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Abstract
In this paper, we propose a ring-type grating in which grating vectors formed by periodic molecular alignments are superimposed radially. Depending on incident polarization states, the anisotropic gratings generate polarization dependencies for the ring-like first order diffraction. To determine the polarization states of incident light for both single and multi-wavelengths, we attempted to extract the optical intensity distribution on the ring-like diffraction from each single-shot diffraction image. In addition, a system that enables the detection of spatial modulation of optical birefringence is proposed. Finally, we concluded that the optical intensity distribution on the circumference of the ring-like diffraction has sufficient potential to be available.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Rotational manipulations are usually required to measure the polarization states of optical waves. The motor-driven analyzer and photodetector are synchronized, and the optical intensity on an elliptical orbit is measured. The sequence requires certain duration, which results in difficulties in dynamic polarization measurements in a temporary changing system. A typical example is ellipsometry, which accurately determines the optical coefficients of flat substances [1,2], in which at least one polarizer should be manipulated rotationally, and the transmitted optical intensity should be measured sequentially. This problem is also closely related to optical birefringence measurements because the polarization measurements are almost the same for analyzing the anisotropy in optical coefficients. The birefringence measurement or birefringence imaging is also an important technique for visually revealing the inherent factors in various fields [3–7].
Therefore, previous studies have utilized several approaches to solve this problem. For example, many kinds of snapshot based, polarization imaging systems have been proposed; in these studies, specifically arranged polarizers have been designed and demonstrated successfully [8–12]. An advanced polarization camera was developed and fabricated, resulting in various applications [13–16]. However, most cases require some analytical calculations to extract or exhibit polarization information. If a spectroscopic polarization measurement system based on simple calculation (or data processing) for single-shot images, without rotational manipulations is developed, it would be applicable to various objects regardless of the field.
Previously, we also reported several polarization measurement systems without rotational manipulation using different type of diffractive gratings [17,18]. Among these studies, the use of anisotropic diffractive gratings exhibited a high potential. In this study, a unique design of anisotropic diffractive gratings is employed to realize spectroscopic polarization detection system without rotational manipulations, based on simple data processing.
2. Concept of a ring-type polarization grating
2.1 Anisotropic diffraction gratings for polarization analysis
Anisotropy in refractive index (i.e., optical birefringence) can be controlled by the alignment of uniaxial molecules, such as in the case of liquid crystal molecules. When the molecular alignment is modulated periodically, it acts as an anisotropic diffractive grating. Figure 1(a) shows a typical periodic molecular alignment formed by random and uniaxial alignments. Consequently, the spatial modulation forms anisotropic diffractive gratings with grating vector k.
Fig. 1. (a) Periodic molecular alignment. (b) and (c) Anisotropic refractive index modulations. (d) Anisotropic diffraction efficiencies. The graphs A, B, and C correspond the calculated results for the conditions of $\mathrm{\Delta }{n_x}$ : $\mathrm{\Delta }{n_y}$ = 0.080 : 0.080, $\mathrm{\Delta }{n_x}$ : $\mathrm{\Delta }{n_y}$ = 0.104 : 0.056, and $\mathrm{\Delta }{n_x}$ : $\mathrm{\Delta }{n_y}$ = 0.136 : 0.024, respectively. (e) Diffraction efficiencies depending on polarization azimuth angle and (f) ring-type anisotropic diffractive gratings.
Figure 1(b) illustrates the anisotropic refractive index modulations when the molecular direction in the uniaxial regions is parallel to k. Here, the orthogonal refractive index modulations ${n_x}(\delta )$ and ${n_y}(\delta )$ exhibit positive and negative amplitudes, respectively, as an anisotropic modulation. In the Raman-Nath diffraction region, the diffracted field corresponds to the Fourier image of the complex amplitude distribution of the optical field passing through the gratings. If the attentional diffraction is limited to the first order, the refractive index modulations can be approximated as fundamental sinusoidal modulations, as shown in Fig. 1(c). For the anisotropic gratings, the first-order diffractions are based on the sinusoidal modulations ${n_x}(\delta )$ and ${n_y}(\delta )$ to the incidence of p- and s-polarized beams, respectively, where $\delta = \pi x/\mathrm{\Lambda }$ and $\mathrm{\Lambda }$ is the grating pitch. In this case, because the diffractions with orthogonally polarized beams are independent of each other, each diffraction efficiency is also characterized using a Bessel function of the first kind [19]. The phase modulations for each polarization are expressed as
where $\mathrm{\Delta }{n_x}$ and $\mathrm{\Delta }{n_y}$ are the absolute values of the amplitudes of ${n_x}(\delta )$ and ${n_y}(\delta )$, respectively, and $\textrm{k}$ and d are the wavenumber and grating thickness, respectively. The diffraction efficiencies of the m-th order diffraction of the p- and s-polarized beams areAs mentioned earlier, the anisotropic diffraction gratings exhibit either characteristic in Fig. 1(e), depending on the incident polarization direction. This means that the incident polarization direction can be detected at a condition of ${\boldsymbol E} \bot {\boldsymbol k}$ or ${\boldsymbol E\; }/{/}\,{\boldsymbol k}$ from the diffraction intensity. In the other conditions, the symmetric characteristics shown in Fig. 1(e) suggest both positive and negative polarization directions. In this study, ring-type anisotropic diffractive gratings, shown in Fig. 1(f), were developed to obtain reasonable polarization directions. Each grating vector ${{\boldsymbol k}_n}$ (${{\boldsymbol k}_{\boldsymbol n}},\; n = 1,2, \ldots .$) consists of periodic molecular alignment, as shown in Fig. 1(a), and they are aligned radially. Therefore, any incident beam fulfills the condition of ${\boldsymbol E} \bot {\boldsymbol k}$ or ${\boldsymbol E\; }/{/}\,{\boldsymbol k}$, and the polarization directions can be detected based on the diffraction intensity distributions.
2.2 Preparation of ring-type polarization gratings
To prepare ring-type anisotropic diffractive gratings, a photo-crosslinkable liquid crystal polymer (PLCP) material was employed [20–22]. Similar PLCPs were used in our previous study. A spin-coated film of PLCP on a glass substrate was prepared at a thickness of approximately 2.9μm. When the PLCP materials are irradiated by linearly polarized ultraviolet (UV) light, axis-selective photo-crosslinking occurs in the parallel direction to the UV light polarization. Subsequently, the film was annealed at the transition temperature from the solid phase to the liquid crystal phase. During annealing, the mesogenic moieties of PLCP are oriented parallel to the crosslinking moieties. As a result, the PLCP molecules can be aligned parallel to the polarization direction of the irradiated UV light.
A radial polarizer and a concentric line/space pattern (12.5/12.5μm) mask were prepared to form a periodic molecular alignment, as illustrated in Fig. 1(f), in the PLCP film. A photograph and polarization optical microscopy (POM) image of the resultant gratings are shown in Fig. 2(a) and 2(b), respectively. The radial alignment can be confirmed from the dark region distributions along the directions of polarizer (P) and analyzer (A) in the POM image. However, concentric periodic modulation cannot be confirmed because of the low magnification of the POM image. As a reference, other gratings that were formed using a relatively wide concentric line/space pattern (50/50μm) mask were prepared and observed, as shown in Fig. 2(c). Alternative modulation of the uniaxial and random molecular alignment was observed along the radial direction.
Fig. 2. (a) Photograph (b) and (c) polarization optical microscope images of ring-type anisotropic gratings, (d) optical setup, and (e) and (f) diffraction patterns for non- and linearly-polarized incident light, respectively.
As another test of the ring-like anisotropic gratings in Figs. 2(a) and 2(b), the diffraction pattern was captured using an optical setup with white light emitted from a tungsten halogen lamp, as shown in Fig. 2(d). The emitted white light passed through a pinhole, and the light was collimated using lens 1. The collimated light was incident on the ring-type gratings. Finally, the output light was Fourier-transformed by lens 2 to generate far-field diffraction at the digital camera plane to acquire the diffraction intensity distribution. Here, a color CMOS camera was used, and a quarter of the diffraction images were captured for non- and linearly-polarized incident white light, as shown in Fig. 2(e) and 2(f), respectively. Rainbow-like and ring-like diffractions were observed. In addition, fractured ring-like diffraction was observed for the linearly polarized light, which suggested the formation of polarization-dependent functional gratings, as expected. In both cases, the higher order diffractions could not be observed distinctly.
3. Results and discussion
3.1 Fundamental specifications for single wavelength
For the optical setup shown in Fig. 2(d), a band-pass filter for a wavelength of 633 nm with a full width at half maximum (FWHM) of 10 nm, a linear polarizer, and a quarter-waveplate (QWP) for 633 nm were introduced behind lens 1. In addition, a monochrome CCD camera with a wide detection area of 1.1 inch was also introduced to capture the entire ring-like first-order diffraction.
When the incident light was adjusted to the linear polarization at 45°, ring-like diffraction was captured, as shown in Fig. 3(a). The ring fracture can be confirmed in the direction of an azimuthal angle of 45°. Therefore, the prepared gratings have the characteristic that the maximum diffraction intensity is obtained at a condition of ${\boldsymbol E} \bot {\boldsymbol k}$, represented by the red curve in Fig. 1(e). The optical intensity distribution on the circumference of the ring-like diffraction was extracted and plotted in Fig. 3(d) using a red circle marker. This intensity distribution on the polarization azimuthal angle θ corresponds to the transmitted optical intensity modulation under rotational manipulation of the polarization analyzer (linear polarizer) to the linearly polarized light, although the peak intensity corresponding to the polarization direction is shifted ±π/2 because of the selected condition of ${\boldsymbol E} \bot {\boldsymbol k}$ for the maximum diffraction intensity.
Fig. 3. (a), (b), and (c) Ring-like diffractions for linearly, elliptically, and circularly polarized light, respectively, and (d) light intensity distributions on the circumferences of ring-like diffractions. White bars in the captured images indicate the scale on digital camera plane.
When the fast-axis of QWP was adjusted to the relative angles of 22.5° and 45.0° to the linearly polarized light set at 45°, the ring-like diffractions were captured, as shown in Figs. 3(b) and 3(c), respectively. Similarly, the optical intensity distributions on their circumferences were extracted and plotted in Fig. 3(d). The elliptic polarization and almost circular polarization can be confirmed. The red, blue, and green solid lines indicate their ideal characteristics based on the conditions of the polarizers used for the preparation of linear, elliptic, and circular polarization states. Each experimental result deviated slightly from their identical distributions. The reason for this is still unclear. We consider that alignment failures would exist locally. This may be associated with deviations in the optical characteristics.
These experimental results demonstrated that the polarization states can be detected from single capture images using ring-type anisotropic diffractive gratings, without any rotational manipulations, although slight deviations exist. (However, the circular polarization state cannot be distinguished between left- and right-handed circular polarizations in the present system.)
3.2 Potential quick polarization detection for multi-wavelength measurement
In Figs. 2(e) and 2(f), the ring-like diffractions showed a rainbow-like coloration, indicating spectroscopic function. To evaluate this function, a linear polarizer and QWP for 633 nm were introduced behind lens 1 in the optical setup shown in Fig. 2(d). The polarizers were set under the same conditions as in the experiment in Fig. 3(c). In other words, if a band-pass of 633 nm is introduced in the optical setup, almost circular polarization would be observed. Under this condition, the diffraction image for the white light incidence was captured by a monochrome digital camera without any filters, as shown in Fig. 4(a). Interestingly, the first order diffraction was distributed elliptically around the zero-order diffraction.
Fig. 4. (a)-(d) Ring-like diffractions for white light, wavelengths of 500, 633, and 750 nm, respectively, in polar coordinates. Similarly, (e)-(h) ring-like diffractions for each light, in rectangular coordinates. (i) Light intensity distributions on the circumferences of ring-like diffractions. (j) and (k) Contour and 3-D polarization information maps in the rectangular coordinates of θ and r, like landscapes.
To investigate the properties, three band-pass filters of 500, 633, and 750 nm were used to capture the diffraction images, and the results are shown in Fig. 4(b), 4(c), and 4(d), respectively. First, the polarization states for 633 nm appeared close to a circular polarization, similar to that in Fig. 3(c). On the other hand, the polarizations for 500 and 750 nm were elliptical, based on the phase shift differences from $\pi /2$, and it appeared that the directions of the ellipses were orthogonal to each other.
To analyze the ring-like diffractions with different diameters systematically, the polar coordinates regarding the azimuthal angle θ and the radius r in Figs. 4(a), 4(b), 4(c), and 4(d) were converted into rectangular coordinates, as shown in Figs. 4(e), 4(f), 4(g), and 4(h), respectively. As a result, the spectroscopic indications for polarization information became clear. First, each optical intensity distribution on the circumferences of the three wavelengths was extracted directly along the θ axes in Figs. 4(f), 4(g), and 4(h), and the intensity is plotted in Fig. 4(i). Almost circular polarization for 633 nm and orthogonal elliptic polarizations for 500 and 750 nm, respectively, were found because of the use of QWP for 633 nm. Therefore, the white light diffraction pattern converted into rectangular coordinate can be redrawn as a landscape indicating the spectroscopic polarization states, as shown in Fig. 4(j). This landscape would support the understanding of features of the measured polarized light, intuitively and quickly. The three-dimensional (topographical) plot in Fig. 4(k) would also be beneficial for understanding the polarization changes depending on the wavelength.
We demonstrated that polarization analysis for continuous multi-wavelength measurement can be performed quickly and simultaneously, based on a single-shot image of ring-like diffraction, through a simple coordinate conversion. This optical function corresponds to a vectorial analysis performed on information that was simultaneously received. Therefore, the proposed anisotropic gratings can be considered as an analog parallel-processing device.
3.3 Spatial distribution detection from single diffraction pattern
The fundamental potential was revealed, as mentioned above, in which each polarization state of an optical pupil including various wavelengths was converted into a two-dimensional map, indicating their polarization states like a landscape. Therefore, various temporal vibrations of optical electric fields (various optical frequencies) with different vectorial features could be expanded two-dimensionally as an optical intensity distribution, which corresponds to a type of frequency-space conversion. In the next stage, we attempted an advanced approach that corresponded to a space-space conversion (or space-frequency-space conversion) using the vectorial analysis function in the ring-type gratings.
Figure 5(a) illustrates the experimental setup, in which the white light emitted from a lamp was divided at every wavelength using a prism pair system, as shown in Fig. 5(b). The polarization state was adjusted for the p-polarization. When a transparent sample was placed on the sample plane, spatial optical modulation was reflected for each wavelength. A stripe of transparent polymer film with a width of approximately 1 mm was attached to a glass substrate, where the machine direction (MD) of the film was aligned at a crossing angle of 45° to the bottom edge of the glass substrate, as shown in Fig. 5(c). When the polymer stripe was placed on the sample plane overlapping with the divided light on the S-axis (i.e., the stripe was placed on the region surrounded by the red dashed line in Fig. 5(b)), the polarization states of the light passing through the stripe was changed at the spatially corresponding wavelength.
Fig. 5. (a) Optical setup, (b) spatial optical-wavelength dispersion produced by a prism pair system, and (c) transparent polymer stripe on a glass substrate.
Figures 6(a) and 6(c) show the initial ring-like diffraction of the incident p-polarized light, as the polarization states expanded on the polar and rectangular coordinates, respectively. We found that all wavelengths were linearly polarized. On the other hand, when the polymer stripe was placed on the sample plane, the ring-like diffraction was modulated clearly, as shown in Fig. 6(b) and 6(d). To evaluate this modulation quantitatively, the differences between the optical intensity distributions of Figs. 6(c) and 6(d), defined as $\mathrm{\Delta }I(\theta )$ were calculated, and the results were plotted at certain wavelengths, as shown in Fig. 6(e). In addition, the amplitude A (i.e., A = max [$\mathrm{\Delta }I(\theta )$]- min [$\mathrm{\Delta }I(\theta )$]) of each graph was calculated and plotted as a function of the wavelength, as shown in Fig. 6(g). It is clear that polarization modulation was generated at wavelengths of around 625 nm. On the S-axis in the sample plane, the wavelength difference of 100 nm corresponds to a spatial distance of 1.4 mm, as confirmed in Fig. 6(f). The bottom lateral axis of Fig. 6(g) indicates that the wavelengths can be rewritten as the top lateral axis, indicating the distance on the S-axis. Therefore, the stripe width was estimated at 1.25 mm, based on the FWHM of the graph. Because the actual width was approximately 1mm, the deviation was at least +25%. This is attributed to the relatively low resolution in the wavelength dispersion in Fig. 6(f). Nevertheless, the proposed approach is successfully demonstrated. The spatially modulated optical anisotropy can be expanded two-dimensionally and analyzed quickly and clearly based on the single-shot optical intensity distribution and preliminary acquired reference data.
Fig. 6. (a) and (c) Ring-like diffraction patterns in polar- and rectangular-coordinates without a sample on the sample plane. Similarly, (b) and (d) Ring-like diffraction patterns for both coordinate system with the polymer stripe on the sample plane. (e) and (g) Optical intensity differences and amplitudes for detecting the sample in (f).
As a result, the function of the space-space conversion was demonstrated using ring-type anisotropic diffractive gratings. In this case, each wavelength was used as the carrier wave to transport the spatially distributed optical modulation. Therefore, we must consider the wavelength dispersion in the optical coefficient of targeted substances to generate the spatially distributed optical modulation.
5. Conclusion
Herein, a ring-type anisotropic diffractive grating consisting of a fundamental modulation of birefringence is proposed as an optical device to reveal the polarization information. For the ring-like first-order diffraction, the optical intensity distribution on the ring circumference corresponds to the optical transmission obtained under the rotational manipulation of a polarization analyzer. Therefore, the use of these ring-type gratings allows omitting not only rotational manipulation but also the use of the polarization analyzer itself for vectorial analysis. In addition, this function can be adopted for multi-wavelength spectroscopy. Therefore, many vectorial information incidents can be processed simultaneously, and the results are expanded (or projected) spatially as a two-dimensional image that is acquired easily using a digital camera. This type of optical function is suitable for the assembled sensing units. At the same time, the frequency-space (or space-space) conversion could be useful for analyses in the various research fields such as materials science and applied physics.
Funding
Japan Society for the Promotion of Science (19H02056, 20H02767).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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