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Jun 15, 2015 - ANTONINO CALA' LESINA,1,2,* LORA RAMUNNO,1,2. AND PIERRE BERINI. 1,2,3 ... has a large second-order nonlinear susceptibility. An en-.
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Letter

Vol. 40, No. 12 / June 15 2015 / Optics Letters

Dual-polarization plasmonic metasurface for nonlinear optics ANTONINO CALA’ LESINA,1,2,* LORA RAMUNNO,1,2

AND

PIERRE BERINI1,2,3

1

Department of Physics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada Centre for Research in Photonics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada 3 School of Electrical Engineering and Computer Science, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada *Corresponding author: [email protected] 2

Received 4 May 2015; revised 23 May 2015; accepted 24 May 2015; posted 26 May 2015 (Doc. ID 238378); published 12 June 2015

A plasmonic metasurface for the enhancement of nonlinear optical effects is proposed. The metasurface can simultaneously enhance perpendicularly polarized electric fields in the same volume. We illustrate application of the metasurface to the production of Terahertz radiation via the parametric process of difference frequency generation in 43m non-centro symmetric materials, e.g., GaAs, which has a large second-order nonlinear susceptibility. An enhancement over bulk of almost two orders of magnitude near the surface supports the use of the proposed structure for thin-film, surface-based, or chip-based nonlinear optical applications for several crystal classes. © 2015 Optical Society of America OCIS codes: (250.5403) Plasmonics; (190.4223) Nonlinear wave mixing;

(190.4400)

Nonlinear

optics,

materials;

(190.4410)

Nonlinear optics, parametric processes; (240.4350) Nonlinear optics at surfaces; (310.6845) Thin film devices and applications.

width w, separated at their closest point by a gap of size g oriented at θ  45°. The metal strips of thickness t are embedded into the surface of a semi-infinite nonlinear optical substrate. As for a dipole antenna, we expect a large enhancement in the gap and tunability based on changing geometrical parameters. We consider a square unit cell for simplicity, but this is not a strict limitation. To illustrate an application of the metasurface, we consider THz generation via DFG. Obtaining THz from DFG is a wellestablished concept [13], and enhancing THz generation by plasmonics is of recent interest [14–16]. We consider a second-order nonlinearity, but higher order nonlinearities could also be used. The crystal axes are aligned with the Cartesian coordinate system of Fig. 1. We assume the system to be excited by two CW laser beams with angular frequencies ω1 and ω2 , propagating along y and impinging on the structure from air. We assume the incident beams are undepleted by the nonlinear interaction. We also assume for the nonlinear material intrinsic permutation symmetry, full permutation symmetry

http://dx.doi.org/10.1364/OL.40.002874

The enhancement of nonlinear optical phenomena via plasmonics has attracted significant interest in recent years [1–3]. Plasmonic nanostructures can concentrate light to subwavelength volumes [4]. The associated field enhancement can enhance nonlinear optical processes such as sum frequency generation (SFG), difference frequency generation (DFG), second-harmonic generation (SHG), third-harmonic generation (THG), and four-wave mixing [5]. Plasmonic metasurfaces have recently been investigated for several applications including nonlinear optics [6–12]. For certain crystal classes, the nonlinear process requires orthogonal driving electric fields aligned with the principal axes of the crystal. For this to be efficient, the simultaneous enhancement of all driving fields within the same volume is required. To this end we propose a dual-polarization plasmonic metasurface. Our metasurface, as sketched in top view in Fig. 1, is periodic in the x and z directions. The unit cell is highlighted in the center; it has length l and is composed of bent metal strips of 0146-9592/15/122874-04$15/0$15.00 © 2015 Optical Society of America

Fig. 1. Top view of the metasurface with the unit cell (center).

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Vol. 40, No. 12 / June 15 2015 / Optics Letters

(valid for lossless media), and Kleinman symmetry (valid for dispersionless media). The nonlinear polarization leading to DFG is described by the matrix equation 3 2 3 2 d 11 d 12 d 13 d 14 d 15 d 16 P x ω3  7 6 7 6 4 P y ω3  5  4ϵ0 4 d 21 d 22 d 23 d 24 d 25 d 26 5 P z ω3 

d 34 d 35 d 36 3  E x ω1 E x ω2  6 7 E y ω1 E y ω2  6 7 6 7  6 7 E z ω1 E z ω2  6 7 6 7; 6 E y ω1 E z ω2   E z ω1 E y ω2  7 6 7 6 E ω E  ω   E ω E  ω  7 4 x 1 z 2 x z 1 2 5 E x ω1 E y ω2   E y ω1 E x ω2  2

×

d 31

d 32

d 33

(1)

ϵr ω  ϵ∞ 

2 X Δϵn ω2pn − iγ p0 n ω ; ω2 − 2iωγ pn − ω2 n1 pn

them in the same volume. Even if we were to excite the structure by only E x ω1  and E z ω2 , the metasurface itself would create E x ω2  and E z ω1  components. The second part of the simulation is the nonlinear THz generation through DFG. The presence in the same volume of two perpendicular electric fields creates a second-order polarization that is perpendicular to the plane of the primary fields. Since we have no y component in the incident fields, the E y ω1  and E y ω2  contributions created by the metasurface are very small, so they are neglected. Thus we investigate from Eq. (1) only the y component of the polarization, i.e., P y ω3 ; ⃗r   4ϵ0 d eff E x ω1 ; ⃗r E z ω2 ; ⃗r   E z ω1 ; ⃗r E x ω2 ; ⃗r :

where ω3  ω1 − ω2 is the generated frequency. We consider a non-centro-symmetric medium of crystal class 43m, e.g., GaAs, which has a large second-order nonlinear optical susceptibility χ 2 . For this crystal class, the only nonzero elements in the d il matrix are d 14  d 25  d 36  d eff [5]. An in-house parallel 3D finite-difference time-domain (FDTD) code [17–20] is used to simulate the optical response of a metasurface consisting of bent gold strips arranged according to Fig. 1 embedded in GaAs. The metasurface is modeled by applying periodic boundary conditions (PBCs) to the unit cell. We use per-component mesh refinement and a uniform grid with a space-step Δx  Δy  Δz  0.5 nm. Convolutional perfectly matched layer (CPML) absorbing boundary conditions [21] are used in y to truncate the simulation domain. We perform a multi-frequency analysis in a single run of the code by in-line discrete Fourier transform (DFT). To this end, a broadband plane-wave pulse excitation is introduced by means of the total-field/scattered-field (TF/SF) method. The auxiliary differential equation (ADE) technique is used to implement the dispersion relations for gold [22] and GaAs [23]. For gold, we use the Drude with two critical points model (Drude+2CP) [24], with experimental data for the complex permittivity originating from [25], and fitting parameters as described in [26]. For GaAs, we use experimental data from [27,28] and the modified Lorentz model [23] (2)

Δϵ1  4.769, Δϵ2  4.529, where ϵ∞  1.837, ωp1  7.152 · 1015 , ωp2  4.639 · 1015 , γ p1  7.737 · 1014 , γ p2  6.395 · 1014 , γ p0 1  6, and γ p0 2  1.963 · 1015 (ωp , γ p , and γ p0 are in rad/s) are the fitting parameters we calculated. The first part of the simulation consists of the metasurface excitation by the two plane-wave laser pulses. We assume ω3 ∕2π  1 THz, and we calculate the DFT at ω1 and ω2 in the wavelength range 200–2000 nm, with ω1 and ω2 such that ω1 − ω2  ω3 . Both laser beams E⃗ inc ω1  and E⃗ inc ω2  have strength of 1 V/m, and they are polarized at θ  45° in the xz plane to maximize the field enhancement in the gap regions; 135° polarization does not produce any enhancement in the gap. This is similar to a dipole antenna irradiated by light polarized perpendicular to its axis. The incident beams contain all the field components E x ω1 , E z ω1 , E x ω2 , and E z ω2 . The metasurface geometry is able to enhance all of

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(3)

For GaAs d eff  370 pm∕V [5]. The amplitude of the generated signal at ω3 is computed in FDTD by X iω3 Ay ω3 ; L  P ω ; ⃗r e −ik3 y−y0  Δy; (4) 2n3 ϵ0 c 0 N x N z GaAs y 3 where L is the penetration depth into GaAs, y0 is the air/GaAs interface position, n3 is the refractive index of GaAs at ω3 , c 0 is pffiffiffiffiffi the speed of light in vacuum, i  −1, and N x and N z are the number of computational grid cells in x and z, respectively. The intensity of the generated signal at ω3 is I ω3 ; L  2n3 ϵ0 c 0 jAy ω3 ; Lj2 :

(5)

The summation in Eq. (4) is optimized when the condition for phase matching is satisfied, i.e., Δk  k 1 − k 2 − k3  0. This may happen for λ > 600 nm, where the GaAs refractive index decreases slowly and smoothly with λ. Modeling dispersion for GaAs allows us to take into account the walk-off between the two lasers and the phase mismatching. The small variation of the refractive index allows us to apply Eq. (1), which is valid for non-dispersive media. At wavelengths greater than 600 nm, the absorption in GaAs is small, so the lossless condition is satisfied. Below 500 nm, GaAs is highly absorptive, and we do not see any DFG. We report as an example case a metasurface with l  200 nm, w  80 nm, t  100 nm, and g  10 nm. We performed our analysis considering a GaAs substrate 1 μm thick. Based on our laboratory experience, we considered some fabrication constraints, such as w∕t ≥ 0.5 and t∕g ≤ 5. We allowed the latter to roam slightly beyond to reflect the fact that this constraint is not a strict technological limit. In Fig. 2, we show a 3D map of the jAy ω3 ; Lj enhancement over bulk due to the presence of the metasurface, as function of the wavelength λ1 (λ2 is implicitly derived) and L. The structure exhibits multi-mode behavior, and the number of supported modes increases as a function of unit cell size. A maximum enhancement over bulk of ∼9.2 is observed at λ1 ∼ 950 nm and λ1 ∼ 1770 nm near the surface, then drops with L, but is greater than one down to a depth L of ∼550 nm. The intensity of generated THz waves, as shown in Eq. (5), is proportional to the square of its electric field, suggesting that a maximum intensity enhancement of ∼85 is possible. In Figs. 3 and 4 we show the absolute value of the fields involved in Eq. (3), i.e., jE x ω1 j, jE x ω2 j, jE z ω1 j, and jE z ω2 j, at λ1  1770 nm (λ2  1780.512 nm), and λ1  950 nm (λ2  953.02 nm), respectively; the xz cut-plane is taken 5 nm into the surface. The peak in the jAy ω3 ; Lj

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Fig. 2. 3D map of the jAy ω3 ; Lj enhancement relative to the substrate without the metasurface for ω3 ∕2π  1 THz.

enhancement at λ1 ∼ 1770 nm in Fig. 2 is due to enhanced fields in the gap (Fig. 3), whereas for the peak at λ1 ∼ 950 nm, the fields outside the gap region also contribute (Fig. 4). A 135° polarization excitation produces only the fields outside the gap, correspondingly, in the 3D enhancement map, and only one peak survives with smaller amplitude and blueshifted to λ1 ∼ 910 nm. In Figs. 3 and 4 we also observe that the hot-spots for the pairs fjE x ω1 j; jE z ω2 jg and fjE z ω1 j; jE x ω2 jg are colocated; this is due to the fact that ω1 ≃ ω2 . In Fig. 5, we plot the phase difference between the LHS and RHS terms in Eq. (3), i.e., Δϕ  ∠E x ω1  − ∠E z ω2  − ∠E z ω1  − ∠E x ω2 : (6) We observe that Δϕ ≃ 0 in all the hot-spots, and the phase relation is maintained along y. Thus the two terms in Eq. (3) add up coherently. Moreover, we found that the phases of these four field components are almost equal in all the co-located hot-spots, this means that

Fig. 3. jE x ω1 j, jE x ω2 j, jE z ω1 j, and jE z ω2 j in units of V/m for λ1  1770 nm (left) and λ2  1780.512 nm (right).

Fig. 4. jE x ω1 j, jE x ω2 j, jE z ω1 j, and jE z ω2 j in units of V/m for λ1  950 nm (left) and λ2  953.02 nm (right).

P y ω3  ≃ 4ϵ0 d eff jE x ω1 jjE z ω2 j  jE z ω1 jjE x ω2 j; (7) which is ultimately responsible for the enhanced generated signal at ω3 . In Fig. 6, we show jE x ω1 j in the middle of the gap region for six λ1 values in the range of analysis; the interface air/GaAs is at y0  50 nm (the other components have qualitatively similar distributions). The gap region acts as a metal-insulatormetal (MIM) waveguide [29] with boundary conditions imposed by the geometry of the structure. The multi-spot field distribution we observe in the gap is typical of a Fabry−Perot cavity, and it depends on cavity dimensions and wavelength [30]. We have proposed a plasmonic metasurface for the enhancement of nonlinear optical phenomena where perpendicular polarizations are required. Applying this metasurface to THz generation via DFG, we found an intensity enhancement over bulk of almost two orders of magnitude at the surface. There exists an enhancement up to L∼550 nm into the substrate, thus the proposed structure is most suitable for thin-film, surface-based, or chip-based nonlinear applications. This may be interesting for AlGaAs that is restricted to thin-films, and has

Fig. 5. Phase difference Δϕ in Eq. (6) in units of rad for λ1  1770 nm (left) and λ1  950 nm (right).

Letter

Fig. 6. jE x ω1 j in units of V/m for various λ1 in the plane cut through the middle of the gap region.

an even higher χ 2 than GaAs [31]. The multi-mode and polarization-dependent properties could also be exploited for sensing applications. Though we focused on the 43m crystal class, the proposed metasurface could be useful for enhancing the induced polarization (P y in our chosen coordinate system) in all crystal classes with at least one nonzero term among d 21 , d 23 , and d 25 . For example, the pairs fE x ω1 ; E x ω2 g and fE z ω1 ; E z ω2 g are in fact enhanced and co-located, and could also be exploited. This opens new possibilities for enhancing nonlinear signal generation and will be explored in future works. Canada Research Chairs (Chaires de recherche du Canada); IBM Canada Research and Development Centre; SciNet (Compute Canada); Southern Ontario Smart Computing Innovation Platform (SOSCIP). REFERENCES 1. M. Kauranen and A. V. Zayats, Nat. Photonics 6, 737 (2012). 2. J. Khurgin and G. Sun, Opt. Express 21, 27460 (2013). 3. P. Genevet, J.-P. Tetienne, E. Gatzogiannis, R. Blanchard, M. A. Kats, M. O. Scully, and F. Capasso, Nano Lett. 10, 4880 (2010).

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