The Special Polarization Characteristic Features of a Three-Dimensional Terahertz Photonic Crystal with a Silicon Inverse Diamond Structure

Recently, THz (terahertz) technologies, the range of which is located halfway between microwaves and infrared lights, have been proceeding in various industrial fields such as non-destructive examinations [1, 2, 3], the body checks for securities, cameras [4], chemical identifications [5] and microscopes [6] etc..
  The three-dimensional (3D) photonic crystals (PC) have the regular periodicity of dielectric materials [7, 8, 9, 10]. They can control waves by forming point defects and line defects on the surface and in the interior and have possibilities of further contribution to photonic devices and new characteristic features in 3D-PC and may also present a new frontier for them.
  In the present paper, the band structure of the Si inverse diamond structure was calculated using plane wave expansion method. The lattice point shape of this structure was vacant regular octahedrons and it was theoretically confirmed that the complete photonic band gap (CPB) existed at around 0.4 THz. The polarization anisotropy, which was the polarization orientation difference between a reflected light and incident one, was studied on the surface (001) at around BGX, X point’s photonic band gap, using the incident light direction [001] whose polarization orientations {I(X,Y)} were {I(1,1), I(1,0), I(1,$-$1), I(0,$-$1)}.

The lattice of the diamond structure in this work is shown as Fig.1(a). The sphere is the lattice point and its shape is the regular octahedrons, see Fig.1(b).

It’s vacant and the dielectric constant ${\epsilon }_a$ is 1.00 (atmosphere). The surrounding materials is pure Si (resistivity $\rho >{10}^4\Omega$ cm) and the dielectric constant ${\epsilon }_b$ is set as 11.9. The lattice constant a = 300 $\mu$m and the length of the regular octahedrons side, L = 150 $\mu$m is specified in this theoretical and experimental work. Fig.2(a) shows the calculated photonic band structure using plane wave expansion method and CPB is seen at around 0.4 THz. The first Brillouin zone is shown as Fig.2(b).

In this work, the direction of the incident light is [001] ¹¹1[001], (001), {100} and I(1,1) etc. are definded on the X-Y-Z coordinate system in Fig1(a). in the real space and it corresponds to $\Gamma$-X direction in the wave number space (K-space). BGX exists between 0.36 and 0.44 THz and its center is 0.40 THz.

The main periodic square patterns (side L = 150 $\mu$m) on both surfaces (001) of Si whose area is 10 $\times$ 10 mm and height is 75 $\mu$m, is etched and periodically arranged along the direction [001]. Two-faced patterns are totally displaced by $(\sqrt{2}a/4,\sqrt{2}a/4,0)$ with mask patterns’ coordinate system. The etching angle of Si {100} surface is ${54.7}^{\circ }$ and the etched vacant shape forms the regular octahedrons between 4 layers. The total number of layered Si chips is 48 layers.
  The conceptual diagram of the measurement system with THz-TDS (time-domain spectroscopy) equipment owned by Nippo Precision co., ltd. is shown as Fig.3.

The polarization orientation of the light source is S polarization (S-p) and it’s parallel to x-axis. The mirror 1 and mirror 2 are coated with Au. S-p launched by the light source is reflected by the mirror 1 and the orientation of polarization is also S-p ²²2The expression of the inversion, phase shifting by $\pi$, of S-p is abbreviated in Fig.3.. The 3D-PC sample is so horizontally set that the layered direction is parallel to z-axis and the incident angle is $7^{\circ }$, which is the perpendicular incidence approximately. Another polarization perpendicular to S-p is P polarization (P-p) and included in the incidence plane. According to Fermat’s principle, the reflection angle is equal to the incident one. When not only S-p but P-p is included in the reflected light of sample, S-p and P-p are naturally included in the reflected light by the mirror 2. Meanwhile, the detection equipment detects only S-p. This optical system can’t be adjusted for rental equipment. Therefore a 1/2 wave plate is used for measurements of the ratio of S-p and P-p included in the reflected light of the sample.
  The material of the 1/2 wave plate is quartz SiO${}_2$. Xc and Zc (crystal axes) are included in the plane of the plate and Yc is parallel to the direction of the thickness, 8.45 mm. The ordinary and extraordinary refractive indices are n${}_o$ = 2.108 (= n${}_{Xc}$, n${}_{Yc}$) and n${}_e$ = 2.156 (= n${}_{Zc}$) at 1 THz, respectively [11].
  Yc is parallel to y-axis. The designed 1/2 wave plate converts S-p into P-p and P-p into S-p at around 0.4 THz when one of the two bisector of Xc and Zc is parallel to x-axis. Meanwhile, when one of Xc or Zc axis is parallel to x-axis, the orientation of the incident polarization, S-p or P-p don’t change though the plate.
  avXZ(f), defined as the average of the transmitted spectra of S-p $\parallel$ Xc and S-p $\parallel$ Zc, is used for the normalization of the characteristic features of the 1/2 wave plate. f is THz frequency. Fig.4 shows the transmittance WS(f)³³3The harmonic in Fig.4 appears from the reflection of the back surface of the 1/2 wave plate. which is the spectrum normalized by avXZ though the 1/2 wave plate at around BGX. 1$-$WS(f) is the ratio of P-p conversion into S-p, inversely. In measurement, the incidence light is S-p and Au plate is set instead of sample and used as a reference of the reflected light.

The sample is rotated in the x-y plane in Fig.3 as a substitute for the rotation of the incident light for measurements of the polarization anisotropy. z-axis corresponds to Z-axis in Fig.1(a). The polarization orientation of the incident light, I(X,Y) is defined with vector (X,Y,0) in Fig.1(a) as follows. For example, it’s called I(1,1) when the incident S-p is parallel to (1,1,0). The polarization spectra of the reflected light are measured for four kinds of orientation {I(1,1), I(1,0), I(1,$-$1), I(0,$-$1)}.
  Fig.5 shows the reflection spectra measured with I(1,1) at around BGX. In Fig.5(a), R11 (f) is the reflectance with no 1/2 wave plate and is normalized by Au reflection spectra. R11W(f) is the reflectance though the 1/2 wave plate and is normalized by avXZ. The reflection spectra of I(1,$-$1) also have similar characteristics. In Fig.5(b), R11(f)$\times$WS(f) is shown with R11W(f). R11(f)$\times$WS(f) is nearly equal to R11W(f). Therefore, this experimental results indicate that R11(f) consists of only S-p.

Next, the reflection spectra of I(1,0) are shown in Fig.6. In Fig.6(a), R10(f) is the reflectance with no 1/2 wave plate and is normalized by Au reflection spectra. R10W(f) is the reflectance though the 1/2 wave plate and is normalized by avXZ. The reflection spectra of I(0,$-$1) also have similar characteristics. In Fig.6(b), R10(f)$\times$WS(f) is not entirely equal to R10W(f) at around BGX. These experimental results are entirely different from those in Fig.5(b).

In Fig.7, $\Delta$R10W(f), which is the difference between R10W(f) and R10(f)$\times$WS(f), corresponds to the contribution from P-p though the 1/2 wave plate. Accordingly, the normalized $\Delta$R10W(f)/{1$-$WS(f)} corresponds to P-p reflected spectrum with no 1/2 wave plate. Especially near 0.38 THz, S-p incident light is almost entirely converted into P-p reflected light. At 0.44THz, the reflected light consists of S-p (about 38%) and P-p (about 64%).

In the Si inverse diamond structure whose lattice point shape is vacant regular octahedrons and that has CPB at around 0.4 THz theoretically, the polarization anisotropy, which means that the polarization orientation of a reflected light is different from that of a incident light, is studied at around BGX (0.36 $-$ 0.44 THz) on the plane (001) using the incident light direction [001] ($\Gamma$-X direction) and four orientations of incident polarization {I(1,1), I(1,0), I(1,$-$1), I(0,$-$1)}. The polarization orientation of the reflected light is parallel to that of the incident light for incident polarization I(1,1), I(1,$-$1) at around BGX. In contrast, that of the reflected light is entirely different from that of the incident light for incident polarization I(1,0), I(0,$-$1). Especially, the former is perpendicular to the latter in the vicinity of 0.38 THz. As far as the 3D-PC in this work is concerned, method of resolution and synthesis of the incident polarization vector isn’t apparently able to apply to the analysis of experimental results.

The author would like to thank Hidekazu Onishi, who fabricated the designed Si inverse diamond structure and Ph.D. Takeshi Sawada, who is a research scientist, Terahertz Project, Second Design Department, Nippo Precision co., ltd.