1. Introduction

Surface plasmon polaritons (SPPs) guided by metal surfaces are widely used for optical spectroscopy. In the visible and near-infrared, SPPs of metal surfaces are well localized near the surface and extend only a fraction of wavelength from it (roughly 640 nm for Au at the wavelength 800 nm). This localization provides high sensitivity of the SPP propagation characteristics to the properties of very thin layers deposited on the metal surface.

SPPs guided by metals become inappropriate for terahertz spectroscopy, because the SPPs lose their localization (e.g., a 1.2 THz SPP extends from Au surface for 1.8 cm [1]), and thus their sensitivity to surface deposits deteriorates. A possible way to confine terahertz SPPs near the surface is to use SPPs on doped semiconductors that have similar electromagnetic properties in the terahertz range to metals in the optical [1]. Moreover, using semiconductors nonlinearity, terahertz SPPs can be directly excited on the surface of a semiconductor with femtosecond optical spots moving along the surface [2, 3, 4]. This allows one to evade coupling terahertz radiation from an external source to SPPs. A drawback of SPPs on semiconductor surfaces is a modest (< 1 cm) propagation length of SPPs related to Ohmic losses in the semiconductor.

Well-confined terahertz surface waves with small propagation losses can be guided by periodically structured – with arrays of holes or grooves – metal surfaces [5]. In the terahertz range, metals resemble perfect conductors. The dispersion properties of the surface waves on the structured surface of a perfect conductor mimic surface plasmon polaritons on the flat surface of a plasma-like medium and can be engineered by tuning the geometrical parameters of the surface corrugation [6, 7]. However, due to weak nonlinearity of metals the above mentioned technique of direct optical excitation of terahertz SPPs becomes inefficient for the surface waves on structured metal surfaces (dubbed “spoof” or “designed” SPPs).

In this paper, we propose a scheme that combines the advantages of direct optical excitation of terahertz SPPs on semiconductors with remarkable waveguiding properties of structured metal surfaces. We propose to deposit a strip of an electro-optic material, for example, a semiconductor, on a structured (with grooves or holes) metal surface [Fig. 1(a)]. Illuminating the strip with an obliquely incident femtosecond laser pulse, focused by a cylindrical lens to a line [Fig. 1(a)], creates an optical spot moving along the strip with superluminal velocity. A nonlinear polarization induced in the spot acts as a superluminal dipole that emits surface (and free-space) terahertz waves via the Cherenkov radiation mechanism.

 

Fig. 1. (a) Excitation scheme. The inset shows the geometry of the surface used in calculations. (b) The frequency ω_s of a partial SPP (blue) and stationary frequency ω ₀ (red) as functions of φ for a/d = 0.5, h = 15 μm, and two values of α(equal to φ at ω_s = ω ₀ = 0).

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2. Model and approach

In further analysis we consider a perfect-conductor surface corrugated with a one-dimensional array of rectangular grooves of width a, depth h, and lattice constant d [Fig. 1(a), inset]. A strip of a semiconductor is deposited on the surface perpendicularly to the grooves [Fig. 1(a)]. We assume that the strip is thin and narrow as compared to the terahertz wavelength and, therefore, the effect of the strip on the generated SPPs can be neglected. We consider the case of above-band-gap excitation when the penetration depth of the laser beam in the semiconductor (~ 1 μm for GaAs excited with Ti: sapphire laser) is smaller than the strip thickness and, thus, there is no reflection of the light from the lower boundary of the strip. We do not account for transient effects at the beginning and end of the strip focusing on the stationary regime of terahertz emission in the infinitely long structure.

Under these approximations, the nonlinear polarization induced in the strip via optical rectification becomes

with δ(y) the usual delta function. The function G(x) describes the transverse profile of the optical beam; the function F(ξ) is the time-dependent envelope of optical intensity, ξ = t − z/V, and V = c/sin α. To specify our final formulas, we will use Gaussian functions F(ξ) = exp(−ξ ²/x ²) and G(x) = exp(− x ²/ℓ ² _⊥), where τ is the pulse duration [the standard full-width at half maximum (FWHM) is ${\tau }_{\mathrm{FWHM}}=2\sqrt{\ln 2}\tau \approx 1.7\tau ]$ and ℓ _⊥ is the laser transverse size $\left({\ell }_{\perp FWHM}=2\sqrt{\ln 2}{\ell }_{\perp }\right)$. The orientation of the amplitude vector p is determined by the polarization of the optical beam and orientation of the cry stallographic axes of the strip. We assume the most efficient (according to our general analysis) configuration: p_x = p_z = 0 and p_y ≠ 0. For p-polarization of the optical beam, which provides higher transmission of the optical electric field to the semiconductor (as compared to s-polarization), the optimal orientation of a (100)-cut zinc-blende crystal (for example, GaAs) maximizing p_y is with [010] and [001] axes in the x,z plane directed at 45° to the z direction. For this orientation, p_y ≈ d ₁₄ ℓ_y E ² _t, where d ₁₄ is the nonlinear coefficient, ℓ_y is the penetration depth of the optical beam in the semiconductor, and E_t is the amplitude of the transmitted optical field in the strip (due to a high refractive index n of the semiconductor, E_t is almost parallel to the z axis even for grazing incidence [2]). For estimations, we will use the parameters of GaAs [8]: d ₁₄ = 66 pm/V, n = 3.7 (at 800 nm), and ℓ_y = 1 μm. We relate E_t to the amplitude E ₀ of the incident optical field via Fresnel formulas.

To find the terahertz fields generated by the moving nonlinear polarization, we use the approach developed in Ref. [2]. We apply Fourier transforms with respect to ξ and x to Maxwell’s equations (ω and g are the Fourier variables which correspond to ξ and x, respectively;^~will denote quantities in the Fourier domain) and solve the resultant equations in the regions y > 0 and y < 0. For y > 0, the solution is a superposition of p-polarized waves with

and s-polarized waves with

where κ = [g ² − ω ² c ⁻²(1 − β ⁻²)]^1/2 and β = V/c = sin^–1 α [2]. Other field components in the waves can be found by using Maxwell’s equations. For y < 0, assuming the period of the surface corrugation d to be small as compared to the terahertz wavelength (ωd/c ≪ 1), we use an approximate solution in the form of the lowest-order mode of the groove waveguide [7]

where k_y = (ω ² c ⁻² − g ²)^1/2. We match the solutions (2), (3), and (4) by the boundary conditions

that arise after integrating the equations across the surface y = 0 [2]. In Eq. (5), G̃(g) = (ℓ_⊥/2√π)exp(−g ²ℓ² _⊥/4), F̃(ω) = (τ/2√π)exp(−ω ² τ ²/4), and the square brackets denote the change of the enclosed quantity at the surface y = 0, for example, [B̃_x] = B̃_x∣_y=0⁺ − B̃_x∣_y=0⁻. When substituting Eq. (4) to Eq. (5), the field Ẽ_z at y = 0⁻ should be averaged over the period d, i.e., taken with a weight of a/d [7].

As a result, we arrive at the expressions for the wave amplitudes

where Λ = κ − k_y(a/d) tan k_yh. With solution (6) and (7) in the Fourier domain at hand, we can transform it to the ξ, x domain by taking inverse transforms such as

3. Properties of the excited SPPs

The fields of SPPs are defined by the residue contributions to the internal integral in Eq. (8) from the poles ω = ±ω_s given by the equation Λ (ω, g) = 0, i.e., κ − k_y(a/d)tan k_yh = 0. This equation is the dispersion equation of a partial (with a given g) SPP (for g = 0, the equation is well-known [7]). Using the relation between the components g and (ω/V of the SPP’s in-plane wave vector and the propagation angle φ [Fig. 1(a)],i.e., g = (ω/V) cot φ φ, the frequency ω_s of a partial SPP can be written as a function of φ:

Figure 1(b) shows the dependence ω_s(φ) for a/d = 0.5, h =15 μm, and two values of β − 1.15 (α = 60°) and 1.0035 (α = 85°). According to Eq. (9), the propagation angle φ varies in the interval φ _min < φ < φ _max, where tan φ_min = β ⁻¹ = sin α and sin φ _max = β ⁻¹, i.e., φ _max = α. For example, 41° < φ < 60° for α = 60° and 45° < φ < 85° for α = 85°. Correspondingly, the frequency ω_s increases from 0 at φ _max to ∞ at φ _min [Fig. 1(b)].

Summing the residue contributions from the poles ω = ±ω_s to the internal integral (8) and changing the integration variable g → ω by use of equation Λ(ω,g) = 0 in the external integral (8), we find the electric field of the excited SPPs (at y > 0)

In Eq. (10), Λ′_g denotes the derivative with respect to g, the function g(ω) should be substituted into Eq. (10) from Λ(ω,g) = 0 (only positive g should be taken). Equation (10) gives the expansion of the electric field E^s_y(ξ,x,y) into plane surface waves that propagate from the strip to x → ±∞ (outgoing waves) and toward the strip (incoming waves). At large ξ, only the outgoing waves survive; the incoming waves interfere destructively. As a result, a farfield pattern of the outgoing radiation will be formed at large ξ. An asymptotic evaluation of the integral (10) for large ξ using the stationary phase method gives

where g″ denotes the second derivative with respect to ω and zero subscripts imply that all variables that depend on ω are taken at the frequency ω ₀ for which

In Eq. (12), φ is a half-apex angle of the cone with its apex on the moving laser pulse [Fig. 1(a)]; evidently, this angle coincides with the propagation angle of a partial plane wave.

Figure 1(b) shows the stationary frequency ω ₀ as a function of φ. Solutions of Eq. (12) exist in the interval φ ⁽⁰⁾ _min < φ < φ _max, where φ ⁽⁰⁾ _min is defined by the equation g″ = 0. For example, φ _min ≈ 39° for a/d = 0.5, h = 15 /μm, and a = 60°, i.e., φ ⁽⁰⁾ _min < φ _min. For a given φ from the interval φ _min < φ < φ _max, there is a single stationary frequency ω ₀ [Fig. 1(b)]. In the interval φ _min < φ < φ _min, the dependence ω ₀(φ) is double-valued [Fig. 1(b)], i.e., Eq. (12) has two roots. However, the contribution from the high-frequency root to the field (11) is negligible - it is suppressed by the Gaussian factor F̃(ω ₀). Additionally, the interval φ ⁽⁰⁾ _min < φ < φ _min, where the upper branch of ω ₀(φ) exists, occupies only a minor part of the whole interval φ ⁽⁰⁾ _min < φ < φ _max. The lower branch of ω ₀(φ) increases with decreasing φ from φ _max to φ ⁽⁰⁾ _min. Thus, the frequency of SPPs arriving at a fixed observation point far from the strip increases with time.

 

Fig. 2. (a) Snapshot of the electric field E_y(ξ,x) at y = 0⁺ Inset: Oscillogram E_y(ξ) at x = 1 mm and y = 0⁺. (b) Spectral density of energy w_ω(ω) (black) and the contributions to w_ω(ω) from the p- (red) and s-polarized (blue) fields and from the groove mode (green). In (a) and (b) a/d = 0.5 and h= 15 μm. (c) The radiated energy W ^s _± as a function of h and α for a/d = 0.5.

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Figure 2(a) shows the snapshot of the electric field E_y(ξ,x) at y = 0⁺ calculated by taking integral (8) numerically. In the calculation, we assumed that a Ti: sapphire laser pulse with τ _FWHM = 150 fs, ℓ_⊥FWHM = 30 μm, and the intensity I ₀ − cE ² ₀/(8π) = 50 GW/cm² is incident on a GaAs strip at α = 60°. In Fig. 2(a), the radiation pattern is symmetric with respect to the z axis. The fields of radiated SPPs are confined to a sector between φ ⁽⁰⁾ _min ≈ 39 and φ _max = 60°, in accord with the prediction of the stationary phase method [Fig. 1(b)]. At the front boundary of the sector φ _max = 60° the wavefronts of SPPs are parallel to the boundary, at the rear boundary φ _min⁽⁰⁾ ≈ 39° the wavefronts are tilted at an angle φ ≈ 50°, i.e., at ≈ 10° with respect to the boundary. This agrees well with Fig. 1(b). Indeed, in Fig. 1(b) the curves ω ₀(φ) and ω_s(φ) coincide at φ ≈ φ _max, therefore, the terahertz field near φ _max = 60° is formed by the partial SPPs with the propagation angle φ ≈ 60°. The terahertz field near φ ⁽⁰⁾ _min ≈ 39° has a frequency ω ₀ ≈ 4 – 5 THz [Fig. 1(b)]. The partial SPPs of such frequency ω_s propagate at φ ≈ 50° [Fig. 1(b)] and form the field near φ ⁽⁰⁾ _min ≈ 39°. The inset of Fig. 2(a) shows the oscillogram E_y(ξ) at x = 1 mm and y = 0⁺; the frequency of the oscillations increases noticeably with time, in accord with Fig. 1(b). The field amplitude is ~ 2 times as high as for SPPs excited on a flat surface of GaAs (for the same x and I ₀) [2].

To find the energy radiated into the SPPs per unit length of the strip, we integrate the ±x-component of the Poynting vector S _±x = ±c(4π)⁻¹(E_yB_z-E_zB_y) over intervals -h < y < +∞ and -∞ < ξ +∞ (only the fields of SPPs are included). Thus, we obtain the energy radiated in ±x directions,

where the spectral density of energy is

In Eq. (14), first term in square brackets comprises the contributions from the p-polarized (term ∝ ω ⁴) and s-polarized [term ∝ (gκcV)²] fields, whereas second term in the brackets corresponds to the groove mode. The contribution from the p-polarized fields dominates at all frequencies [Fig. 2(b)] and provides 84% of the SPPs energy W ^s _± for the parameters of Fig. 2(b), the contributions from the s-polarized fields and the groove mode equal 6% and 10%, respectively. The portion of energy carried by the groove mode increases with h (for example, up to 37% at h = 30 μm). In Fig. 2(b), the spectrum is centered at ≈ 3 THz and spreads from 1 to 6 THz. The position and height of the spectrum’s maximum change only slightly in the wide range of the parameters a, h, and a/d. For example, the maximum frequency grows from ≈ 2.5 THz to ≈ 3.5 THz with decreasing h from 30 to 5 μm at a/d = 0.5 or with decreasing a/d from 1 to 0.1 at h = 15 μm.

Figure 2(c) shows the energy W ^s _± as a function of h and a for a/d = 0.5. The energy reaches maximum at a ≈ 60° and h ≈ 15 μm. The position and height of the maximum depend only slightly on a/d. For example, the maximum shifts from h ≈ 20 μm at a/d = 0.1 to h ≈ 10 μm at a/d = 1, the maximum angle a ≈ 60° practically does not change. However, the parameter a/d affects significantly the localization of the SPPs near the guiding surface – the localization deteriorates with decreasing a/d. For example, the localization of a 3 THz SPP deteriorates from 43 μm to 160 μm with decreasing a/d from 0.5 to 0.1 at h = 15 μm and α = 60°. Similarly, decreasing h from 15 to 5 μm deteriorates the localization from 43 μm to 190 μm at a/d = 0.5 and α = 60°. For comparison, the localization of a 3 THz SPP on a flat surface of GaAs with parameters from Ref. [2] is 140 μm. In Fig. 2(c), the maximum energy is more than an order of magnitude as high as for SPPs excited on a flat surface of GaAs [2]. This can be attributed to a higher efficiency of the nonlinear source moving above the waveguiding surface in the present scheme as compared to the subsurface source in Ref. [2].

Since p_y ∝ ℓ_y, the generated terahertz energy should increase with increase in ℓ_y as W ^s _± ∝= ℓ ² _y (until ℓ_y remains small compared to the extent of the SPP in vacuum). However, using below-band-gap excitation [4] would be inefficient due to reflection of the laser light from the strip’s bottom and formation of a standing optical wave in the strip. Nevertheless, if the reflection is suppressed, for example, via putting an absorbing layer at the strip’s bottom, using a wideband-gap material with high optical nonlinearity, such as LiNbO₃, instead of GaAs would allow one to improve significantly the efficiency of the present scheme.

We studied a stationary regime of terahertz emission in the infinitely long structure. If we account for the fact that the laser pulse scans a finite distance, which we denote L, along the semiconductor strip, the generated Cherenkov cone acquires a finite size. The size is defined by the interference between the cone and a transiently excited SPP propagating from the starting point on the strip [9]. For a 3 THz SPP, the length of the Cherenkov cone is estimated as Δz ≈ 0.5L.

4. Conclusion

We have proposed a method for direct optical excitation of terahertz SPPs on a structured metal surface. In practical implementation, the excitation scheme may be asymmetric – with a slab of an electro-optic material placed near the boundary of a structured metal surface. For detection, sampling in another electro-optic slab placed near the opposite boundary of the surface can be used. The proposed technique can be a useful tool to probe the properties of materials deposited on the surface.