1. Introduction

The development of new methods to confine and control light in the terahertz (THz) region (0.1–10 THz) of the spectrum is a major area of current photonic research [1, 2]. The expansion of the wireless communication spectrum, for example, promises multi-Gbit data transfer rates as the carrier wave is pushed into the THz band [3–6]. Integration of several technologies into a functional system typically requires confinement and isolation of light via guided modes, as opposed to free space optics that are susceptible to environmental conditions and requires a large footprint. For THz light, a parallel plate waveguide (PPWG) operating in the TEM mode provides a low loss platform for the integration of photonic components [7], with several passive [8–11] and active [12–15] devices having recently been demonstrated. Periodicity of dielectric and metallic structures embedded within the waveguide modifies the photonic band-structure, modulating the amplitude and phase of transmitted light. Another interesting degree of freedom in such structures is the possibility of breaking time-reversal symmetry through dynamic modulation of the local dielectric function via photoinjection of free charges [2, 16]. In the optical regime, interesting effects such as all-optical stopping [17] and storage of light [18], a photonic Aharonov-Bohm effect [19], an effective magnetic field for photons [20], and a photonic Haas-van Alphen effect [21] have been demonstrated using dynamically modulated photonic structures and resonators. Nonlinear optical effects such as the frequency shifting of light are also possible, where the dynamic modulation induces an effective χ⁽³⁾ interaction [22, 23].

At THz frequencies, wavelengths are large (1 THz = 300 µm) and so spatial modulations of sub-wavelength dimension can be easily achieved through illumination by an optical pulse, down to the diffraction limit. Using femtosecond pulses to modulate the dielectric function, one can photoexcite on a sub-cycle time scale as well. This new possibility allows one to dynamically modulate the photonic band structure in the extreme non-adiabatic limit by sudden introduction of spatial periodicity on a sub-wavelength scale. Furthermore, the possibility of an optically generated metal-dielectric structure through spatial patterning eliminates the need for fabrication of permanent photonic structures, making the device fully-reconfigurable and reversible in nature. Here we demonstrate the dynamic creation of a THz metal-dielectric photonic crystal inside a PPWG on a time scale much shorter than the pulse transit time. The sudden creation of a resonant structure faster than the THz pulse transit time permits the capture of energy within the resonator volume, delaying the arrival of THz light by an amount determined by the group delay. The more closed a cavity, the longer it can store energy however the more difficult it is to couple in light. By switching on the entire structure faster than the transit time, we transiently increase the coupling into the structure compared to a static resonator, as verified by experiments and finite-difference time domain (FDTD) simulations. It is important to note here that no pre-existing structure is present inside the waveguide as is often the case in dynamic photonic experiments [2, 23, 24]. We show the onset of enhanced transmission peaks relative to the background absorption as evidence for the dynamically created resonance and selective temporal delay of THz photons up to 10.8 ps.

2. Experimental

The time-domain THz spectrometer used in this work is based on optical rectification in a LiNbO₃ prism of 800 nm, 40 fs pulses from a Ti:sapphire fs laser amplifier [25–28] using the tilted-pulse front method and electro-optic sampling in an undoped (110) cut GaP crystal of 200 µm thickness optically bonded to a 4 mm inactive GaP (100) cut substrate to avoid reflections that can limit the spectral resolution. The THz pulses are collimated exiting the LN prism and teflon lenses with focal length of 100 mm ensure THz coupling in and out of a tapered aluminium PPWG, schematically drawn in Fig. 1(a). Coupling into the transverse electro-magnetic (TEM) mode of the waveguide provides minimal waveguide losses and dispersionless propagation in the enclosed 147 µm-thick high-resistivity silicon wafer (ρ > 20,000 Ω-cm). The Si-filled region of the PPWG is back-coated with a 200 nm-thick layer of gold and top-coated with several µm-thick indium tin oxide, a transparent conductor. Both coatings provide < 1Ω/sq sheet resistance to confine the THz pulse while the transparency of ITO is sufficient to allow for optical pumping as depicted in Fig. 1(b). The THz peak electric field is lowered using a wire-grid polariser pair preventing inter-valley scattering of the photoinjected carriers inside the waveguide. The pump pulses are centered at 1050 nm, produced by an optical parametric amplifier. The wavelength was chosen to be above the silicon indirect bandgap to homogeneously inject charge carriers through the thickness of the waveguide with penetration depth > 100 µm. A 100 nm-thick gold shadow mask lithographically deposited atop the ITO layer allows spatial definition of the d₁ = 10 µm wide photoconductive regions with sub-THz-wavelength precision. The travel time of the NIR pulse across the Si waveguide restricts the creation time of the resonator to 1.7 ps, comparable to the travel time of the THz pulse between two adjacent photoexcited regions. Furthermore, the creation time is an order of magnitude smaller than the transit time of the pulse through the entire structure (20.5 ps). Thus, a resonator structure can be dynamically and all-optically created inside a silicon waveguide at a moment where it entirely encompasses a THz pulse, as depicted in Fig. 1(b). The turn-off time of the resonator is ultimately limited by the recombination time of the charge carriers; ∼ 100 µs in silicon. However, charge diffusion will contribute appreciably to the cavity deterioration once charges have diffused over a distance comparable to the smallest length scale of the photonic crystal. Assuming a critical diffusion distance equal to 10% of the photoexcited wall thickness (d₁ = 10 µm) the diffusion time is given by t_d = (0.1d₁)²/D ≃ 4.3 ns. Here, the diffusion coefficient is given by D = µ_ehk_BT/e = 2.33 cm²/s where k_B is the Boltzmann constant, T is the temperature in Kelvins, e is the elementary charge and given the ambipolar mobility of electrons and holes in silicon µ_eh ≃ 90 cm²/Vs [29, 30].

 

Fig. 1 (a) Schematic of the experimental setup showing the THz pulse coupled into the Si-filled waveguide via a tapered aluminium PPWG. A window allows pumping of the silicon-filled region using a near infrared (NIR) pump pulse. (b) Schematic representation of the periodic metal-dielectric array (gray rectangles) being photoinjected on a sub-transit timescale. (c) Electric field amplitude (arbitrary units) transmitted through the unexcited (E_ref) and photoexcited sample (E_pump), well after the pump pulse arrives. The a = 60 µm lattice periodicity of the structure introduces a resonance at 0.728 THz, indicated by the arrow. (d) Photonic band structure of the created cavity (red) and of the unpumped waveguide (black) in the vicinity of the first resonance for the Bragg wavevector q. (e) Corresponding imaginary part of the wavevector and (f) electric field transmission t(ω) = |E_pump (ω)|/|E_ref(ω)|.

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The signature of a metal-dielectric photonic crystal is a resonant transmission enhancement, relative to the background absorption, when an integer multiple of half-wavelengths inside the media is approximately equal to the periodicity of the stack, with small corrections due to the finite thickness of the metal film [31]. This occurs at a frequency f_m = mc₀/(2n_Sia), where n_Si = 3.418 is the refractive index of Si at THz frequencies, a is the structure periodicity and m is an integer. Figure 1(c) shows the THz transmission spectrum through both an unexcited silicon slab, with the THz pulse arriving well before the pump (E_ref), and the spectrum when the THz pulse arrives well after the pump, in the quasi-steady state (E_pump). The 60 µm periodic metallo-dielectric structure exhibits a marked transmission enhancement at a frequency of 0.73 THz, in excellent agreement with first order (m = 1) Fabry-Pérot resonator physics, as well as a broadband attenuation due to Drude absorption. We also note that the onset of oscillations in both data sets at frequencies greater than 1 THz is due to multi-modal interference due to partial coupling into the TM₂ geometric mode, caused by slight misalignment into the tapered PPWG [15, 32].

The band diagram of the structure created is calculated using the one-dimensional metal-dielectric stack dispersion relation: [33]

Here, q represents the complex-valued Bloch wavevector, d₁ and d₂ the thickness of the pumped and unpumped regions, respectively and a = d₁ + d₂ is the periodicity of the stack. The modulation is introduced in the model via the complex wavevector of the photoexcited region $k_1=\frac{\omega }{c}\sqrt{n_{\text{Si}}^2+i\sigma \left(\omega \right)/{\epsilon }_0\omega }$ and using n_Si to define the real-valued k-vector of the unpumped region k₂ = n_Siω/c. A scattering time τ_D = 50 fs and an effective mass m^* = 0.26m_e are used to define the complex conductivity of the photoexcited region through the Drude model $\sigma \left(\omega \right)=\frac{n{e}^2{\tau }_D/m^{*}}{1-i\omega {\tau }_D}$, for the photoexcited carrier density n ≃ 18¹⁰ cm⁻³. The calculated band structure in the presence (red) and absence (black) of photoexcitation is presented in Fig. 1(d) in the vicinity of the m = 1 resonance. Without photoexcitation, propagation occurs on the light line representing the dispersionless propagation of the TEM mode in the waveguide. Upon photoexcitation, Bragg scattering occurs at the lattice wavevector π/a, with a zero-energy gap and a reversal of group velocity above and below resonance. On resonance, the imaginary part of the wavevector, governing loss, shows a marked reduction, shown in Fig. 1(e), interpreted as a resonant tunnelling phenomena. Transmission spectra can be calculated easily by neglecting reflections at the beginning and end of the structure, which is a good approximation due to the soft Gaussian onset of photoconductivity, and assuming Beer’s Law for the electric field transmission coefficient t = e^−Im(q)L. A L=1.8 mm thick effective medium equivalent to the 12-pair, 60 µm periodic, metal-dielectric structure is created here. On resonance a transmission peak appears upon photoexcitation with a linewidth of 11 GHz, shown in Fig. 1(f). This represents the fundamental limit to the linewidth, governed by the conductivity of the metallic constituents in the absence of disorder, with an upper Q-factor for the resonator of ν₀/Δν = 66.

3. Results and discussion

Several transmission peaks can be arranged within the bandwidth of the THz pulse simply by increasing the periodicity. Transmission spectra for various pump-probe time-delays (τ) are presented in Fig. 2(a) (open circles) for 12 photoexcited lines of 20 µm width separated by 140 µm unpumped Si. The m = 1, 2 and 3 resonances are evident. The zero time τ = 0 is defined as the delay where the THz pulse just exits the photoexcited region when the pump pulse arrives, identified by the earliest change in THz transmission. Negative pump-probe time delays are defined as a late arrival of the NIR pump, and the THz pulse transit time through the total structure is approximately 21 ps. FDTD simulations taking into account the dispersion of the Drude conducting regions are carried out for a simplified one-dimensional system using a freely available code written by Larsen et al. [34], with the results showing excellent agreement with the data in Fig. 2 (solid lines). The experimentally observed low-frequency roll off is attributed to the portion of the electric field existing outside the waveguide, in the window region, due to finite plate conductivity [35,36]. Accordingly, the same behaviour is also present for a uniform non-periodic photoexcitation of the waveguide as shown by the dashed red curve in Fig. 2(a). Pump energy is used as the sole floating parameter and fixed to the value which best fits the static (τ > 27 ps) photoexcitation transmission. FDTD simulations further reveal sharply peaked features in the reflected electric field amplitude on resonance. This behaviour appears counter-intuitive when considering a lossless dielectric resonator where minima in reflection are usually accompanied by transmission maxima. The observed reflection and transmission maxima occurs here due to the lossy nature of the metallic constituents of the 1D array. Further investigation reveals that when the imaginary part of the complex refractive index (κ) of the photoexcited silicon regions reaches the same order of magnitude as the real part (n), the device will begin to exhibit simultaneously enhanced transmission and reflection on resonance. For a Drude metal, this occurs for frequencies f ≤ 1/(2πτ_D) or ≤ 3 THz for silicon, which encompasses the entire THz pulse bandwidth. At late pump-probe time delays the resonator acts as a broadband absorber for off-resonant frequencies. Static regime results are also well-reproduced by analytical transfer-matrix method calculations in both reflection and transmission (not shown here). The broadband absorption and reflection peak is highest for τ = 20 ps, when the THz pulse is traversing the entire photoexcited structure. A noticeable narrowing of the reflection peak is shown in Fig. 2(b) as one moves from a situation where the photonic structure is created while the THz pulse is inside, to a pre-existing structure that it transmits through in its entirety. Broadening of the reflected peaks is attributed to the decreasing number of reflective interfaces on which the THz pulse is incident at earlier τ, reducing the number of reflected THz electric field cycles thus broadening the reflection peaks.

 

Fig. 2 (a) Experimental data (open circles) and FDTD simulation results (solid lines) of the field transmission amplitude t(ω,τ) = |E_pump (ω,τ)|/|E_ref (ω)| for various pump-probe time delays (τ) for a 12 metal-dielectric pairs. (b) Corresponding simulated reflection amplitude r(ω,τ) = |E_refl (ω,τ)|/|E_inc (ω)|. The dashed red curve presented in (a) shows the transmission for a homogeneous photoexcitation, plotted below 0.5 THz for clarity. Lattice periodicity is set to 160 µm and pump energy kept constant at 5.6 µJ over the 1.8 mm structure length.

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Two dimensional pump-probe spectroscopy performed on the a = 60 µm sample provides insight on the dynamic effect of sub-transit-time resonator creation. The intensity modulation ( $\mathrm{\Delta }\mathrm{T}/{\mathrm{T}}_0=\left(t^2-t_0^2\right)/t_0^2$) of the 1^st order peak taken with respect to the off-resonant THz transmission baseline (T₀) is presented in Fig. 3(a). For a constant pump energy of 5.6 µJ, a single crest arises, steadily peaked at 0.73 THz. Shoulder-like features at frequencies adjacent to the resonance are attributed to inhomogeneities in pump intensity distribution as they are present even in the static regime; excluding the possibility of dynamic sideband generation. The absence of additional features in the spectrum further confirms that the 1.7 ps creation time of the resonator is insufficient to induce any observable nonlinearities at these frequencies. Fitting of the time-averaged static excitation portion (τ > 27 ps) to a Lorentzian lineshape, shown in Fig. 3(b) indicates a 27 GHz linewidth (Δν₀) frequency-centered at 0.728 THz. The modulation is peaked at 56% and exhibits a lifetime (t_c) of 6.0 ± 0.4 ps at large τ and a Q-factor of 27. Presented in Fig. 4 is the electric field transmission, phase change and group delay induced by the resonator for varying pump energies. Electro-optic sampling provides direct access to both amplitude and phase of the transmitted electric field allowing us to extract the group index $n_g\left(\omega \right)=n_{\varphi }\left(\omega \right)+\omega \frac{\partial n_{\varphi }\left(\omega \right)}{\partial \omega }$. The phase change induced by the structure can be observed in Fig. 4(b). The phase index is given by $n_{\varphi }\left(\omega \right)=n_{Si}+\frac{c_0\mathrm{\Delta }\varphi \left(\omega \right)}{\omega L}$ where Δϕ (ω) = ϕ_pump (ω) − ϕ_ref (ω) and d is the length of the resonator. We can thus calculate an effective group delay assuming an effective media of length L with a refractive index n_g and knowing that the group velocity is given by v_g (ω) = c₀/n_g (ω). An average over all τ > 27 ps of the group delay induced for a 1.8 mm wide structure made of 30 periodically repeating 60 µm index modulations induced using 5.4 µJ pump energy reveals a maximum of 10.8 ± 0.7 ps group delay on resonance. The results of a single pump-probe delay are shown in Fig. 4(c).

 

Fig. 3 (a) Intensity modulation (ΔT/T₀) as a function of frequency and pump-probe time delay (τ). (b) Lorentzian fit of the intensity modulation in the static regime (τ > 27 ps). The pump energy is kept constant at 5.6 µJ with a 60 µm periodic structure.

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Fig. 4 (a) Measured electric field transmission t(ω), (b) measured phase change Δϕ (ω) = ϕ_pump (ω)/ϕ_ref (ω) and (c) extracted group delay for a 60 µm periodic sample for various pump energies. The data is recorded in the static regime (τ > 30 ps).

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Lorentzian fit parameters extracted from single pump-probe time delay steps indicate no evolution in the transmitted resonance linewidth, or lifetime, within error margins, once the resonance becomes appreciable (τ > 5 ps). Peak intensity modulation amplitude increases with time delay over the 20.5 ps transit time (Fig. 5(a)) due to the increased number of photoinjected interfaces in the THz path. Accordingly, the intensity transmission baseline, presented in Fig. 5(c), decreases over the same timescale to a minimum of 20 %. The performance of this device is in this case limited by the available pump energy, by the substantial reflection losses undergone by the NIR pump at the shadow mask and by the low transparency of the thick ITO layer.

 

Fig. 5 Intensity modulation as a function of (a) pump-probe time delay (τ) and (b) pump energy with a sigmoidal fit curve shown as a red dashed line. Intensity baseline as a function of (c) pump-probe time delay (τ) and (d) pump energy. The data is presented for the 60 µm periodic sample with a baseline defined at 0.6 THz. (a) and (c) are measured at 5.6 µJ pump energy; (b) and (d) are measured in the static case (τ > 30 ps).

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Adjustment of the pump power provides additional tunability over the intensity modulation and baseline transmission of this optically-induced resonator. Peak modulation intensity data presented in Fig. 5(b) demonstrates a superlinear increase of the peak intensity modulation ratio, ΔT/T₀, with pump energy. A sigmoidal fit to the data is shown as a red dashed line, describing the data reasonably well while respecting the upper and lower bounds of 1 and 0, respectively. This fit indicates that the modulation would saturate above 90% at a pump energy of 8 µJ and has a 10% threshold of 2.5 µJ. The associated intensity transmission baseline, shown in Fig. 5(d), indicates a linear decrease with increasing pump energy. This result is well described by the linear increase of the conductivity with pump fluence predicted by the Drude model. No appreciable change in center frequency or peak linewidth is observed with varied pump intensities, again differentiating this device, where the metallic regions are thinner than the THz skin depth, from a lossless dielectric photonic crystal for which the lifetime is characterized by the interface reflectance.

4. Conclusion

In conclusion, we have demonstrated an all-optically injected and dynamically created resonator inside a PPWG at THz frequencies in a homogeneous semiconductor slab. The turn-on time of the cavity, 1.7 ps in this case, is largely limited by the transit time of the NIR pulse through the thickness of the waveguide which can be reduced by using a thinner wafer. Dynamic creation of resonator structures encompassing a THz pulse on timescales faster than the transit time (20.5 ps) demonstrates potential for introducing frequencies otherwise not allowed inside a structure. The 60 µm periodicity sample exhibits a resonant lifetime of 6.0 ± 0.4 ps at 0.728 THz corresponding to just above 8 electric field cycles. The measured Q-factor of 27 can ultimately be improved by a factor 2.45 according to analytical predictions by increasing the array size and general homogeneity of the pump excitation and by reducing the waveguide plate losses, dominated here by the ITO. Phase analysis additionally reveals a group delay of 10.8 ps in the static regime. These resonant effects can be tuned to any arbitrary frequency via the structure periodicity, and incorporation of a spatial light modulator should enable video rate switching between structures.