Phenomenological modeling of nonlinear holograms based on metallic geometric metasurfaces

Weimin Ye; Xin Li; Juan Liu; Shuang Zhang

doi:10.1364/OE.24.025805

1. Introduction

With their superior capability of controlling the phase, amplitude and polarization of light at the nanoscale with low manufacturing complexity, metallic metasurfaces [1–4] with thickness far less than the wavelength of light have opened up new opportunities for planar photonics and miniaturization of optical devices. Owing to the strong field enhancement and large nonlinear polarizability of the metal nanostructure, nonlinear optical properties of metasurfaces have recently attracted tremendous attentions [5, 6]. Second-harmonic generation (SHG) from metasurfaces constituted by split-ring resonator arrays have been reported [7–10]. It has been remarkably shown that metasurfaces consisting of plasmonic elements of certain rotational symmetries can generate symmetry-selective high order harmonic generations [11, 12]. The concept of gradient metasurfaces, which plays an important role in manipulating the wavefront of light, has also been extended to nonlinear domain to control the nonlinear emission from metamaterials for linearly [13–15] and circularly [16–18] polarized fundamental waves.

Among various types of applications, metasurface holography [19] has caught growing attention due to their powerful control of the phase of light. Combining precise local phase control of metasurfaces with the development of algorithms for computer generated holography (CGH) [20–22], phase-only holograms based on metasurfaces are capable of recording and reconstructing wavefronts with high image quality. With circularly polarized incident light, by making use of geometric Pancharatnam-Berry phase [23] that is encoded into the orientation angle of the constituent meta-atoms, three-dimensional [24] and high efficient holograms [25] have been realized by geometric metasurface constituted by identical meta-atoms with spatially varying orientations. Very recently, metasurface holography has been extended to nonlinear optics, with successful demonstration of nonlinear holographic images in both second (SHG) and third harmonic generation (THG) [26, 27].

Most reported theoretical modelings of metasurface holography are based on scalar theory. Owing to subwavelength unit-cell size of metasurfaces, the scalar theory fails to predict the optical performance of metasurface holograms, such as the distributions of output electromagnetic fields and efficiencies of holograms. On the other hand, full-wave numerical simulations on metasurface holograms are nearly impossible because of the large number of different meta-atoms contained in the metasurface. We have recently developed a phenomenological model to precisely describe the far field diffraction of a metasurface without involving extended full scale numerical simulations of the metasurface [28]. In this paper, we extend our phenomenological modeling of geometric metasurfaces to the nonlinear optical region, and propose a general theory to model nonlinear metallic metasurfaces with arbitrarily distributed nonlinear phase profiles.

2. Theoretical modeling of nonlinear geometric metasurface

Here we restrict the discussion of nonlinear optics to SHG, though similar methods can be readily extended to calculate higher order harmonic generations. Same as our previous phenomenological modeling of linear metasurfaces [28], only geometric metasurfaces made up of achiral and metallic meta-atoms with thickness far less than the wavelength of light are considered. Focusing only on the electric polarizability and ignoring the coupling between meta-atoms, the meta-atom located at (x_m, y_n) can be approximated as a point electric dipole described by a local polarizability tensor, which is combination of the local linear polarizability tensor $α_{m n}^{L}$ and second-order polarizability tensor $α_{m n}^{(2)}$ , which can be generally expressed as

α_{m n}^{L} = α_{S S} e_{m n}^{S} e_{m n}^{S} + α_{P P} e_{m n}^{P} e_{m n}^{P},

α_{m n}^{(2)} = α_{P P P} e_{m n}^{P} e_{m n}^{P} e_{m n}^{P} + α_{P S S} e_{m n}^{P} e_{m n}^{S} e_{m n}^{S} + α_{S P S} e_{m n}^{S} e_{m n}^{P} e_{m n}^{S} + α_{S S P} e_{m n}^{S} e_{m n}^{S} e_{m n}^{P} .

where the local unit vectors

e_{m n}^{S}

and

e_{m n}^{P}

are perpendicular and parallel to the symmetry axis of the achiral meta-atom, respectively. It is worth noticing that

α_{S P S}

is equal to

α_{S S P}

for SHG. By introducing the circular basis

e_{\pm} = (e_{x} \pm i e_{y}) / \sqrt{2}

, the local polarizability tensors can be rewritten as

α_{m n}^{L} = α^{+} (e_{+} e_{-} + e_{-} e_{+}) + α^{-} (e^{- i 2 φ_{m n}} e_{+} e_{+} + e^{+ i 2 φ_{m n}} e_{-} e_{-}),

\begin{array}{l} α_{m n}^{(2)} = i [κ^{+} e^{i 3 φ_{m n}} e_{-} e_{-} e_{-} + ν e^{i φ_{m n}} e_{-} e_{+} e_{-} + κ^{-} e^{- i φ_{m n}} e_{-} e_{+} e_{+}] \\ - i [κ^{+} e^{- i 3 φ_{m n}} e_{+} e_{+} e_{+} + ν e^{- i φ_{m n}} e_{+} e_{-} e_{+} + κ^{-} e^{i φ_{m n}} e_{+} e_{-} e_{-}], \end{array}

α^{\pm} = \frac{α_{S S} \pm α_{P P}}{2}; κ^{\pm} = \frac{α_{P S S} - α_{P P P} \pm 2 α_{S P S}}{2 \sqrt{2}}; ν = \frac{α_{P S S} + α_{P P P}}{\sqrt{2}} .

where the orientation angle φ_mn of the meta-atom is defined by the angle between the local unit vector

e_{m n}^{S}

and the positive direction of x axis. Thus, the linear and second-order susceptibility tensor of the geometric metasurface can be written as the summation of linear and second order susceptibilities of meta-atoms, respectively. That is,

χ^{L} (X) = \sum_{m, n} δ (x - x_{m}) δ (y - y_{n}) α_{m n}^{L} = \sum_{p, q} e^{i \frac{2 p π}{L_{x}} x + i \frac{2 q π}{L_{y}} y} {\bar{α}}^{L} (p, q),

χ^{(2)} (X) = \sum_{m, n} δ (x - x_{m}) δ (y - y_{n}) α_{m n}^{(2)} = \sum_{p, q} e^{i \frac{2 p π}{L_{x}} x + i \frac{2 q π}{L_{y}} y} {\bar{α}}^{(2)} (p, q),

{\bar{α}}^{L} (p, q) = S^{- 1} \sum_{m, n} e^{- i \frac{2 p π}{L_{x}} x_{m} - i \frac{2 q π}{L_{y}} y_{n}} α_{m n}^{L}, {\bar{α}}^{(2)} (p, q) = S^{- 1} \sum_{m, n} e^{- i \frac{2 p π}{L_{x}} x_{m} - i \frac{2 q π}{L_{y}} y_{n}} α_{m n}^{(2)} .

where L_x and L_y are the smallest periods of the metasurface along the direction x and y directions, respectively. S = L_x L_y is the area of the metasurface. p and q are two integers.

Without loss of generality, we consider a metallic metasurface sandwiched between the homogenous and isotropic cover and substrate, as depicted in Fig. 1. $E_{(x, y)}^{ω, S (C) \pm}$ ( $E_{(x, y)}^{2 ω, S (C) \pm}$ ) denotes tangential component of fundamental (second-harmonic) electric fields propagating along ± z direction at the interface between the substrate (cover) and metasurface. Here we consider that the fundamental wave is incident on the metasurface only from the substrate side.

Fig. 1 Schematic illustration of the model of nonlinear metasurface.

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Because the metasurface has a negligible thickness, the tangential components of electric fields are continuous across the metasurface. Thus, the tangential components of the electric and magnetic fields on the interfaces of the cover and substrate satisfy [28]

E_{(x, y)}^{S +} + E_{(x, y)}^{S -} = E_{(x, y)}^{C +} = E_{(x, y)} = [\begin{matrix} E_{x} \\ E_{y} \end{matrix}],

e_{z} \times (H_{(x, y)}^{C +} - H_{(x, y)}^{S +} - H_{(x, y)}^{S -}) = σ_{d} = ε_{0} \frac{\partial}{\partial t} (χ \cdot E_{(x, y)}) .

where

χ

is the electric susceptibility tensor of the metasurface. For a plane wave with tangential wave vectors

(k_{x}^{i n}, k_{y}^{i n})

incident from the substrate side, fundamental electric fields at the interfaces of the cover and substrate can be expanded as the sum of various spatial frequency components. That is

E_{(x, y)} (x, y, ω) = \sum_{p, q} E_{(x, y), (p, q)}^{ω} \exp (i k_{p} x + i k_{q} y),

k_{p q} = k_{p} e_{x} + k_{q} e_{y} = (k_{x}^{i n} + 2 p π / L_{x}) e_{x} + (k_{y}^{i n} + 2 q π / L_{y}) e_{y} .

Combining Eqs. (4)a), (4b) and (5a), the Fourier expansions of transmitted and reflected electric fields at the interfaces satisfy

E_{(x, y), (m, n)}^{ω, C +} = 2 {[Π^{ω}]}_{(m, n), (0, 0)}^{- 1} Y^{S} (k_{x}^{i n}, k_{y}^{i n}, ω) \cdot E_{(x, y), (0, 0)}^{ω,S+},

E_{(x, y), (m, n)}^{ω, S -} = 2 {[Π^{ω}]}_{(m, n), (0, 0)}^{- 1} Y^{S} (k_{x}^{i n}, k_{y}^{i n}, ω) \cdot E_{(x, y), (0, 0)}^{ω,S+} - E_{(x, y), (0, 0)}^{ω, S +} δ_{m, 0} δ_{n, 0} .

here

Π_{(p, q), (m, n)}^{ω} = [Y^{C} (k_{p}, k_{q}, ω) + Y^{S} (k_{p}, k_{q}, ω)] δ_{p, m} δ_{q, n} + i ω ε_{0} {\bar{α}}^{L} (p - m, q - n, ω),

Y^{S(C)} (k_{x}, k_{y}, ω) = \frac{- \sqrt{ε^{S(C)} / μ^{S(C)}}}{ω n_{S(C)} \sqrt{ω^{2} n_{S(C)}^{2} - (k_{x}^{2} + k_{y}^{2}) c^{2}}} [\begin{matrix} ω^{2} n_{S(C)}^{2} - k_{y}^{2} c^{2} & k_{x} k_{y} c^{2} \\ k_{x} k_{y} c^{2} & ω^{2} n_{S(C)}^{2} - k_{x}^{2} c^{2} \end{matrix}] .

In Eq. (7)b),

ε^{S(C)}

,

μ^{S(C)}

and

n_{S (C)}

are respectively the permittivity, magnetic permeability and refractive index of the substrate (Cover).

Similar as the fundamental wave, second-harmonic (SH) electric fields at the interfaces can also be expanded as the sum of various spatial frequency components.

E_{(x, y)} (x, y, 2 ω) = \sum_{p, q} E_{(x, y), (p, q)}^{2 ω} \exp (i {k^{'}}_{p} x + i {k^{'}}_{q} y),

{k^{'}}_{p q} = {k^{'}}_{p} e_{x} + {k^{'}}_{q} e_{y} = (2 k_{x}^{i n} + 2 p π / L_{x}) e_{x} + (2 k_{y}^{i n} + 2 q π / L_{y}) e_{y} .

Utilizing the fundamental electric fields, the Fourier expansions of SH electric fields at the interfaces of the cover and substrate satisfy

E_{(x, y), (m, n)}^{2 ω, S -} = E_{(x, y), (m, n)}^{2 ω, C +},

E_{(x, y), (m, n)}^{2 ω,C+} = {[Π^{2 ω}]}_{(m, n), (p, q)}^{- 1} \cdot (- i 2 ω ε_{0}) [\sum_{l, h} {\bar{α}}^{(2)} (p - l, q - h, 2 ω) : {(E_{(x, y)}^{ω, C+} E_{(x, y)}^{ω, C+})}_{(l, h)}],

{(E_{(x, y)}^{ω, C+} E_{(x, y)}^{ω, C+})}_{(l, h)} = \sum_{m, n} E_{(x, y), (l - m, h - n)}^{ω,C+} E_{(x, y), (m, n)}^{ω, C+} .

Thus, SHG efficiency or second-harmonic conversion efficiency

η_{S H G}

can be defined by the z-component of Poynting vector S_z of second-harmonic wave and the fundamental plane wave. That is

η_{S H G} = \frac{\sum_{p, q} S_{z, 2 ω}^{+} ({k^{'}}_{p}, {k^{'}}_{q})}{S_{z, ω}^{+} (k_{x}^{i n}, k_{y}^{i n})} = \frac{\sum_{p, q} 0.5 Re (E_{x, (p, q)}^{2 ω, C +} H_{y, (p, q)}^{2 ω, C +}^{*} - E_{y, (p, q)}^{2 ω, C +} H_{x, (p, q)}^{2 ω, C +}^{*})}{0.5 Re (E_{x, (0, 0)}^{ω, S +} H_{y, (0, 0)}^{ω, S +}^{*} - E_{y, (0, 0)}^{ω, S +} H_{x, (0, 0)}^{ω, S +}^{*})} .

Based on Eqs. (6)a), (6b), (9a-c) and (10), we can study SHG of metallic metasurface with arbitrarily distributed phase profiles such as that of metasurface holography. In addition, the above theory of nonlinear metallic metasurface could be directly extend to a reflective multilayer design where the metallic meta-atom layer is on top of a homogeneous dielectric substrate layer of finite thickness and a thick metal ground plane (28).

3. Retrieval of local polarizability tensors of individual meta-atom

For a meta-atom of a specific geometry, the local polarizability tensors defined in Eqs. (1)a), (1b) can be retrieved by fitting the full wave simulation of a simple periodic array of the meta-atoms with our analytical theory. Here, COMSOL Multiphysics finite-element-based electromagnetic solver is used for the simulation. We consider the metallic metasurface consisting of a periodic array of identical meta-atoms with orientation angles φ equal to zero in a square lattice with period a. For the wavelength much greater than the period, we only need to consider the contribution of the zero-order Fourier component of the polarizability tensors in Eq. (3)a) and electric fields in Eqs. (6)a), (6b) and (9a-c). That is, both p and q in Eqs. (3)a), (5a) and (8a) are equal to zero. For a X or Y-polarized plane-wave normally incident upon the periodic metasurface, we can derive the effective linear polarizability of the meta-atom from the transmission coefficient t_xx and t_yy by using Eqs. (6)a) and (6b):

\frac{α_{S S} (ω)}{a^{2}} = \frac{c}{i ω} (n_{S} + n_{C} - \frac{2 n_{S}}{t_{x x}}), \frac{α_{P P} (ω)}{a^{2}} = \frac{c}{i ω} (n_{S} + n_{C} - \frac{2 n_{S}}{t_{y y}}),

t_{x x} = \frac{E_{x, (0, 0)}^{ω, C +}}{E_{x, (0, 0)}^{ω, S +}}, t_{y y} = \frac{E_{y, (0, 0)}^{ω, C +}}{E_{y, (0, 0)}^{ω, S +}} .

where subscript x and y denote the X and Y components, respectively.

To obtain second-order polarizabilities, we use the total surface SH current component of a meta-atom obtained from the hydrodynamic model of electrons in the metal [9, 13], which is written as

I_{d}^{2 ω} \cdot e_{x (y)} = - i 2 ω P_{x (y)}^{2 ω} = - i 2 ω ε_{0} \frac{ε_{0} ε_{r Au}^{2}}{2 e n_{0}} \sum_{m} \iint d S^{(m)} [\begin{array}{l} \frac{(3 ω + i γ_{e})}{2 (2 ω + i γ_{e})} E_{n, ω}^{(m) 2} (e_{n}^{(m)} \cdot e_{x (y)}) + \\ E_{n, ω}^{(m)} E_{t1, ω}^{(m) 2} (e_{t1}^{(m)} \cdot e_{x (y)}) + E_{n, ω}^{(m)} E_{t2, ω}^{(m) 2} (e_{t2}^{(m)} \cdot e_{x (y)}) \end{array}] .

Superscript m denotes the different surface of the meta-atom.

e_{n}^{(m)}

,

e_{t 1}^{(m)}

and

e_{t2}^{(m)}

are local unit vectors normal and parallel to the mth surface, respectively. n₀ and γ_e are the number density and damping frequency of electrons in the metal. In Eq. (12), the fundamental electric fields on the surfaces can be calculated by COMSOL simulation. On the other hand, according to the definition of local second-order polarizability tensor of a meta-atom in Eq. (1)b), we have

P_{x}^{2 ω} = 2 ε_{0} α_{S P S} E_{y, (0, 0)}^{ω,C+} E_{x, (0, 0)}^{ω,C+},

P_{y}^{2 ω} = ε_{0} α_{P S S} E_{x, (0, 0)}^{ω, C+} E_{x, (0, 0)}^{ω, C+} + ε_{0} α_{P P P} E_{y, (0, 0)}^{ω, C+} E_{y, (0, 0)}^{ω, C+} .

Combining Eqs. (12), (13a) and (13b), we can obtain the effective second-order polarizability of the meta-atom from the fundamental electric fields. That is

\frac{α_{P S S}}{a^{2}} = \frac{P_{y}^{2 ω} (e_{x}^{In})}{a^{2} ε_{0} t_{x x}^{2}}, \frac{α_{P P P}}{a^{2}} = \frac{P_{y}^{2 ω} (e_{y}^{In})}{a^{2} ε_{0} t_{y y}^{2}}, \frac{α_{S P S}}{a^{2}} = \frac{P_{x}^{2 ω} (e_{45^{0}}^{In})}{a^{2} ε_{0} t_{x x} t_{y y}} .

where

P_{y}^{2 ω} (e_{x}^{In})

and

P_{y}^{2 ω} (e_{y}^{In})

denote the y component of the surface SH polarization produced by a normally incident plane wave with electric fields equal to e_x and e_y, respectively, and

P_{x}^{2 ω} (e_{45^{0}}^{In})

denotes the x component of the surface SH polarization produced by a normally incident plane wave with electric field described by

(e_{x} + e_{y}) / \sqrt{2}

.

Here, we choose to study a metasurface consisting of periodic gold U-shaped split ring resonators (SRR) array [7–10, 13] due to their large second harmonic conversion efficiency. The structure of SRR is shown in Fig. 2(a). The semi-infinite cover and substrate are air and SiO₂ with refractive indexes of 1 and 1.46, respectively. In the simulation, the relative permittivity of gold is taken from [29], with the fitted electron number density n₀ and damping frequency γ_e being 5.9 × 10²² cm⁻³ and 1.208 × 10¹⁴Hz, respectively. Figure 2(b) shows the linear transmission spectra of X-polarized and Y-polarized light normally incident from the substrate side onto the metasurface. Two resonance dips are observed for X-polarized incident beam, which correspond to the two odd modes of the SRR shown in Figs. 2(d) and 2(f) at wavelengths 693nm and 1185nm. The resonance dip for Y-polarized incidence results from the even mode of the SRR shown in Fig. 2(e) at the wavelength of 710nm.

Fig. 2 Optical properties of the reference metasurface constituted by periodic array of gold U-shaped split ring resonators. (a) The unit-cell structure of metasurface. The lattice sizes for the metasurfaces are a = 360nm along the x-axis and y-axis directions. Length, width and thickness of U-shape SRR are 180nm, 120nm and 30nm, and the split size is 60nm × 60nm. (b) Linear transmission spectra of X-polarized and Y-polarized light normally incident from substrate onto the metasurface. (d-f) Transverse electric filed vector distributions of three eigenmodes of the SRR on the plane with 2nm above the surface of SRR. Where, the small arrows denote directions of transverse electric filed vectors, the different color hue denotes relative amplitudes of them. (g) SHG efficiency spectra of X-polarized and Y-polarized fundamental light with power density 13MWcm⁻² normally incident from substrate onto the metasurface. (c, h, i) The effective linear and second-order polarizability components of the SSR retrieved from the COMSOL simulations, respectively.

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Based on the COMSOL simulation, the effective linear and second-order polarizability tensors of the SRR can be obtained from Eqs. (11)a) and (14), which are shown in Figs. 2(c), 2(g), and 2(i), respectively. Due to the strong field enhancement for X-polarized incident light at the fundamental mode wavelength (1185 nm), the absolute values of effective linear and second-order polarizability components $α_{S S} / a^{2}$ , $α_{P S S} / a^{2}$ and $α_{S P S} / a^{2}$ reach the maxima ~1.1 × 10⁻⁶ m, 9.8 × 10⁻¹⁸ m²V⁻¹ and 5 × 10⁻¹⁹ m²V⁻¹, respectively. However, the absolute values of $α_{P P} / a^{2}$ (~6.1 × 10⁻⁸ m) and $α_{P P P} / a^{2}$ (~2.8 × 10⁻²⁰ m²V⁻¹) remain relatively small because of absence of resonant response to the X-polarized fundamental mode. Combining Eq. (6)a) with Eq. (9)b), SHG efficiency (or second-harmonic conversion efficiency) of the metasurface shown in Fig. 2(g) can be calculated by setting the pump intensity of the normally incident X-polarized and Y-polarized fundamental light to a realistic value of 13MWcm⁻². It is obvious that the SHG is drastically enhanced for X-polarized fundamental light due to the resonance of the fundamental mode. The maximum SHG efficiency is ~1 × 10⁻⁹, which is in good agreement with previous reports on SHG from uniform SRR arrays [9, 13].

4. Simulations of nonlinear holograms based on a geometric metasurface

We next move to the investigation of nonlinear optical holograms based on the geometric metasurface, which is constituted by an array of 101 × 101 identical gold SRR shown in Fig. 2(a) with designed orientation angles φ(x, y). It has been shown previously that for a fundamental incident beam of circular polarization with spin state σ, where σ = ± 1 stands for left- (LCP) or right circular polarization (RCP), respectively, the SHG with spin σ and –σ acquire a nonlinear Berry phase of σφ and 3σφ, respectively [17, 18, 27]. The different geometric phases provide two channels for encoding distinct information in geometric metasurfaces to realize spin-multiplexed nonlinear holography. Here we focus on only a single holographic image of the Chinese characters for “Left” or “Right” reconstructed at k-space by the transmitted SHG with spin σ or –σ. The classical Gerchberg–Saxton algorithm [20] is employed to obtain the desired orientation angle distribution of arrays of gold SRRs. The dimension of the array is determined by the number of pixels (101 × 101) of the target images. To be consistent with the dimension, the Fourier expansion order p and q of the susceptibility tensor in Eq. (3)a) and electric fields in Eqs. (5)a) and (8a) range from −50 to 50 in our calculations. Both L_x and L_y in Eq. (3)a) are equal to 101a.

The far-field spatial pattern of transmitted light is given by the distribution of z-component of Poynting vector S_z in k-space - S_z (k_x, k_y), with the desire spin. The overlay coefficient $η$ [30] between the SHG holographic image and the corresponding target image is introduced to evaluate the quality of the simulated holographic images.

η = \frac{\sum_{p, q} f_{0} (p, q) \cdot S_{z, 2 ω}^{+} ({k^{'}}_{p}, {k^{'}}_{q})}{\sqrt{\sum_{p, q} f_{0} {(p, q)}^{2}} \sqrt{\sum_{p, q} S_{z, 2 ω}^{+} {({k^{'}}_{p}, {k^{'}}_{q})}^{2}}} .

where f₀ (p, q) denotes the grey values of pixel (p, q) of the target image. The z-component of Poynting vector of SHG

S_{z, 2 ω}^{+} ({k^{'}}_{p}, {k^{'}}_{q})

is defined in Eq. (10).

First, we consider co-circular polarized SHG hologram. With the fundamental RCP plane-wave normally incident on the geometric metasurface, only the spatial pattern of the transmitted SH-RCP is designed to reconstruct the target image of Chinese characters for ‘“Left”. The designed orientation angle distribution φ(x, y) of the gold SRRs is shown in Fig. 3(a). Using the retrieval of local polarizability tensors of the gold SRR in Eq. (11)a) and Eq. (14), the transmitted fundamental wave and SHG in k-space can be solved from Eqs. (6)a-b) and (9a-9c), respectively. Figure 3(b) shows that around the fundamental mode wavelength, the efficiency of SH-RCP reaches the maximum ~3.2 × 10⁻¹⁰. But, the efficiency of SH-image appears a small dip (~6 × 10⁻¹¹). Especially, the overlay coefficient in Fig. 3(c) between the SH-image and the target image exhibits a dip with a minimum ~0.42. This dip is associated with poor quality of the simulated holographic images around fundamental resonance, which is shown in Fig. 3(d). On the other hand, when the wavelength of incident RCP is 1140nm, slightly detuned from the fundamental resonant wavelength, the efficiency and overlay coefficients of SH-image are improved, reaching 7.7 × 10⁻¹¹ and 0.93, respectively.

Fig. 3 Theoretical results of co-circular polarized SHG hologram based on the geometric metasurface to reconstruct Chinese character for ‘“Left” in the far field. (a) Orientation angle distribution φ(x, y) of 101 × 101 arrays of gold SRRs is designed to generate the holographic image. (b) The optical efficiency spectra of the transmitted SH-RCP (denoted by ■) and the holographic image carried by it (denoted by ●) for a fundamental RCP plane-wave normally incident on the designed metasurface with average pump intensity 13 MWcm⁻². (c) The overlay coefficient between SHG holographic images and the target image of Chinese character for the incident fundamental wave with different wavelengths. (d) The spatial patterns of the transmitted SH-RCP with different wavelengths. The most right panel shows the corresponding target image.

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We further investigate the cross-circular polarized SHG hologram. For a fundamental RCP plane-wave normally incident on the geometric metasurface, only the spatial pattern of the transmitted SH-LCP is designed to reconstruct the target image of Chinese characters for ‘“Right”. Figure 4(a) shows the designed orientation angle distribution. Similar as co-circular polarized SHG hologram, around the fundamental mode wavelength, the efficiency of SH-LCP and the quality of the holographic image are both low, as shown in Figs. 4(c) and 4(d).

Fig. 4 Theoretical results of cross-circular polarized SHG hologram based on the geometric metasurface to reconstruct Chinese character for ‘“Right” in the far field. (a) Orientation angle distribution φ(x, y) of 101 × 101 arrays of gold SRRs is designed to generate the holographic image. (b) The optical efficiency spectra of the transmitted SH-LCP (denoted by ▲) and the holographic image carried by it (denoted by ▼) for a fundamental RCP plane-wave normally incident on the designed metasurface with average pump intensity 13 MWcm⁻². (c) The overlay coefficient between SHG holographic images and the target image of Chinese character for the incident fundamental wave with different wavelengths. (d) The spatial patterns of the transmitted SH-LCP with different wavelengths. The most right panel shows the corresponding target image.

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Finally, we discuss the reason of the poor quality of the SH-holographic images around the fundamental resonance wavelength. Intuitively, due to the lack of rotational symmetry of the SRR structure, the CP incident fundamental beam is partially converted into its opposite circular polarization upon interaction with the SRRs. This conversion to the cross polarization reaches maximum at the resonance frequency of SRRs. As such, a second harmonic nonlinear photon can arise from either the combination of two fundamental frequency photons of the same spin that has the designed phase of 3φ and contribute to the holographic image, or two fundamental frequency photons of opposite spins with a phase φ and contribute to the background noise. Equation (9)b) shows that the Fourier components of SH electric fields are mainly determined by the Fourier expansions of surface SH current density, which is the last item on the right side of Eq. (9)b), defined as

σ_{d (p, q)}^{2 ω} = (- i 2 ω ε_{0}) [\sum_{l, h} {\bar{α}}^{(2)} (p - l, q - h, 2 ω) : {(E_{(x, y)}^{ω, C+} E_{(x, y)}^{ω, C+})}_{(l, h)}] .

Among the Fourier components of transmitted fundamental electric fields from the metasurfaces with thickness far less than the wavelength of light, the zero-order one denoted by subscript (0, 0) with the identical spin as the incident light is the largest. Thus, Eq. (16) can be approximates as

σ_{d (p, q)}^{2 ω} \approx (- i 2 ω ε_{0}) [\sum_{(l, h)} {\bar{α}}^{(2)} (p - l, q - h, 2 ω) : E_{(x, y), (0, 0)}^{ω, C+} E_{(x, y), (l, h)}^{ω,C+}] .

For the cross-circular polarized SHG hologram with fundamental polarization e₊ and SH polarization e_-, combining Eqs. (2)b), (3c) and (17), the surface SH current density can be written as

\begin{array}{l} e_{+} \cdot σ_{d (p, q)}^{2 ω} \approx \frac{2 ω ε_{0}}{S} \sum_{m, n} e^{- i \frac{2 p π}{L_{x}} x_{m} - i \frac{2 q π}{L_{y}} y_{n}} κ^{+} e^{i 3 φ_{m n}} (E_{(x, y), (0, 0)}^{ω,C+} \cdot e_{-}) (E_{(x, y), (0, 0)}^{ω, C+} \cdot e_{-}) \\ + \frac{2 ω ε_{0}}{S} \sum_{m, n} e^{- i \frac{2 p π}{L_{x}} x_{m} - i \frac{2 q π}{L_{y}} y_{n}} ν e^{i φ_{m n}} [E_{(x, y)}^{C+} (x_{m}, y_{n}, ω) \cdot e_{+}] (E_{(x, y), (0, 0)}^{ω, C+} \cdot e_{-}) . \end{array}

Based on Eq. (2)a) and Fig. 2(c), the conversion efficiency of cross-circular polarized fundamental light increases around the fundamental mode wavelength. Meanwhile, Fig. 2(i) shows that the second-order polarizability component

ν

in Eq. (2)c) is much larger than

κ^{+}

. Thus, in Eq. (18), the second term of the expression of SH current density could be comparable with the first term. Hence, the second term introduces the obvious noise in the SH-holographic images close to the fundamental resonance wavelength of the SRR. To eliminate this problem, one may employ plasmonic structures that have C₃ rotational symmetry such that there is no conversion of circular polarization at fundamental wavelength [16].

5. Conclusion

In summary, we have derived a generally theoretical modeling for nonlinear metallic metasurfaces based on the assumptions that every meta-atom could be described by localized linear and nonlinear polarizability tensors and the tangential electric fields on both sides of the metasurface are continuous. Applying it to nonlinear holograms based on geometric metasurfaces, we demonstrate theoretically that the classical Gerchberg–Saxton algorithm could not efficiently improve the quality of SH-holographic images at the fundamental resonance frequency of the SRR because the resonance of the fundamental light enhances the conversion of cross-circular polarizations and second-order polarizability of the metasurface. A simple way to achieve SH-holographic images with high efficiency and quality is with the wavelength of incident light reasonably detuned from the fundamental resonant wavelength. Our theoretical modeling provides an efficient method to design and model optical devices based on nonlinear metasurfaces of complex phase profiles.

Funding

EPSRC (Grant No. EP/J018473/1); Leverhulme Trust (Grant No. RPG-2012-674); National Natural Science Foundation of China (Grant Nos. 11374367, 61420106014, 61235002); National Basic Research Program of China (973 Program Grant No. 2013CB328801).

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Phenomenological modeling of nonlinear holograms based on metallic geometric metasurfaces

Abstract

1. Introduction

2. Theoretical modeling of nonlinear geometric metasurface

3. Retrieval of local polarizability tensors of individual meta-atom

4. Simulations of nonlinear holograms based on a geometric metasurface

5. Conclusion

Funding

References and links

Cited By

Figures (4)

Equations (32)

Optics Express