# Azimuthally Varying Guided Mode Resonance Filters

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{th}diffracted order to propagate in the original direction. The period of the SWG is chosen such that all of the diffracted orders, except the 0

^{th}, are evanescent, meaning that the scale of the grating period is smaller than the incident light wavelength. The resonance of the GMRF is a function of the optical properties of the dielectric materials used: the thickness of the dielectric layers, the period of the SWG, and the fill-fraction (duty-cycle) of the SWG. The sensitivity of GMRFs to the various structural and optical parameters has been exploited to create various types of GMRFs where the resonance condition can be tuned by modifying one or more of these parameters [9,10,11,12]. The resonance of a GMRF can also be tuned by adjusting the angle of incidence of the light source [13]. In addition, the sensitivity of the resonance condition of GMRFs to the structural and optical parameters can be applied to sensors [14].

## 2. Simulation Results

_{x}N

_{y}) and Silicon Oxide (SiO

_{x}), respectively. A three-dimensional representation of a conventional GMRF, with a fixed spatial period and grating duty cycle, is shown in Figure 2. The period of the grating is 1.15 µm; the thickness of the silicon oxide is 240 nm; and the thickness of the wave-guiding layer is 375 nm. The SWG is etched 170 nm into the silicon oxide layer. The substrate is a 1 mm thick fused silica wafer.

**Figure 3.**RCWA results of a fixed period and duty-cycle GMRF. The R

_{x}and R

_{y}(and T

_{x}and T

_{y}) represent the reflection (and transmission) spectra from the polarizations in the x and y directions.

_{x}and R

_{y}(and T

_{x}and T

_{y}) represent the reflection (and transmission) spectra from the polarizations in the x and y directions, as shown in Figure 2, respectively. The slight difference in the spectra between the two polarization states is a result of numerical errors in simulating the unit cell geometry.

^{th}diffracted order can affect the properties of the reflected beam. Therefore, the decay length of the leaky modes was studied by approximating the hexagonal GMRF as a 1D ruled grating GMRF using effective medium theory. A one-dimensional structure was constructed with the same period and layer thicknesses as the original GMRF, but the fill-fraction and depth of the SWG was modified to match the behavior of the 2D GMRF. The method of lines was used to calculate the decay rate of the fundamental mode from the S

_{21}scattering coefficient [16]. Based on this analysis, the decay length was found to be 30 µm.

**Figure 5.**3D conceptual drawing of an azimuthally varying hexagonal GMRF. The hexagonal period, the SWG thickness, and the waveguide thickness are all uniform; however, the fill fraction of the lattice varies with the azimuth angle.

**Figure 7.**Simulated Transmission and Reflection Beam profiles. (

**a**) reflected beam for wavelength of 1,540 nm; (

**b**) transmitted beam for wavelength of 1,540 nm; (

**c**) reflected beams for wavelength of 1,545 nm; and (

**d**) transmitted beam for wavelength of 1,545 nm.

## 3. Fabrication Methods

_{4}) and an Ammonia (NH

_{4}) gas chemistry. The gas chemistry for the silicon oxide was Silane and Nitrous Oxide (N

_{2}O). The silicon nitride and silicon oxide were deposited to thicknesses of 375 nm and 240 nm, respectively.

_{3}) to oxygen (O

_{2}). The SWG was etched to a depth of 170 nm.

#### 3.1. Analog Lithography for Fill-Fraction Modulation

^{th}order to propagate, which provides the variation in the intensity of the light incident upon the photoresist as a function of the fill-fraction of the grating. The diffraction efficiency of the 0

^{th}order as a function of fill fraction is given by Equation (1) [19]. If the fill-fraction of the phase grating is slowly varied across the mask, then the relative intensity of the exposure will also be varied across the device on the wafer. Previous research has shown that this method, of creating an analog intensity profile, can be used to fabricate three dimensional optical elements, as well as, spatially varying the fill fraction or duty cycle of a grating [17,19].

**Figure 11.**(

**Left**) Desired analog intensity; (

**Right**) SEM of an hexagonal grating with an azimuthal fill fraction variation.

## 4. Experimental Results and Discussion

## 4. Conclusions

## Acknowledgments

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**MDPI and ACS Style**

Roth, Z.A.; Srinivasan, P.; Poutous, M.K.; Pung, A.J.; Rumpf, R.C.; Johnson, E.G.
Azimuthally Varying Guided Mode Resonance Filters. *Micromachines* **2012**, *3*, 180-193.
https://doi.org/10.3390/mi3010180

**AMA Style**

Roth ZA, Srinivasan P, Poutous MK, Pung AJ, Rumpf RC, Johnson EG.
Azimuthally Varying Guided Mode Resonance Filters. *Micromachines*. 2012; 3(1):180-193.
https://doi.org/10.3390/mi3010180

**Chicago/Turabian Style**

Roth, Zachary A., Pradeep Srinivasan, Menelaos K. Poutous, Aaron J. Pung, Raymond C. Rumpf, and Eric G. Johnson.
2012. "Azimuthally Varying Guided Mode Resonance Filters" *Micromachines* 3, no. 1: 180-193.
https://doi.org/10.3390/mi3010180