1. Introduction

In recent years, it has been demonstrated that radially and azimuthally polarized light and the radial and azimuthal polarization converters are very useful for lots of optical applications [1–5]. Some of them are listed here; first, the light with radial and azimuthal polarization could be used to fabricate LC lens with reduced aberrations [1]. The polarization converters that could change the light with certain polarization state to radial and azimuthal polarization state could be used as polarization axis finder [2]. Another attractive feature of radial and azimuthal polarization converters is the formation of light beam with orbital angular momentum that is of great interest for beam shaping applications [3–8]. It has also been disclosed that light with radial polarization could have a sharper focus that is good for optical trapping [9, 10]. Moreover, radially polarized light is proposed to use for laser cutting to increase the cutting speed [11]. There are many more applications for such light polarizations and the polarization converters.

Recently, several methods have been proposed to convert radially and azimuthally polarized light from certain polarization states of the light. One of the methods is using structured sub-wavelength gratings [7, 12], but the method is characterized by complicated fabrication process and because of sub-wavelength grating period, it is difficult to use this structure for small wavelength such as UV range. Another method is to utilize electro-optics of the ceramics [8] such devices are proposed to have fast switching time, but they have large defects at the center area and moreover it needs large electric field. Recently, LCs, because of their easy tunability of refractive index by mean of electric field, temperature or patterned alignment, have become very important for such polarization converters [13–19], but such LC based polarization converters should be axially symmetric that can be done by axial symmetric alignment either by circular rubbing or by photo-alignment [16–18]. The way of using axially symmetric electric field has also been reported [15]. Among these methods, photoalignment technology [20] is most commonly used due to its nano scale controllability, simply fabrication and high optical quality. The commonly used structures of the LC polarization converter are based on twisted nematic (TN) effect [13–15], the effect of half-wave plate (HWP) [2, 4, 15–18] or dichroic LCs [19]. The polarization converters based on TN structure shows defect line at the boundary of the opposite twisted domains [2], whereas the polarization converter made of HWP offers good optical quality however such system are characterized for the slow response time typically in the range of few ms [20]. While the current and future application demands high speed photonic elements, thus in this context, here we are proposing a polarization converter based on QWP which has advantages such as the spiral phase wave front, less retardation, higher switching speed and no defect line. The benefits of using QWP condition for the making of such polarization converters has also been shown for devices made of ceramic [8] and structured sub-wavelength grating [7]. In this paper, we disclose the radial and azimuthal polarization converter, based on photo-patterned LC QWP that shows high optical quality, less film thickness, fast switching speed and relatively simple fabrication procedure.

2. Theory

The properties of the radially and azimuthally polarized light can be described as in Fig. 1 . Considering polar coordinates (r, θ) with the polar axis L, the angle between polarization direction and the polar axis L is defined as φ at the point (r, θ), shown in Fig. 1(a). Thus, the properties of the axially symmetric polarization can be described as follows [2]

 

Fig. 1 Description of radially and azimuthally polarized light. (a) polar coordinates model (b) radially polarized light, (c)azimuthally polarized light.

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First, it is important to discuss the optical properties for circularly polarized light passing through the uniform QWP. If the fast optical axis of the QWP has an angle $\beta$ with the x axis, shown in Fig. 2 , and the input light is LHC or RHC polarized light, shown in Figs. 2(a) and 2(b). Then the output light property can be described by Jones vector in Eq. (2) (for input light with LHC polarization) and Eq. (3) (for input light with RHC polarization) [21]:

 

Fig. 2 (a) LHC and (b) RHC polarized light passing through the QWP.

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Thus it is clear that the output light is linearly polarized light with angular shift of $\pi /4$ from the fast optical axis of the QWP in clockwise direction (for input light with LHC polarization) and in anticlockwise direction (for input light with RHC polarization), and with a phase shift of $\exp (-j\beta )$ and $\exp (j\beta )$ separately.

Therefore, when circularly polarized light passes through a QWP with fast optical axis spatially variable, the output light will be linearly polarized light with its polarization direction spatially shifted from the fast axis with angle of $\pi /4$. Thus, for generating the radially or azimuthally polarized light, we can simply design the patterned QWP with its fast axis $\pi /4$shifted from the target polarization direction. The fast optical axis of the QWP can be denoted as:

So, to construct the axially symmetric polarization light with $q=1,\phi {}_0=0or\pi /2$, the axially symmetric QWP with $q=1,\alpha {}_0=\pi /4$can be used. When the input light is LHC or RHC polarized, the output light will be radially or azimuthally polarized, as shown in Figs. 3(a) and 3(b) respectively. Therefore, a patterned QWP with axial symmetry of $q=1,\alpha {}_0=\pi /4$can be used as both, radial and azimuthal polarization converter. The polarization of the output light i.e. radial or azimuthal can be determined by input light with LHC or RHC polarization.

 

Fig. 3 Patterned QWP with property of $q=1,\alpha {}_0=\pi /4$working as (a) radial polarization converter and (b) azimuthal polarization converter.

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3. Experimental details

To fabricate the LC patterned QWP with axial symmetry with $q=1,\alpha {}_0=\pi /4$, as shown in Fig. 3, the most efficient way is to control the easy axis of the alignment layer. Usually, the fast axis of LC wave plate is determined by the azimuthal angle of the easy axis of alignment. The traditional method for LC alignment (i.e. rubbing) offers fixed azimuthal angle of the easy axis same as the rubbing direction [2]. However, for the easy axis with azimuthal axial symmetry$q=1,\alpha {}_0=\pi /4$rubbing method is not feasible because of its fabrication complexity. The photo-alignment is a fast developing technology that decreases alignment process complexity and is relatively simpler for multi-domain alignment, in the comparison of rubbing method [20]. There are several photo-alignment materials are available these days. In present work, we used sulfonic azo dye SD1 (Dainippon Ink and Chemicals Inc. (DIC)) as the photoalignment layer that aligns the LC molecules in the direction perpendicular to the incident light polarization direction [20].

Interaction of SD1 molecules with a pumping linearly polarized UV light strongly depends on orientation of the azo-dye molecules relative to the polarization vector of the actinic light [20]. Such angular dependence produces the light-induced anisotropy of the orientational distribution of azo-dye molecules that can be described as the photoinduced orientational ordering that can generally occur by a variety of photochemically induced processes. The model describing the light-induced reorientation of azo-dye molecules in SD1 films as rotational Brownian motion governed by the light intensity-dependent mean-field potential was developed in [20].

First, a glass substrate was coated with 0.5% wt/wt solution of SD1 in N.N. dimethyl formamide (DMF) and afterwards it was baked at 100 °C for 5 min to evaporate excess solvent. Then after, the glass substrate was fixed on to a rotating stage as in the experimental setup shown in Fig. 4 . A UV lamp (Omnicure S1000) was used to illuminate the alignment layer. Two lenses were introduced in the light path to broaden the beam size and then a narrow light beam has been projected onto the substrate by a line mask with a cylindrical lens. Then after a wire-grid polarizer with axis at $\pi /4$ to the x-axis (shown in Fig. 4) was placed in the light path to provide linearly polarized light. The rotating stage with SD1 substrate was rotated at the speed of one degree per minute. Thus, the half circle of rotation (i.e. 180°) of the SD1 substrate provides the alignment, which aligns the easy axis at the azimuthal angle with the axial symmetry of$q=1,\alpha {}_0=\pi /4$. Then after, the LC polymer (LCP) UCL 017 (DIC) was coated on the top of the SD1 layer at the speed of 3000 rpm for 30 seconds, thus constructed layer thickness of LCP gives the retardation equal to the QWP for wavelength 450nm. The SD1 layer aligns the LCP spatially with the same axial symmetry. Then the substrate was exposed by UV light to polymerize and fix the orientation of the LCP and thus, the polarization converter was fabricated. However, for the switchable polarization converter, first the LC cell was assembled by two SD1 coated glass plate for the cell thickness 0.8 μm, afterwards the cell was fixed on the same rotating stage and aligned in the same manner. The beauty of this method is that no precise adjustment is required here for both patterns (i.e. on the top and bottom glass plate) because both of them were aligned simultaneously in a single step process that makes the fabrication simple and accurate. Then after, LC A0138 (DIC) has been filled in the cell. The response time has been measured by placing the LC cell between two crossed polarizers and illuminated by a He-Ne laser. Afterwards, the electro-optical response has been recorded and processed on the digital oscilloscope Tektronix TDS1002.

 

Fig. 4 Experiment setup for the patterned photo-alignment.

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4. Results and discussion

Thus fabricated patterned QWP made of LCP layer and LC cell filled with A0138 show similar optical quality. For optical analysis, the patterned QWP made of LCP layer is used for measurement and results are shown in Fig. 5 . In Figs. 5(a), 5(b) and 5(c), different optical setups to measure the patterned QWP are disclosed. In Figs. 5(d), 5(e) and 5(f), the experimental results are shown under the optical setups described in Figs. 5(a), 5(b) and 5(c) respectively. To analyze the optical performance of the patterned QWP, the same polar coordinates model as shown in Fig. 1(a) was used, and the origin of the polar coordinates is designed coinciding with axially symmetric point. In Fig. 5(a), the patterned QWP was placed between two crossed polarizers, therefore, when the fast optical axis of the polarization converter is in the direction of optical axis of either polarizers, the light will be blocked by the polarizer and black state appears. So it shows black when $\theta \text{=}\pi /4,3\pi /4,5\pi /4\text{and}7\pi /4$ in Fig. 5(d). Whereas, if the impinging light is LHC or RHC polarized light, then after passing through the polarization converter, the output light becomes radially and azimuthally polarized light, respectively. Now to investigate the performance and output light properties, an analyzer in L-axis direction of the polar coordinates was placed after the polarization converter. For the radially and azimuthally polarized output light, after passing through the analyzer, the transmittance can be calculated independently in Eqs. (5) and (6) as:

 

Fig. 5 Illustration of the working principle of the proposed patterned QWP polarization converter. (a), (b) and (c) represent different optical setups for measurements, where (a) the polarization converter placed under the crossed polarizers (b) the polarization converter has been illuminated by LHC, and in (c) the polarization converter has been illuminated by RHC. Here P and A represent the polarizer and analyzer respectively. The (d), (e) and (f) represent experimental images for the top view of (a), (b) and (c) here the arrow shows the direction of optic axis of analyzer and white marker is equal to 2mm. (g) and (h) represent the angular transmittance profile, along the yellow circles shown in (e) and (f) respectively, for the radially and azimuthally converted polarized light. The open legends represent the experimental data while the solid red line represents the best theoretical fit.

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In order to compare the experimental data with the theory, The MATLAB was used to get the angular dependence of transmittance data from Fig. 5(e) and Fig. 5(f). First, we try to obtain the position of the axial symmetric point using MATLAB. The process is as follows. First, we chose any point at position (a, b) on the experimental image shown in Fig. 5, and if t(a, b) represent the transmittance data at the point (a, b), then we can use MATLAB to minimized the Z(a, b) value which is defined in Eq. (7). When Z value is minimized for a certain a and b, the axial symmetric point is at the position of (a, b).

Then after, the transmittance profile at the periphery of a 3mm radius circle was calculated along the yellow circle shown in the respective figures with the axial symmetric point as the center. The normalized angular dependence of transmittance profile for both converted radially and azimuthally polarized light has been shown in the Figs. 5(g) and 5(h) respectively. The open legends represent the experimental data while, the theoretical data (according to Eqs. (5) and (6)) is shown by the solid red line in Figs. 5(g) and 5(h). It can be seen in figures that the experimental data is in the good agreement with the theory; therefore, it confirms the working principle and good optical quality of the proposed polarization converter.

For the application of the patterned QWP as polarization converter, the input light should be circularly polarized light. Moreover, if the patterned QWP, with axial symmetry of $q=1,\alpha {}_0=\pi /4$, is inverted, the same QWP shows the axial symmetry of $q=1,\alpha {}_0=-\pi /4$. So when the patterned QWP is inverted, for the same input circularly polarized light, it works as radially polarization converter and azimuthally polarization converter independently.

According to the Eqs. (2) and (3), apart from the rotation of the polarization, there is a phase factor $\exp (-j\beta )$or $\exp (j\beta )$for input LHC or RHC polarized light respectively, then after passing through the patterned QWP (with axial symmetry of $q=1,\alpha {}_0=\pi /4$) the phase factor will be $\exp (-j(q\theta +\pi /4))=\exp (-j\theta )\exp (-j\pi /4)$or$\exp (j\theta )\exp (j\pi /4)$ respectively. Neglecting the constant phase change of $\exp (-j\pi /4)$ and $\exp (j\pi /4)$, the phase factor $\exp (-j\theta )$ and $\exp (j\theta )$will give an spiral phase to the beam, thus the output radial and azimuthal polarization light beam will have orbital angular momentum $l=-1$ or $l=1$, which has also been shown in [7]. Therefore, the patterned QWP could convert the impinging circularly polarized light to radially or azimuthally polarized light with orbital angular momentum, at the same time. However, the HWP could only give output light either with orbital angular momentum or with vortex (radial or azimuthal) polarization independently, depending on the input light polarization [3]. This is one of the important advantages of the proposed patterned QWP polarization converter over the polarization converters based on the HWP. For the applications where only output light polarization is concerned, the patterned QWP and HWP will have the same performance.

Moreover, the QWP incorporate half of the LC layer thickness than HWP, thus from the point of switching such systems are much faster than the polarization converters based on HWP. At the electric field of 25V/μm, the switching ON time and switching OFF time revealed for the proposed system is 20μs and 1.3ms respectively. Optimization of the LC material parameters and application of higher electric field can further improve the Switching OFF and ON time. However, less LC layer thickness limits the application of higher electric field and cause short-circuiting. To avoid any short-circuiting one may deploy the dielectric layers on glass plates before coating the photoalignment layer.

Furthermore, the QWP requires the half of the retardation than the HWP. For the visible range of the light both of the technologies are efficient but for the longer wavelength applications like telecommunication and THz regions, the HWP polarization converter becomes less efficient additionally the fabrication becomes rather complicated because of the higher film thickness. In addition, QWP structure offers fewer losses and comprises relatively simpler fabrication procedure.

5. Conclusions

In conclusion, we disclosed a radial and azimuthal polarization converter by deploying the patterned LC QWP. Axially symmetric photoalignment was used to fabricate the patterned QWP. The radially and azimuthally polarized output light can be generated with input light with circular polarization. Moreover, because of thinner LC thickness, the proposed device reveals very fast switching time. In addition to this, the proposed polarization converter does not show any defect line like axial polarization converter based on TN structure at the boundary of opposite twist [2]. Therefore, the proposed device has immense potential for application in optical trapping systems, to increase the trapping force, in laser cutting machine to increase the cutting speed and could find application in several other electro-optical and telecommunication elements.