# Generation of over 1000 Diffraction Spots from 2D Graded Photonic Super-Crystals

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fabrication and Simulation Methods

_{th}, and air when I(r) > I

_{th}. A finite-difference time-domain (FDTD) open-source software tool MIT MEEP program [24] was used for the simulation of the reflection from Al GPSC and of E-field. The simulations were performed through cloud-based parallel computations at Amazon Web Service using 36-core virtual machines at the Simpetus Electromagnetic Simulation Platform. For the reflection, a broadband plane wave was used. For the E-field, we used 532 nm wavelength.

## 3. Results

_{1}and others (blue lines Figure 1a) with a cone angle of 2θ

_{2}. The interference intensity as a function of location r can be calculated from Equation (1):

^{o}, sin(θ)sin(45 + n × 90)°, cos(θ)), n = 0,1,2,3, where θ = θ

_{1}for beams represented by red line and θ = θ

_{2}for those by blue lines in Figure 1a. When θ

_{1}= 22.1 degrees and wavelength = 532 nm, the small period Λ

_{S}equals approximately 2π/(2ksin(θ

_{1}) × sin45) = 1000 nm. The big period can be approximately estimated by Λ

_{L}= 2π/(ksinθ

_{2}$\sqrt{2}$) = 12,100 nm when θ

_{2}= 1.79 degrees in Figure 1a.

_{i}and ∆R

_{j}in Figure 1c. Although there is a gradual change ∆R

_{i}for two types of lattices in the supercell, the unit supercell in Figure 1b repeats itself periodically in the GPSC. The Bloch theorem is still valid for the supercell, however, fractional orders of Bloch waves may occur [18,25]. The first Brillouin zone is still determined by the lattice constant a and indicated by red dashed square in Figure 1d. The fractional orders of momentum in k-space are determined by the supercell size. For the unit supercell size of 12a × 12a in Figure 1b, the fractional orders are increased by β=2π/(12a) in k

_{x}-direction as shown in Figure 1d. The electromagnetic field in GPSC can be expanded as the sum of a series of Bloch waves, but the reciprocal lattice vector needs to include m × β (m is an integral number between −6 and 6) for the fractional order waves.

^{2}. The red dashed square indicates the unit supercell of 12a × 12a. Figure 2b shows the diffraction of 532 nm laser from the GPSC after the Al deposition. Diffraction orders of (−3, 1), (−2, 0) and (−1, −1) are labelled in the figure. There are 12 × 12 diffraction spots in Figure 2b in a square with vertices of (1,1), (1,−1), (−1,1) and (−1,−1), corresponding to the number of the fractional orders of momentum in k-space in Figure 1d. By comparing the measurement and calculation through Bragg diffraction equation, the pattern in Figure 2b can be understood by the diffraction from lattices with a period a and a weak coupling between two types of lattices indicated by the red and blue arrows in Figure 1c. We are not able to observe the first order diffractions of (±1,0) or (0,±1), indicating that the light scattering coupling between small and large holes in x or y-direction is weak. This effect might be due to the different phases of diffracted light from the small and large holes. This is similar to a case where a different phase was observed in diffracted light from a spatial light modulator coded with high and low grey levels in a checkerboard-like format [26]. Between the (1,0) plane and its next nearest neighbor with a plane distance of 2a in Figure 1c, second order diffraction occurs following Equation (2):

^{th}order of 5.36 and 8.30 cm, respectively, comparing with the measured values of 5.4 and 8.0 cm. The diffraction equations were tested by other sample with a period of 1/$\sqrt{2}$ = 0.707 μm. The calculated distances for (−1,−1) and (−2,0) spots were 8.30 and 15.08 cm, respectively, comparing with the measured data of 8.3 and 15.1 cm.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**(

**a**) Schematic of eight beams arranged in a cone geometry. (

**b**) Designed patterns for E-beam lithography for the generation of GPSC with a unit cell size of 12a × 12a. (

**c**) Enlarged view of the pattern in (

**b**) with labels of lattice planes. (

**d**) Fractional orders of momentum increasing by 2π/(12a) in the reciprocal space.

**Figure 2.**(

**a**) SEM of fabricated GPSC in PMMA. (

**b**) Diffraction pattern from the GPSC using 532 nm laser.

**Figure 3.**Simulated reflection as a function of wavelength from Al-coated GPSC with periods of 1000 and 400 nm.

**Figure 4.**(

**a**) E-field intensity in a cross section in yz plane. (

**b**–

**d**) E-field intensity in a cross section in xy plane through z-location at 1, 2, and 3 labeled in (

**a**), respectively.

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## Share and Cite

**MDPI and ACS Style**

Hassan, S.; Jiang, Y.; Alnasser, K.; Hurley, N.; Zhang, H.; Philipose, U.; Lin, Y.
Generation of over 1000 Diffraction Spots from 2D Graded Photonic Super-Crystals. *Photonics* **2020**, *7*, 27.
https://doi.org/10.3390/photonics7020027

**AMA Style**

Hassan S, Jiang Y, Alnasser K, Hurley N, Zhang H, Philipose U, Lin Y.
Generation of over 1000 Diffraction Spots from 2D Graded Photonic Super-Crystals. *Photonics*. 2020; 7(2):27.
https://doi.org/10.3390/photonics7020027

**Chicago/Turabian Style**

Hassan, Safaa, Yan Jiang, Khadijah Alnasser, Noah Hurley, Hualiang Zhang, Usha Philipose, and Yuankun Lin.
2020. "Generation of over 1000 Diffraction Spots from 2D Graded Photonic Super-Crystals" *Photonics* 7, no. 2: 27.
https://doi.org/10.3390/photonics7020027