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Abstract
We demonstrate the emergence of slow-light in dual-periodic dielectric one-dimensional photonic crystals with self-similar features at different length scales. Specifically, using numerical modelling, we explore self-similar photonic crystals which are formed as effective combinations of dual periodic stacks of dielectric layers and show that the emergent photonic band diagram can be widely designed by different structural parameters. The width and the position of bandgaps can be designed to work over a wide range of bands and frequencies. The proposed design also leads to the emergence of flat bands and major slow-light regimes, with possible group refractive index of light as large as 103 and in a range of bands.
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1. Introduction
Slow light can be classified as the light for which the group refractive index $n_g$ is much larger than the refractive index of the material in which the light propagates [1–3]. There has been extensive interest in slow light systems due to interesting properties, such as spatial signal compression in systems with minimal dispersion, and spatial dispersion in slow light systems with strong dispersion. Applications are proposed in the fields of long distance telecommunications [4,5], waveguides [6], optical signal processing [7], all-optical switches [8], and optical sensors [9]. Slow light was observed in systems with electromagnetically induced transparency, in cold gas/atom systems [10], and in solid state matter [11].
Photonic crystals – materials where a selected dielectric structure is repeated periodically – have proven to be a distinct class of materials where it is possible to engineer high group refractive indices [1,12–18]. For example, modified photonic crystal waveguides [11,13,19–21] were shown to exhibit theoretical values of the group refractive index $n_g$ as high as $100$ [20,22]. Theoretical predictions of flat bands with low band widths and slow group velocities have been reported in dual periodic dielectric multilayer structures [14,23] but notably relying on multiple different-valued dielectric layers, amongst which some require a continuously varying dielectric profile along their thickness, which is experimentally challenging to realise.
Optical properties of photonic crystals depend on the geometry and the optical properties of their material components within the basic unit cell [24]. Therefore, the design of photonic crystals’ photonic response is crucially controlled by the design of the dielectric profile. With the desire to engineer task-specific optical response, parallel to experiments, methods of numerical optimisation and modelling [25–33] and neural-network-based design toolkits [34,35] are being developed.
Self-similar structures show promising design routes for photonic crystals, with selected parts of the unit cell being similar to the unit cell itself, notably on selected length-scales [36]. Quasi-periodic 1D photonic crystal structures, where two different dielectric layers appear in an arranged sequence, like Cantor, Thue-Morse, Rudin Shapiro and Fibonacci can perform as optical filters [37], mirrors [38,39], negative refractive index materials [40] and as ultra-slow light devices [41]. 2D self-similar photonic crystals have also been proposed [36].
In this work we demonstrate one-dimensional dielectric photonic crystals based on dual-periodic self-similar layers, with designable and customizable optical response, exhibiting prominent flat bands and high group refractive index. Notably, the photonic crystal comprises of only two dielectric materials, which is a major advantage for possible experimental realisation. Simplified structure, compared to quasi-periodic photonic crystals used to achieve slow light [41], allows for precise design of widths and positions of photonic band gaps without the use of additional optimisation techniques. We show that by changing the length scale at which the dielectric profile of the unit cells is self-similar, we can design the width and the position of band gaps over a wide range of bands and frequencies. Moreover, in the explored self-similar structure we can also design the group refractive index of light in a wide range of bands with slow-down factors of up to $10^3$. More generally, this work is a contribution towards the realisation of customizable slow light photonic crystals.
2. Theory and methods
The photonic response of the dual-periodic self-similar dielectric multilayered films can be determined by the wave-equation (i.e. Helmholtz equation, for a given frequency $\omega$) in the form:
We design the self-similar periodic dielectric multilayer films, with the goal to realise photonic flat band regions with high group refractive indices (see Fig. 1(a)). The multilayer structure is constructed of (only) two dielectric materials with different refractive indices (Fig. 1(b), Fig. 1(c) and Fig. 1(d)). The core of the design of the self-similar photonic structure is the introduction of two different thicknesses of the layers, leading to two length-scales of the layered material: (i) the overall length of the unit cell of the single-periodic structure $a$ (which corresponds to thicker layer thickness $a/2$), and (ii) the period of the finer sub-unit cell modulation $d$ (which corresponds to thinner layer thickness $d/2$). Examples of dual periodic unit cells with different ratios of $a/d$ are shown in Fig. 1(d). A single periodic unit cell with the same average dielectric constant in each half of the unit cell is also shown for comparison. $\varepsilon _{A}$ and $\varepsilon _{B}$ are the dielectric permittivities of two different dielectric materials and $\Delta \varepsilon=|\varepsilon_{A}-\varepsilon_{B}|/2$. Overall, the structure is a combination of two self-similarly organised dielectric bi-layer films which differentiate in their thicknesses and dielectric constants.
Fig. 1. Design of dual-periodic self-similar dielectric multilayered films (a) Example of dielectric structure in unit cells (divided by red dashed lines). (b) Dielectric profile in a unit cell for $\Delta \varepsilon = 4.0$ and $a/d = 5.0$. (c) Effective self-similar design of the dielectric profile of the photonic crystal with structure parameters $\Delta \varepsilon = 2.0$, $\overline {\varepsilon } = 4$ and $a/d = 5.0$. (d) The single periodic dielectric structure and the dielectric structures with structural parameters $\Delta \varepsilon = 2$, $\overline {\varepsilon } = 4$, $a/d = 2, 6,$ and $10$.
3. Band structures of self-similar multi-layer films
Calculated band structures for $a/d=10, 20$ and $30$ are shown in Fig. 2(a). In the first band diagram, corresponding to a single structure, we observe gaps between every set of bands, which is a characteristic feature of such multilayer Bragg-resonator type films with two layers in the unit cell [24]. However, interesting behaviour, characteristic of dual periodic films, emerges when a finer structure ($a/d>1$) is incorporated in the unit cell, as shown in Fig. 1(c). Distinctly, band gaps (marked in orange) in selected frequency ranges become significantly wider, whereas the remainder of the band structure is mainly unchanged. Consequently, bands (black lines) in the regions of wide gaps flatten, which is the prime characteristic sought-for in the design of materials for slow light applications. Moreover, note that positions of wide-bandgap regions can be extensively designed by the material structure parameter $a/d$.
Fig. 2. Photonic band structures of self-similar multi-layer films (a) The band diagrams of the single periodic structure and structures with $a/d = 10, 20, 30$. The bands are drawn as black lines and the band gaps are drawn as orange lines, the width of which corresponds to the width of the band gap. For all structures $\varepsilon _{A} = 2$ and $\varepsilon _{B} = 6$. (b) Eigenmodes of the electric field in the 16th ($n = 16$) and 26th ($n = 26$) band at $k=0$. The eigenmodes in the 16th or 26th band are drawn in red or purple, which are colour-matched with the marked eigenmodes in (a).
The photonic modes for two selected bands in the $k = 0$ point are shown in Fig. 2(b). Note, how modes of the 16th band are similar for very different structures of $a/d=1$ and $a/d=30$. Such behaviour occurs because when the wavelength is large enough compared to the period of the finer structure, the finer structure does not affect the light and the band diagrams for those frequencies remain unaltered when the finer structure is introduced. However, differently, for example, the mode of the 26th band for $a/d=30$, has a significant amplitude drop in the region of the finer structure, which indicates strong coupling with the dual-scale structure. Namely, the modes are localized in the wide layers without the fine structure. This is the opposite of the mode in the 16th band for $a/d=20$, where the mode is mostly localized in the region with the finer structure. The 16th mode for $a/d=10$ is about evenly distributed across the unit cell, which indicates weak coupling with the dual-scale structure, similar to the 26th mode and the 26th mode for $a/d=20$. An indicator of how strongly the modes of light are coupled to the dual-periodic structure is how different the band diagram near the specific eigenfrequency of a mode is compared to the band diagram of the single periodic structure.
The origin of the wide-bandgap regions can be explained by considering two substructures of the actual dielectric profile – the single-periodic and the fine-scale structure – which effectively combine to give the wide-bandgap photonic response (see Fig. 3). The first structure is the single-periodic structure, with the unit cell consisting of two equally thick layers with different dielectric constants (here $\varepsilon _A=2$ and $\bar {\varepsilon }=4$), the unit cell length $a$ and characteristic band gaps between every set of bands at either the edge of the Brillouin zone or its center (Fig. 3(a)). The sizes and positions of the band gaps depend on the dielectric constants of the layers and the ratio of their thicknesses (here fixed at $1:1$). The second sub-unit cell is similar to the single-periodic structure, if taken independently (i.e. is a multilayer film with two layers in the unit cell, see Fig. 3(b)), but with different minimal and maximal values of dielectric constant (here $\varepsilon _A=2$ and $\varepsilon _B=6$) and smaller layer thickness $d$. Its corresponding band diagram would resemble the band diagram of the single-periodic structure if the frequency would be plotted in the units of its own unit cell ($2\pi c/d$). However, in order to be able to compare both diagrams, we plot the frequency in the units of the unit cell of the single-periodic structure $2\pi c/a$ (Fig. 3(b)), which then needs rescaling of the frequency by a factor of $a/d$.
Fig. 3. Origin of the wide-bandgap regions. In the top row, we show the dielectric profiles of the substructures (a,b) that effectively combine to give the response of the actual dual-periodic self-similar multilayered films (c). (a) The base dielectric structure — a single periodic multilayer film with two equally thick layers in the unit cell with the period $a$ — and its corresponding band diagram. (b) The finer dielectric structure with the period $d$, which is present in one half of the unit cell of the hierarchical structure and its corresponding band diagram. The band diagram is rescaled and plotted in the units of ($2\pi c/a$). (c) The hierarchical dielectric structure consists of a thick layer with a thickness of $a/2$ in one half and $a/d$ thin layers in the second half of the unit cell and its corresponding band diagram. Note how the positions and widths of the wide band gap regions in (c) match with the positions and widths of the band gaps in (b).
Finally, we can obtain the band structure of the hierarchical pattern by combining both band diagrams (Fig. 3(c)). The band structure of the base dielectric structure, which determines the unit cell size of the hierarchical system, is altered at the frequencies within the band gaps of finer structures shown in Fig. 3(b). At these frequencies band gaps of the single-periodic structure widen and the bands become flatter — hence we name them wide-bandgap regions. The light is (partially) reflected from the finer structure, leading to destructive interference. Therefore, a band gap should exist at the same frequencies as in Fig. 3(b). However, in the case of the hierarchical structure, the finer structure only exists in half of the unit cell, leading to reflected modes being localized in the wide layers without fine structure. The system can be essentially seen as a stack of coupled resonators, where the fine structure in each unit cell acts as a Bragg mirror for the frequencies in the areas of wide-bandgap regions. Due to the finite size of Bragg mirrors, modes that are localized in thick regions remain coupled through evanescent field and can slowly propagate between thick layers, which causes the bands to flatten. In other words, the flat band modes can be seen as coupled thick-layer defect modes in the fine photonic crystal structure.
4. Design of wide-bandgap regions
The photonic band structure of the demonstrated dual-period self-similar photonic crystals depends on dielectric constants $\varepsilon _A$ and $\varepsilon _B$ and the periodicity of the second structure, compared to the single-periodic structure $a/d$, which can be used to design the wide-bandgap regions. Figure 4 shows how changing these parameters affects the width of the band gaps, and consequently the positions of wide-bandgap regions and the number of band gaps that are widened.
Fig. 4. Tuning of wide-bandgap regions by structure parameters. Type I (a) and Type II (b) approaches for changing the dielectric structure. Type I modulation of the dielectric structure of the unit cells corresponds to the variation of the value of $\Delta \varepsilon = |\varepsilon _B - \varepsilon _A|/2$ while keeping the values of $a/d$ and $\overline {\varepsilon } = (\varepsilon _B + \varepsilon _A)/2$ constant. (a) shows two dielectric structures in the unit cell, one with $\varepsilon _A = 2$, $\varepsilon _B = 6$ (black line) and the other with $\varepsilon _A = 3$, $\varepsilon _B = 5$ (orange line). Type II modulation of the dielectric structure in the unit cells corresponds to changing the value of $a/d$ while keeping the values of $\Delta \varepsilon$ and $\overline {\varepsilon }$ constant. (b) shows two dielectric structures in the unit cell, one with $a/d = 4$ (black line) and the other with $a/d = 6$ (orange line). (c) and (d) show the variation of band gap widths (orange regions) with respect to Type I (c) and Type II (d) modulation of the dielectric structures and their respective mid-gaps. The black lines run along the lower and upper frequencies of each band gap, a close-up of which is shown in the blue inset in (d). The widths of the band gaps increase as $\Delta \varepsilon$ increases (c). By increasing $a/d$ the mean frequency of the wide-bandgap region increases linearly by a factor of $a/d$ (d).
We focus on two different types of modulations. In the Type I modulation (Fig. 4(a) and Fig. 4(c)), we vary the contrast between $\varepsilon _A$ and $\varepsilon _B$, as given by $\Delta \varepsilon$, while keeping $\overline {\varepsilon }$ fixed. As shown in Fig. 4(c), increasing the contrast increases the widths, but does not significantly change the location of the wide-bandgap regions (regions of wider orange lines). Note, that in this modulation the value of the dielectric constant averaged over the entire unit cell increases, which causes changes in the location of all bands. For example, if we change $\Delta \varepsilon$ from $1$ to $2$ for a structure with $a/d = 16$ and $\overline {\varepsilon } = 4$, the band gap between the 14th and 15th band will increase by a factor of $\sim 38$. Similarly, if we change $a/d = 0$ to $a/d = 16$ for a structure with $\Delta \varepsilon = 2$ and $\overline {\varepsilon } = 4$ the band gap between 14th and 15th band changes by a factor of $\sim 8$. In the Type II modulation (Fig. 4(b) and Fig. 4(d)), we increase $a/d$ ratio by decreasing the thickness of the layers in the fine structure, thus changing the structure periodicity. Such modulation linearly increases the mean frequency of wide-bandgap regions, namely by a factor of $a/d$, compared to the frequency of the single-periodic structure. To generalise, by tuning the dielectric constants of the materials for each layer and the thicknesses of the base and fine structure, one can extensively control the positions of wide-bandgap regions where larger gaps between bands and flatter bands will occur.
4.1 Slow light
We have shown that band gaps widen by introducing the self-similar substructure and that the band structure outside those regions remains unaltered which directly implies that bands within the wide-bandgap regions have flattened. Frequency of a flat band mode changes slowly with respect to the wave vector, compared to the modes of the free space (outside wide-bandgap regions), which can be observed from the calculated band diagrams (for example Fig. 5(a)). Therefore the group velocity of flat bands in wide-bandgap regions $v_g$, which is obtained as $\frac {d \omega (\mathbf {k})}{d \mathbf {k}}$, is much lower than the speed of light in a vacuum. Namely, the group velocity is reduced by a factor of $n_g$ — the group refractive index.
Fig. 5. Emergence of slow light in dual-periodic self-similar dielectric multi-layer films. (a) The band diagram of a structure with $a/d = 19$, $\varepsilon _A = 2$ and $\varepsilon _B = 6$. (b) The density of states and (c) group velocity for the eigenfrequencies of the structure used in (a). In (a), (b) and (c) the grey dotted lines indicate the lower and upper frequencies of band gaps in (a). (d) The plots of $\mathrm {log}_{10}(n_g)$ with respect to the midgap $\omega$ and structural parameter $a/d$ for different values of $\varepsilon _A$ and $\varepsilon _B$.
The group velocity of eigenmodes at specific reciprocal vectors is calculated with an MPB build-in method, which is based on the Hellman-Feynmann theorem [46]; the results are shown in Fig. 5(c).
For each structure in the phase spaces, we calculated the group velocity of states in the band diagram with $k \in (0, 0.5)$ up to the 40th band. This allows us to better understand the relationship between the width of the band gap and the group velocity of states in the adjacent bands. Group velocity is related to the density of states $\frac {d N}{d\omega }$ [1,12] and is obtained by calculating the probability density function of states in a specific band diagram with respect to their eigenfrequencies. The obtained probability density function is normalized so that the integral over the range of all eigenfrequencies in the specific band diagram equals one. The results are shown in Fig. 5(b). We calculated the density of states in each band diagram for every structure in the phase spaces in order to show that there is a strong relation between the width of the band gap, the group velocities of the adjacent bands, and the density of states in the adjacent bands (see Fig. 5(a), (b), (c)). The group velocity is low where the density of states is high and the band is adjacent to a wide band gap.
Finally, we obtain the group refractive index as $n_g=c_0/v_g$, where $c_0$ is the speed of light in a vacuum. The dependence of the lowest group refractive index in each band with respect to the period of the finer structure $a/d$ for selected values of dielectric constants is shown in Fig. 5(d). Interestingly, we observe a significant slowing down of light by increasing $a/d$. Actually, the group refractive index only depends on $a/d$ once the dielectric constants are selected.
5. Discussion
The introduced self-similar dielectric structures allow for design of the photonic response, so that incident light with specific wavelengths will propagate with slow group velocities. Frequencies of the modes with the highest group refractive indices grow linearly with $a/d$, roughly following the relation $\omega = K a/d$, where $K$ is the coefficient that describes the slope in Fig. 5(d) (for normalisation $\omega$ is taken in units $2\pi c/a$). In fact, $K$ is determined by the position of the first band gap in the band diagram of the single periodic fine structure with the layer thickness $d/2$, shown in Fig. 3(b) for $d=a$. This mid-gap frequency depends on the material parameters $\varepsilon _A$ and $\varepsilon _B$ [24]. For $\varepsilon _A=2$ and $\varepsilon _B=6$ used in the examples here, $K$ is approximately 0.25. Actually, we can relate the structural parameter $a/d$ with the ratio $a/\lambda$ ($\lambda$ is the wavelength of light) for which we want to realise a certain group refractive index, as:
We reach the above equation by inserting $\omega = K a/d$ into $2 \pi c = \lambda \omega 2 \pi c/ a$, which is the expanded form of $c = \nu \lambda$, where $c$, $\nu$, $\lambda$ are the vacuum speed of light, frequency and wavelength.
High group refractive indices are achieved for higher values of $a/d$, and once the value of that parameter is chosen it also determines the unit cell size for the selected wavelength of light. The thinnest layer one must be able to manufacture for a given wavelength can be estimated as:
where the unit cell is then constructed by manufacturing $a/d$ pairs of thin layers of materials with $\varepsilon _A$ and $\varepsilon _B$ and one thick layer of material with $\varepsilon _A$ and thickness $a/d$-times larger than the thin layer.For example, if we want to construct a photonic crystal with $\varepsilon _A = 2$, $\varepsilon _B = 6$ for green incident light ($\lambda = {512}\;\textrm{nm}$) so that it will experience the largest group refractive index for the selected $a/d$ ($K = 0.25$) the smallest feature we must be able to manufacture will be ${64}\;\textrm{nm}$. For light in the telecom C-band [47] ($\lambda = {1550}\;\textrm{nm}$) the thinnest layer would have to be $\approx {200}\;\textrm{nm}$ thick. Estimated $a/d$ and sizes of the unit cell for different values of $n_g$ with respect to the wavelength are given in Table 1. The group refractive indices in Table 1 represent the lower estimate of the group refractive index in the respective mode, where the exact value depends on the $k$ vector for the respective $\omega$ of the incident light. Overall, in order to realize a photonic crystal with large group refractive indices we should be able to create relatively large unit cells in comparison to the thickness of the thinnest layer.
Table 1. Group refractive index for selected system design ($a/d$ and $\lambda$). The table shows what $a/d$ (row 3) is needed to achieve the desired group index $n_g$ (row 4) for a structure with given $d$, $\varepsilon _A = 2$ and $\varepsilon _B = 6$.
The possible drawback of the proposed design is that it becomes increasingly harder to couple light into the structure that supports higher group refractive indices. Increasing $a/d$ increases the $n_g$ by making the layered structure a better Bragg mirror, meaning that light will more likely reflect on the incoming plane. Possible solutions to the issue could be coupling through the evanescent field by sending light through a waveguide close to the photonic crystal in a direction parallel to the layering of the photonic crystal [48,49] or by creating light within the system using lasing [50,51].
All dielectric materials naturally absorb light to some – negligible or non-negligible – degree. To test the role of absorption – introduced as the imaginary part of the dielectric constant – we performed additional numerical calculations (not shown here) and found that the band structures of presented photonic crystals remain mostly unchanged at lower eigenfrequencies even with substantial material losses ($Im(\varepsilon )/Re(\varepsilon ) \geq 0.5$). We have performed additional simulations where we modified the unit cell structures so that the borders between regions with $\varepsilon _A$ and $\varepsilon _B$ were not sharp but rounded, as they would be for example due to imperfect manufacturing. The more gradual transition between regions with different dielectric constants does contribute to the reduction of group refractive indices of certain modes, but typically this reduction is in the order of a few percent.
6. Conclusions
In this paper, we demonstrate one-dimensional photonic crystals based on self-similar dual-periodic unit cells of two different dielectric material layers, which cause selected modes of light to propagate with slow group velocities. We construct the unit cells from two sets of Bragg mirror photonic crystals with different periodicities $a$ and $d$ and show how the band diagrams of the Bragg mirror structures are reflected in the band diagrams of self-similar dual-periodic structures. By increasing the contrast between the dielectric constants of the two different materials, we can control the number of slow-light bands in the band diagrams of our self-similar structures and the group refractive indices in those bands. The frequencies of the slow-light bands are determined by the geometric ratio of the unit cell sizes $a/d$ and to the mean of the dielectric constants. Since only two different materials (i.e. different values of dielectric permittivity) are used, such structures could possibly allow for precise and highly effective design of desired group refractive indices of slow light for specific wavelengths, only by controlling the layer thicknesses and consequently the structural parameter $a/d$. Additionally, the use of only two different materials could also possibly simplify the manufacturing process. As applications, the proposed materials could be used as customizable optical filters or be part of multi-channel slow light devices.
Funding
European Research Council (Grant agreement ID: 884928); Javna Agencija za Raziskovalno Dejavnost RS (J1-2462, N1-0195, P1-0099).
Acknowledgement
P.R. acknowledges helpful discussions with Jeroen Beeckman and Frédéric Monet.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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