Controlling the polarization of electromagnetic fields is important, e.g., in optical information technology, laser material processing, microscopy, and quantum communication [1–6]. Usually, this is achieved with state-of-the-art polarization components which act on the optical field uniformly across the aperture, such as photonic crystals, metamaterials for controlling the polarization state, dielectric subwavelength gratings, and various integrated optical polarization elements [7–14]. In recent years, advanced micro- and nano-structuring techniques have opened up the path to novel hardware [15]. Here we consider the use of nano-structuring to demonstrate a space-variant polarization converter. The converter consists of a 2D array of several million hollow waveguides with sub-wavelength lateral dimensions (Fig. 1). The main advantage lies in the fact that for the same wavelength, the minimal structure size is larger than for comparable metamaterial-based elements (for comparison, see [16]). This facilitates fabrication, in general, and specifically allows one to use for example two-photon polymerization, as we demonstrate it here.

Artificial birefringence occurs due to the rectangular shape of the openings and suitable design of the geometry. This can be described mathematically by

 

Fig. 1. Scanning electron microscopy image showing one hollow waveguide from the top view with the geometric conditions $w_x$ as well as $w_y$, which are the sizes of the holes. Inset shows a schematic drawing of a side view with the height $h$. In the main image, the brighter areas indicate the surrounding metallic sidewalls and the black area in between is the hollow waveguide.

Download Full Size | PDF

For the fabrication, a combination of 3D direct laser writing (3D DLW) and electroplating is used. 3D DLW (Nanoscribe GmbH, type Photonic Professional GT2) in a positive tone photoresist (Microchemicals GmbH, type AZ3027) yields a negative of the hollow waveguide array (HWA) grid structure. In a subsequent process, the negative structure is filled with gold by electrochemical deposition. The dimensions of the fabricated element are in excellent agreement with targeted values: using scanning electron microscopy, we measured an average deviation of $\pm$ 5 nm for lateral dimensions, by means of tactile profilometry, and the average thickness across the array was found to be 1979 nm $\pm$ 115 nm. Tolerances were uniform over the entire element and did not depend on the waveguide dimensions. A detailed description of the fabrication process and about 3D direct laser writing in general can be found in Refs. [19–21].

The size of the complete hollow waveguide array is 4 mm $\times$ 4 mm. For performing spatially resolved polarization conversion, the HWA was divided into individual sections with different geometric conditions $w_y$ of the hollow waveguides and, consequently, a different phase delay ($\Delta \Phi =k_0\Delta n_{eff}h$) in each section. Some of the individual sections were divided further into smaller sections. The largest sections had a size of 100 µm in the $x$- and $y$-directions (orange colored areas in Fig. 2). The waveguide dimension $w_x$ = 1550 nm does not change across all sections. For a wavelength of $\lambda$ = 1550 nm, a change in $w_y$ in the different sections along the vertical direction leads to a phase delay of $\Delta \Phi _{(1034)}$ = 90 $^\circ$, $\Delta \Phi _{(953)}$ = 125 $^\circ$, $\Delta \Phi _{(872)}$ = 180 $^\circ$, and $\Delta \Phi _{(801)}$ = 270 $^\circ$. Please note, the sections $w_{y(872)}$ and $w_{y(801)}$ for the orange colored areas are not shown in Fig. 2. To the right of the orange areas, there are four smaller sections (blue areas) each with an edge length of 50 µm. In the vertical, these smaller sections are arranged repeatedly without changing the geometric conditions. The next sections along the $x$-direction of the hollow waveguide array (yellow areas) consists of 16 sub-sections, each with an edge length of 25 µm of the square. Within each sub-section, $w_x$ remains constant at 1550 nm and $w_y$ is reduced by 15 nm starting at 1034 nm to 809 nm. The 16 sub-sections are also arranged repeatedly along the vertical without changing their geometric conditions. This entire area of 300 µm in the $x$-direction and 400 µm in the $y$-direction extends iteratively over the entire area of 4 mm $\times$ 4 mm of the HWA.

 

Fig. 2. Scanning electron microscopy image of a space-variant polarization converter. In this overview, the sub-sections are highlighted in different shades of gray. (a) The numbers in the sub-sections correspond to the waveguide width $w_y$ in nanometers and (b) the numbers correspond to the phase delay in degree, whereas $w_x$ = 1550 nm is identical for all sub-sections. White lines have been added to the figure for better visualization of the individual sections. The inset shows a detailed top view of some hollow waveguides.

Download Full Size | PDF

A schematic drawing of the setup for the polarization analysis is shown in Fig. 3(a). For illumination, an expanded and collimated beam from a laser diode emitting at $\lambda$ = 1550 nm is used, which is linearly polarized in the horizontal direction. A Kepler telescope [not shown in Fig. 3(a)] with $20\times$ angular magnification images the HWA onto a CMOS camera. An analyzer is mounted on a rotation stage directly in front of the camera and is rotated by the angle $\gamma$ with increments of 10 $^\circ$. This results in 37 camera pictures for a full rotation from $\gamma$ = 0 $^\circ$ to 360 $^\circ$. Based on these images, the spatially resolved evaluation of the ellipticity and the azimuth angle is conducted. To minimize the effect of ambient condition fluctuations, each of the 37 images consists of the mean values of 150 individual images taken for each angular position of the analyzer.

 

Fig. 3. (a) Schematic drawing of the setup for the space-variant measurement. Here $E_{(t)}$ represents the input polarization state of the electromagnetic field. (b) Curve fitting (top dashed line) of the measurement values (rhombuses). The other two dashed curves represent graphically the individual terms for the underlying curvefit.

Download Full Size | PDF

To evaluate the ellipticity and the azimuth angle, the normalized transmitted power $P_{(\gamma )}$ as a function of the analyzer angle is obtained for each camera pixel. Figure 3(b) shows an example of $P_{(\gamma )}$ (rhombuses). The values for calculating the ellipticity and the azimuth angle are determined by a curve fit (top dashed line) using the least squares method [22]. The other two curves also drawn in Fig. 3(b) are intended to represent the procedure for determining the fitting equation. In the figure, the blue curve corresponds to a $\cos ^2$ progression, and the prefactor $a$ coincides with the maximum of the fitted curve. The gray curve has a $\sin ^2$ course whose prefactor $b$ is equal to the minimum of the curve fit. Both prefactors $a$ and $b$ correspond to the length of the two semi-axes of a polarization ellipse and therefore the ellipticity can be calculated by $\tan \chi$ = $\pm b/a$ for each pixel. The azimuth angle is equal to the position of the first peak of the fitted curve. In Fig. 3(b), the position of the azimuth angle is marked with $c$ and can be determined by expanding the $\cos ^2$ and $\sin ^2$ curve by this factor. This results in

Figure 4(a) shows the ellipticity resulting from a spatially resolved polarization conversion. Depending on $w_y$, the ellipticity for the orange as well as for the blue areas, see Fig. 2, changes from $\chi$ = 43.21 $^\circ$ $\pm$ 0.18 $^\circ$ for a phase delay of $\Delta \Phi$ = 90 $^\circ$, to $\chi$ = 23.09 $^\circ$ $\pm$ 0.27 $^\circ$ for $\Delta \Phi$ = 125 $^\circ$, to $\chi$ = 0.46 $^\circ$ $\pm$ 0.66 $^\circ$ for $\Delta \Phi$ = 180 $^\circ$, and to $\chi$ = -43.02 $^\circ$ $\pm$ 1.08 $^\circ$ for a phase delay of $\Delta \Phi$ = 270 $^\circ$. These results are in good agreement with results known from literature for the same phase delays [23]. A phase delay of $\Delta \Phi$ = 90 $^\circ$ and $\Delta \Phi$ = 270 $^\circ$ changes horizontal linear polarized light into circular polarized light. The ellipticity for complete circular polarized light is $\pm$ 45 $^\circ$. In contrast, for a phase delay of $\Delta \Phi$ = 180 $^\circ$, the linearly polarized input field remains linearly polarized, and only the polarization direction changes. For this, the input polarization as well as the output polarization are linearly polarized, the prefactor $b$ is zero, the prefactor $a$ = 1, and therefore $\chi$ = 0 $^\circ$. A phase delay of 125 $^\circ$ results in elliptical polarized light and the ellipticity is between $\pm$ 45 $^\circ$ and 0 $^\circ$. Apart from minor deviations, the measured values agree well with the theoretical values. Also for the areas consisting of 16 sub-sections, a continuous change in ellipticity is observed.

 

Fig. 4. Measurement of the polarization conversion dependent on position: (a) ellipticity and (b) standard deviations of the ellipticity of the two marked sub-areas.

Download Full Size | PDF

The standard deviation of the measured ellipticity increases with phase delay. This is illustrated by a false color representation in Fig. 4(b) for a phase delay of $\Delta \Phi$ = 90 $^\circ$ and $\Delta \Phi$ = 270 $^\circ$. The plots indicate that the standard deviations of the ellipticity are larger for a phase delay of 270 $^\circ$ compared with the standard deviation for $\Delta \Phi$ = 90 $^\circ$. This can be assigned to the nonlinear progression of the phase delay as a function of the waveguide dimension $w_y$, shown in Fig. 5 and calculated by $\Delta \Phi =k_0\Delta n_{eff}h$ with Eq. (1). For smaller $w_y$, the change in phase delay is more dependent on the geometric condition than for larger waveguide dimensions $w_y$. Figure 6 shows the phase delay resulting from each section. The distribution of the phase delays determined in the experiment agrees very well with the theoretical values shown in Fig. 2(b) and Fig. 5. In comparison with Fig. 4(a), the functionality of the space-variant polarization converter can be also seen in this figure. The deviations between the sections with the same phase delay are negligible and shows the high quality of fabrication.

 

Fig. 5. Phase delay $\Delta \Phi$ as a function of $w_y$.

Download Full Size | PDF

 

Fig. 6. Measurement of the position dependent distribution of the phase delay $\Delta \Phi$.

Download Full Size | PDF

Compared with the ellipticity and the phase delay, the azimuth angle does not change during the measurement. The changes shown in Fig. 7 are marginal and are all within the measurement uncertainty. This effect could be expected because the polarization converter is not rotated during the measurement. Only a change in the angle of the hollow waveguide array results in a shift of the azimuth angle. Therefore, the measurement result confirms the previously made assumptions.

 

Fig. 7. Measurement of the position dependent distribution of the azimuth angle.

Download Full Size | PDF

In conclusion, we have shown a position dependent polarization converter using a hollow waveguide array. The converter allows changing the polarization state of optical radiation at 1550 nm spatially resolved down to areas with 25 µm size in the $x$- and $y$-directions. The ellipticity, the phase delay, and the azimuth angle are calculated from transmitted power measurements. The predicted polarizing properties are comparable to non-space-variant polarization converters.