1. Introduction

Historically, diffraction gratings have consisted of a periodically patterned surface with infinite extent in one direction [1–3 ]. Metasurfaces provide a new method of wavefront control which is poised to change this two hundred year paradigm. The genesis of this change, initiated by early observations that hybridizing or structuring materials could illicit desirable macroscopic properties [4, 5], is the relaxation of the surface-symmetry through the patterning of surfaces in both spatial directions. Metasurfaces manipulate the wavefront using tiled sub-wavelength nano-structures [6, 7]. Only three years after their first demonstration [6], metasurfaces have performed functions previously reserved to bulky optical elements including dual polarity lensing [8], generation of optical vortex beams [9], efficient coupling of propagating waves to surface waves [10], quarter [11] and half [12] waveplate phase retardation, and axicon lensing [13]. Here we consider how metasurfaces can complement the performance of conventional diffraction gratings. Larouche and Smith [14] showed that meta-surfaces can be engineered to act like diffraction gratings and this behavior has been observed [6, 13, 15–17 ]. However, to this point, the efficiencies and mechanisms of a metasurface based grating have not been compared to a more traditional grating.

We compare the response of a traditional ruled-grating to a meta-surface based grating, or meta-grating, with design criterion that are challenging for either system: A visible light grating with 1800 lines/mm. For a ruled-grating this requires steep blaze angles that are known to induce anomalies [3] while, for meta-gratings, the small grating constant limits the size per meta-atom. We first discuss the mechanism of phase control and the design of the ruled-diffraction grating. Next, we describe the design of our 1800 lines/mm visible meta-grating. Finally, we compare the simulated response of the two devices. The more established technology narrowly outperforms the new technology, but the increased manufacturability, possibility of independent polarization control [18], and the prospect of completely eliminating phase accumulated through propagation indicate that meta-gratings have a bright future.

2. Investigated device

The performance of a diffraction grating is determined by the wavefront modulation imparted by the surface. Engineering a planar surface that obeys the grating equation,

We will concentrate on maximizing the efficiency in the m = +1 order for an 1800 lines/mm (L_x=556 nm) reflection grating operating with a 400 nm bandwidth around 650 nm. According to scalar diffraction theory this is accomplished by a wavefront modulation, f(x) = e ^{−im2πx/L_x}, with constant amplitude and a linear phase ramp [14]. Traditionally, the wavefront modulation has been realized using blazed ruled-gratings with triangular groove profiles [3] as shown in the left panel of Fig. 1. For gratings where the period is on the order of a wavelength, scalar diffraction theory can only provide a qualitative description [19]. Light with transverse magnetic (TM) polarization interacts with surface plasmon modes causing Rayleigh and resonance anomalies while light with transverse electric (TE) polarization is well approximated by scalar diffraction theory. Thus, the full vector Maxwell’s equations must be used when simulating the diffraction efficiency [3].

 

Fig. 1 Schematic detailing the unit cell of the ruled-grating (left) and the meta-grating (right). The ruled-grating is constructed by periodically repeating in the x direction the cross section shown on the bottom left. The meta-grating is constructed by tiling the cuboidal unit cell illustrated by the top view shown on the top right and the central cross section shown on the bottom right. The unit cell is built of four meta-atoms consisting of silver nano-antennas patterned on a magnesium floride/silver substrate. The meta-atoms that make up the unit cell are numbered in order of decreasing phase. A plane wave is incident on both structures at angle α measured from the grating normal. The relevant parameters are L_x = 555.5 nm, χ=35.8°, L_y = 221 nm, T_b = 130 nm, T_m = 75 nm, T_t = 30 nm, l ₁ = 84.6 nm, w ₁ = 105 nm, l ₂ = 47.7 nm, w ₂ = 105 nm, l ₃ = 177 nm, w ₃ = 50 nm, l ₄ = 150 nm, and w ₄ = 105 nm.

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The ruled-grating used as a standard to compare to the meta-grating is shown in Fig. 1. The blaze angle, χ, of 35.8° is set based on the Littrow blaze condition, λ = 2L_x sin(χ) [20]. From the perspective of scalar diffraction theory, this should maximize the efficiency at the center of our bandwidth (650 nm) for incident angles that satisfies α = β. The ruled-grating is made of silver for consistency with the meta-grating.

The phase control provided in meta-gratings is fundamentally different than that of ruled-gratings. The phase ramp across the unit cell is stepwise approximated using an array of meta-atoms (as depicted in Fig. 1), each of which has its own reflection phase and amplitude. To successfully approximate f(x) the meta-atom reflection phase must vary across most of the phase range from −π to π. This has been accomplished using V-shaped antennas [6], using the Pancharatnam-Berry phase [8, 9], or with the so-called metal-insulator-metal structures which consist of nano-antennas above metallic back planes [16]. We use the latter technique because of the functional similarities to ruled-grating: They work in reflection, are polarization maintaining, and have shown to exhibit high efficiency [16]. In these devices the π phase change provided by the nano-antenna resonance is supplemented by multiple reflections in the dielectric to achieve full 2π phase coverage [11]. Hence, unlike ruled-gratings, the phase profile is not generated purely by light propagation. The diffraction efficiencies of meta-gratings can be predicted by applying scalar diffraction theory to a stepwise approximation of f(x) with steps given by the complex reflection coefficient of the individual meta-atoms [14].

The 1800 lines/mm meta-grating was designed to have a wide bandwidth that ranges into the visible. The right panel in Fig. 1 shows the four meta-atoms that make up the unit cell and provides the dimensions of the device. The smallest feature of the proposed device is the 34 nanometers that horizontally separate both nano-antennas 1 and 2 and nano-antennas 4 and 1. Based on scalar diffraction theory, limiting the device to four meta-atoms limits the efficiency upper-bound to 80%. This was outweighed by the less restrictive phase constraints and the increased lateral space afforded per meta-atom. To choose meta-atoms that would complement each other over a wide bandwidth we first performed simulations of the reflection of y-polarized normally incident light off a periodic array of identical meta-atoms as a function of wavelength, nano-antenna length and width, and MgF₂ thickness. A commercial-grade simulator based on the finite-difference time-domain method was used to perform the calculations [21]. We then selected four meta-atoms that made a satisfactory four step approximation to f(x) over a wide spectral bandwidth. The amplitude and phase of the reflection coefficients of these four devices is shown in Fig. 2(a) and (b). The light, dotted black line in Fig. 2(b) corresponds to the ideal, 2π/4, phase difference between adjacent meta-atoms using the fourth meta-atom as a baseline. All four devices exhibit amplitudes greater than 80% for wavelengths longer than 575 nm.

 

Fig. 2 The individual optical responses of the four meta-atoms used to construct the unit cell plotted as a function of wavelength. Plotted in (a) is the amplitude and in (b) is the phase of the complex reflection coefficient, r = |r|e^iϕ, for a normally incident plane wave on a uniform periodic array of the four meta-atoms. The legend inset in (a) indicates color and line style used for each meta-atom. The dotted gray lines in (b) show the ideal scenario where the phase of each subsequent meta-atom is shifted by 2π/4. (c) The absolute efficiencies predicted based on the reflectivities are greater than 50% over a wide bandwidth.

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In Fig. 2(c) scalar diffraction theory is used to predict the absolute diffraction efficiencies based on a meta-grating unit cell composed of these four meta-atoms. The low expected efficiency in the blue is due to the reduced reflectivity amplitude of each meta-atom. This is caused by the resonance of the meta-atom nano-antennas which absorb light during each reflection in the MgF₂ [11]. For longer wavelengths, on the other hand, the expected efficiency decreases is due to the meta-atom phases. The phase response of the first meta-atom does not maintain the optimal 2π/4 phase separation from the second meta-atom. Overall, this meta-grating is expected to have greater than 50% efficiency for wavelengths greater than 525 nm.

3. Results

Both grating modalities perform well, but the different mechanisms of phase modulation result in different performance characteristics. Figure 3 shows the simulated m=+1 absolute grating efficiencies for both gratings. Each curve represents the incident angle dependent grating response for a single wavelength and spans only those incident angles that result in diffraction angles less than 90°. Simulations were performed by illuminating the periodic surface with a plane wave with incident angle α and then projecting the reflected fields into the far field to determine grating efficiencies [21]. On a quick comparison, both grating types perform fairly well for TE polarized light. The finer detail differentiates the two systems.

 

Fig. 3 Comparison of the m=+1 simulated absolute grating efficiencies for the ruled-grating (first-row) and the meta-gratings (second-row) as a function of polarization (columns), incident angle, and wavelength. Curves of different color correspond to different wavelength light.

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The ruled-grating exhibits the characteristics expected for a high blaze angle grating [22]. As is typical, the maximum efficiency for TE polarized light is blue shifted relative to the 650 nm blaze wavelength. In fact, the efficiency increases with decreased wavelength throughout the spectral range. Each wavelength, in turn, has an efficiency maximum at the incident angle for which diffracted light reflects back along the incident light (α = β). This angle shifts with wavelength because of the wavelength dependence of the grating equation. These wavelength and incident angle dependencies arise because the phase modulation is accrued through propagation. Due to grating anomalies, the response of the ruled-grating to TM polarized light is markedly different. Through the visible, the efficiency peak broadens in angle and increases in amplitude as the wavelength increases then plateaus at the 650 nm blaze wavelength. The flat incident angle response for wavelengths longer than the blaze wavelength is characteristic of high blaze angle ruled-gratings [23]. At lower wavelengths the shape of the angle dependence is similar to the TE response with the addition of a surface plasmon-resonance dip [23]. The different polarization responses complicate polarization dependent spectral intensity measurements, usually requiring a polarization analyzer to precede the grating.

The meta-grating exhibits a more extreme difference in polarization response. On the one hand, the TE polarized efficiency is high because the meta-atoms were designed to interact with this polarization. On the other hand, the electric field of TM polarized light is not aligned along the nano-antennas and, thus, doesn’t excite the resonances that create the wavefront modulations. Rotating the grating tunes this polarization response. Recently for 350 lines/mm meta-gratings, Pors and coworkers demonstrated that the diffraction constants and efficiencies of TE and TM polarized light could be controlled independently [18]. In the future, it is possible that this type of control might extend to the more spatially restrictive 1800 lines/mm gratings. Here we will concentrate on the TE-polarized response. The meta-grating response does not exhibit the anomalies that complicate the interpretation of the blazed grating response. The wavelength dependent efficiency of the meta-grating qualitatively agrees with the predicted efficiencies in Fig. 2: Efficiencies tend to increase as wavelength increases from 400 nm but tails off at higher wavelengths. The efficiency increases toward the red as the individual meta-atom efficiency increases then decreases for the longer wavelengths where the phase spacing between meta-atoms worsens. Surprisingly, the angle dependencies are similar in shape to the TE-polarized ruled-grating efficiencies and the maximum follows the condition α = β. This suggests that the origin is the variation in accrued propagation phase. Future meta-atoms might be able to eliminate any dependence on propagation phase and flatten the incident angle dependence. Overall, meta-grating TE-polarized efficiencies are higher than the ruled-grating TE polarized efficiencies for wavelengths longer than 600 nm and is competitive with the TM polarized efficiencies for wavelengths between 500 and 650 nm.

4. Conclusion

In summary we compared the response of a ruled diffraction grating to a metasurface based diffraction grating in a difficult application space. Even though meta-gratings are a new concept only realized in 2011 [6], the performance is already comparable to a blazed ruled-gratings. Possible future advances such as the ability to engineer the polarization response for smaller grating periods or the total elimination of propagation phase in the phase manipulation may further advance this technology. Finally, it is important to note the manufacturability of meta-gratings. Meta-gratings exchange the difficult height profile and period control necessary to manufacture a blazed grating with the two-dimensional binary surface control needed for meta-gratings.