We present a low-cost terahertz wave plate based on form birefringence fabricated using ordinary paper. Measurements of the transfer function of the wave plate between polarizers closely agree with predictions based on the measured complex indices of refraction of the effective medium. For the design frequency, the dependence on wave plate angle also agrees with theory.
© 2011 OSA
The terahertz region of the electromagnetic spectrum has seen significant technological advances during the last few decades . These advances have mainly been driven by applications in sensing , although this frequency range is also very useful for fundamental science [3–6]. The development of generation and detection methods based on ultrafast lasers, including photoconductive switches  and nonlinear crystals , have been key to the rapid advances. However, one technological area has been lagging, namely the development of components for manipulating polarization, for example wave plates, at these frequencies. The challenge of fabricating wave plates that work in this frequency range has been recognized since before the development of terahertz technology based on ultrafast lasers . Methods used in the optical regions of the spectrum, such as, birefringence in crystalline material  or liquid crystals , can be adapted to terahertz frequencies. Alternatively, materials that are not usable in the optical regions of the spectrum, for instance wood , can be used to construct wave plates at THz frequencies. However, the dramatic difference in length scales between the optical and terahertz regions of the spectrum can dramatically change the evaluation of various technologies. One example of this is the concept of form birefringence. Form birefringence results from an anisotropic periodic structure, such as a grating. However, the grating period needs to be subwavelength , which makes fabrication at optical frequencies very difficult, whereas it is as not difficult at terahertz frequencies.
The use of form birefringence to fabricate phase retarders was first demonstrated in the optical region of the spectrum by etching sub-wavelength gratings into dielectrics [14,15]. However the fabrication of these devices with subwavelength length scales and either high dielectric contrast or high aspect ratio is challenging and costly. Recently, efforts have been made towards low cost, but laborious, fabrication of form birefringent quarter-wave plates . Form birefringence has been demonstrated at terahertz frequencies by etching gratings in silicon  and stacked polymer layers .
Another approach to fabricating wave plates is based on the birefringence provided by metamaterials [19,20]. However, this approach is more complex and difficult compared to the approach described below.
In this paper, we describe the fabrication and characterization of wave plates for terahertz frequencies that are fabricated out of ordinary paper. As compared to our previous work , this approach is very low cost and simple to implement. In particular no sophisticated raw materials such as TiO2 doped polyethylene are needed. Moreover, we perform a detailed investigation of the angular dependent transmission and discuss the polarization dependent attenuation, which modifies the optimum angle for the half-wave plates. Furthermore, the paper based wave plates have a reduced reflection coefficient in comparison to the structure in . The demonstrated performance of the wave plate exceeds more expensive approaches and agrees with theoretical predictions based on the measured properties of the materials.
2. Fabrication of device
The paper terahertz wave plates are fabricated by simply cutting strips of standard office paper (for all measurements shown in this paper, we use Xerox Business TCF 80 g/m2, thickness 120 µm) of appropriate dimension and then stacking them so that air gaps between the individual layers result. A guillotine cutter was used to cut the sheets to the width matching the desired wave plate thickness. Some of the resulting sheets are cut to a shorter length to be used as spacers. As shown in Fig. 1(b) , the shorter spacer sheets are placed at each end of the wave plate to create the appropriate air gaps between the longer sheets. The wave plates produced consisted of 150 to 200 paper/air pairs. Once stacked, the paper was held in place using standard binder clips or rubber bands.
Measurements on wave plates, built from different paper types with different quality have shown that the paper quality slightly affects the properties of the wave plates. However, wave plates made from different paper brands with comparable quality show very similar results.
For a half wave plate, the width of the paper strips d (see Fig. 1(a)) for a desired design frequency is where f is the design frequency of the wave plate, c is the speed of light and Δn is the birefringence. We found that 0.15 is a reasonable first estimate for the birefringence of our wave plates. This value agrees well with the experiments in the next section. We estimate that the upper limit on the half wave frequency is around 450 GHz for manually fabricated paper wave plates. This value is determined by two factors. The first is the constraint that the stacking period has to be smaller than the wavelength . Secondly, manual fabrication of the wave plates by cutting and stacking sets a mechanical limit to the range of operation.
3. Measured indices of refraction
The indices of refraction and absorption coefficients of the wave plate were measured as a function of frequency for both polarizations using a conventional terahertz time domain spectrometer . The results are shown in Fig. 2 .
Figure 2(a) shows the indices of refraction for the two polarizations. The p-polarized THz wave experiences a higher effective refractive index than an s-polarized wave. The birefringence of the wave plate is relatively constant over the measured frequency range. We determine the exact value at the half wave frequency of the 4.2 mm thick wave plate to be Δn0.244THz = 0.146. This value is determined by the filling factor of the paper-air-composite and applies for all wave plates that we produced and measured regardless of their thickness d. The graph in Fig. 2(b) shows the absorption coefficient, calculated with the method described in [22,23]. We find that it is higher for p-polarization than for s-polarization.
The measurements of the indices of refraction were compared to the theoretical expressions given by Scheller et al. , which are shown as solid lines in Fig. 2(a). For the theory, the index of refraction for the air gaps was assumed to be 1 and that of the paper to be 1.6, while their respective thicknesses were q = 180 µm and p = 120 µm. We attribute the variance between the thickness of the air gap and the thickness of the spacer layer to the stack not being completely pressed together. However, these thickness values are consistent with the total measured thickness of the stack. These values yield good agreement between the measurements and theory.
4. Transfer function
To characterize the performance of the paper wave plates, the same terahertz time domain spectrometer was used with wire grid polarizers inserted to obtain clean polarization states for the generated terahertz radiation and the detected radiation. The paper wave plate was inserted between the polarizers. Using this system, the frequency dependence of the transfer function of parallel polarizations was measured with the wave plate adjust to rotate the polarization by 90° at the half-wave frequency. The dependence of the transfer function on the wave plate angle was measured for generation and detection co- and cross-polarized.
4.1 Frequency dependence
A half-wave plate set with its principal axes at 45°with respect to the polarization direction of an incident linearly polarized field will rotate the polarization by 90°. However, the presence of polarization dependent loss, as displayed in Fig. 2(b), causes the wave-plate angle required for a 90° polarization rotation to deviate from 45°. The wave plate angle was optimized by adjusting the azimuthal angle to minimize the transfer function at the designed half-wave frequency. The detailed angle dependence is presented and discussed for the 4.2 mm wave plate in the next section. For the 4.2 mm, 5 mm and 7.08 mm wave plates, the optimum angles were found to be 53°, 52° and 53°, respectively. These values are consistent with simulations taking into account the measured absorption coefficients. For frequencies around 0.25 THz the overall transmission losses due to absorption are less than 5 dB. The measured frequency dependence of the transfer functions for wave plates of the 3 different thicknesses are shown in Fig. 3 . The transfer function is defined as , where Eα is the electric field polarized with an angle α as defined in Fig. 1(a) and Ey is the reference electric field polarized along the y-axis which means that it is s-polarized. The cusps correspond to the half wave frequencies of the wave plates so that the transmitted terahertz radiation is cross polarized with respect to the detection polarizer.
Using the measured indices of refraction and absorption coefficient, the the transfer functions of the three wave plates were calculated and are plotted in Fig. 3 as lines. The calculated transfer functions agree quite well with the measured ones. The contrast ratio is limited by the dynamic range of the measurement system, nevertheless, the 40 dB contrast ratio shown in these measurements demonstrates that the wave plates are producing a pure polarization state and a depolarizing is negligible.
4.2 Angular dependence
To further characterize the performance of the 4.2 mm thick paper wave plate, the transfer function at the half wave frequency of 0.244 THz was measured as function of the azimuthal angle α (see Fig. 1(a)) of the wave plate for the detector polarizer parallel (Fig. 4(a) ) and perpendicular (Fig. 4(b)) to the source polarizer. As expected, nulls are observed when the wave plate rotates the polarization of the transmitted terahertz radiation to be cross-polarized with respect to the detection polarizer. The observed pattern (symbols) agrees with the calculated pattern (lines) and confirms that the wave plate can be used to rotate a linearly polarized state while preserving the purity of the polarization.
4.3 Beam profile
We investigate to what degree the wave plate affects the beam spatial profile. To this end, we determined the profile of the beam when propagating through air and through the wave plate, respectively, by x-y-scanning of the detector (see  for a description of the apparatus). In this experiment the wave plate was positioned at α = 52°, i.e., the wave plate rotates the polarization by 90°. The results are shown in Fig. 5 . For the measurement with the wave plate, the detector and source polarizer are oriented perpendicular to each other. For the reference measurement, they are oriented parallel to each other. In this figure, we plot the intensity of the Fourier transformed time domain signal, integrated between 200 GHz and 300 GHz, i.e. in the frequency window for which the paper stack acts as a half wave plate. One can see that the beam quality is only weakly affected by the wave plate as the beam profile still exhibits a Gaussian shape.
We have described wave plates for terahertz frequency radiation that are fabricated from ordinary office paper. The indices of refractions and absorption coefficients of the wave plates were measured. To characterize their performance as wave plates, the frequency and angular dependence of their transfer functions when inserted between polarizers was measured. The measured transfer functions agree well with calculations based on the measured indices of refraction and absorption coefficients. The large contrast ratio confirms that the wave plates are not depolarizing the transmitted terahertz radiation.
These wave plates have the advantage of being extremely inexpensive and easy to fabricate. They show excellent performance at their design frequency. Of course, terahertz radiation often has extremely large fractional bandwidth, which presents a challenge for polarization control based on phase shifts from propagation in birefringent material. Achromatic wave plates consisting of multiple plates of varying thickness can be constructed and have been demonstrated at terahertz frequencies . Constructing a paper achromatic wave plate is clearly the next step.
B. S. acknowledges financial support from the Friedrich Ebert Stiftung. S. T. C. would like to acknowledge funding from the Alexander von Humboldt Stiftung.
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