal [21, 22]. We assume that the upper interface of this layer is the plane $z=h_c$ ($h_c>-h_1$), so the thickness of the first layer amounts to $h_1+h_c$. The electric field over the interface corresponds to an evanescent plane wave:

Figure 1(b) shows the BSW dispersion curves (effective refractive index $n_{\text{eff}}$ vs. $h_c\in \left[-120,120\right]\text{nm}$) for two modes of adjacent orders at the following parameters: $\lambda =800\text{nm}$, ${\epsilon }_d=1$, ${\epsilon }_1=7$ (which is close to the dielectric permittivity of several high-index materials such as TiO₂ or SiC), ${\epsilon }_2=2.25$, $h_1=120\text{nm}$, $h_2=200\text{nm}$. These parameters match those of the example considered in the authors’ previous work [30]. Horizontal dotted lines in Fig. 1(b) depict the boundaries of the photonic band gaps of the PhC (outside the band gaps, the PhC eigenmodes are not decaying in the crystal and BSW does not exist). Vertical dashed lines show the $h_c$ values corresponding to the band gap boundaries (0 nm and 17 nm), as well as the value $h_c=32\text{nm}$corresponding to the beginning of two-mode regime, in which two BSW with different propagation constants exist at a fixed $h_c$.

3. Phase-shifted Bragg gratings for BSW

A conventional PSBG for plane electromagnetic waves consists of two symmetric Bragg gratings separated by a defect layer and is shown in Fig. 2(a). For illustrative purposes, only two periods of each of the Bragg gratings are depicted in Fig. 2.

 

Fig. 2 (a) Geometry of a PSBG for plane electromagnetic waves. (b) the refractive index profile of the considered PSBG. (c) Analogue of PSBG for BSW. The length scales in (c) correspond to the considered example, except the ridge height increased by 3 times for clarity.

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In the simplest case, the period of a Bragg grating consists of two homogeneous layers with equal optical thickness:

The proposed PSBG for Bloch surface waves consists of a set of dielectric ridges located on the PhC surface and is shown in Fig. 2(c). We assume the ridges to have the same dielectric permittivity as the upper crystal layer. Let us note that the approach based on the change of the local height of the PC upper layer is not limited to the binary structures shown in Fig. 2(c). However, in the present work we restrict our consideration to such structures to enable an easy comparison with realistic PSBG for plane electromagnetic waves. According to Fig. 1(b), the effective refractive index of the BSW depends on the PhC upper layer thickness $h_c$. Let us choose the heights $h_{\text{gr,}\text{1}}$ and $h_{\text{gr,}\text{2}}$ so that the BSW effective refractive indices in the ridge and groove regions are equal to the corresponding refractive indices $n_1$ and $n_2$ of the conventional Bragg grating. The ridge and groove lengths are equal to the thicknesses $l_1$, $l_2$ of the layers of the corresponding conventional Bragg grating as shown in Fig. 2.

Let us study the performance of such PSBG with the parameters ${\lambda }_{\text{B}}=800\text{nm}$, ${\theta }_0=0$, $n_1=n_{\text{def}}=1.6$, $n_{\sup }=n_2=1.44$. For the chosen values of $n_1$ and $n_2$, $l_1=125n\text{m}$, $l_2=139\text{nm}$, and $l_{\text{def}}=250\text{nm}$. The parameters of the used photonic crystal are given above in Section 2. It follows from Fig. 1(b) that the BSW refractive index takes the values $n_1$ and $n_2$ at $h_{c,1}=-66\text{nm}$ and $h_{c,2}=-83\text{nm}$, respectively. In this case, the upper layer thickness values in the ridge and groove regions amount to $h_{\text{gr,}\text{1}}=h_1+h_{c,1}=54\text{nm}$ and $h_{\text{gr,}\text{2}}=h_1+h_{c,2}=37\text{nm}$. Outside the PSBG, $h_c=h_{c,2}=-83\text{nm}$. Hence, the effective refractive index of the incident BSW is equal to $n_{\text{eff,inc}}=n_2=1.44$. Let us note that the values $h_{c,1}$, $h_{c,2}$ correspond to the single-mode regime. It is the case when the BSW diffraction is expected to be close to the plane wave diffraction [30].

We consider the PSBG consisting of two Bragg gratings with the number of periods $N_{\text{per}}=4$ and $N_{\text{per}}=5$. The total number of layers in the structure amounts to 17 and 21, respectively.

Dashed lines in Fig. 3 show the absolute values and phases (arguments) of the complex reflection coefficient vs. frequency for the conventional (plane-wave) PSBG at $N_{\text{per}}=4$ (red curves) and $N_{\text{per}}=5$ (green curves). For this structure, the reflection coefficient vanishes at angular frequency ${\omega }_{\text{B}}=\text{2}.\text{3546}\cdot {\text{10}}^{15}{\text{s}}^{-1}$ corresponding to the wavelength ${\lambda }_{\text{B}}=800\text{nm}$. At the same time, the phase of the reflected wave abruptly changes by $\pi$ due to the change of sign of the reflection coefficient. Note that the increase in the number of periods of the Bragg gratings leads to an increase in the quality factor of the resonance and, consequently, to the decrease of the width of the reflection dip.

 

Fig. 3 Absolute values (left vertical axis) and phases (right vertical axis) of the PSBG complex reflection coefficient vs. angular frequency. Solid lines show the values for a BSW grating at $N_{\text{per}}=4$ (red curves) and $N_{\text{per}}=5$ (green curves). Dashed lines show the corresponding values for a conventional PSBG for plane electromagnetic waves.

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Solid lines in Fig. 3 show the absolute values and phases of the complex reflection coefficient of the PSBG for BSW, calculated using an in-house implementation of the aperiodic rigorous coupled-wave analysis technique [31] at $N_{\text{per}}=4$ (red curves) and $N_{\text{per}}=5$ (green curves).

According to Fig. 3, the reflection spectra of a conventional PSBG and a PSBG for Bloch surface waves are in good agreement: the Pearson correlation coefficient between two spectra exceeds 0.99. At the same time, the spectra are shifted with respect to each other. The reflection coefficient of the PSBG for Bloch surface waves reaches its minimum not at the angular frequency ${\omega }_{\text{B}}=\text{2}.\text{3546}\cdot {\text{10}}^{15}{\text{s}}^{-1}$ corresponding to the design wavelength ${\lambda }_{\text{B}}=800\text{nm}$, but at the value ${\tilde{\omega }}_{\text{B}}=\text{2}\text{.3587}\cdot {\text{10}}^{15}{\text{s}}^{-1}$ (${\tilde{\lambda }}_{\text{B}}=798.6\text{nm}$). Moreover, the minimal reflection coefficient value is not strictly zero and amounts to 0.0013 and 0.0021 at $N_{\text{per}}=4$ and $N_{\text{per}}=5$, respectively. Note that the energy fraction contained in the reflected BSW is defined as the squared modulus of the reflection coefficient and for the considered examples is of about ${10}^{-6}$. The observed differences in the reflection spectra are caused by the parasitic scattering of the BSW. Indeed, upon the BSW diffraction at the PSBG the propagating PhC eigenmodes and the modes of the dielectric medium over the crystal are generated along with the reflected and transmitted surface waves. At $N_{\text{per}}=4$ and ${\tilde{\lambda }}_{\text{B}}=798.6\text{nm}$, the squared absolute values of the reflection and transmission coefficients characterizing the energy fractions in the corresponding surface waves amount to ${\left|R\right|}^2=1.69\cdot {10}^{-6}$ and ${\left|T\right|}^2=0.984$. Thus, the scattering losses $S=1-\left({\left|R\right|}^2+{\left|T\right|}^2\right)$ in the considered example are relatively low (less than 1.6%). With the increase in the number of grating periods, the scattering losses and the minimum reflectance increase as well.

Figure 4(a) shows the calculated field distribution $\left|E_y\left(x,z\right)\right|$ generated upon the BSW diffraction at the considered PSBG at $N_{\text{per}}=4$ and ${\tilde{\lambda }}_{\text{B}}=798.6n\text{m}$. The propagation direction of the incident BSW is shown with a white arrow. It is evident from Fig. 4 that in the BSW incidence region (to the left of the grating) there is no typical interference pattern of the incident and reflected BSW since the reflection coefficient is close to zero. The transmission coefficient is almost unity, and the field of the transmitted BSW (to the right of the grating) is visually identical to the field of the incident BSW. In the grating area, the field is typical for a mode localized around the defect layer. This confirms the resonant nature of the reflectance minimum. For comparison, Fig. 4(b) shows a similar field distribution generated at a Bragg grating without a defect layer consisting of 9 periods. In this case, ${\left|R\right|}^2=0.543$, ${\left|T\right|}^2=0.445$ and the interference pattern formed by the incident and reflected surface waves is clearly visible.

 

Fig. 4 The field distribution $\left|E_y\left(x,z\right)\right|$ generated upon BSW diffraction at a PSBG $\left(N_{\text{per}}=4\right)$ (a) and at a Bragg grating without defect having 9 periods (b). Grating ridge height is shown tripled (51 nm instead of 17 nm) for visual purposes.

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4. Temporal differentiation of BSW pulses

One of the promising applications of the conventional PSBG consists in temporal transformations of the optical signals. In particular, it was previously demonstrated that a conventional PSBG can perform temporal differentiation of the envelope of the incident optical pulse in reflection [3]. In the present section, we consider the application of the designed PSBG for the differentiation of BSW pulses.

The transformation of the envelope $P\left(t\right)$ of the incident pulse performed by the grating can be described in terms of transformation of the signal $P\left(t\right)$ by a linear system with the transfer function (TF) [3, 9, 10]

Let us now show that the TF (6) also describes the transformation of the BSW pulse envelope. The field of a monochromatic BSW propagating in the x direction can be written as

If the PSBG is located at $0\le x\le L$, where L is the total PSBG length, the field of the reflected BSW pulse corresponding to the incident pulse in Eq. (8) can be written as

It follows from Eqs. (9) and (11) that the transformation of the incident pulse envelope $P_{\text{inc}}\left(0,0,t\right)$ to the reflected pulse envelope $P_{\text{refl}}\left(0,0,t\right)$ corresponds to the transformation by a linear system with the TF described by Eq. (6).

It is evident from the reflection spectra of the conventional PSBG (dashed curves) and the PSBG for Bloch surface waves (solid curves) in Fig. 3 that the reflection coefficients in the vicinity of the angular frequencies ${\omega }_{\text{B}}$ and ${\tilde{\omega }}_{\text{B}}$ are close to the transfer functions of a differentiating filter up to a linear phase. This linear phase difference causes the delay of the reflected pulse and does not affect the differentiation quality.

We now consider the performance of the PSBG for an incident BSW pulse with Gaussian envelope. In this case, $P_{\text{inc}}\left(0,0,t\right)=\exp \left(-t^2/{\sigma }^2\right)$ and $G\left(\omega \right)~\exp \left(-{\sigma }^2{\omega }^2/4\right)$. As the central frequency of the pulse, we choose the value ${\tilde{\omega }}_{\text{B}}=\text{2}\text{.3587}\cdot {\text{10}}^{15}{\text{s}}^{-1}$ (${\tilde{\lambda }}_{\text{B}}=798.6\text{nm}$) at which the absolute value of the reflection coefficient reaches its minimum. Figure 5(a) shows the absolute value and phase of the TF described by Eq. (6) and corresponding to the reflection coefficient of the surface-wave PSBG at $N_{\text{per}}=4$. This TF was obtained from the reflection spectrum shown in Fig. 3 with solid red curves. Black dotted curve in Fig. 5(a) depicts the normalized spectrum $G\left(\omega \right)$ of the incident pulse envelope at $\sqrt{2}\sigma =0.2\text{ps}$. For this pulse, the $1/{\text{e}}^{\text{2}}$ widths of the incident pulse envelope $P_{inc}\left(0,0,t\right)$ and spectrum $G\left(\omega \right)$ amount to $T=\sqrt{2}\sigma =0.2\text{ps}$ and $2\Omega =0.8\cdot {\text{10}}^{14}{\text{s}}^{-1}$, respectively. According to Fig. 5(a), both the amplitude and phase of the transfer function of the PSBG are almost linear at $\left|\omega \right|\le \Omega$. Thus, in the considered case high-quality differentiation can be expected. Figure 5(b) shows the incident pulse envelope and the absolute values of the envelope of the reflected pulse calculated using Eq. (11) and the analytically calculated derivative of the Gaussian function. It is evident from Fig. 5(b) that the studied structure enables high-quality differentiation of an optical pulse. For the considered example, the Pearson correlation coefficient between the reflected pulse envelope and the exact derivative exceeds 0.999. The maximal amplitude of the reflected pulse in this case amounts to 0.285. Note that the 20 fs shift between the optically computed and exact derivatives can be interpreted as the time at which the differentiation operation is performed.

 

Fig. 5 (а) Normalized spectrum of the incident BSW pulse (dotted black curve); amplitude and phase of the PSBG transfer function at $N_{\text{per}}=4$ (solid red curves). (b) Envelope of the incident BSW pulse (solid black curve, right vertical axis), absolute value of the envelope of the reflected pulse (solid blue curve, left vertical axis), and absolute value of the analytically computed derivative of the Gaussian function (dashed green curve, left vertical axis). The exact derivative is normalized so that its amplitude is equal to the amplitude of the reflected pulse.

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Let us note that the increase in the number of periods of the Bragg gratings constituting the PSBG leads to the increase in the resonance quality factor, and, as a consequence, to the decrease in the TF linearity range. Figure 6 depicts the same quantities as in Fig. 5(b), but at $N_{\text{per}}=5$. In this case, the differentiation quality decreases insignificantly: the correlation coefficient between the optically computed and exact derivative is 0.997. At the same time, maximal amplitude of the reflected pulse increases to 0.42. Thus, the choice of the number of periods $N_{\text{per}}$ provides a way to achieve a required trade-off between the differentiation quality and the reflected pulse energy.

 

Fig. 6 Envelope of the incident BSW pulse (solid black curve, right vertical axis), absolute value of the envelope of the reflected BSW pulse (solid blue curve, left vertical axis) at $N_{\text{per}}=5$, and absolute value of the analytically computed derivative of the Gaussian function (dashed green curve, left vertical axis).

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In Table 1, the values of correlation coefficient and maximal amplitude of the reflected pulse calculated for different durations of the incident pulse are presented. As previously, two cases are considered: $N_{\text{per}}=4$ and $N_{\text{per}}=5$. It follows from Table 1 that the proposed PSBG are able to perform the temporal differentiation of BSW pulses with durations from 0.1 ps to 3 ps. The correlation coefficient for the considered cases exceeds 0.98. Note that with the increase in the pulse duration (and decrease in the spectral width) the correlation coefficient first increases but then begins to decrease again (two last rows of Table 1). This is due to the fact that the PSBG transfer function does not vanish at $\omega =0$ as the TF of the exact differentiator does. Thus, with the narrowing of the spectrum in the vicinity of $\omega =0$ the differentiation quality deteriorates.

Table 1. Pearson correlation coefficient r between exact and optically computed derivatives and maximal amplitude of the reflected pulse $P_{\text{max}}$ for incident pulses with different durations $T$.

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5. Conclusion

In the present work, we proposed an analogue of the phase-shifted Bragg grating for Bloch surface waves propagating along the interface between a one-dimensional photonic crystal and a homogeneous dielectric medium. It was shown using rigorous electromagnetic simulations that the reflection spectrum of the investigated structure is in good agreement with the reflection spectrum of a conventional PSBG for plane electromagnetic waves. For the considered example, the correlation coefficient between the spectra exceeds 0.99. The proposed structure was found to be able to perform high-quality temporal differentiation of BSW pulses of picosecond and subpicosecond duration.

The application area of the studied PSBG is not limited to temporal differentiation of the envelope of BSW pulses. The authors believe that the structure can also be used for temporal integration of BSW pulses as well as for spatial differentiation and integration of BSW beams. The proposed approach can most likely be extended to the modes of other planar structures such as silicon-on-insulator slab waveguides.

The presented results can be applied for the design of the prospective on-chip systems for all-optical analog computing.