# Holographic Fabrication and Optical Property of Graded Photonic Super-Crystals with a Rectangular Unit Super-Cell

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Description of Experimental Methods and Formation of Graded Photonic Super-crystal

^{2}with 1920 × 1080 pixels. The pixel size of the SLM is 8 × 8 μm

^{2}(“P” is the side length of a pixel square = 8 μm, which has been used in this paper). The laser is linearly polarized along the longer side of the active area and is incident onto the SLM with an incident angle of four degrees, relative to the normal of the SLM. As shown in Figure 2a, the diffracted beams from the phase pattern (Figure 2b) are displayed in the SLM and are collected through lens 1 and selected by a Fourier filter at the Fourier plane.

_{1}is the first order diffraction angle due to the periodic array of the gray levels of (128 and 254) or (190 and 254), and can be obtained by setting D = 2P in Equation (1). For a large period in x- and y-directions, D in Equation (1) equals 18P and 24P, and α

_{i}equals α

_{3}and α

_{2}, respectively. The distance L

_{sp}, L

_{x}, and L

_{y}between the +1 and −1 order diffractions due to small period (SP) 2P arrays, 18P arrays in x-direction, and 24P arrays in the y-direction, and can be measured in Figure 2, respectively. The diffraction condition in Equation (1) was tested by verifying L

_{sp}= 2 × f

_{1}tan(α

_{1}), L

_{x}= 2 × f

_{1}tan(α

_{3}), and L

_{y}= 2 × f

_{1}tan(α

_{2}), as shown in Figure 2a,c. Theoretically, L

_{sp}/L

_{x}= 9 and L

_{sp}/L

_{y}= 12. As measured in Figure 2c, L

_{sp}/L

_{x}= 9.20 and L

_{sp}/L

_{y}= 12.23. The agreement between the measured and theoretical values is high, indicating the correct assignment for D.

_{1}= 400 mm and f

_{2}= 200 mm for lens 1 and lens 2, respectively, were used. The 4f setup, as shown in Figure 2a, was used. The graded photonic super-crystal was fabricated by exposing the dipentaerythritol penta/hexaacrylate (DPHPA) mixture to the inference pattern with similar spin-coating, exposure, and development conditions, as in reference [7,10,11].

## 3. Results

#### 3.1. Holographic Fabrication Results

_{1}and θ

_{2}(zenith angle) are the interfering angles of the outer beam and inner beams in Figure 2, respectively; 45° and β are the azimuthal angles for outer and inner beams in Figure 2, respectively; and $\varphi $ is the initial phase of the beam. When the eight beams are overlapped, the intensity distribution in the interference pattern is determined by the following:

_{s}= 2π/(k sin(θ

_{1}) cos(45)), where θ

_{1}is determined by tan(θ

_{1}) = f

_{1}tan (α

_{1}) ×$\text{}\sqrt{2}$/f

_{2}. Thus, Λ

_{s}= (f

_{2}/f

_{1}) 2P. The period Λ

_{x}in x-direction in the interference among beams 5–8 is different from period Λ

_{y}in the y-direction, and are calculated as follows: Λ

_{x}= 2π/(k sin(θ

_{2}) cos(β)) and Λ

_{y}= 2π/(k sin(θ

_{2}) sin(β)), where sin(θ

_{2}) cos(β) ≃ (Di/f

_{2})(0.5 L

_{x}/Di) and sin(θ

_{2}) sin(β) ≃ (Di/f

_{2})(0.5 L

_{y}/Di), as shown in Figure 2c,d. Thus, Λ

_{x}= (f

_{2}/f

_{1}) 18P and Λ

_{y}= (f

_{2}/f

_{1}) 24P.

_{x}, Λ

_{y}) is labeled for the size of the unit super-cell in the holographic structure in Figure 3a. Although both the square (12 × 12) and rectangular (6 × 12) sub-unit cells are used in Figure 2b, the sub-unit cell, as indicated by the dashed red line, is in a rectangular shape in Figure 3a. The unit super-cell in Figure 3a has a ratio of length over width of Λ

_{y}/Λ

_{x}= 24/18, which is obtained from (12 + 12)/(12 + 6), in Figure 2b. The design of the phase pattern is flexible for obtaining a rectangular unit super-cell in a holographic structure. For example, the unit super-cell in the phase pattern can have a sub-unit of k × k and k × m pixels. The obtained holographic structure can have a rectangular unit super-cell with a ratio of side lengths defined by 2k/(k + m).

#### 3.2. Simulation of Light Extraction Efficiency

_{max}were used. The results for the dipole polarization of E in the y-, x-, xy-, and z-directions are shown in Figure 5a–d, respectively. Overall, the extraction efficiency is between 70% and 80% for the dipole polarization in the xy plane, except for the exposure threshold of 30% I

_{max}for the E

_{y}dipole polarization. The extraction efficiency is 86.8% for the E

_{y}dipoles for the 30% I

_{max}threshold, while they are 76.6% for the E

_{x}dipoles at 524 nm. It is reasonable, because the graded intensity is modulated in a higher number of steps along a length of the rectangle in the y-direction, rather than the x-direction. If a polarization effect is needed in an OLED [30], a grade photonic super-crystal with a rectangular unit super-cell can help to reach the goal. For the laser exposure threshold of the 30% I

_{max}, the maximum extraction efficiency is at the infrared range beyond 800 nm. Further study in the discussion section will shift the wavelength for the maximum efficiency toward the visible range. The extraction efficiency for the E

_{z}dipoles in Figure 5d is not significantly dependent on the exposure threshold, as the groove depth of 40 nm in Figure 4a is fixed in the simulation. An overall light extraction efficiency, ρ, can be calculated using the average efficiency of the dipoles polarized in x − y (parallel dipole) and z (perpendicular dipole) [8,29], as follows:

_{x}

_{−y}and ρ

_{z}are also the average efficiency for ten dipoles. The extraction efficiency is calculated to be 71.5% at 563 nm and 73.6% at 633 nm. Under all of the exposure threshold conditions, the lowest overall extraction efficiency is 64.9% at 434 nm in the visible range.

## 4. Discussion

_{z}dipoles occurred at 400 or 760 nm. The maximum efficiency of the E

_{z}dipoles in Figure 5 was in the infrared range beyond 800 nm. If we scale down the lattice period from 1000 nm to 750 nm, the wavelength for the maximum efficiency is expected to be shifted from 800 to 800 × 750/1000 = 600 nm. Figure 6 shows the simulated extraction efficiency for OLED, where the cathode is patterned with a graded photonic super-crystal with a sub-unit cell size of 9a × 12a (a = 750 nm) with a similar groove depth of 40 nm. There are no large differences in the extraction efficiencies for the E

_{x}, E

_{y}, and E

_{xy}dipoles in Figure 6a. As shown in Figure 6b, the wavelength for the maximum extraction efficiency is 633 nm, close to the expected wavelength. In the range between 608 and 690 nm, the extraction efficiency is above 75%.

_{eff}is the effective refractive index of the graded photonic super-crystal, λ is the wavelength in free space, and R is the reciprocal lattice vector. The effective refractive index is related to the filling fraction f by Equation (13), as follows:

_{metal}and n

_{org}are the refractive index of the metal and organic material in the graded photonic super-crystal, respectively. Due to the graded filling fraction in the graded photonic super-crystal, the coupling condition can be met by many wavelengths simultaneously.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Image of a graded photonic super-crystals: the lattices can be grouped by blue and red dots. The size of the basis becomes smaller along the blue and red arrows, and then becomes larger after a quarter of “Period 2”. The filling fraction of the dielectric material is higher inside the dashed green rectangle than the solid green rectangle. The graded photonic super-crystals have a unit super-cell as indicated by Period 2 in x-direction.

**Figure 2.**(

**a**) Schematic of the optical setup for the holographic fabrication. The spatial light modulator (SLM) is used to display the phase patterns. The diffracted beams from the SLM are filtered at the Fourier plane and form interference patterns through a 4f imaging system of lens 1 and lens 2. α

_{1}, α

_{2}, and α

_{3}are the first order diffraction angles due to the periodic array of 2P (“P” is the side length of a pixel square = 8 μm, which has been used in this paper) pixels in x and y directions, 24P pixels in y direction, and 18P pixels in x-direction in the phase pattern in (

**b**), respectively. θ

_{1}and θ

_{2}(zenith angle) are the interfering angles of the outer beam and inner beams in (

**c**), respectively. (

**b**) An enlarged view of the designed phase patterns. A unit super-cell is indicated by the dashed red square. Inside the unit-cell, there are two 12 × 12 square pixel patterns and two 12 × 6 square pixel patterns. The gray levels of (190 and 254) correspond to the dashed green and blue regions, while (128 and 254) correspond to the remaining regions inside the unit super-cell. (

**c**) The laser diffraction pattern from the phase pattern in (

**b**) at the Fourier plane. A Fourier filter is used to allow the diffraction spots inside the red circles passing through. (

**d**) Schematic of eight beams corresponding to the outer and inner beams in (

**c**) for the interference lithography. β is an azimuthal angle for one of inner beams in (

**c**).

**Figure 3.**(

**a**) Simulated eight-beam interference pattern; (

**b**) the charge-coupled device (CCD, attached to an optical microscope) image of the fabricated graded photonic super-crystal in dipentaerythritol penta/hexaacrylate (DPHPA); (

**c**) diffraction pattern of a fabricated sample from 532 nm laser.

**Figure 4.**(

**a**) Schematic of the organic light emitting device (OLED) where the cathode (Al) is patterned with the graded photonic super-crystal that has a rectangular unit super-cell; (

**b**) output of the structure design from the simulation software MIT Electromagnetic Equation Propagation (MEEP); (

**c**) electric-field intensity in the glass substrate in OLED at the location 740 nm away from indium tin oxide (ITO) layer; (

**d**) the fraction of the total emitted power in glass substrate, both in the surface plasmonic mode and in the waveguide.

**Figure 5.**(

**a**) Fraction of total emitted power (light in glass substrate over total emitted power) as a function of the wavelengths for different exposure thresholds of 25% I

_{max}, 30% I

_{max}, 35% I

_{max}, and 40% I

_{max}for the E

_{y}dipoles (

**a**), E

_{x}dipoles (

**b**), E

_{xy}dipoles (

**c**), and E

_{z}dipoles (

**d**).

**Figure 6.**Fraction of total emitted power as a function of wavelengths with an exposure threshold of 35% I

_{max}for the E

_{x}, E

_{y}, and E

_{xy}dipoles (

**a**), and the E

_{z}dipoles (

**b**) for the OLED patterned with the graded photonic super-crystal with a sub-unit cell size of 9a × 12a (a = 750 nm).

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**MDPI and ACS Style**

Hassan, S.; Sale, O.; Lowell, D.; Hurley, N.; Lin, Y.
Holographic Fabrication and Optical Property of Graded Photonic Super-Crystals with a Rectangular Unit Super-Cell. *Photonics* **2018**, *5*, 34.
https://doi.org/10.3390/photonics5040034

**AMA Style**

Hassan S, Sale O, Lowell D, Hurley N, Lin Y.
Holographic Fabrication and Optical Property of Graded Photonic Super-Crystals with a Rectangular Unit Super-Cell. *Photonics*. 2018; 5(4):34.
https://doi.org/10.3390/photonics5040034

**Chicago/Turabian Style**

Hassan, Safaa, Oliver Sale, David Lowell, Noah Hurley, and Yuankun Lin.
2018. "Holographic Fabrication and Optical Property of Graded Photonic Super-Crystals with a Rectangular Unit Super-Cell" *Photonics* 5, no. 4: 34.
https://doi.org/10.3390/photonics5040034