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We report on microcavities comprising para-magnetic garnet and electro-optic films (MPMEO) for modulation of the polarization rotation angle of light at near-UV wavelengths with a slight intensity change, with applying a low voltage. The MPMEO are composed of para-magnetic garnet and electro-optic films sandwiched between two Bragg mirrors. The microcavity states in MPMEO are split and yield both the large rotation angle and high optical efficiency. Significant enhancement and modulation by applied voltages are verified through a conventional matrix calculation approach. High optical efficiency (>90%) and large modulation (~90 degree) of the polarization rotation are proved.
©2013 Optical Society of America
1. Introduction
The non-reciprocal rotation of polarized light is one of the unique characteristics of magnetooptical (MO) materials that cannot be observed in other material systems. Hence, the MO is widely used and studied in optical isolator [1, 2] or modulators [3, 4] for optical circuit [5], laser processing [6], etc. From a macroscopic view, MO effects originate from a refractive index difference between right- and left-circularly-polarized (RCP and LCP) light beams in these materials [4]. The well-known MO material is rare-earth-substituted yttrium iron garnet (R:YIG), has a large refractive index difference (off-diagonal permittivity elements) caused by the Fe3+ ion transition in blue light and at shorter wavelengths (<450 nm) [7]. However, their large absorption hinder their potential use in devices, so for short-wavelength modulation bulk para-magneto-optic (para-MO) materials, e.g. terbium gallium garnet (Tb3Gd5O12, TGG) or terbium aluminum garnet (Tb3Al5O12, TAG), are extensively used [6, 8] because of high transparency in spite of the large external magnetic field. These para-MO bulk are several centimeter thick to obtain sufficient polarization rotation angle for device applications. This size requirement is one of the largest issues, so any short-wavelength modulation using para-MO materials in nano or micro scale devices have not been reported so far. In addition, a high temperature risk caused by a large external magnetic field generated by an electromagnet would occur in the case where the magnetization in the para-MO material is switched.
To address these two issues, we propose the use of microcavities with MO films, so called one-dimensional magnetophotonic crystals (MPCs) [9–11]. The microcavity modes are spectrally located inside the photonic band gaps and are associated with sharp transmission and reflection peaks in the spectra. These localized modes are strongly coupled to the magnetic constituents of the MPCs, and can enhance their linear and nonlinear MO responses because of the multiple reflections within the microcavities. Therefore, MPCs comprising para-MO materials can also enhance the MO responses; in other words, it can reduce the thickness of para-MO material. In actual, a microcavity comprising TGG was fabricated by sputtering and it showed ~50 times enhanced Verdet constant (ς = –8170 deg/T•cm) at a wavelength of 435 nm [12].
Second, to overcome the high temperature risk caused by a large external magnetic field generated by an electromagnet, MPCs with inserted electro-optic (EO) films are considered to be effective following theoretical [13] and experimental approaches [14]. The EO film stacked on the MO film can change its refractive index with the applied voltage, which can vary the existing mode not only inside the EO film but also inside the MO film because of the interference, resulting in a slight MO modulation with the applied voltage [15]. However in this case, the working wavelength is limited at the near-IR region because of the small rotation angle or the large absorption. To improve the performances, the use of the microcavity mode was proposed. The MO-EO microcavity shows high polarization angle modulation and low power consumption with a slight change in reflectivity. Such performance was achieved with the combination of non-reciprocity and spectral peak shift from the MO and EO effects [13]. A microcavity mode in MPC, shown as a spectral reflection peak within the photonic band gap, is split because of the MO effect, but a polarization rotation angle spectral peak is not. This results in high reflectivity with large rotation angle at the center of the split spectral reflection peaks, i.e. at the rotation angle spectral peak. The spectral positions for the whole peaks are shifted by EO effect with the applied voltage (the refractive index of the EO layer can be controlled with the voltage applied in parallel to the incident light direction). This is the working principle of the MO-EO microcavity.
In this paper, a para-magnetic garnet was used in the MO-EO microcavity. We optimized the structures and revealed the characteristics of microcavities with para-magnetic garnet and EO materials (hereafter MPMEO) as a pixel of spatial light modulators (SLMs) [16–19], Fig. 1, using a conventional calculation method: a 4 × 4 matrix approach [9]. The results given here allow us to conclude that the MPMEO can modulate short-wavelength light extensively with only a small voltage.
Fig. 1 Sketch of SLM with a microcavity comprising para-magnetic garnet and electro-optic layers (MPMEO).
2. Calculation method: 4 × 4 matrix approach
The optical and MO responses were calculated with a conventional 4 × 4 matrix approach. The incident light entering perpendicular to the film at z = z0 is linearly polarized along the x direction, Fig. 2, and has a plane waveform. Kato et al. described the detailed of this method in [20]. To deal with the combination of the para-MO and EO effects, several variables were modified.
2.1 MO material
We used TGG as a MO material in the calculations because of its large Verdet constant and low absorption. The permittivity tensor of the magnetized isotropic MO material is written as:
where with the refractive index n, the extinction coefficient κ (the complex refractive index). The refractive index difference between RCP and LCP lights is , and the extinction coefficient difference is .The Verdet constant ς and the refractive index difference have a proportional relationship, can be described like:
with the rotation angle θ, the wavelength of the light in a vacuum λ0, the film thickness d, the external magnetic field H. As the para-MO film, we used TGG, and the optical and MO parameters were obtained from [8, 21, 22]. TGG magnetized along the z direction (parallel to the light beam) is a uniaxial crystal with 4 (C4) point group symmetry. At a wavelength of 405 nm, the film’s Verdet constant is ς = –213 (deg/T•cm) [8]. Note that the direction of the positive Verdet constant is a clockwise rotation. The value of TGG was derived from Eq. (4) and (5), and was shown in Table 1. The external magnetic field was assumed as H = 10 kOe.
Table 1. Material parameters used in the calculations
2.2 EO material
We used lithium tantalate (LiTaO3, LTO) as the EO material because it has a large linear EO effect and a low extinction coefficient at the working wavelength [23, 24]. Unpolarized LTO is a uniaxial crystal with 3m (C3v) point group symmetry [25]. We assumed the c axis, i.e. extraordinary axis, of LTO was parallel to z axis, and LTO was polarized along the uniaxial z direction with the voltage applied along the z direction. In this study, the incident light is also parallel to the z axis, then the ordinary refractive index no of LTO in [23] was used, Table 1.
Yariv described the EO effects in [25]. The change in these coefficients by applied electric field E (where E1 = Ex, E2 = Ey, E3 = Ez) is
The rij is the 6 × 3 electro-optic tensor for the EO material. The EO tensor for LTO is:As mentioned above, the refractive index in the x and y directions are the ordinary one no, and the refractive index in the z direction is the extraordinary one ne. We choose the direction of the applied filed E parallel to the z axis, so Ex = Ey = 0. Then the refractive index ellipsoid isIn our configuration, the incident angle of the light beam is perpendicular to the surface, so we can neglect the effects from nz. The refractive index with applying field along the z direction Ez can be described as:With using a series expansion, this equation can be described as:which can be also described:with the applied voltage V in the z direction, and the LTO thickness dLTO. In the calculation, we used the Pockels constant r13 = 7.62 × 10−12 (m/V) at room temperature [26]. The Verdet constant of LTO (ς = –4.4 deg/T•cm) [27] was 100 times smaller than TGG, so in this study this was not taken into account.2.3 Structures
Figure 3 shows the model of the MPMEO used in this study. The microcavity has defect layers and two Bragg mirrors (BM) made of Ta2O5 (T) and SiO2 (S). The thicknesses of each layer in the BMs were set so that their optical thicknesses were equal to λ0/4 for the working wavelength of λ0 = 405 nm. The defect layer was composed of TGG and LTO films. Here, the optical thicknesses (physical thicknesses) of the TGG and LTO films were chosen to be 10 × λ0/2 (~1002 nm) and λ0/2 (~89 nm), respectively. The LTO film was sandwiched between two transparent indium tin oxide (ITO) electrodes. The optical parameters of T, S, and the substituted gadolinium gallium garnet [(GdCa)3(GaMgZr)5O12, SGGG] substrate were determined by ellipsometry measurements, Table 1. On the incident side of the substrate, there was an anti-reflection (AR) coating. The whole structure can be described like: AR/SGGG/(T/S)f/TGG/LTO/(S/T)r, with the repetition numbers f and r for front and rear BM. For simplicity in the present analysis, we did not take any change in the optical parameters of the transparent electrodes into account.
Two options were available for the use of the MPMEO: transmission or reflection type microcavities (hereafter T-type and R-type). To compare the two types, a figure of merit (FOM) was defined using the polarization rotation angle θ and whichever was the larger value of the light intensity I between transmissivity and reflectivity as:
Here the ellipticity of light was not included because of the fact that both transmitted and reflected light show the ellipticity of 0 at a resonant wavelength [28].3. Transmission and reflection types
The calculation results, with a matrix approach written in C++ language showed the transmission (or reflection), rotation angle, and FOM spectra. Figure 4 shows the results of the T-type and R-type MPMEO as functions of the repetition numbers f of the BMs. For T-type, f equals r, but for R-type, r is set at 20 so that all of the light can be reflected back in the incident direction. The absolute value of rotation angle is increased as the increasing of BM repetition numbers. Although the FOM of the T-type tends to decrease, the FOM of the R-type tends to increase as the repetition numbers increase. This tendency can be understood when we look at the spectra in the vicinity of the localized wavelengths, Fig. 5. The split transmission and reflection spectral peaks produce decreasing and increasing light intensities, respectively. This is because of the refractive index difference between the RCP and LCP light beams [13]. The increased Q factor of the microcavities makes the peak separation clear. Overall, the R-type was found to be better than the T-type, and f = 9 is the closest condition to obtain the 90 degree rotation angle, which is required in modulators. R-type’s performances were similar (except the resonant spectral position) in case where the incidence of light was oblique from 90 (normal incidence) to 50 degree.
Fig. 4 (a) Transmissivity, polarization rotation angle and FOM of T-type microcavities versus the repetition number f ( = r) of BMs. (b) Reflectivity, polarization rotation angle and FOM of R-type microcavities versus the repetition number f of the front BMs.
Fig. 5 Transmission and polarization rotation angle of spectra in T-type microcavities, AR/SGGG/(T/S)f/TGG/LTO/(S/T)f, where f = 8–10. (b) Reflectivity and polarization rotation angle of R-type microcavity, AR/SGGG/(T/S)f/TGG/LTO/(S/T)20, where f = 8–10.
4. Defects thicknesses
With regard to further optimization, the contribution of TGG and LTO thicknesses in R-type MPMEO were comprehensively examined, Fig. 6. We cannot see a large contribution from the LTO thickness. That layer thickness was thus determined to have a minimum value of 89 nm, which equals an optical thickness of λ0/2, to decrease the required voltage for its electronic polarization.
Fig. 6 Reflectivity (a) and polarization rotation angle (b) dependences on the TGG and LTO thicknesses in the microcavities with AR/SGGG/(T/S)9/TGG/LTO/(S/T)20 structure.
When the LTO thickness was fixed at 89 nm, the reflectivity and rotation angle dependence on the TGG thickness were then examined and are shown in Fig. 7. The reflectivity does not changed with the TGG thickness, so we can optimize that only with the rotation angle magnitude. For instance, a SLM requires a 90 degree rotation angle to produce its maximum contrast. Hence, the optical thickness (physical thickness) of the TGG layer was set at 8 × λ0/2 (802 nm), which gives a rotation angle of 93 degree.
5. Responses for the applied voltage
We finally determined the entire structure of the MPMEO, AR/SGGG/(T/S)9/TGG/LTO/(S/T)20 with optical thicknesses (physical thicknesses) of 8 × λ0/2 (~802 nm) and λ0/2 (~89 nm) for TGG and LTO, respectively. When a small voltage is applied to the LTO, the alternating optical path length in the microcavity caused by the EO effect can shift the localized peak to a shorter wavelength. In Fig. 8, one can see polarization rotation angle modulation versus a low applied voltage, with negligibly small change of reflectivity.
Fig. 8 Change of reflectivity and polarization rotation angle with the applied voltage to the LTO film in the microcavity with ~802 nm thick TGG and ~89 nm thick LTO.
6. Conclusion
We have reported that MPMEO comprising TGG and LTO achieve high-performance modulation of blue and shorter wavelength light (<450 nm). It should be noted that we do not need to change the direction of magnetization, but can control the polarization rotation angle using a small voltage. The performance shown here cannot be accomplished by using any other system.
These calculated results open the availability of paramagnetic garnets in micro or nano scale devices as potential polarization modulators for blue and near-UV light beams.
Acknowledgments
TG acknowledges the Japan Society for the Promotion of Science (JSPS) Postdoctoral Fellowships for Research Abroad.
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