Ⅴ. Electrically Tunable Beam Manipulation from Intersubband Polaritonic Metasurfaces
5.3 Simulation and Calculation of Optical Responses
FDTD calculation and simulation are performed by using a commercial software (Lumerical FDTD solutions, 2020a). The optical properties for the one-dimensional grating nanocavity are calculated by applying boundary conditions with anti-symmetric along the x-direction, symmetric along the y- direction, and perfectly matched layer (PML) along the z-direction at normal incidence. The farfield projection from the grating superlattice is applied with periodic boundary conditions along the x, y- directions, and PML along the z-direction. The reflection spectra and phase responses, under incident x-polarized plane wave, are extracted from S-parameter Analysis Group, and farfield projection data is extracted from the 2D Frequency-domain field and power monitor. The unit grating nanocavity and grating superlattice are simulated with global mesh level 4, and up to 1×10-5 of auto shutoff min to optimize runtime, collected data, and precise calculations. The Drude parameters of gold was used as
‘Au (Gold) – Palik’ in an embedded material group.
Under TM-polarized incident wave, optical responses from the polaritonic grating nanocavity with change of an external voltage bias are calculated by using numerical FDTD simulation. Figures 5.3.1 (a) shows the simulated polaritonic reflection spectra with change of different voltages. The grating metasurface is designed to be strongly coupled between its plasmonic resonance and intersubband absorption of the MQWs, showing clear evidence of the Rabi splitting. It leads to achieve the broadly spectral shifts from the positive to negative centering around zero voltage as shown in green semitransparent windows. The polaritonic absorption peaks show the movement in direction to the short wavelength entirely by the QCSE from negative to positive voltages, indicating in good agreement with the trend of IST energy level shifts. However, when the applied voltage bias is greatly increased or decreased, the confined Ez nearfields induced in the top metal surfaces of grating nanocavities cannot sufficiently support the IST due to the Stark shifts, gradually disappearing the clear evidence for Rabi splitting. The polaritonic phase responses are shown in Figure 5.3.1 (b). The vertical distance between two red dot points represents the most similar 180 degrees of relative phase difference at V2a = -2 V and V2b = +2 V or vice versa, and red vertical arrow points an operating wavelength of 6.55 μm for beam diffraction that produces a maximum efficiency. The vertical distance between two blue dot points at V3a = -1 V and V3b = +1 V represents the similar ±120 degrees of phase difference from the phase at zero bias, and blue vertical arrow points an operating wavelength of 6.37 μm for beam steering that generates a maximum efficiency. The phase gradient metasurface for selective beam steering provides a trinary digit system of the relative phase sequence of 'positive bias (2 : +120 degrees)’, ‘negative bias
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(1 : 0 degrees)’, and ‘no bias (0 : -120 degrees)’ or vice versa, so maximum efficiency can be achieved at a total phase difference of 240 degrees with an interval of 120 degrees.
Figure 5.3.1. FDTD simulation results. (a) Under TM-polarized incident plane wave, the optical responses of the polaritonic metasurface with change of external DC voltage bias (V1) from -2 V to +2 V with +1 V (across +50 kV/cm in MQWs) step.
The polaritonic reflection spectra added vertically by 0.5 value each to better display visually. The red dashed lines indicate the position of the polaritonic double absorption peaks. (b) The polaritonic phase responses under the external voltage bias.
The distance between red dot points indicates the phase difference of 180 degrees for the condition of V2a = -2 V and V2b = +2 V or vice versa, and red dashed line points an operating wavelength of beam diffraction. For the condition of V3a = -1 V and V3b = +1 V or vice versa, the distance between blue dot points indicates the phase difference of ±120 degrees from the phase at zero bias, and blue dashed line points an operating wavelength of beam steering.
-2 V
4 5 6 7 8 9 10
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
Phase (p)
Wavelength (μm)
-2 V -1 V 0 V +1 V +2 V
(a)
4 5 6 7 8 9 10
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Reflection
Wavelength (μm)
(b)
-1 V 0 V +1 V +2 V
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Figure 5.3.2 (a) shows the beam diffraction intensities normalized to each peak value. Under the voltage condition of V2a = V2b = 0 V (bottom panel), only the 0th order beam appears, and the beam diffraction intensities for ±1st order or more are almost suppressed due to the unit grating cavity in size of the subwavelength scale compared to an operating wavelength of 6.55 μm. For the beam diffraction simulation, the metasurface is applied to an electrical bias condition of V2a = -2 V and V2b = +2 V. This condition represents the phase difference of 180 degrees at an operating wavelength, efficiently supporting beam diffraction signals for both side lobes. It shows the possibility to switching beam diffraction intensities by inducing the phase difference through an external bias. In this condition, the phase difference of 180 degrees can be obtained, but the 0th order diffraction beam intensity cannot be completely suppressed because the reflection intensities are different under the different voltage conditions, unlike the metasurfaces based on the PB phase elements. The diffraction angle is defined as the grating equation.
𝜃𝑚= sin−1(sin 𝜃𝑖±𝑚𝜆
Γ𝑠)
(5.2) Where the θi, θm, m, and Γs means the incident angle, the diffraction angle, and diffraction order, and the length of superlattice, respectively. The diffraction angle for ±1st order is calculated as ±24.8 degrees.
Figure 5.3.2 (b) shows the beam steering intensities normalized to each peak value. For the voltage condition of V3a = V3b = 0 V (bottom panel), as in the previous beam diffraction, only the 0th order beam of the scattered patterns is characterized. For the selective beam steering simulation, the metasurface is applied to an electrical bias condition of V3a = -1 V and V3b = +1 V (top panel), which achieve the phase difference of ±120 degrees at an operating wavelength of 6.37 μm, efficiently supporting beam steering response on the left side nearly. For the opposite voltage condition of V3a = +1 V and V3b = -1 V (middle panel), the beam steering is characterized almost on the right side by an opposite phase gradient sequence which form an opposite wavefront. Actually, in the case of a beam steering based on a ternary digit system that we employ, a phase difference of up to 240 degrees with an interval of 120 degrees is required to achieve the perfect beam steering on the one side only. Figure 5.3.2 (c) shows the relative phase sequence for beam diffraction and selective beam steering under different voltages. The superlattice of both metasurfaces is equally designed to 15.6 μm (Γs = Γg). The diffraction angle at normal incidence is defined as the generalized Snell’s law.33
𝜃 = ±sin−1(𝜆
Γ𝑔)
(5.3)
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Figure 5.3.2. The simulated farfield profiles as a function of the polar angle for beam diffraction (a) and selective beam steering (b). Each diffraction intensities are normalized to each peak intensity value. The farfield profile for beam diffraction is characterized with the condition of V2a = -2 V and V2b = +2 V (top panel) at an operating wavelength of 6.55 μm, and the farfield profile for beam steering is characterized with the condition of V3a = -1 V and V3b = +1 V (top panel) or vice versa (middle panel) at an operating wavelength of 6.37 μm. Superlattice of both metasurfaces are Γs = Γg = 15.6 μm. (c) Relative phase sequences for electrical tunable beam diffraction (left), selective beam steering (middle and right).
(a)
-40 -20 0 20 40
0.0 0.5 1.0 0.0 0.5 1.0
Polar Angle (degree)
V2a = 0 V V2b = 0 V
Normalized Intensity
V2a = -2 V V2b = +2 V
-40 -20 0 20 40
0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0
Polar Angle (degree)
V3a = 0 V V3b = 0 V
Nomalized Intensity
V3a = +1 V V3b = -1 V V3a = -1 V V3b = +1 V
(b)
Voltage (V)Phase (degree)
0Γs 0
Voltage (V)Phase (degree)
0
Voltage (V)Phase (degree)
0
1Γs 0Γg 1Γg 0Γg 1Γg
1 1 1 1 1 1 0 0 0 0 0 0 2 2 2 2 1 1 1 1 0 0 0 0
V2a> 0
V2b< 0
V3b> 0 V3= 0
V3a< 0 V3= 0 V3b< 0
V3a> 0
(c)
0 0 0 0 1 1 1 1 2 2 2 2
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