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• Abstract

SECTION I

## INTRODUCTION

A diffraction grating acts as a prism and disperses different wavelengths of light into a spectrum. The grating spacing or groove determines the dispersion angle. Using a commercial replicated holographic grating of 2400 grooves/mm $(\hbox{dispersion} = 0.33\ \hbox{nm/mrad})$ as an example, the dispersion angle from a light wavelength of 400 to 700 nm is 52°. Recently, numerous studies have focused on using photonic crystal structures to create a large-angle light dispersion phenomenon [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. They have reported a maxima dispersion angle of approximately 50 ° [13].

This paper presents a 68° white-light dispersion angle (360 to 700 nm) using 2-D subwavelength gratings. The hexagonal gratings of a 500-nm period were fabricated using a holographic polymer-dispersed liquid crystal (H-PDLC) technique that used a single prism as an optical interference setup [14]. Typically, H-PDLC gratings do not provide sufficient refractive index (RI) contrast when fabricating 2-D or 3-D subwavelength periodic structures. However, adding a moderate amount of solvent increases the RI contrast to 0.2 because of the RI difference between the polymer and nanopore regions [15]. The results of this paper offer alternatives for fabricating low-cost efficient optical components used for spectroscopic purposes.

SECTION II

## EXPERIMENT

Fig. 1 shows the optical setup for the holographically fabricated samples. A specially designed prism was used to achieve three-beam interference from single laser exposure. A description of the specially designed prism and corresponding interference patterning technique can be found in [14]. The holographic writing laser was expanded and collimated to 40 mm in diameter and exposed to the top of the prism. The 3-$\mu \hbox{m}$-thick sandwiched sample was placed on the bottom of the prism using index-matching oil. The prepolymer syrup consisted of 50-wt.% monomer, dipentaerythritolhydroxypenta acrylate; 20-wt.% toluene; 7-wt.% cross-linking monomer, N-vinylpyrrollidone; 1-wt.% photoinitiator, rose bengal; and 2-wt.% coinitiator, N-phenylglycine, procured from Sigma-Aldrich, and 20-wt.% nematic LC, MDA3461 procured from Merck (Taiwan). The sandwiched sample was exposed to a 514-nm-Ar+ laser for 3 min and cured under a table light for 24 h. During holographic interference patterning, the monomer fluids were cross-linked and became polymer in the high light-intensity regions; at the same time, the LC/toluene mixture were phase-separated in the low light-intensity regions. The sandwiched sample must be opened to allow the solvent vapor to evaporate to achieve optical characteristics. Fig. 1(d) shows an image of the fabricated structure under an optical microscope in dark-field operation mode. Dark-field operation enhances the image contrast between polymer (dark) and nanopore (bright) regions.

Fig. 1. Diagrams of (a) the optical setup, (b) the specially designed prism $(\phi = 60^{\circ})$, and (c) the three beam interference angle ($\theta = 25.3^{\circ}$ and $\varphi = 60^{\circ}$) created by single writing laser exposure. (d) Dark-field optical image of the fabricated 2-D hexagonal periodic structure.
SECTION III

## RESULTS AND DISCUSSION

Fig. 2(a) shows the schematic setup for recording dispersion angles and spectra from the 2-D subwavelength transmission grating, and Fig. 2(b) shows the corresponding dispersion spectra images at different white-light incident angles $(\theta)$. The dispersion angle is calculated by measuring the dispersion distance (d) from 360 to 700 nm. The dispersion angle increases when the incident angle increases, reaching a maximal value of 68° at an incident angle of 20°. The dispersion angle decreases when the incident angle is larger than 20°, and the dispersed color vanishes when the incident angle is larger than 60°. This dispersion behavior is similar to that of commercial holographic gratings, where the dispersion angle increases with the incident angle and then decreases (see Fig. 3). The holographic transmission grating dispersion angle is calculated using the following equation: TeX Source $$\beta (\lambda) = \sin^{-1}\left({\lambda_{i} \over \Lambda} - \sin \alpha \right)\eqno{\hbox{(1)}}$$ where $\beta(\lambda)$ is the dispersion angle corresponding to incident wavelength $\lambda{\rm i}$, $\alpha$ is the incident angle, and $\Lambda$ is the grating period. Using a 1-D transmission grating with a period of 522 nm as an example, the calculated dispersion angles corresponding to 360 and 700 nm wavelengths are approximately 20° and 90 °, respectively, at an incident angle of 20 °. The calculated dispersion angle for a 1-D grating between 360 and 700 nm is 70°, which is similar to the proposed 2-D hexagonal grating.

Fig. 2. (a) Experiment setup for measuring the white light dispersion angle using the 2-D subwavelength grating. (b) Color dispersion images from unpolarized white light (probe beam) recorded at different probe beam incident angles. The inset shows the dispersed spectra from 360 to 700 nm at different positions.
Fig. 3. Tunable dispersion spectra of 2-D subwavelength holographic transmission gratings dependent on the incident angle of unpolarized white light when the detector is fixed at position 1, as shown in Fig. 2(b).

Dispersion spectra shifts were recorded at Detector-Fixed Position 1 (see Fig. 2) by changing the probe beam incident angle. The peak position was tuned by 90 nm between a sample rotation of 25° and 50°, providing a spectral resolution of 3.6 nm per degree.

The transmission spectra for the sample at different probe beam incident angles were recorded and analyzed, as shown in Fig. 4. The experiment results in Fig. 4(a) and the simulated results in Fig. 4(b), obtained using DiffractMOD (RSoft Co.), show that the RI of the fabricated structure is 0.2, which is the same as that obtained from previously fabricated 1-D reflection gratings. A grating period of 522 nm was used in the simulation. By considering the scattering loss from the H-PDLC samples, the simulated results are matched to the experimental results.

Fig. 4. (a) Measured transmittance responses and (b) simulated spectra for the 2-D hexagonal subwavelength transmission grating with different incident angles of s-polarized probe beam. The scattering loss was considered to be 40%, 50%, 55%, and 65% corresponded to rotating angle of 40°, 45°, 50 °, and 55°.
SECTION IV

## CONCLUSION

This paper demonstrates large-angle color dispersion using a 2-D subwavelength holographic transmission grating. When an unpolarized white light is incident to the sample, a dispersion angle of 68 ° is observed within a continuous spectrum distributed from 360 to 700 nm. Rotating the sample tunes the detected spectrum and achieves a spectral resolution of 3.6 nm per degree. The results could be used for spectrometric applications. Future studies should investigate methods of tuning the spectrum without rotating the samples by adding or infiltrating photoactive molecules, such as azobenzene deriatives.

## Footnotes

This work was supported in part by the National Science Council, Taiwan, under Project 100-2628-E260-003-MY3 and in part by AOARD under Project FA2386-12-1-4023. Corresponding author: V. K. S. Hsiao (e-mail: kshsiao@ncnu.edu.tw).

Y.-Y. Lo, Y.-C. Su, and V. K. S. Hsiao are with the Department of Applied Materials and Optoelectronic Engineering, National Chi Nan University, Nantou 54561, Taiwan.

J. Yu and Z. Chen are with the Key Laboratory of Optoelectronic Information and Sensing Technologies of Guangdong Higher Educational Institutes, Jinan University, Guangzhou 510632, China.

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