Three different nano-grating structures are designed as phase retarders that can transform linearly polarized light to circularly polarized emission for the wavelengths of 488 nm, 532 nm and 632.8 nm, respectively. Gold based nano-grating structures with various periods are fabricated by utilizing laser interference lithography. The ellipticity of all circularly polarized emission can reach around 90% such that the structure has great potential in the applications of three-dimensional (3D) display. The effects of the slit width and metal thickness modulations are simulated by rigorous coupled wave analysis (RCWA) method. Besides, the field intensity and phase of the transmitted TM and TE waves are also simulated to understand their polarization characteristics.
© 2014 Optical Society of America
The polarized light has been applied in scientific studies of photo-chemical or photo-biological reactions [1–3] recently, and attracts much attention for its special role in three dimensional (3D) liquid crystal display. In the past, linearly polarized emission light with large polarization ratio can be obtained by passing an unpolarized light through a grating structure [4–8]. The grating structure functions as a polarizer to select the transverse magnetic (TM) wave and reflect transverse electric (TE) wave in the wavelength range of 400-700 nm . However, circularly polarized light have drawn much attention recently due to its special optical characteristics [10,11]. The electric field of the circularly polarized light does not change amplitude but only changes direction in a rotary manner. As a result, it cannot be filtered out by only using a linear polarizer. To filter out a right-handed circularly (RHC) polarized light from a left-handed circularly (LHC) polarized light, a quarter-wave plate and an associated linear polarizer are needed. The selective transmission of RHC and LHC polarized light are considered important in the realization of the 3D display. Previously, many efforts have focused on designing quarter-wave plates by dielectric subwavelength gratings [12–18]. However, the design based on metallic subwavelength gratings is rare . In this paper, the laser interference method is used to fabricate a metallic nano-grating structure on the glass. The gold based nano-grating structures with various periods and slit widths are designed as quarter-wave plates which can transform linearly polarized light to circularly polarized light at the wavelengths of 488 nm, 532 nm and 632.8 nm, respectively. The corresponding red, green and blue circularly polarized emission light can be directly used in laser projector 3D system due to the great thermal durability of nano-grating as compared to polymer materials which have been widely used to be quarter wave plates in 3D display system. It is noted that the metallic grating sample is able to endure high temperature (200 °C) without any damage. Moreover, the detailed transmission spectra, phase and amplitude distribution of the transmitted TM and TE waves are simulated by using commercial rigorous coupled wave analysis (RCWA) software (Rsoft, DiffractMOD 9.0). The ellipticity of the obtained circularly polarized light can be designed to reach around 90% and the cross-talk of those are lower than 7% which is low enough for the 3D display applications [20–22].
Figure 1(a) displays the schematic of the nano-grating structure and the propagation properties for the TE and TM waves. The incident wave is located in region I at point z = 0µm and the transmitted waves are located in region IV at point z = 302 µm. The physical definition of the ith order transmission (Ti) is the ratio of the power density of ith order transmitted wave to that of the incident wave. To calculate the transmission using software (e.g. Rsoft), the simulated model will be first simplified into three mediums (region I~III). The permittivity in the grating region (II) can be expanded in a Fourier series and expressed as
After defining the electric field in each region, the software solves the Maxwell-Faraday equation in all regions, resulting in a series of coupled differential equations . Using the known boundary conditions, the transmitted and reflected waves can be solved. For analysis of the incident TM wave case, the normalized total electric field (Ef) of the forward diffracted waves can be expressed as
The transmission (Ti) of the ith order transmitted wave can be calculated by24] to solve the transmission in region IV, and thereby the ith order transmission at point z = 302 µm can be obtained.
In order to design a phase retarder at certain wavelength, the 0th order transmission intensities of TM and TE waves are required to be close. However, before choosing the slit width, the period and metal material of nano-grating structure need to be decided. Wood’s anomaly which is related to the grating period and the effective refraction index will result in dips   in the spectrum and thereby need to be avoided at the targeted wavelength. Therefore, grid periods of 600 nm (sample A), 300 nm (sample B) and 500 nm (sample C) are chosen to achieve the targeted wavelengths of 488 nm, 532 nm and 632.8 nm, respectively. To obtain a large birefringence characteristic, the metal material of nano-grating structure are chosen to be gold and the thickness(d) is initially set to be 150 nm . Table 1 lists the detailed parameters of nano-grating structure including the period(a), grid width(w), slit width(s) and targeted wavelength(λ).
Figure 2 show the simulation results of 0th order transmission spectra for both TE and TM waves under different slit widths. In Fig. 2(a), the transmission intensities of sample A at a wavelength 488 nm for TE and TM wave are close as the width of grating slit is chosen to be around 180 nm. From Figs. 2(b) and 2(c), the slit width of sample B at 532 nm and sample C at 632.8 nm should be selected to be around 200 nm and 280 nm, respectively, to obtain equal transmissions for TM and TE waves. However, the slit width of sample A and C are finally selected to be 200 and 300 nm for their larger birefringent characteristic. It is noted that the above phenomena are not predicted in the past since it is believed that the nano-grating structure will reflect the TE wave and only the TM wave can pass through . As a matter of fact, this view is only correct when the slit is narrow [7–9] and the TM wave still exhibits high transmission due to the propagation of surface plasma wave . Here, as the slit width is widened, the coverage of gold film decreases. Therefore, the transmission of TM and TE waves are both enhanced  .
Figures 3(a)–3(c) display the simulation results of 0th order transmission spectra and phase difference for samples A, B and C with various metal thicknesses at the wavelengths of 488 nm, 532 nm and 632.8 nm, respectively. The slit widths of samples A, B and C are 200, 200 and 300 nm, respectively. The phase difference (ΔΦ) is defined as the difference between the transmitted TE and TM waves in region IV as compared to incident wave in region I. The upper figures are the transmission spectra and the lower figures are the phase difference. For the grating here working as birefringent material, the thickness of metal will determine the phase delay between TE and TM waves. The phase difference is described by , where is the magnitude of birefringence, λ is the incident wavelength and d is the metal thickness. In Fig. 3(a), the λ is 488 nm and d is selected to be 200 nm. As a result, the theoretical is around and the corresponding was 0.61. It’s noticed that the result will be even better if the metal thickness are selected to be 210 nm. However, the thickness of the photoresist limits the maximal thickness of gold layer . It is hard to lift off the metal when the thickness of metal is above 150 nm and the critical thickness in our work is 200 nm. In Fig. 3(b), the λ is 532 nm and d is selected to be 160 nm. Therefore, the theoretical was 0.83. In Fig. 3(c), the λ is 632.8 nm and d is selected to be 160 nm. As a consequence, the theoretical was 0.99 which is well matched to the previous reported value . Again, it’s noticed that we also used finite-difference time domain (FDTD) method to simulate the phase information of nano-grating structures. The phase difference of TM and TE modes for samples A, B and C are 85°, 92° and 90° respectively, which are close to the simulated results from RCWA method.
To identify the theoretical ellipticity of the transmitted light from the nano-grating in region IV, the electric field amplitude of TE and TM waves in all regions are simulated. Figure 4(a) displays the schematic of the nano-grating structure and the propagation properties of the TE and TM modes. The basic definition of the direction of electric field, the propagation direction of the incident wave, and diffraction angles are illustrated. Figures 4(b)–4(d) show the electric field amplitude for the TE and TM waves of samples A, B and C in all regions. In Figs. 4(b)–4(d), sample B is the truly subwavelength grating while Sample A and sample C have 3 transmitted propagating orders (−1, 0, 1) in the glass (region III). The mode number and the diffraction angle for the each sample can be identified by Eq. (4). For normal incidence in sample B, the value of m can only be zero and thereby only the 0th order transmission exists. As for sample A and sample C, the value of m ranges from −1 to 1, and therefore 3 transmitted propagating orders (−1, 0, 1) exist. We also calculate the diffraction angles of sample A and sample C. For m = 1, the diffraction angle of sample A and sample C are around 34 o and 60 o, implying that there will be a total internal reflection at the interface between region III and region IV for sample C, but for sample A the non-zeroth order transmission will still exist in region IV, which is well matched with our simulated results.
The non-zeroth order transmission for sample A will be cut off by the aperture when the grating samples are used in laser projection system. Therefore, we only consider the transmission and phase difference of the 0th order transmission.
The amplitude distributions between the slit are compared at the same relative position in region IV of each sample to calculate the ellipticity. From the amplitude of TE mode, Ey, and Ex from the TM mode. The theoretical ellipticity of circularly polarized emission should be close to the ratio of transmission (RT), which is calculated by (), where , are the 0th order transmission of TM and TE modes. It is noticed that the is the physical definition [28,29] but RT indicates the cross-talk performance which is more important in the 3D display system. In addition, it’s obvious that the TE mode results in peak intensity at the middle of slit and the TM wave generates peak intensity at the boundary which agrees with previous studies .
Figures 5(a)–5(c) show the associated phase of the TE and TM waves for sample A, B and C, respectively. It can be seen that the phase delay only happen at the grating region due to the birefringent characteristic. The phase information of the TE and TM waves are extracted at point z = 302 µm to calculate the phase difference used in Figs. 3(a)–3(c). For sample A, the ellipticity is 87.1% and RT is 94.3%. For sample B, ellipticity is 91.2% and RT is 99.4%. For sample C, ellipticity is 99.8% and RT is 99.6%. The further corresponding cross-talk is measured with the associated polarizer and quarter-wave plate to identify the possibility for 3D applications. It is noted that the phase difference and the ellipticity calculated from region IV are close to results from region III, which make sense as both glass and air are isotropic mediums.
Figures 6(a)–6(c) show the scanning electron microscope (SEM) image of the nano-grating structure with various periods and grid widths. The nano-grating patterns are first formed by laser interference lithography on an anti-reflective coating (ARC) layers. The ARC (XHRiC-11, Brewer Science) layers are deposited to prevent the photo-resist pattern from the damaging of the reflected light . Then the photo-resist patterns and AR-coating are etched by the reactive ion etching (RIE) process. Chromium (Cr) as adhesion layers (2 nm) and gold layers are evaporated and lifted-off on the sample. Finally, the samples are put into the H2O2-NH4OH-H2O solution at 80°C for 5 minutes to remove the ARC layers. To further identify the parameters of nano-grating structures, the samples are measured by atomic force microscopy (AFM) as shown in the inserted of Fig. 6. Sample A is a nano-grating with a period of 600 nm, a metal thickness of 200 nm and a grid width of 400 nm. Sample B is a nano-grating with a period of 300 nm, a metal thickness of 160 nm and a grid width of 100 nm. Sample C is a nano-grating with a period of 500 nm, a metal thickness of 160 nm and a grid width of 200 nm.
4. Results and Discussion
Figure 7 shows the setup for measuring ellipticity of the circularly polarized emission light. The testing optical components are arranged according to Fig. 7(a). In Fig. 7(a), the linearly polarized light composed of a half of TE and TM waves are incident onto the nano-grating structures. The intensities of the light with various angles can be measured as the polarizer rotates from θ = 0 to 360 degrees. As a result, the ellipticity can be calculated from Figs. 7(b)–7(d). In addition, in Figs. 7(b)–7(d), the black lines represent the ideal circularly polarized light which exhibits the same amplitude with various angles. The red lines are the experimental results which exhibits the similar optical characteristic as the black line. The experimental ellipticity of samples A, B and C are 96%, 98% and 97%, respectively. Here, the slight difference between the theoretical and experimental results may be due to the deviations of each optical component.
Figure 8 shows the setup for measuring cross-talk of the circularly polarized light. The testing optical components are arranged according to Fig. 8(a). The 0th order transmitted light from sample will be transformed to circularly polarized light and the following quarter wave plate will again transform the circularly polarized light back to linearly polarized light which can be tested by the last polarizer. The unexpected light leakage will become the noise in 3D system [12–14], and therefore the cross-talk can be measured and calculated by (Ileakage/Icorrect), where Ileakage is equal to I (θ = 0) and Icorrect is equal to I (θ = 90). Figures 8(b)–8(d) show the intensities of the light with various angles. Again, the black lines are the ideal cases and the red lines are the experimental results. The polar figures exhibit the linearly polarized properties and the cross-talk of samples A, B and C are 3%, 4%, and 7% respectively, able to be used in commercial 3D display system [20–22] as a polarized source or glass.
Table 2 lists the optical performance of nano-grating structure in detail. By using sample A, B and C, the average cross-talk of 3D system is 4.67%. To further understand the relation between the image quality and cross-talk, a 3D display system with the polarized glass is modeled in LightTools, an optical modeling software based on Monte Carlo ray trace method. The left and right visions are taken from 3D camera which has two shots. The mixed images of two visions are combined by using MATLAB and display in LightTools as shown in Fig. 9(a). The effects of cross-talk are taken into account on the materials attached periodically to the display in the model. The simulated visions for right and left eyes are shown in Figs. 9(b) and 9(c), respectively. It can be seen that 3D system with cross-talk of 5% can form the clear visions in both eyes which agree with previous researches [20–22]. However, in our simulation, the visions will become unclear as the cross-talk increases from 5% to 10%.
In conclusion, nano-grating structures as the phase retarder are designed step by step, and fabricated by using laser interference lithography. The effects of the slit width and metal thickness modulations are simulated and studied. The polarized characteristics of transmitted light are estimated theoretically and experimentally. The theoretical ellipticity of circularly polarized emission for all samples can reach around 90% and the cross-talk of those of samples are smaller than 7%, which implies great potentials in 3D display system.
The authors would like to thank the National Science Council of the Republic of China, the Center for Emerging Materials and Advanced Devices, and the Photonic Advanced Research Center of the National Taiwan University, for financial support under contracts of NSC 100-2120-M-002-014, 10R80908-4; 10R7b07-4, NSC 102-2120-M-002-003-, 103R890942, 100-2221-E-002-054-MY3; NSC-102-2221-E-002-205-MY3, 102R70607-4; 102R3401-1, NTU-CESRP-102R7607-2 and NTU-ICRP-102R7558, 100-2221-E-002-161-MY2.
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