The resonant properties of planar dielectric multilayers in sustaining and guiding optical modes of different kinds are well known [1]. Certain surface modes, known as Bloch surface waves (BSWs) [2,3], have attracted particular recent interest mainly because of the advantages offered by the narrow resonances (in both energy and momentum) associated with evanescent field distributions, which are well exposed to interact with an external medium [4]. Most of the research on BSW excitation has focused on the spectral, momentum, and spatial features related to the coupling and manipulation of BSWs, while substantially neglecting possible temporal effects. However, the latter can be particularly relevant in applications wherein BSW excitation is used to trigger ultrafast light–matter interactions [5,6]. It is well known that the spectral and temporal properties of pulses can be greatly altered in the presence of photonic bandgaps [7] and in the vicinity of resonances [8], leading to phenomena such as pulse splitting, temporal widening, and tailing effects.

Provided this framework, we consider the effects of BSW resonances on pulsed incident fields, assuming that the spectral width of the incident pulse and the BSW resonance are of the same order of magnitude. We show, in particular, that the sharp spectral phase variation around the BSW resonance leads to profound temporal modulation of the reflected pulse. For clarity, we restrict the discussion to plane-wave pulses.

The structure we are concerned with is a multilayer consisting of $N$ identical high-/low-index bilayers with an extra top layer separating a substrate (refractive index $n_{\rm i}$) and a superstrate (refractive index $n_{\rm t}$) as shown in Fig. 1(a). The bilayers have refractive indices $n_{\rm H}/n_{\rm L}$ and thicknesses $h_{\rm H}/h_{\rm L}$, while the index and thickness of the top layer are denoted by $n_{\rm T}$ and $h_{\rm T}$, respectively. State-of-the-art deposition techniques provide a large choice of materials suitable for fabricating multilayers sustaining BSWs across a broad spectral range, from mid-IR [9,10] to near-IR [11,12], visible [13,14], and UV [15]. Let us first assume that the structure is illuminated by a TE-polarized (monochromatic) plane wave with angular frequency $\omega$, incident at an angle $\theta _{\rm i}$. The incident wave with wave vector ${\mathbf k}_{\rm i}$ produces, in the half-space $z<0$, a reflected wave with wave vector ${\mathbf k}_{\rm r}$. Figure 1(b) shows the band structure of an infinitely periodic high-/low-index stack, calculated from Eqs. (25) and (27) of Ref. [1]. As usual, the horizontal axis represents the transverse component of the wave vector, $k_x = (\omega /c)n_{\rm i}\sin \theta _{\rm i}$. The pink and white regions represent the allowed and stop bands of the infinitely periodic stack, respectively. Considering the finite structure in Fig. 1(a), the theory of stratified media provides the solid line in the stop band, i.e., the BSW dispersion line.

 

Fig. 1. (a) Geometry and (b) band structure of the infinitely periodic stack and the BSW dispersion line with parameters $h_{\rm H} = 106$ nm, $h_{\rm L} = 161$ nm, $n_{\rm H} = 2.442$, $n_{\rm L} = 1.646$, $h_{\rm T} = 31$ nm, $n_{\rm T} = 2.366$, $n_{\rm i} = 1.457$, $n_{\rm t}=1$, and $N=4$.

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We next assume a plane-wave pulse arriving from the substrate side at an angle of incidence $\theta _{\rm i}$ that exceeds the critical angle for the substrate/superstrate interface. In TE polarization the incident spectral electric field can be expressed as $\mathbf {E}_{\rm i}(x,z;\omega ) = -\mathbf {\hat {y}}E_{\rm i}(x,z;\omega )$, where

The $y$ component of the reflected field in the half-space $z<0$is

Figure 2 illustrates the spectral response of the finite stack considered in Fig. 1(a) when illuminated by a transform-limited Gaussian pulse:

 

Fig. 2. Spectral line shapes. Blue: normalized spectrum of the incident pulse, $\Omega \approx 0.0134$ rad/fs, $\Omega _{\rm L}\approx 0.00315$ rad/fs, and $\delta \omega = 0.1083$ rad/fs. Black: BSW resonance line shape $|r(\omega )|^2$. Dashed green: reflected normalized spectrum.

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The temporal field representations are obtained by Fourier-transforming Eqs. (1)–(3) according to

Denoting $\omega ^\prime = \omega -\omega _0$ and defining the retarded time as

Finally, for the transmitted field, we define the retarded time as

This is the evanescent tail of the pulsed BSW in the superstrate.

For the Gaussian spectral pulses defined in (4), the envelope of the incident pulse has the form

Figure 3 illustrates the spectral and temporal characteristics of the reflected and transmitted pulses for the stack in Fig. 1. We assume $\omega _0$ and $\Omega$ as in Fig. 2, and choose $\omega _{\rm BSW}=\omega _0$. The normalized spectra of the incident, reflected, and transmitted fields are shown in Fig. 3(a). As expected on the basis of Fig. 2, the reflected spectrum is split while the transmitted spectrum narrows due to the resonance. The spectral phases $\Phi _j(x,z;\omega ) = \arg [E_j(x,z;\omega )]$, $j={\rm i},{\rm r},{\rm t}$, are illustrated in Fig. 3(b) at $(x,z)=(0,0)$ for the incident and reflected fields, and at $(x,z)=(0,h)$ for the transmitted field. The BSW resonance causes a phase modulation in the transmitted and reflected spectral fields. In the case of the transmitted field the phase difference between the asymptotic limits at $\omega \gg \omega _0$ and $\omega \ll \omega _0$ is $\sim \pi$ rad, whereas that of the reflected field is $\sim 2\pi$ rad (equivalent to zero).

 

Fig. 3. (a) Normalized spectra and (b) spectral phases of the incident, reflected, and transmitted fields for lossless stacks when $\omega _{\rm BSW} = \omega _0$. (c) Temporal intensities and (d) temporal envelope phases. The distributions of $S(\omega )$ and $I(t)$ are normalized to their peak values.

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The consequences of the BSW resonance in the time domain are shown in Figs. 3(c) and 3(d), where we plot distributions of the normalized temporal intensity $I_{j}(x,z;t) = |A_j(x,z,t)/A_{0}|$ and the temporal envelope phase $\Phi _j(x,z;t)=\arg [A_j(x,z;t)]$ at the same reference points as above. The temporal intensity profile of the transmitted (BSW) pulse is delayed in time in comparison with the incident Gaussian pulse, which is affected by choice of the reference point as $(0,h)$ instead of $(0,0)$. The BSW pulse is also broadened and it features an exponentially decaying tail due to the spectral phase modulation. The intensity profile of the reflected pulse is split in two temporally separate parts, with a zero at a certain value $t=t_{\rm c} < 0$. The leading part of the pulse, at $t>t_{\rm c}$, has approximately the same duration as the incident pulse, although its peak is shifted. The trailing part at $t<t_{\rm c}$ is temporally wider than the incident pulse and has an exponentially decaying tail. The temporal phase of the reflected pulse is modulated by $\sim \pi$ rad at around $t_{\rm c}$.

The strong temporal modulation effects may appear surprising, given that the temporal width of the incident pulse is far larger than the time it takes to travel through the stack. Indeed, in nonresonant conditions only a small fraction of the pulse interacts with the stack at any given time, and one would only expect a small delay arising from the finite stack thickness. At resonance, however, the physical situation is dramatically different. A narrow resonance implies strong multiple reflections within the stack, causing the pulse to effectively stay inside it considerably longer and to leak out only gradually, thus producing the observed tail of the trailing part. The splitting that originates from the rapid spectral phase change at the resonance forces the leading part of the reflected pulse forward in time (within the duration of the incident pulse). This leading part is due to nonresonant spectral components in Fig. 2, whereas the trailing part is due to the resonance.

In Fig. 4 we demonstrate the dependence of the (normalized) temporal intensity profile $I_{\rm r}(0,0;t)/I_{\rm i}(0,0;t)$ on the frequency offset $\delta \omega$ between the BSW resonance and the center frequency of the incident spectral pulse. The general behavior is illustrated in Fig. 4(a), from which we can see that the effect of the BSW resonance diminishes as the offset (due to angular detuning) gradually takes the BSW resonance out of the spectral range of the input pulse. Figure 4(b) demonstrates this behavior more quantitatively. The dip at $t=t_{\rm c}$ reaches zero only when $\delta \omega =0$, but it persists as long as the detuning is moderate. In out-of-resonance conditions the tail gradually disappears and the reflected intensity distribution reduces (in functional form) to that of the incident field.

 

Fig. 4. (a) Temporal intensity profile $I_{\rm r}(0,0;t)$ of the reflected pulse as a function of the spectral detuning parameter $\delta \omega$. (b) Intensity profiles at specific values of $\delta \omega$. Black: $\delta \omega _{\rm 0} = 0$. Blue: $\delta \omega _{\rm 1} = 0.00375$ rad/fs. Red: $\delta \omega _{\rm 2} = 0.00815$ rad/fs. Green: $\delta \omega _{\rm 3} = 0.0119$ rad/fs.

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The spectral width of the BSW resonance, as compared with that of the incident pulses, has a decisive effect on the character of the temporal modulation phenomena studied here. If the spectrum of the incident field is effectively confined within the BSW resonance, which is the case for sufficiently long pulses, the leading contribution of the doubly peaked reflected field becomes small. In the opposite case (short pulses), the leading contribution dominates since only a small part of the incident spectrum is subjected to the BSW resonance. For fixed values of the refractive indices and layer thicknesses, the spectral width $\Omega _{\rm L}$ of the BSW resonance depends critically on the number $N$ of bilayers in the stack [17]. The effect of $N$ on the temporal characteristics of the reflected and transmitted pulses is illustrated in Fig. 5. With $N=4$, the trailing part of the reflected pulse dominates since the incident spectrum falls mostly within the BSW resonance. At larger values of $N$, when the BSW resonance becomes sharper, this is no longer the case. With $N=5$, the weights of the leading and trailing parts of the reflected pulse are roughly equal, and with $N=6$ the leading part dominates. Corresponding results hold for other stack structures if we adjust the angle of incidence according to the resonance frequency.

 

Fig. 5. Reflected temporal pulses for different values of the number of bilayers in the stack, $N$, for the structure defined in the caption of Fig. 1 and $\Omega = 0.000261$ rad/fs.

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The final issue that we wish to address here is the role of absorption in the temporal pulse shapes. Adding reasonable values of the imaginary part $n'$ of the refractive index of the stack materials [18], we observe a smoothing of the spectral phase gradient near the BSW resonance as illustrated in Fig. 6(a). When the loss is increased above a certain value, which is $n'\approx 1.605\times 10^{-3}$ with the present parameters, the character of the spectral phase changes. However, this has no dramatic effect on the intensity profile of the reflected pulses, as illustrated in the logarithmic plot shown in Fig. 6(b) and the phase in Fig. 6(c). The exponential tail of the pulse remains clear, even though the temporal width of the reflected pulse is reduced by the loss.

 

Fig. 6. (a) Spectral phase of the reflected pulse as the loss $n^\prime$ is increased. (b) Reflected temporal intensities in logarithmic scale. (c) Temporal phases.

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To conclude, we have analyzed the effect of BSW resonances on temporal properties of the evanescent-wave pulses above the top surface of a multilayer structure and, in particular, on the reflected pulses. Provided specific multilayer design and illumination conditions, temporal splitting of the reflected pulse is expected from our theoretical model. This effect can have interesting implementations in BSW-based sensing, where a time-gated detection can be employed to improve the useful signal collection and filter out undesired background.

We restricted the discussion to plane-wave pulses, but the analysis can be straightforwardly extended to spatiotemporally confined pulses. This extension, to be discussed elsewhere, allows us to consider the spatial and temporal behavior of the pulses on top of and inside the multilayer structure, to study the gradual leak-out of the surface pulse, and to analyze the spatiotemporal nature of the associated Goos–Hänchen effect.