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In this paper, an achromatic quarter waveplate scheme at terahertz frequency range is proposed. The waveplate is based on a silicon grating with large fill factor and period close to the wavelengths of the terahertz wave. Owing to the grating structure, the frequency dependence of waveplate is compensated. An achromatic QWP is fabricated and experimentally demonstrated to show an excellent wideband feature at the range of [0.47, 0.8] THz, which is more than 6 times wider than a conventional QWP working at single frequency 0.5 THz, by assuming ± 3% variance of phase retardation.
© 2015 Optical Society of America
1. Introduction
Terahertz radiation is electromagnetic waves with frequencies ranging from 100 GHz to 10 THz. Recently some breakthroughs have pushed terahertz technologies into a wide range of applications including security, biomedicine and others sensing systems. In such applications, some crystals and biological tissues were reported to exhibit THz birefringence and polarization dependent loss (PDL), these characteristics may be considered as their “polarization fingerprint” in the THz band [1, 2]. Therefore, polarization related measurement at the THz band is one of hot topics and polarization devices, such as waveplates, are important components at the THz band.
Natural materials can be utilized to fabricate waveplates in THz range, such as quartz crystal, and liquid crystal [3, 4]. Form birefringence materials, which are usually grating structure with larger birefringence than natural materials, are also widely studied for the THz waveplate, such as one-dimensional photonic crystals [5], stacked paper [6], polyethylene (HDPE) [7], transparencies [8], and silicon wafers [9]. Such waveplates usually present strong frequency dependences, i.e. these waveplates only work in very narrow frequency ranges, which does not meet the achromatic requirements in real applications.
Achromatic waveplate is well developed and commercially available in visible and near infrared range by combining two materials with different birefringence features. In THz range, achromatic waveplates has been realized by stacking several normal waveplates together which are with different, rigorous orientations [10, 11]. However, the waveplates setups are bulky and lossy due to multi-layer stacking. Metamaterial is another solution for achromatic waveplate [12–15], but it still needs several layers combination.
In this paper, we propose an achromatic terahertz waveplate scheme based on form birefringence on high resistivity silicon. Its grating microstructure has a large fill factor and period close to the wavelengths of the terahertz wave. With this design, the frequency dependence of waveplate is compensated. An achromatic silicon-grating-based QWP, operating at [0.47, 0.8] THz, is experimentally demonstrated by assuming ± 3% variance of phase retardation. Its bandwidth is 6 times wider than a conventional QWP.
2. Principle
In Fig. 1, a grating on silicon substrate consists of alternating silicon and air components, with refractive indices n1 = 3.415 and n2 = 1, width l1 and l2, and period L. When an incident wave propagates along the direction k, as shown in Fig. 1, the grating presents effective birefringence which is depicted as TE and TM. When the wavelength of the incident wave is much larger than the period of the grating, a quasi-static effective medium theory (EMT) is utilized to describe the refractive indices along TE and TM directions [5],
where η is the fill factors of the silicon components and η = l1/L. When the period of grating is close to and less than the wavelength, the quasi-static EMT is not accurate enough, and 2-nd order EMT should be used to describe the refractive indices as [5],The nTE and nTM turn out to be functions of fill factor η and period to wavelength ratio L/λ denoted as R. When the incident direction k is perpendicular to the surface of the grating, Eq. (2) is valid when [16, 17] (but in some paper strict limitation is used [18]),Hence the 2-nd EMT is valid when R < 0.3 for a silicon grating. Regarding an achromatic QWP, the phase retardation between TE and TM mode should maintain at 90 degree regardless of THz frequency changing. Thus,where c is the speed of light in vacuum, and f is the operation frequency of the QWP. Then,Therefore, for an achromatic QWP with a given grating structure (L and d), the products on the left should be a constant in the following equations,which means Δn should be decreasing with the frequency f or period to wavelength ratio R to compensate the frequency dependence of waveplate.
Fig. 1 Structure of the grating (L: period of grating; l1: width of silicon component; l2: width of air component; d: depth of grating; k: propagate direction of wave).
Assuming a ± 3% variance of 90°- phase retardation, we have
Figure 2(a) shows the dependence between Δn∙R and R (corresponding to frequency f), with fill factor varying from 0.65 to 0.9. The shadow area indicates invalid conditions when R>0.3. When the fill factors are smaller than 0.7, the curves rise straightly in the range of R<0.3; however, with the increasing of the fill factor from 0.75, the curves change slowly around their peaks. According to Eq. (7), given a fill factor 0.85, the peak point ‘a’ corresponds to the upper boundary, i.e. . The points corresponding to lower boundary can be calculated out as points ‘b’ and ‘c’, which also correspond to period to wavelength ratio and . Hence the frequency range of the achromatic QWP can be described as,The center frequency should be,Hence Eq. (8) can be further deducted as,where . The Rr can be interpreted as a band factor that determinates the bandwidth of the QWP once the center frequency fc is specified. So far an achromatic QWP is realized with center frequency fc and bandwidth 2fcRr. The dimension of the grating will be deducted in next section.Fig. 2 (a) relation between Δn∙R and R when fill factor is varying from 0.65 to 0.95; (b) the relationship between [R1, R2, Rr] and fill factor.
Figure 2(b) shows the simulated R1, R2 and Rr with different fill factors. Due to the EMT is only valid when R<0.3, the valid range for fill factor is 0.81~0.98. Considering the difficulty in fabrication, the fill factor is the smaller the better. In order to avoid the invalid range, 0.82 is an optimum value for the fill factor.
Considering a conventional QWP working at single frequency, the phase retardation has a linear relationship with frequency. Hence, its bandwidth can be roughly estimated as 0.06fc, assuming ± 3% variance. Referring Rr in Fig. 2(b), the bandwidth of proposed achromatic QWP is more than 6 times wider than that of the conventional QWP.
3. Grating design
The process to design a specific wideband QWP is given:
- 1) Specify center frequency, for example fc = 1 THz.
- 2) Choose an optimized fill factor which is 0.82. The corresponding Rr is 0.0225. According to Eq. (10), the bandwidth is 0.45 THz, which means [0.775, 1.225] THz.
- 3) From Eq. (9), period can be deducted as = 71.1 μm.
- 4) From Eq. (7), the depth can be deducted as = 106.2 μm.
- 5) Table 1shows the specification of wideband QWPs centered at difference frequencies
Table 1. Specifications of wideband QWPs centered at various frequencies with fill factor 0.82.
4. Fabrication and experiment result
A 1 mm-thick, orientation <100>, double side polished silicon wafer with resistivity larger than 10 Ω*cm was used to fabricate an 0.7-THz-centered achromatic QWP described above. The wafer was cleaned with acetone, iso-propanol (IPA) and deionized water (DI water). 2-μm-thick AZ7220 photo resist was spin-coated on the wafer. The wafer was then soft baked at 100 °C for 100 s, and exposed using a mask plate in a Suss MA8 aligner for 7 s at 17-mW/cm2 power and developed in MF-320 developer for 60 s. After development, the wafer was hard baked at 120 °C for 30 min in an oven. An MUC-21 etching system was used in deep reactive ion etching (DRIE). The silicon wafer was etched in SF6 (200SCCM, 10s, 3Pa) and C4F8 (85SCCM, 3s, 2Pa) cycling for 60min. The coil and plate powers were 600Wattt and 15Watt respectively. Then the photoresist mask was stripped and cleaned using hot acetone, IPA, and DI water.
In our preceding measurements, the real result is found deviating a bit from the theoretically model, due to the influence from fabrication variance and testing environment, as well as the accuracy of EMT. After optimization, we get a grating shown in Fig. 3(a)&(b), with period of 100μm, fill factor of 83.3% and etching depth of 160.8μm.
Fig. 3 The microscope view of the fabricated grating: (a) cross section view; (b) top view. (c) The schematic figure of TDS measurement system.
The fabricated QWPs are measured using a polarimetric terahertz time domain spectrometer system shown in Fig. 3(c) [19–21]. The incident THz beam is firstly linear polarized by polarizer 1, and then goes through the QWP sample. Polarizer 2 acts as a polarization analyzer by rotating with different angles. In our measurement, the principle direction of QWP grating is defined as along TE direction, and the grating is located at 45° with respect to the direction of polarizer 1. In this case, incident wave is equally spitted into TE and TM components. After transmitting through the QWP, once phase difference between TE and TM components equals to 90°, the output signal becomes a circularly polarized beam. The definition, measurement and calculation formula of the Stokes parameter can be found in reference [19–21].
In terms of normalized Stokes parameter (S1, S2, S3), (1, 0, 0) and (0, 0, 1) stand for horizontal linear and right circularly polarization states respectively. Figure 4 shows the simulated Stokes parameter of the outgoing beam, after transmitting through a QWP with the same dimensions as the fabricated grating. Considering ± 3% variance of phase difference, the Stokes parameter is close to (0, 0, 1) at frequency range [0.53, 0.83], which means a 0.3- THz-wide achromatic QWP. A conventional QWP work at only 0.68 THz is also simulated for comparison. Obviously, the frequency range of our proposed QWP is much wider than normal one. Figure 5(a) shows the real measured Stokes parameter of the outgoing beam after transmitting through the fabricated achromatic QWP. Considering ± 3% variance of phase difference, the Stokes parameter of output beam is quite close to circular polarization state of(0, 0, 1) in the frequency range of [0.47, 0.8] THz, which shows an excellent performance of QWP in a 0.33-THz wide band. The slight drift of center frequency and broaden of bandwidth is due to small error from theoretic model and refractive index of silicon. The result is basically consistent with the simulation result in Fig. 4. Comparatively, a conventional QWP at 0.5THz done in previous work [22] is referred and its measurement result through the same polarimetric THz-TDS system is compared with this achromatic QWP. Assuming ± 3% variance of phase retardation, the referred conventional QWP only have a 0.046-THz wide band which can convert linear polarization state of (1,0,0) to circular polarization state of (0,0,1). Comparing the two QWPs, obvious achromatic feature can be observed in our proposed QWP.
Fig. 4 Simulated Stokes parameter for the achromatic QWP (solid line) centered at 0.68 THz compared with that for a conventional 0.68 THz QWP (dot line).
Fig. 5 Stokes parameter performance of a (a): proposed achromatic QWP; (b) conventional QWP at 0.5 THz.
5. Conclusions
We propose a scheme for achromatic waveplate in THz range, based on silicon grating with large fill factor and period close to wavelength. An achromatic QWP is experimentally demonstrated and shows a good achromatic feature at the terahertz range of [0.47, 0.8] THz. The proposed achromatic QWP has more than 6 times wider band than that of a conventional QWP at 0.5 THz. This scheme is also suitable for form birefringence-based waveplates using other materials for the whole infrared range.
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