1. Introduction

Optical bistability is an essential means to realize key functions for all-optical signal processing such as an all-optical switching and an optical memory [1–3]. Optical bistable devices based on resonant structures made of nonlinear materials have been widely investigated, which include photonic crystals [4, 5], slab waveguide gratings with guided mode resonance [6], and surface plasmons [7]. These resonant structures can achieve a low switching intensity by virtue of their high quality-factors (Q-factor) and strong field enhancement, which, however, inevitably increases their response times because they are proportional to Q-factor [6]. In order to overcome this trade-off between a switching power and a response time of an optical bistable device, non-resonant schemes for optical bistability are needed. Recently, a non-resonant optical bistable device based on effective permittivity variation of a metal-dielectric multilayer structure has been proposed and a response time of ~1 ps has been demonstrated theoretically [8]. However, systematic investigation on its performance optimization was not performed. Moreover, a unique hyperbolic dispersion feature of the metal-dielectric multilayer structure enables alternative optical bistability schemes, which have not considered at all. The metal-dielectric multilayer structure has attracted a lot of interest as one possible implementation of hyperbolic metamaterials (HMMs) [9–11]. The unique hyperbolic dispersion feature of the HMMs finds many interesting applications such as spontaneous emission enhancement, negative refraction, hyperlenses, and sensing [9–15]. While linear-optic applications of the HMMs have been widely studied, their applications combined with optical nonlinearity have rarely been studied.

In this work, various multilayer HMM-based optical bistability schemes were considered and their systematic performance optimization was performed. A multilayer structure shows topological transitions in their isofrequency contours depending on a fill fraction and an operating wavelength, which results in a sharp transmission variation. From this sharply varied transmission, we can realize high-performance optical bistability, and this proposed non-resonant scheme can overcome the trade-off between response time and switching power in the resonance based optical bistability schemes. All calculations in this paper have been performed by using the finite-difference time-domain (FDTD) method.

2. Transmission in hyperbolic metamaterials

Figure 1(a) shows a schematic of a multilayer structure consisting of alternating layers of metal and dielectric with thicknesses of d_m and d_d, respectively. When the thicknesses of metal and dielectric layers are much smaller than wavelength (d_m and d_d << λ), an effective medium method can be applied and the effective permittivities in the parallel (x and y axis) and perpendicular (z axis) directions are described as [16, 17]

 

Fig. 1 (a) Schematic of the multilayer. d_m and d_d represent the thickness of the metal and dielectric layer, respectively. (b) A summarized optical phase diagram. P₁, P₂, and P₃ represent the transitions from Type II HMM to Type I HMM, from Type II HMM to an effective dielectric, and from an effective metal to Type II HMM, respectively.

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The effective permittivity of a multilayer structure is anisotropic and the signs of permittivities in the parallel and the perpendicular directions are varied with a wavelength or a fill-fraction [17]. Figure 1(b) illustrates all possible characteristics of the multilayer structure as a function of a wavelength and a fill-fraction: it exhibits Type I HMM (ε_xx > 0, ε_zz < 0), Type II HMM (ε_xx < 0, ε_zz > 0), effective dielectric (ε_xx > 0, ε_zz > 0), or effective metal (ε_xx < 0, ε_zz < 0) characteristics depending on the wavelength and the fill-fraction.

We investigated light propagation along the z direction when the characteristic of the multilayer changes from Type II HMM to Type I HMM (P₁), from Type II HMM to an effective dielectric (P₂), and from an effective metal to Type I HMM (P₃). For the effective dielectric, the permittivity components with the same signs resulting in an ellipsoidal isofrequency contour. In contrast, Type I HMM and Type II HMM with opposite signs of the permitivities exhibit hyperboloidal isofrequency contours along the vertical (z axis) and the horizontal direction (x axis), respectively. For the effective metal case, there is no real k vector satisfying a dispersion relation. Figures 2(a)-2(d) show the change of the isofrequency contour topology for P₁, P₂, and P₃ transitions. The interesting difference of the Type II HMM and the effective metal compared with other cases is that light propagation along the vertical (z) direction is prohibited because the k vector state does not exist for k_x = 0 in the isofrequency contour. Therefore, when light is vertically launched into the multilayer structure, the incident light is totally reflected if the multilayer structure shows Type II HMM and the effective metal characteristics. On the other hand, if the multilayer behaves as Type I HMM, or an effective dielectric, the incident light can propagate because k_x = 0 state exists. Thus, the topological transition of the isofrequency contour can make a sharp transmission variation.

 

Fig. 2 Isofrequency contour variations in multilayer structure from (a) Type II HMM to Type I HMM, (b) Type II HMM to an effective dielectric, and (c) an effective metal to Type I HMM.

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In order to demonstrate the transmission variation due to the topological transitions of the isofrequency contour, multilayer structures composed of the Ag and dielectric layers have been designed, and transmission spectra have been calculated for three transitions: Type II HMM to Type I HMM (P₁), Type II HMM to an effective dielectric (P₂), and an effective metal to Type I HMM (P₃). For fair comparison, multilayer structures have been designed in such a way that they show respective transitions at the same wavelength. Figures 3(a)–3(c) show the effective permittivities for the designed multilayer structures for P₁, P₂, and P₃ transitions, respectively. The thickness of the metal and the dielectric layers are d_m = 15 nm and d_d = 15 nm (p = 0.5) for P₁ transition, d_m = 15 nm and d_d = 45 nm (p = 0.25) for P₂ transition, and d_m = 30 nm and d_d = 10 nm (p = 0.75) for P₃ transition. The permittivity of Ag is described by Drude model [18], ε_Ag (ω) = 1 – ω_p/(ω_p + iγω) with ω_p = 11.5 fs⁻¹, γ = 0.083 fs⁻¹, and the refractive indices of the dielectric layer for P₁, P₂, and P₃ transitions are n_d = 2.38, 1.375, and 4.2, respectively. It should be noted that we adopted the different dielectric materials to match the wavelength (λ = 425 nm) at which those three transitions occur. As shown in Figs. 3(a)–3(c), for the longer wavelength region than the transition wavelength, the designed multilayers behave as Type II HMM (P₁, P₂) or the effective metal (P₃), and for the short wavelength region, the multilayer structures behave as Type I HMM (P₁, P₃), or the effective dielectric (P₂). Figure 3(d) shows the calculated transmission spectra along the z direction for the multilayer structure designed for P₁, P₂, and P₃ transitions, where the total thicknesses of the multilayer structures are d = 420 nm, 720 nm, and 600 nm, respectively. The total thicknesses of multilayer structure were optimally chosen to exhibit a sharp transition property, which will be discussed later in detail. As expected from the isofrequency contours, high transmission can be obtained for the short wavelength region where the multilayer behaves as Type I HMM or effective dielectric. For the long wavelength region, on the other hand, the transmission is almost zero due to the Type II HMM or the effective metal characteristics not allowing the state of k_x = 0. Note that the transmission fluctuations in the short wavelength range come from Bragg interference between the waves reflected from the interfaces of the structure [8]. While the sharp transmission variations are observed for P₁ and P₂ transitions, P₃ transition show rather slow transition variation. So, P₁ and P₂ transitions are expected to perform better for optical bistability that P₃ transition.

 

Fig. 3 Real and imaginary parts of the parallel (ε_x) and the perpendicular (ε_z) permittivities for the multilayer structure with fill fraction (a) p = 0.5 (P₁ transition) and (b) p = 0.25 (P₂ transition) (c) p = 0.75 (P₃ transition), and the transmission spectra for various transition types: Type II HMM – Type I HMM (black line), Type II HMM – effective dielectric (red line), and effective metal – Type I HMM (blue line).

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Figures 4(a) and 4(b) show transmission spectra for various thicknesses of multilayer for P₁ and P₂ transitions, respectively, where the multilayer structure is treated as homogeneous slab with the permittivity modeled by effective medium method to investigate the effect of the total thickness, and the same fill-fractions and the refractive indices of the dielectric materials as the previous calculations are assumed. As shown in Figs. 4(a) and 4(b), the total thickness of the multilayer affects the transmission variation. As the thickness of the medium increase, the transmission is reduced to almost zero for longer wavelength than the transition wavelength because the effect of the Type II HMM is enhanced. As a result, the sharper transmission variation can be obtained near the transition wavelength. However, the loss of the medium by metal also increases, resulting in decrease of transmission for the short wavelength region. Therefore, there is an optimal thickness of a multilayer structure to achieve a sharp transmission variation for high optical bistability performance. According to the linear transmission calculation, the optimal thicknesses appear to be around d = 450 nm and d = 700 nm for P₁ and P₂ transitions, respectively since both the slope of the transmission variation and the transmission value at the short wavelength are relatively high. However, it is difficult to decide the optimal thickness for bistability performance only from the linear transmission spectra. So, in the next section, in order to find out the optimal structural parameters, the bistability performance is directly investigated for various layer thicknesses with both the number of layers and the period varied.

 

Fig. 4 Transmission spectra for various thicknesses of the medium modelled by effective medium approach for (a) P₁ transition (p = 0.5, n_d = 2.38) and (b) P₂ transition (p = 0.25, n_d = 1.375).

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3. Optical bistability in hyperbolic metamaterials

3.1 Optical bistability based on P₁ transition (transition from Type II HMM to Type I HMM)

We numerically investigated the optical bistability by P₁ transition. Figure 5(a) shows the transmission spectra for the various numbers of metal-dielectric pairs in the multilayer structure from 14 pairs (d = 420 nm) to 17 pairs (510 nm), where the thicknesses of the Ag layer and the dielectric layer are 15 nm, respectively, and the refractive index of the dielectric is 2.38. The multilayer structure is designed to have transition from Type I HMM to Type II HMM at a wavelength of 425 nm as a wavelength increases. For a more quantitative investigation on the transmission variation, the transmissions and their wavelength derivatives are plotted together in Fig. 5(b). As the number of pairs decreases, the maximum derivative value increases, and at the same time, its wavelength shifts to shorter wavelength, where the transmission value is high. To lower the switching intensity of an optical bistability, a large derivative value occurring at an operating wavelength, where a transmission value is low (close to zero), is needed. So, in terms of a low switching intensity of an optical bistability, a large number of pairs, that is a thick multilayer structure, seems to be desired. However, for a larger number of pairs, the transmission value for a short wavelength range (high transmission region) decreases resulting in a low contrast of transmission variation. This may degrade the optical bistability performance.

 

Fig. 5 (a) Transmission spectra for the number of pairs of the multilayer designed for the P₁ transition. The multilayer consists of Ag (d_m = 15 nm) and dielectric (d_d = 15 nm) layers. (b) Enlarged transmission spectra (solid line) of (a) and their wavelength differential values (dash line) near the transition region for the number of pairs: 14 pairs (d = 420 nm), 15 pairs (d = 450 nm), 16 pairs (d = 480 nm), and 17 pairs (d = 510 nm).

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We have calculated and plotted optical bistability behaviors for the designed multilayer structure with the transition from Type II HMM to Type I HMM (P₁) for the various numbers of pairs in Figs. 6(a)–6(d): (a) 14 pairs (d = 420 nm), (b) 15 pairs (d = 450 nm), (c) 16 pairs (d = 480 nm), and (d) 17 pairs (d = 510 nm). We considered only the nonlinearity of Ag (whose is nonlinear susceptibility is χ₃ = 2.49 × 10⁻⁸ + 7.16 × 10⁻⁹i esu [19]) because the nonlinearity of the dielectric material is negligible compared to Ag. The operating wavelengths are set to the wavelength where the transmission is T = 0.01 for all cases for fair performance comparison. As the number of pairs decreases, the contrast ratio and the hysteresis width increase due to high derivative value of transmission. As expected from the linear transmission investigation, the lower switching input intensity is observed for the larger number of the pairs. However, as the number of pair increases, the optical bistability behavior becomes less clear, that is the hysteresis width becomes very narrow because of the small derivative value of transmission. Note that we can hardly recognize as optical bistability the cases above 17 pairs (d = 510 nm), which is not shown here. Considering overall optical bistability performance in terms of the contrast ration and the switching intensity, the multilayer structure of 16 pairs (d = 480 nm) with switching intensity of ~135MW/cm² appears to be the best.

 

Fig. 6 Optical bistability by P₁ transition in the multilayer consisting of Ag (d_m = 15 nm) and dielectric (d_d = 15 nm) layers with the number of pairs. (a) 14 pairs (d = 420 nm), (b) 15 pairs (d = 450 nm), (c) 16 pairs (d = 480 nm), and (d) 17 pairs (d = 510 nm). The operation wavelengths are (a) 440 nm for 14 pairs, (b) 437.8 nm for 15 pairs, (c) 436.2 nm for 16 pairs, and (d) 434.8 nm for 17 pairs.

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We have also investigated the optical bistability performance dependence on the thickness of Ag and dielectric layers with fixed fill-fraction (p = 0.5) and total thickness of multilayer (d = 480 nm). The linear transmission and optical bistability for d_m = d_d = 5 nm, 10 nm, and 15 nm are shown in Figs. 7(a)–7(d), respectively. Since the transmission spectra for d_m = d_d = 10 nm and d_m = d_d = 15 nm are similar to each other, the performances of the optical bistability are almost the same for both cases. For d_m = d_d = 5 nm, on the other hand, the poor performances of the bistability are obtained compared with the others due to the weak increase of transmission near the transition wavelength. Since the performances of optical bistability for the d_m = d_d = 10 nm and 15 nm are not different much, this structure appears to have a good fabrication tolerance.

 

Fig. 7 (a) Transmission spectra for various thicknesses of Ag and dielectric layers with fixed thickness of multilayer (d = 480 nm). Optical bistability for various thicknesses of Ag and dielectric layers with fixed thickness of the multilayer (d = 480 nm): (b) Ag (d_m = 5 nm) and dielectric layer (d_d = 5 nm), (c) Ag (d_m = 10 nm) and dielectric layer (d_d = 10 nm), and (d) Ag (d_m = 15 nm) and dielectric layer (d_d = 15 nm). The operation wavelengths are (b) 449.7 nm for d_m = 5 nm and d_d = 5 nm, (c) 441.9 nm for d_m = 10 nm and d_d = 10 nm, (d) 442.2 nm for d_m = 15 nm and d_d = 15 nm,.

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3.2 Optical bistability based on P₂ transition (transition from Type II HMM to an effective dielectric)

We also investigated the optical bistability based on the transition from Type II HMM to an effective dielectric (P₂). Figure 8(a) shows the transmission spectra for the various numbers of metal-dielectric layer pairs from 10 pairs (d = 600 nm) to 13 pairs (d = 780 nm) of the multilayer structure consisting of d_m = 15 nm and d_d = 45 nm. The refractive index of dielectric layer is assumed to be 1.375 for matching the transition wavelength with the P₁ transition case. The enlarged transmission spectra near the transition wavelength and their derivatives are plotted in Fig. 8(b). It is very similar to the P₁ transition case that as the thickness of the structure decreases, the maximum derivative value increases and the wavelength of the maximum derivative shifts to the shorter wavelength. The contrast of the transmission variation decreases for the larger number of pairs.

 

Fig. 8 (a) Transmission spectra for the number of pairs of the multilayer designed for the P₂ transition. The multilayer consists of Ag (d_m = 15 nm) and dielectric (d_d = 45 nm) layers. (b) Enlarged transmission spectra (solid line) of (a) and their differential values (dash line) near the transition region for the number of pairs: 10 pairs (d = 600 nm), 11 pairs (d = 660 nm), 12 pairs (d = 720 nm), and 13 pairs (d = 780 nm).

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The calculated optical bistability curves for the multilayer structure of a various numbers of pairs are plotted in Figs. 9(a)–9(d): (a) 10 pairs (d = 600 nm), (b) 11 pairs (d = 660 nm), (c) 12 pairs (d = 720 nm), and (d) 13 pairs (d = 780 nm). The operating wavelengths are set as the same as the previous case (T = 0.01). As the total thickness of the structure increases, both the contrast ratio and the switching power decrease. These properties are the same as the P₁ transition case, but the bistability performance change by the thickness variation is more distinct than the P₁ transition case. Moreover, for the overall optimum optical bistability performance structure of 12 pairs (d = 720 nm), the switching intensity is ~100 MW/cm², which is rather lower than the P₁ transition case. It is surmised that this switching intensity reduction stems from the increase of the total thickness of the multilayer structure.

 

Fig. 9 Optical bistability by P₂ transition in the multilayer consisting of Ag (d_m = 15 nm) and dielectric (d_d = 45 nm) layers with the number of pairs. (a) 10 pairs (d = 600 nm), (b) 11 pairs (d = 660 nm), (c) 12 pairs (d = 720 nm), and (d) 13 pairs (d = 780 nm). The operation wavelengths are (a) 446 nm for 10 pairs, (b) 443.7 nm for 11 pairs, (c) 442nm for 12 pairs, and (d) 441 nm for 13 pairs.

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The Ag thickness effect on the optical bistability performance has also been investigated. Linear transmission spectra and optical bistability curves were calculated for various Ag thicknesses with the fill fraction (p = 0.25) and total thickness (d = 720 nm, 12 pairs) fixed and plotted in Fig. 10. Overall tendency is the same as the P₁ transition case: as Ag becomes thicker, both the switching intensity and the switching contrast increase slightly. The optimal thickness of Ag in this case appears to be ~15 nm.

 

Fig. 10 (a) Transmission spectra for various thicknesses of Ag and dielectric layers with fixed thickness of the multilayer (d = 720 nm). Optical bistability for various thicknesses of Ag and dielectric layer with fixed thickness of multilayer (d = 480 nm): (b) Ag (d_m = 5 nm) and dielectric layer (d_d = 15 nm), (c) Ag (d_m = 10 nm) and dielectric layer (d_d = 30 nm), and (d) Ag (d_m = 15 nm) and dielectric layer (d_d = 45 nm). The operation wavelengths are (b) 440.5 nm for d_m = 5 nm and d_d = 15 nm, (c) 439 nm for d_m = 10 nm and d_d = 30 nm, (d) 445.7 nm for d_m = 15 nm and d_d = 45 nm,.

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3.3 Optical bistability based on P₃ transition (transition from an effective metal to Type I HMM)

We calculated and plot the transmission spectra and optical bistability based on the transition from an effective metal to Type I HMM (P₃ transition) in Figs. 11(a) and 11(b), respectively. The multilayer structure consists of d_m = 30 nm and d_d = 10 nm, and the total thickness and refractive index of dielectric layer are set to be d = 600 nm (15 pairs) and n_d = 4.2 respectively. As mentioned before, this structure composed of a high metal ratio is unsuitable for high-state of bistability due to the high loss of metal and the resulting in slow variation of transmission near the transition wavelength, and the optical bistability can be hardly achieved by P₃ transition as shown in Fig. 11(b).

 

Fig. 11 (a) Transmission spectra for the multilayer consisting of Ag (d_m = 30 nm) and dielectric (d_d = 10 nm) layers designed for the P₃ transition, (b) optical bistability curves. The operating wavelength is 453 nm.

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4. Temporal response of optical bistability based on P₁ and P₂ transitions

Figures 12(a)–12(d) show the temporal responses of optical bistabilities based on the transition from Type II HMM to Type I HMM (P₁), where the thicknesses of the Ag and dielectric layers are d_m = d_d = 15 nm and the thickness of the multilayer is varied from d = 420 nm (14 pairs) to d = 510 nm (17 pairs). In the temporal response analysis, we first set the incident intensity below the switching intensity (low state) for enough time, and then increase the intensity to high state. The magnitude of the input and output electric field is shown in Fig. 12 as blue dash and red lines, respectively. The numerically estimated response times are approximately 250 fs, 300 fs, 350 fs, and 360 fs for the total multilayer thicknesses of 420 nm, 450 nm, 480 nm, and 510 nm, respectively. In this multilayer structure, optical bistability is realized based on the non-resonant transition of the dispersion property, so that in large, the response time appears to increase proportionally to the total multilayer thickness, and the shorter total length simply makes the optical bistable device the faster. Whereas, the short device length tends to increase the switching intensity aforementioned because of weakened HMM property. Therefore, the HMM based optical bistability also appear to have a weak tendency of the trade-off between the switching intensity and the response time basically. This trade-off, however, is not as strict as the resonance-based case. In the resonance-based case, both the switching intensity and the response time are directly associated with a quality factor of the resonator, the trade-off between them is a fundamental property. On the other hand, in the HMM-based case, the weakening of HMM property is caused only for a certain short range of the device length, so that this trade-off tendency is valid only for very short devices.

 

Fig. 12 Temporal response of the multilayer structure consisting of Ag (d_m = 15 nm) and dielectric (d_d = 15 nm) layers with (a) 14 pairs (d = 420 nm), (b) 15 pairs (d = 450 nm), (c) 16 pairs (d = 480 nm), and (d) 17 pairs (d = 510 nm). The operation wavelengths are (a) 440 nm for 14 pairs, (b) 437.8 nm for 15 pairs, (c) 436.2 nm for 16 pairs, and (d) 434.8 nm for 17 pairs.

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Figure 13 shows the temporal response associated with bistability based on the transition from Type II HMM to an effective dielectric (P₂), where the thicknesses of the Ag and dielectric layer are d_m = 15 nm, d_d = 45 nm, and the thickness of the multilayer is varied from d = 600 nm (10 pairs) to d = 780 nm (13 pairs). The response times are approximately 580 fs, 750 fs, 760 fs, and 350 fs for the total multilayer thicknesses of 600 nm, 660 nm, 720 nm, and 780 nm, respectively. The response time of bistability by P₂ transition is longer than P₁ transition due to the longer total thickness of the structure.

 

Fig. 13 Temporal response of multilayer structure consisting of Ag (d_m = 15 nm) and dielectric (d_d = 45 nm) layers with (a) 10 pairs (d = 600 nm), (b) 11 pairs (d = 660 nm), (c) 12 pairs (d = 720 nm), and (d) 13 pairs (d = 780 nm). The operation wavelengths are (a) 446 nm for 10 pairs, (b) 443.7 nm for 11 pairs, (c) 442 nm for 12 pairs, and (d) 441 nm for 13 pairs.

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5. Conclusion

We numerically investigate the optical bistability behavior in multilayer consisting of Ag and dielectric layers. The unique dispersion property of hyperbolic metamaterials and its topological transition by the nonlinear material enable realization of optical bistability. We considered all possible transitions in the multilayer structure, and transitions from Type II HMM to Type I HMM and Type II HMM to an effective dielectric provide high performance optical bistable devices. Since the proposed HMM-based optical bistability scheme is a non-resonant system, the response time is simply determined by the device length, and the switching intensity is affected by the device length only for a certain very short range of the length. Therefore, the trade-off between response time and switching intensity is not strict. The designed structure for the transition from Type II HMM to Type I HMM shows a switching intensity of 135MW/cm² and a response time of 350 fs, and the structure for the transition from Type II HMM to an effective dielectric shows a switching intensity of 100MW/cm² and a response time of 760 fs. The transition from an effective metal to Type I HMM is unsuitable for optical bistability due to high metal loss. The proposed scheme may lead to a new class of non-resonant optical bistability for reducing a response time.