We propose a novel broad-angle polarization beam splitter utilizing the spatial dispersion of a special multi-layered dielectric periodic structure. The equal-frequency contours of this structure are flat lines for TE polarization but curved lines for TM polarization at a designed frequency. This special multi-layered structure has a fixed optical thickness for TE polarization for all incident angles. A polarization beam splitter working over a broad range of angle (from 0° to 70°) is achieved by stacking two such multi-layered structures of finite length (in the normal direction) with a half-period shift in the transverse direction.
©2007 Optical Society of America
A polarization beam splitter (PBS) is used widely in many applications such as optical communications, data storage, image processing, display and some interference measurements. A traditional PBS is based on either natural crystal birefringence or the thin film theory of a multi-layered structure. The former requires a large thickness (and thus is not of compact size) in order to obtain enough walk-off distance between the two polarizations as the birefringence of a naturally anisotropic material is always small. The latter has two types: (1) For the first type, transmission for TM polarization is high when the incident angle is around the Brewster angle (around 45°) and high reflection for TE polarization at this angle is achieved by designing the thickness of each high/low index thin film in the multi-layered structure. It also requires a prism on each side of the multi-layered structure. (2) The other type is based on the different bandwidths of high reflection (due to the interference) for different polarizations. These traditional PBSs are sensitive to the incident angle and thus can not be used as broad-angle PBSs.
In the present paper we propose a novel PBS working over a broad range of angle (from 0° to 70°) based on a multi-layered dielectric periodic structure. Two-dimensional periodic structures (also called photonic crystals [1, 2]) have been used for PBSs by utilizing different bandgaps for two polarizations [3–6], anisotropies for two polarizations , or negative refraction [8, 9]. The present PBS is based on spatial dispersion characteristics of a special one-dimensional dielectric periodic structure, and thus has a much simpler structure and is much easier for the fabrication and integration with other planar lightwave circuits.
2. Spatial dispersion of a multi-layered structure
We design a polarization beam splitter based on a multi-layered structure with alternative high and low refractive index materials (with a period of a), which has a finite length d along z axis [see Fig. 1(a)]. In all our numerical simulations, we choose high refractive index n=3.4 (silicon; with thickness w=0.25a for each layer; see the black strips in Fig. 1), and low refractive index n 0=1 (air). The plane of incidence is x-z plane, and θ is the incident angle.
This dielectric period structure has an interesting spatial dispersion property. The equal-frequency contours (EFCs) of the first band for this structure are shown in Figs. 2(a) and 2(b) for TE polarization and TM polarization, respectively. The EFCs for both polarizations become flat (along kx direction) gradually as the normalized frequency increases. The EFCs for TM polarization are much denser at small frequencies and become flat much slower as compared with those for TE polarization. At normalized frequency ωa/2πc=0.42, the EFC for TE polarization has already become flat (indicating that the energy of the light can propagate only along z direction in this structure) while the EFC for TM polarization is still a curve, see Fig. 2(c). This property will be exploited for the application of a polarization beam splitter in the present paper. Note that flat EFCs for some two-dimensional photonic crystals have been utilized for the application of collimation [10–12]. Some Bloch mode analysis for a finned dielectric structure has been made in Ref.  for the application of waveguides.
3. Working principle for the present PBS
Since the EFC for TE polarization is flat at normalized frequency ωa/2πc=0.42 and the group velocity is always along the normal direction of the EFC, the Poynting vector (i.e., group velocity) of the refractive wave (transmitted into the multi-layered structure) for TE polarization (at this frequency) will always be along z axis regardless of the incident angle (this is schematically shown in Fig. 3). Note that for different incident angles the wave vector of the refractive TE waves will have different x components (this explains why the reflected TE wave in Fig. 5(a) below is not along the normal direction) but the same z component kza/2π=1.035 (for ωa/2πc=0.42). This is verified numerically in Fig. 4 for a TE Gaussian beam (cf. Fig. 7 below for the size for the Gaussian beam) impinging on such a multi-layered structure. To obtain a maximal reflection of TE polarization from such a multi-layered structure of finite length d, we choose d satisfying 2kzd=(2M+1)π, where M=0, 1,2,3... (this is similar to the conventional theory for a highly-reflective quarter-wavelength homogeneous thin film). Here we choose M = 5 (if M is too small the strips will be too short and some surface effect may degrade the performance).
Figure 4(a) shows the snapshot distribution of Ey at a certain time in a region [indicated by the red rectangular frame in Fig. 1(a)] for the case of normal incident angle (incident angle θ=0°), and Fig. 4(b) gives the time averaged result of ∣Ey∣ for Fig. 4(a). Figure 4(c) and 4(d) show the time averaged distributions of ∣Ey∣ when the incident angle is θ=20° and θ=45°, respectively. The incident TE waves with different incident angles will have the same energy-flowing directions and form standing waves of similar distribution pattern in the structure. In Fig. 4 we see that kz is always the same regardless of the incident angle (consequently there always appear about five and half periods of oscillation inside the slab). In this figure we can also see that there is no phase shift during the reflection at the back interface of the slab. Some distortions appearing at the boundaries for the oblique incident cases are induced by the scattering at the grating-like interfaces (however, this does not disturb the general pattern of the internal field distribution).
For TE polarization the structure of Fig. 1(a) has a fixed optical thickness for all incident angles (such a structure will be called meta-structure hereafter). On the other hand, for TM polarization the refractive angle will change as the incident angle varies since the EFC for TM polarization is a curve at normalized frequency ωa/2πc=0.42 (similar to the refraction of light at the interface of two homogeneous materials). In order to ensure large enough transmission of TM polarization over a broad range of incident angle for a slab of certain homogeneous dielectric medium (as an approximation of the meta-structure), we can try to achieve large enough transmission under the worst situation. The worst (minimal transmission) situation for TM polarization is the normal incidence on a quarter-wavelength (thickness) thin film because (i) a quarter-wavelength thin film gives the minimal transmission for TM polarization; (ii) over a broad range of angle (from 0° to at least the Brewster angle, which is always larger than 45°) normal incidence gives the minimal transmission. A simple calculation shows that the transmission for the normal incidence on a quarter-wavelength thin film is more than 90% if the refractive index of the thin film is less than 1.4 (this should be fulfilled in our design for the meta-structure). For the present meta-structure, the equivalent refractive index for TM polarization at the normal incidence is about 1.24 [estimated by kz(kx=0)/k 0]. Therefore, the transmission of TM polarization for the present meta-structure should be quite high over a broad range of incident angle.
To increase the reflection for TE polarization, in the present paper we put together two structures of Fig. 1(a) in a stack with a half-period shift in x direction [along the interface the center of each high refractive index strip is positioned at the center of a low refractive index strip; see Fig. 1(b)]. This can be explained as follows: For TE polarization, the light intensity inside the meta-structure is mainly concentrated around the high refractive index strips [(see Fig. 4). Therefore, a large reflection will occur at the interface of the two meta-structures when there is a half-period shift (like the reflection caused by a mismatch in the eigen-modal fields of two waveguides). Furthermore, all the multi-reflected waves will be added in phase at the front surface when the second meta-structure has the same length d designed previously for the first meta-structure. Therefore, such a stack of two meta-structures will enhance the reflection of TE wave greatly. Obviously the transmission of TM waves through this stack of two mea-structures will remain high. Figure 5 gives a schematic illustration for the working principle of the present polarization beam splitter.
Through numerical simulation we found that the reflection for TE polarization can be improved further when the value of M is adjusted slightly from 5 to 5.2 (this could be due to some interface effect). The corresponding high reflection (for TE polarization) and high transmission (for TM polarization) for such a stack of two meta-structures over a broad range of incident angle (from 0° to 70°) are shown in Fig. 6 by the circles and triangles, respectively. (The extinction ratio for small incidence angles can be improved by increasing the number of meta-structures stacked together with a half-period shift.) Figs. 7(a) and 7(b) show the time averaged distributions of ∣Ey∣ (or ∣Hy∣) when the incident angle (of the Gaussian beam) is θ=45° (the performance for any incident angle between 0° to 70° will be similar) for TE polarization and TM polarization, respectively.
We have designed a meta-structure and studied its properties of spatial dispersion for both TE and TM polarizations. For TE polarization the meta-structure has a fixed optical thickness for all incident angles. Utilizing this property, we have designed a broad-angle (from 0° to 70°) polarization beam splitter. Our PBS works in a frequency range where the EFCs are flat or approximately flat and the kz component of the refractive wave vector is of similar value (so that equation 2kzd=(2M+1)π can be fulfilled roughly in this frequency range). Although our PBS is designed at normalized frequency ωa/2πc=0.42, it works in a range of normalized frequency from 0.4 to 0.45. One way to improve the bandwidth is to stack together several PBS structures with different length d (so that each of them will work at different bandwidth). However, such stacking may increase the complexity of the structure and reduce a bit the transmission of TM wave. Besides the simplicity, this novel polarization beam splitter has many advantages: (i) it works over a broad range of incident angle; (ii) the incident light can be launched from a fiber, which can be positioned anywhere on the flat front-surface; (iii) it can be fabricated on a wafer and integrated with other planar lightwave circuits.
This work is supported partially by the National Basic Research Program (973) of China (2004CB719801) and the National Natural Science Foundation of China (grant number 60688401).
References and links
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