1. Introduction

Hyperbolic metamaterials (HMs) [1, 2] provide great flexibility for tailoring the optical properties of surfaces and are utilized in a wide variety of optical devices [3–6]. Furthermore, the high density of states associated with the hyperbolic isofrequency wavevector surface can greatly enhance spontaneous emission [7, 8], enhance near-field thermal energy transfer [9, 10], lead to enhanced absorption processes [11, 12], and generate negative refraction [13–16].

At mid-infrared frequencies, highly doped semiconductors behave like metals. Thus, a highly doped layer can be used as a “metal” layer, and, along with an undoped layer, can form the unit cell of a semiconductor hyperbolic metamaterial (SHM) [13, 17–19]. We have recently shown in Ref [19]. that it might be possible to use ultrafast optical excitation to generate high densities of photocarriers in nominally undoped quantum wells [20, 21] of semiconductor superlattices. With sufficient pumping strength, the wells will become metallic in the frequency range of interest, and an ultrafast topological transition to the SHM state will occur. This might in turn give rise to ultrafast appearance of the very large density-of-states signatures and enable optical gating and modulation for many applications, such as imaging, lensing, near-field energy transfer, negative refraction, and all-optical switching.

In this paper, we investigate the optical properties of finite-thickness semiconductor superlattices under transient excitation. As a first step, we consider uniform distribution of pump light within the semiconductor superlattice to understand the attainable optical properties; more refined models will be analyzed in the future. However, to a first approximation, a uniform excitation can be obtained by choosing the pump wavelength such that the absorption per quantum well is small. We observe that the combination of filling factors, superlattice thickness, and materials allows for the presence of impedance matching conditions, Fabry-Perot resonances, and epsilon-near-zero conditions that give rise to reflectionless features under plane wave illumination. Remarkably, these conditions can be exploited to obtain ultrafast, wide-angle, all-optical switches and directional perfect absorbers. We also show that we can obtain negative refraction on ultrafast timescales.

2. Modeling of finite-thickness superlattices under plane wave incidence

Consider the experimentally achievable [22], finite-thickness semiconductor superlattice made of alternating InAs and undoped GaSb layers (Fig. 1). For sufficiently thin layers, compared to the operating wavelength, such a structure can be homogenized as a uniaxial material with a uniaxial permittivity tensor of the kind ${\underset{¯}{\epsilon }}_{\text{HM}}={\epsilon }_t\widehat{t}+{\epsilon }_l\widehat{l}$, where ${\epsilon }_t=\frac{{\epsilon }_m{d}_m+{\epsilon }_d{d}_d}{d_m+d_d}$ is the transverse dielectric constant along the transverse direction $\widehat{t}$ (parallel to the layers) and ${\epsilon }_l={\left(\frac{{\epsilon }_m^{-1}{d}_m+{\epsilon }_d^{-1}{d}_d}{d_m+d_d}\right)}^{-1}$ is the longitudinal dielectric constant along the longitudinal direction $\widehat{l}$ (perpendicular to the layers). In these expressions, ${\epsilon }_m$ and $d_m$ are the permittivity and thickness of the InAs layers, and ${\epsilon }_d$ and $d_d$ are the permittivity and thickness of the GaSb layers. Although this slab could be simply modeled considering the superlattice implementation, the homogeneous interpretation provides a better insight on the physical origin of the observed optical properties and may thus lead to device design guidelines.

 

Fig. 1 Ultrafast transition to the hyperbolic state. (a) A superlattice of alternating lightly doped (10¹⁸ cm⁻³) InAs and undoped GaSb layers with permittivities ${\epsilon }_m$ and ${\epsilon }_d$ and thicknesses $d_m$ and $d_d$, respectively. The upper and bottom spaces have permittivities ${\epsilon }_{\text{up}}$ and ${\epsilon }_{\text{bot}}$, respectively. (b) The effective medium permittivities ${\epsilon }_t$ and ${\epsilon }_l$ of the superlattice in (a). (c) By photopumping the structure in (a), carriers are generated within InAs, which now exhibits a doping concentration of 2x10¹⁹ cm⁻³. (d) The effective medium permittivities ${\epsilon }_t$ and ${\epsilon }_l$ of the transient SHM in (c). The colored boxes indicate a type II hyperbolic dispersion (T-II) region, an elliptic dispersion (E) region, and a type I hyperbolic dispersion (T-I) region. In (a) and (c), the black dashed line indicates the reference plane for reflectance computations; and the upper ($Z_{\text{up}}$) and bottom ($Z_{\text{down}}$) impedances are explicitly indicated.

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For the case of lightly doped InAs wells (Fig. 1(a)), $\mathrm{Re}\left({\epsilon }_m\right)$ is positive in the frequency range of interest and the homogenized superlattice acts as an anisotropic dielectric (Fig. 1(b)). If intense optical excitation with above bandgap photons creates a sufficiently large population of carriers (~2x10¹⁹ cm⁻³) in the InAs quantum wells (Fig. 1(c)), $\mathrm{Re}\left({\epsilon }_m\right)$ becomes negative and the homogenized superlattice acts as a hyperbolic material (Fig. 1(d)). After initial pumping, the superlattice will remain in the hyperbolic state for a brief period (~1 ns) before relaxation processes deplete the photocarriers and the system returns to the anisotropic dielectric state. This represents the basic working principle that might enable interesting optical properties on ultrafast timescales.

Let us now explicitly analyze the structures shown in Fig. 1(a, c) via a transmission line formalism [23, 24] in which each layer can be modeled using the ABCD matrix that relates the input electric and magnetic fields to the output electric and magnetic fields as

The total ABCD parameters $A_t$, $B_t$, $C_t$, and $D_t$ for the finite-thickness superlattice are given by the matrix multiplication

The reflection coefficient is evaluated as

We note that the reflectance of the homogenized structures can also be obtained using the transmission line formalism, however in this case $A_t=D_t=\cos \left(k_{l}t\right)$, $B_t=-i{Z}_{\text{homog}}\sin \left(k_{l}t\right)$, $C_t=-i\sin \left(k_{l}t\right)/Z_{\text{homog}}$, with $\frac{k_t^2}{{\epsilon }_l}+\frac{k_l^2}{{\epsilon }_t}=k_0^2$ and $Z_{\text{homog}}$ is the impedance of the homogenized superlattice, being $Z_{\text{homog}}=k_l/\left(\omega {\epsilon }_t{\epsilon }_0\right)$ for TM polarization, and $k_t^2+k_l^2=k_0^2{\epsilon }_t$ and $Z_{\text{homog}}=\omega {\mu }_0/k_l$ for TE polarization.

3. Reflectionless features on ultrafast timescales

We show in this section that the reflection and transmission properties obtained using the explicit superlattice and the homogenized slab are in excellent agreement. We analyze both the TE and TM properties of the superlattice. Although in the literature the TE polarization is seldom considered for SHMs as it does not couple to the hyperbolic branch, we find it is important for the purpose of gathering a complete understanding of the optical properties of transiently excited superlattices. We consider in this section a superlattice with total thickness t = 600 nm (10 pairs). The unbounded media are considered to have a relative permittivity equal to unity (i.e. ${\epsilon }_{\text{up}}={\epsilon }_{\text{bot}}=1$). [Note that results similar to those shown in this paper can be achieved by using a bottom space with ${\epsilon }_{\text{bot}}={1.4}^2$, e.g. a barium fluorite substrate, making the results here discussed experimentally achievable.]

Before presenting the transmission and reflection properties, we describe two important impedances that dictate the reflection behavior of the SHM stack. The first is the “material impedance”, defined by the properties of the homogenized superlattice as $Z_{\text{mat}}=k_l/\left(\omega {\epsilon }_t{\epsilon }_0\right)$ for TM polarization and $Z_{\text{mat}}=\omega {\mu }_0/k_l$ for TE polarization. Note that $Z_{\text{mat}}$ is a bulk material property of the effective medium and does not depend on the overall thickness. The slab will appear reflectionless when $Z_{\text{mat}}$ is matched to both the upper and bottom impedances. The second impedance is the “slab impedance”, which depends on both superlattice and bottom space impedance as $Z_{\text{slab}}=\frac{A_t{Z}_{\text{bot}}+B_t}{C_t{Z}_{\text{bot}}+D_t}$. This impedance depends on all of the parameters of the superlattice, including the layer thickness and number of layers. All the parameters were defined in Sec. 2 for both TE and TM polarizations.

The reflectance and transmittance under TE plane wave incidence for the structures of Fig. 1(a) and Fig. 1(c), evaluated using the effective medium approximation, are shown in Fig. 2. For the unpumped case, one can notice that around 40 THz, and for wide range of incidence angles, the reflectance (transmittance) is fairly high (low), meaning that most of the incoming signal will be reflected. In contrast, for the photopumped case, a reflectionless, mainly transmissive feature appears on ultrafast timescales around 40 THz. The origin of this latter feature is due to a material impedance match with the effective transverse permittivity [25], i.e. $Z_{\text{mat}}=Z_{\text{up}}$, and can be tuned by changing the materials and filling fractions of the superlattice. Interestingly, the feature at 40 THz also corresponds to the m = 0 Fabry-Perot mode of the SHM stack. This correspondence arises only because the slab is very thin.

 

Fig. 2 The TE reflectance and transmittance versus frequency and incidence angle for the (a, c) unpumped and (b, d) photopumped structure of Fig. 1, obtained using the effective medium description. The white dashed lines in (a, b) indicate the dispersion of the m = 0, 1 Fabry-Perot modes of the SHM stack.

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We also notice for both unpumped and photopumped cases an angle-independent reflectionless feature around 70 and 80 THz, respectively. This is due to an m = 1 Fabry-Perot resonance of the SHM stack that can be tuned with thickness. To show the correspondences with Fabry-Perot resonances, we compute the angular dependence of the m = 0, 1 resonances using $2N\left(d_m+d_d\right)\mathrm{Re}\left(k_l\right)+2\arg \left(\Gamma \right)-2m\pi =0$. The results are shown as dashed lines in Fig. 2. We observe that the dispersions of these modes are rather independent of the incidence angle. While for the feature at 40 THz this is because $Z_{\text{mat}}=Z_{\text{up}}$, for the m = 1 resonance this can be justified by rewriting the dispersion relation for TE waves below Eq. (3) as

Thus we see that if the effective transverse permittivity ${\epsilon }_t$ is large in comparison to the term ${\epsilon }_{\text{up}}{\sin }^2\theta$, the $\theta$ term dependence in Eq. (4) can be neglected, leading to the angle-independent properties of the features shown in Fig. 2. This is indeed the case, as ${\epsilon }_{\text{up}}=1$ and ${\epsilon }_t\cong 13.5$ at ~70 THz for the unpumped case and ${\epsilon }_t\cong 10.5$ at ~80 THz for the photopumped case.

The reflectance and transmittance under TM plane wave incidence for the structures of Fig. 1(a) and 1(c), evaluated using the effective medium approximation, are shown in Fig. 3. Features similar to those observed in Fig. 2 are observed for both unpumped and photopumped cases. The reflectionless feature at 40 THz for the photopumped case is due to a slab impedance match, i.e. $Z_{\text{slab}}=Z_{\text{up}}$ [25], and can be tuned by changing the materials and filling fractions of the superlattice. Interestingly, the feature at 40 THz also corresponds to the m = 0 Fabry-Perot mode of the SHM stack, which ends at the material impedance match with the effective transverse permittivity for $\theta =0$, i.e. $Z_{\text{mat}}=Z_{\text{up}}$. The features around 70 and 80 THz for unpumped and photopumped cases are due to the m = 1 Fabry-Perot resonance of the SHM stack. We thus overlap on the reflectance maps of this figure the dispersion of the Fabry-Perot resonances retrieved with m = 0, 1. We observe that the dispersion of the m = 1 mode is rather independent of angle of incidence. This can be justified in a manner similar to the TE case.

 

Fig. 3 The TM reflectance and transmittance versus frequency and incidence angle for the (a, c) unpumped and (b, d) photopumped structure of Fig. 1, obtained using the effective medium description. The white dashed lines in (a, b) indicate the dispersion of the m = 0, 1 Fabry-Perot modes of the SHM stack.

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To demonstrate the excellent agreement between the superlattice and effective medium models, we show in Fig. 4 the reflectance of the photopumped state for both TE and TM plane wave incidence calculated using both models. To avoid unnecessary duplication, we will utilize the effective medium model for the remainder of this paper.

 

Fig. 4 Reflectance versus frequency and incidence angle for the (photopumped) structure of Fig. 1(c): (a, b) TE incidence and (c, d) TM incidence. These results were computed using (a, c) the homogenized and (b, d) the superlattice models.

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It is interesting to note that the transient optical properties presented above can be used to demonstrate an all-optical switch. We show in Fig. 5 the TE and TM reflectances for the “OFF” (photopumped) and “ON” (unpumped) states of the reflection mode switch for an incidence angle of 20 degrees. These curves correspond to horizontal cuts of the maps in Fig. 2 and Fig. 3. It is remarkable that near 40 THz, and for both polarizations, the ON state exhibits a reflectance of about 80% while the OFF state is almost 0%. We can define a modulation parameter given by $M=R_{\text{ON}}-R_{\text{OFF}}$, $0<M<1$, indicated by an arrow in Fig. 5 around 40 THz, to describe the quality of the switching device. For TE polarization, $M\approx 0.75$; for TM polarization, $M\approx 0.7$, indicating a good switching performance for these semiconductor superlattices. However, we note that such performance could also be obtained through appropriate photopumping of a single semiconductor layer as has been demonstrated for bulk semiconductor switches [26–29].

 

Fig. 5 (a) TE and (b) TM reflectances of the OFF (photopumped) and ON (unpumped) states of the structure of Fig. 1 working as a reflection mode switch. These results were computed using the effective medium model.

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4. Directional perfect absorption on ultrafast timescales

In Sec. 3, we have observed the presence of a zero transmittance feature for TM incidence in the photopumped state. This feature corresponds to an epsilon-near-zero condition and may lead to directional perfect absorption [30, 31]. We thus show in Fig. 6 the absorption under TM plane wave incidence for the structures of Fig. 1(a) and Fig. 1(c), evaluated using the effective medium approximation. It is peculiar that near 70 THz the nearly zero absorption of the unpumped state becomes almost unit absorption for the photopumped state. This ultrafast directional perfect absorption is due to the occurrence of an epsilon-near-zero transition of ${\epsilon }_l$ in the photopumped state.

 

Fig. 6 Absorption versus frequency and incidence angle for the (a) unpumped and (b) photopumped structure of Fig. 1 under TM incidence. These results were computed using the effective medium model.

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5. Negative refraction on ultrafast timescales

We analyze in this section the possibility of obtaining negative refraction on ultrafast timescales. We consider two cases: (i) the interface between a medium with unit permittivity and an infinitely extended superlattice; and (ii) a superlattice with total thickness t = 6 µm (100 pairs). The unbounded media are considered to have a relative permittivity equal to unity in case (ii), i.e. ${\epsilon }_{\text{up}}={\epsilon }_{\text{bot}}=1$. The dispersion diagrams of the superlattice for unpumped and photopumped conditions at 68 THz are reported in Fig. 7(a) and Fig. 7(b), respectively. One can notice a transition from elliptic to hyperbolic dispersion upon carrier excitation. In the hyperbolic state, the group velocity will now point inward, allowing for negative refraction, as explicitly indicated in Fig. 7(b).

 

Fig. 7 (a, b) Dispersion diagram at 68 THz for the (a) unpumped and (b) photopumped superlattices, computed using Bloch theory. Notice the transition between an ellipse and a hyperbola. (c, d) Real part of the phasor of the transverse magnetic field at 68 THz for an interface between air and (c) unpumped and (d) photopumped superlattices. Notice the transition between positive and negative refraction. (e, f) As in (c, d), but for a superlattice slab with finite thickness. Notice the transition between positive and negative refraction within the slab. In (c-f), the dashed black lines depict the boundaries of the superlattice.

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To show such a feature, we use full-wave simulation [32] to analyze the refraction of a Gaussian beam in case (i). The unpumped condition is shown in Fig. 7(c), where it is apparent that phase and group velocities are along the same direction, a signature of a standard positive refraction. On the contrary, the photopumped condition of Fig. 7(d) nicely shows that phase and group velocities are in opposite directions, a signature of a negative refraction. Thus, using this approach, negative refraction can now be accessed on ultrafast timescales. Similar results can be achieved also for the situation of case (ii) as shown in Fig. 7(e) and Fig. 7(f) for unpumped and photopumped conditions. Notice the transition between positive and negative refraction within the slab.

6. Conclusion

In conclusion, we have analyzed the optical properties of transiently-excited semiconductor superlattices. Under ultrafast optical excitation of photocarriers, we can optically induce electrons to populate the quantum wells [20, 21] of the superlattice, thereby transforming undoped semiconductor superlattices into SHMs and allowing the rise of interesting properties on ultrafast timescales. These intriguing results suggest that transient excitation of SHMs might enable the ultrafast appearance of the very large density-of-states that are characteristic of hyperbolic metamaterials. This will provide a route to transiently enable phenomena relying on the hyperbolic isofrequency surfaces such as imaging, lensing, near-field energy transfer, and negative refraction. Finally, we stress that experimental verification of such a photopumping scheme will be extremely challenging, but should be achievable.