Simultaneous detection of polarization states and wavefront by an angular variant micro-retarder-lens array

Toshitaka Wakayama; Toshitaka Wakayama; Akane Zama; Yudai Higuchi; Yuta Takahashi; Kohei Aizawa; Takeshi Higashiguchi

doi:10.1364/OE.509574

1. Introduction

Vector beams and optical vortices, which are spatially controlled polarization states and wavefronts, hold the promise of new discoveries in optical sciences based on their unique interactions between light and matter in the fields of astrophysics, material science, and molecular biology [1–3]. Vector beams have been applied to longitudinal electric fields in particle sorting, ultra-small spot creation in super resolution microscopy, and sharpening electric fields in laser processing [4–7]. Optical vortices have opened new avenues for chirality generation with the transcription of orbital angular momentum of light, and orbital angular momentum multiplexing communication with topological charge control [8–12]. To precisely control the polarization states and wavefront of light, meta surfaces fabricated by subwavelength structures and axially symmetric waveplates have been proposed [7,12–14]. To obtain the optical properties of vector beams and optical vortices with tailored polarization states and wavefront, it is necessary to evaluate both the polarization states and wavefront of light.

Polarization imaging has been historically achieved using a rotating polarizer and quarter waveplate [15]. A single shot imaging polarimeter with a spatially controlled polarization array enables polarization imaging without any mechanical control [13–20]. Such an imaging polarimeter has been used for material evaluation in the roll-to-roll process and short pulse laser processing, and also for biopsy imaging for medical diagnosis [16,21,22]. Interferometry has been the dominant method for wavefront detection. Polarization optics using subwavelength structures have been significantly improved by recent advancements in interferometry. Cyclic changes of polarization states in polarization optics allow for the generation of geometric phases [12,23,24], which in turn provide phase shifts to determine the wavefront. Thus, polarization optics help realize both polarization imaging and wavefront imaging. However, interferometric technologies are not suitable for wavefront imaging with a direct light source because interferometry needs interference between two light paths. To address this, a Shack–Hartmann sensor allows direct detection of a wavefront without interferometry [25–27]. The Shack–Hartmann sensor has been applied to astigmatism in laser refractive surgery and cataract surgery [27,28]. Ptychography, which is a reconstruction method that uses multiple interference scattering patterns obtained from an object, has been developed for evaluating x rays and extreme ultraviolet radiation or electrons [29]. As mentioned above, the simultaneous detection of the polarization states and wavefront of light is required. In recent years, the Shack–Hartmann sensor using meta surface optics has promised to achieve the determination of polarization states and wavefront [30]. However, this sensor is limited to a spectral range in the near infrared region. In astrophysics, material science, and molecular biology, it is necessary to measure both polarization states and the wavefront in the visible spectrum.

For this background study, we propose single shot simultaneous detection of the polarization states and wavefront using an angular variant micro-retarder-lens array. The birefringence of an angular variant micro-retarder-lens is distributed axially symmetrically on the circumference of the micro-retarder-lens, like a vortex retarder. In other words, this phenomenon works as a spatial rotating compensator method. Meanwhile, the micro-retarder-lens also acts as a micro lens. A micro-retarder-lens array will allow the detection of polarization states by the analysis of the intensity profile along the circumference of the micro-retarder-lenses and the wavefront of the light from the changes of the focusing spot.

2. Principle of simultaneous detection of polarization states and wavefront

The concept of our method for the simultaneous detection of the polarization states and wavefront of light is to combine a single shot imaging polarimeter and a Shack–Hartman wavefront sensor. Figure 1(a) shows the optical layout for the simultaneous detection. Light from a light source is uniformly distributed with regard to the polarization states and wavefront of the light at point A. After passing through an optical element, the polarization states and wavefront are changed at point B. In Fig. 1(b), the light passes through an angular variant micro-retarder-lens array, which is our unique optical component controlled by birefringence distribution, such as (b1) retardance and (b2) azimuthal angle. Note that, outside the lens, the retardance varies radially, while the azimuthal angle exhibits axial-symmetric changes in Fig. 1(c1). The distributions of the birefringence around the micro-lens provides polarization modulation in the spatial distribution of the polarization states altered by the optical element. After passing through a sheet polarizer, the intensity distribution captured by a two-dimensional detector contains the polarization change of the object from the polarization modulation. Therefore, the polarization change caused by the object can be determined by analyzing the intensity distribution around the micro lenses, as shown in Fig. 1(d1). The wavefront of the light at point B can also be determined by a micro lens array based on the Shack-Hartmann principle. The light is incident onto the lens as shown in Fig. 1(c1) and the focusing spot changes based on the wavefront of the light. For the focusing spot indicated by (x, y) as shown in Fig. 1(d1), the wavefront of light is measured.

Fig. 1. Principle of simultaneous detection of polarization and wavefront by an angular variant micro-retarder-lens array. (a) Optical setup. (b) Birefringence properties of micro lenses such as (b1) retardance and (b2) azimuthal angle. (c1) Focusing by angular variant micro-retarder-lens. (c2) Relationship between focusing position and wavefront of light for a planer wave (blue) and a distorted wave (red). (d1) Captured image by 2D sensor. (d2) Intensity distribution on a circle at radius r₀.

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2.1 Polarization measurement

Let us first explain the details for measuring the polarization states. A Stokes vector of the incident light at point B is given by S_in =(s₀(i), s₁(i), s₂(i), s₃(i))^T, where i represents the number of the micro lens where the light is incident. When the angular variant micro-retarder-lens array has a retardance of 90° at a radius r₀ and the azimuthal angle smoothly changes with a function of angle θ, as shown in Fig. 1(b), a Mueller matrix of the angular variant micro-retarder-lens is equivalent to a Mueller matrix Q of a rotating quarter waveplate. For the Mueller matrix calculation, Stokes vectors of output beam, which is given by S_out =(s’₀(i), s’₁(i), s’₂(i), s’₃(i))^T, the can be written as

(1)$${{\boldsymbol S}_{\textrm{out}}} = {\boldsymbol P}\cdot {\boldsymbol Q}\cdot {{\boldsymbol S}_{\textrm{in}}}, $$

where, P indicates the Mueller matrix of a polarizer. The intensity distribution $I({i,\theta } )$ is given by

(2)$$\begin{aligned} s{\mathrm{^{\prime}}_0}(i )&= I({i,\theta } )\\ &= \frac{1}{4}({2{s_0}(i )+ {s_1}(i )- 2{s_3}(i )\sin 2\theta + {s_1}(i )\cos 4\theta + {s_2}(i )\sin 4\theta } ). \end{aligned}$$

As shown in Fig. 1(d2), this intensity changes as a function of angle θ. From a Fourier series, the intensity distribution in the function of angle θ around the micro lens can be rewritten as

(3)$$I({i,\theta } )= \mathop \sum \nolimits_{k = 0}^\infty ({{a_k}(i )\cos k\theta + {b_k}(i )\sin k\theta } ), $$

where k indicates natural numbers and a_k and b_k are Fourier coefficients represented as

(4)$${a_k}(i )= \frac{1}{\pi }\mathop \smallint \nolimits_0^{2\pi } I(\theta )\cos k\theta , $$

(5)$${b_k}(i )= \frac{1}{\pi }\mathop \smallint \nolimits_0^{2\pi } I(\theta )\sin k\theta .$$

The Stokes parameter of the incident light from s₀ to s₃ can be determined from the Fourier coefficients of a₀, b₂, a₄, and b₄ [14,20], as follows.

(6)$${s_0}(i )= \frac{{4{a_0}(i )- {s_1}(i )}}{2}, $$

(7)$${s_1}(i )={-} 2{b_4}(i ), $$

(8)$${s_2}(i )= 4{a_4}(i ), $$

(9)$${s_3}(i )= 4{b_2}(i ). $$

The ellipticity and azimuth of the incident light can be written by [15]

(10)$$\varepsilon (i )= \frac{{{s_3}(i )}}{{{s_0}(i )+ \sqrt {{s_1}{{(i )}^2} + {s_2}{{(i )}^2}} }}, $$

(11)$$\varphi (i )= \frac{1}{2}{\tan ^{ - 1}}\left( {\frac{{{s_2}(i )}}{{{s_1}(i )}}} \right). $$

The determination of the Stokes parameters is limited to cases where the retardation of the angular variant micro-retarder-lens is 90°. When the retardation varies spatially, a calibration method is required to measure the polarization states.

2.2 Calibration of polarization measurement

To calibrate the obtained Stokes parameters, we used a pseudo inverse matrix. The linear system in our imaging sensor, where the input matrix S_in is converted into the output matrix S_out by a transformation matrix M derived from our measurement, can be expressed as follows.

(12)$${{\boldsymbol S}_{\textrm{out}}} = {\boldsymbol M}\cdot {{\boldsymbol S}_{\textrm{in}}}. $$

An input matrix with an ideal polarization of S_in is represented using Stokes vectors of linear polarization at 0°, 45°, 90°, and 135°, and circular polarization with right- and left-hand as ideal polarization states as

(13)$${{\boldsymbol S}_{\textrm{in}}} = \left[ {\left( {\begin{array}{c} {\begin{array}{c} 1\\ 1 \end{array}}\\ {\begin{array}{c} 0\\ 0 \end{array}} \end{array}} \right)\left( {\begin{array}{c} {\begin{array}{c} 1\\ 0 \end{array}}\\ {\begin{array}{c} 1\\ 0 \end{array}} \end{array}} \right)\left( {\begin{array}{c} {\begin{array}{c} 1\\ { - 1} \end{array}}\\ {\begin{array}{c} 0\\ 0 \end{array}} \end{array}} \right)\left( {\begin{array}{c} {\begin{array}{c} 1\\ 0 \end{array}}\\ {\begin{array}{c} { - 1}\\ 0 \end{array}} \end{array}} \right)\left( {\begin{array}{c} {\begin{array}{c} 1\\ 0 \end{array}}\\ {\begin{array}{c} 0\\ 1 \end{array}} \end{array}} \right)\left( {\begin{array}{c} {\begin{array}{c} 1\\ 0 \end{array}}\\ {\begin{array}{c} 0\\ { - 1} \end{array}} \end{array}} \right)} \right]. $$

Since the ideal input matrix of S_in is a 4 × 6 matrix, we defined a 6 × 6 square matrix using a 6 × 4 transposed matrix of S_in^T to create an inverse matrix. Equation (12) is rewritten as follows.

(14)$${{\boldsymbol S}_{\textrm{out}}}\cdot {{\boldsymbol S}_{\textrm{in}}}^\textrm{T} = {\boldsymbol M}\cdot ({{{\boldsymbol S}_{\textrm{in}}}\cdot {{\boldsymbol S}_{\textrm{in}}}^\textrm{T}} ). $$

To obtain the inverse transformation matrix M^-1, we defined a 6 × 6 pseudo inverse matrix of ${({{{\boldsymbol S}_{\textrm{in}}}\cdot {{\boldsymbol S}_{\textrm{in}}}^\textrm{T}} )^{ - 1}}$.

(15)$${{\boldsymbol S}_{\textrm{out}}}\cdot {{\boldsymbol S}_{\textrm{in}}}^\textrm{T}\cdot {({{{\boldsymbol S}_{\textrm{in}}}\cdot {{\boldsymbol S}_{\textrm{in}}}^\textrm{T}} )^{ - 1}} = {\boldsymbol M}\cdot ({{{\boldsymbol S}_{\textrm{in}}}\cdot {{\boldsymbol S}_{\textrm{in}}}^\textrm{T}} )\cdot {({{{\boldsymbol S}_{\textrm{in}}}\cdot {{\boldsymbol S}_{\textrm{in}}}^\textrm{T}} )^{ - 1}}. $$

For our calculation, the inverse transformation matrix is given by

(16)$${{\boldsymbol M}^{ - 1}} = {[{{{\boldsymbol S}_{\textrm{out}}}\cdot {{\boldsymbol S}_{\textrm{in}}}^\textrm{T}\cdot {{({{{\boldsymbol S}_{\textrm{in}}}\cdot {{\boldsymbol S}_{\textrm{in}}}^\textrm{T}} )}^{ - 1}}} ]^{ - 1}}. $$

To determine the input matrix with the desired polarization, Eq. (12) can be rewritten as

(17)$${{\boldsymbol S}_{\textrm{in}}} = {{\boldsymbol M}^{ - 1}}\cdot {{\boldsymbol S}_{\textrm{out}}}. $$

According to our calculation, our results of the polarization measurement using angular variant micro-retarder-lens array can be calibrated precisely.

2.3 Measurement of wavefront

To measure the wavefront of the light, we used algorism of a Shack-Hartmann wavefront sensor. When a light beam with the desired wavefront is incident onto the i-th micro lens, as shown in Fig. 1(c), the light beam with intensity of I(x, y) is focused on the point (x, y) in Fig. 1(d1). Based on an analysis of the centroid of the intensity distribution, we obtained the focusing points at (x_i_{_s}, y_i_{_s}) as

(18)$$({{x_{i\_s}},{y_{i\_s}}} )= \left( {\frac{{\mathop \smallint \nolimits_0^{N - 1} \mathop \smallint \nolimits_0^{N - 1} x\cdot I({x,y} )dxdy}}{{\mathop \smallint \nolimits_0^{N - 1} \mathop \smallint \nolimits_0^{N - 1} I({x,y} )dxdy}},\frac{{\mathop \smallint \nolimits_0^{N - 1} \mathop \smallint \nolimits_0^{N - 1} y\cdot I({x,y} )dxdy}}{{\mathop \smallint \nolimits_0^{N - 1} \mathop \smallint \nolimits_0^{N - 1} I({x,y} )dxdy}}} \right), $$

where N means the number of pixels in the x-y coodinates used to find the centroid of the intensity distribution. When a plane wave is used as a reference input and the focusing point is denoted as (x_i_{_0}, y_i_{_0}), Δz_i as depicted in Fig. 1(c2) can be expressed as

(19)$$\mathrm{\Delta }{z_i} = \frac{{\sqrt {{{({{x_{i\_s}} - {x_{i\_0}}} )}^2} + {{({{y_{i\_s}} - {y_{i\_0}}} )}^2}} \cdot d}}{f} = \frac{\lambda }{{2\pi }}\mathrm{\Delta }\phi ,{\; }$$

where d, f, l, and Δϕ mean the diameter of the micro lens, focus length, wavelength of the light beam, and wavefront of the light. We also determined the incident angle of ϑ as

(20)$${\vartheta _i} = {\tan ^{ - 1}}\left( {\frac{{{y_{i\_s}} - {y_{i\_0}}}}{{{x_{i\_s}} - {x_{i\_0}}}}} \right). $$

According to these calculations, we can obtain two-dimensional distributions for the polarization states and wavefront of light.

3. Experimental results

3.1 Birefringence properties on laser processing mark

The key to separately handling the polarization states and wavefront is the micro-retarder-lens array. We will now explain our fabrication method for the angular variant micro-retarder-lens array. We used polymethyl methacrylate dyed sky blue as the micro-retarder-lens substrate, processed by a laser diode with a maximum laser average power of 0.4 W at a wavelength of 650 nm. The laser diode, passing through an objective lens with a magnification of x10 and NA = 0.14, included digital modulation for laser energy control. By controlling the pulse width by a functional generator, we evaluated the diameters of the processing marks. Figure 2(a) shows the optical setup for the laser processing. Focusing the laser beam on the substrate created a laser processing mark, as shown in Fig. 2(b). Part of the energy of the laser beam was absorbed by the dye of the substrate. The melted substrate solidified upon cooling to a room temperature of 25°C. During the solidification at the melted spot, stress birefringence resulted in axial symmetry. Figure 2(b) shows microscopic images under unpolarized and cross-polarized conditions at laser energies of 0.32 J and 1.28 J, showing marks with diameters of 435 µm and 903 µm, respectively, while Fig. 2(c) shows the laser energy dependence of the diameters of the processing marks over the range of laser energies used. Note that the processing marks are slightly ellipsoidal since the laser diode used has a higher order transverse mode. These results indicate that the processing marks perform as micro-retarder-lenses with axial-symmetric polarization around the lens.

Fig. 2. Fabrication of micro-retarder-lens. (a) Optical setup of laser processing. (b) Micro lenses fabricated at laser powers of 0.32 J and 1.28 J observed by a microscope and a crossed polarizer. (c) Laser power dependance of processing mark diameter. The scale bars in (b) indicate 0.5 mm, respectively.

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3.2 Angular variant micro-retarder-lens array

To demonstrate our method of simultaneous detection, we fabricated a 7 × 7 angular variant micro-retarder-lens array using a two-dimensional stage. A laser beam was collimated to a Gaussian profile by an anamorphic prism pair, as shown in Fig. 2(a), because of the ellipsoidal shape of the laser diode output. We used a laser diode at a laser energy of 0.66 J. We evaluated the fabricated micro-retarder-lens array birefringence properties using a rotating compensator [14]. Figures 3(a) and 3(b) show the retardance and azimuthal angles. The intervals of the micro-retarder-lens array with diameters of 620 µm was 1.1 mm. Using a micrometer to measure the thickness of the micro-retarder-lens, we obtained it to be 50 µm. The areas in the black squares in Figs. 3(a1) and 3(b1) are enlarged in Figs. 3(a2) and 3(b2), respectively. The retardance around the micro-retarder-lens exhibited a concentric change along the radius direction in Fig. 3(a2), while the azimuthal angle around the micro-retarder-lens is axial-symmetrically distributed in Fig. 3(b2). Figures 3(a3) and 3(b3) show the profiles of the retardance and the azimuthal angle on the dashed circles in Figs. 3(a2) and 3(b2), respectively, with the red highlight area indicating the range between the maximum and minimum values across 5 pixels. The retardance varied slightly from 49° to 86° along the circumferential direction. Note that the employed laser diode had a higher transverse mode. The azimuthal angle changed smoothly between −90° to 90°. Based on these birefringence properties, for the polarization measurement in Section 2.1, the micro-retarder-lens array also requires the calibration method in Sec. 2.2.

Fig. 3. Birefringence properties of micro-retarder-lens array. (a1) Two-dimensional distribution of retardance. (a2) Enlarged view of area in dashed black square in (a1). (a3) Retardance distribution around the dashed circle in (a2). (b1) Two-dimensional distribution of azimuthal angle. (b2) Enlarged view of dashed black square in (b1). (b3) Retardance distribution around the circle in (b2). Scale bars indicate 1.2 mm, respectively.

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3.3 Simultaneous detector for polarization states and wavefront of light

To detect both the polarization states and wavefront of light, we incorporated our angular variant micro-retarder-lens array into a CMOS sensor (STC-MBCM401U3 V, Omron-Sentech Co., Ltd.) as a two-dimensional sensor. Figures 4(a) and 4(b) show the front and side views of the CMOS sensor. The pixel size of the CMOS sensor was 5.5 µm × 5.5 µm. The focusing length of the micro-retarder-lens was f = 3.8 mm. We fixed a sheet polarizer onto the substrate.

Fig. 4. Detector for polarization states and wavefront of light. (a) Front view and (b) side view. Each white scale bars indicate 5 mm, respectively

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The optical setup designed for the simultaneous detection of polarization states and wavefront is shown in Fig. 5(a). As a light source, we employed a laser diode with a laser power of 3 mW at a wavelength of 532 nm. A rotating diffuser was used to reduce speckle patterns. A lens system consisting of an objective lens and a collimated lens expanded the laser beam to a diameter of 12 mm. The incident polarization was controlled with polarization optics consisting of a polarizer and a quarter waveplate. Using the polarization optics, we first calibrated the polarization states of the light.

Fig. 5. Simultaneous detection of the polarization states and wavefront of light. (a) Optical setup for evaluation of both polarization states and wavefront. (b) Image captured by a CMOS camera with the incident beam at a linear polarization (LP) of 0°. (c) Enlarged images of micro-retarder-lens with linear polarization at (c1) 0°, (c2) 45°, (c3) 90°, and (c4) 135°, and (c5) right- and (c6) left-hand circular polarization. (d1) Combined left and right circular polarization images. The bright rings were formed around the micro-retarder-lenses. (d2) Enlarged view of (d1). (e) Intensity profiles of the bright ring for linear polarizations of (e1) 0°, (e2) 45°, (e3) 90°, and (e4) 135°, and (e5) right- and (e6) left-hand circular polarization. Scale bars in (b) and (d1), as well as (c) and (d2) are 1 mm and 0.1 mm, respectively.

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With the incident light linearly polarized at 0°, we captured an image by the CMOS camera, as shown in Fig. 5(b). An enlargement of a micro-retarder-lens is shown in Fig. 5(c1). Using linear polarizations of 0°, 45°, 90°, and 135°, and right- and left-hand circular polarizations, we captured the images shown in Fig. 5(c2)-(c6). The intensity profile at radius r₀, as shown in Fig. 1(d1), is required for polarization measurements. Figure 5(d1) shows a combination of the intensity distributions with left- and right-hand circular polarizations to create a prominent brightness ring profile around the edge of the micro lens. Figure 5(d2) shows an enlarged image from Fig. 5(d1). The width of the brightness ring profile was 10 µm. The intensity profiles for Figs. 5(c1)–5(c6), obtained by registering the coordinates on the brightness ring in Fig. 5(d1), are displayed in Figs. 5(e1)–5(e6). Using these intensity distributions at each linear and circular polarization, we performed the calibration. The Stokes parameters before and after the calibration are shown in Table 1, showing a significant correction. As a result, an accuracy of the polarization detection of ε = 0.01 in ellipticity and φ = ± 1.0° in azimuth was obtained.

Table 1. Stokes parameters before and after calibration.

View Table

We next evaluated the performance of our wavefront measurement using the angular variant micro-retarder-lens array. For this, we controlled the wavefront of the light by varying the optical path difference using a tilted microscopic glass slide. Figure 6(a) illustrates the experimental setup for checking the resolution of wavefront measurement. We placed a microscope glass slide in front of the simultaneous detector for polarization states and wavefront. The glass slide was rotated between 0° and 2.0° in increments of 0.2°. For the rotation, the optical path difference was changed with t / cosθ, where t represents the thickness of the glass slide, and θ denotes the angle of rotation. Figure 6(b) shows the experimental results for resolution check of the wavefront. Note that the wavefront Δz was measured with reference to the state before the glass slide was inserted. Δz, the change in the wavefront, varied smoothly as a function of the tilt angle of the glass slide. Following the approximate formula cosθ ≈ 1 - (1/2)θ², the variation in Δz related to thickness was better represented by a quadratic function.

Fig. 6. To evaluate our wavefront measurement, we used a tilted microscope glass slide as shown in (a). (b) Δz as a function of the tilt angle. Red circles and broken line represented the experimental and theoretical results, respectively.

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The broken line drawn in Fig. 6(b), representing the theoretical values of wavefront Δz, was closely matched the experimental results. Our method resolved the change in wavefront caused by a 0.2° glass slide tilt. We determined that the resolution of Δz with respect to the tilt angle was 25 nm from the magnitude of the detected Δz between the data.

3.4 Structured beam evaluation

We conducted an experiment to evaluate structured light beams, which have spatially distributed polarization states and wavefronts. We inserted a vortex retarder (WPV10L-532, Thorlabs Inc.) as a sample, as shown in Fig. 5(a), and evaluated four structured beams with spatially controlled polarization states and wavefronts. We set the spatial polarization of the light, such as radial and azimuthal polarization for an in-phase wavefront, as well as right- and left-hand vortex wavefronts with left- and right-hand circular polarizations. We controlled the incident polarization by setting the angle of the quarter waveplate (see Fig. 5(a)). Figure 7 shows the results of our evaluation for a (a) radially polarized beam, (b) azimuthally polarized beam, (c) right-hand optical vortex with left-hand circular polarization, and (d) left-hand optical vortex with right-hand circular polarization. Their theoretical polarization states and wavefronts were illustrated in Fig. 7(a1)-(d1), respectively. We evaluated (a2)-(d2) the ellipticity, (a3)-(d3) the azimuth, (a4)-(d4) Δz and (a5)-(d5) the angle of the wavefront, as well as (a5)-(d5) its vector map was illustrated in Figs. 7(a)–7(d), where the lengths of the arrows indicate the magnitudes of Δz.

Fig. 7. Evaluation for four types of structured beam: (a) in-phase radial polarization, (b) in-phase azimuthal polarization, (c) right- and (d) left-hand optical vortex. (a1)–(d1) Theoretical results of polarization states and wavefronts, (a2)–(d2) Ellipticity, (a3)–(d3) azimuth, (a4)–(d4) Δz, (a5)–(d5) angle, respectively. The white scale bars indicate 1 mm, respectively.

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4. Discussion

We have developed a new imaging sensor that simultaneously measures the polarization states and wavefront of light by employing the axisymmetric birefringence distribution around a mark produced by laser processing. Figure 2 shows that the diameter of a micro-retarder-lens was as small as 435 µm at 0.32 J. The ultimate size is limited by the power of the laser diode used; as the output power is reduced, the micro-retarder-lens becomes smaller. However, the retardance also decreases because the laser processing weakens the molecular orientation. Precise birefringence control, enabling the fabrication of smaller micro-retarder-lenses, is expected to be possible by introducing a pulsed laser with high energy and by adding substrate cooling.

Figure 3 shows the birefringence distribution caused by the laser processing. Close inspection reveals that the beam was slightly distorted, resulting in an elliptical shape instead of a concentric lens. Therefore, the retardance is elliptical in the circumferential direction. This is caused by the use of a multimode laser diode. We anticipate that using a laser diode with a regular circle will generate a uniform pattern.

Polarization analysis was performed by observing the polarization change that occurs around a micro-retarder-lens, the wavefront analysis was performed using the principle of a conventional Shack-Hartmann wavefront sensor. We also proposed a calibration method to compensate for the birefringence variation of the angular variant micro-retarder-lens, detailed in Section 3.3. Using this calibration, the accuracy of the polarization measurement was greatly improved, as shown in Table 1. However, since this method requires the introduction of six known polarizations as input beams, the measurement accuracy depends on the quality of the input polarization. Therefore, the measurement includes a slight error due to the fact that the right- and left-hand circular polarizations were not ideal. The accuracy of our polarization measurement was determined from Table 1 to be ± 0.01 in ellipticity and ± 1° in its azimuth. To enhance the accuracy of the measurement, we can prepare two approaches: using more known polarizations for calibration and improving the precision of circular polarization generation. Moreover, the use of sensors with smaller pixel size for wavefront measurements will further improve the resolution. Even based on our estimation, resolution of the wavefront measurement would achieve 5 nm using a sensor with 1 µm per pixel.

Let us discuss spatial resolution of our image sensor. The spatial resolution of our measurement method depends significantly on the size and arrangement of the micro-lenses. As mentioned earlier, small micro-retarder-lens can be produced using laser processing, incorporating high NA objective lenses and pulsed lasers. However, there is a concern that high-energy generation might lead to the burning of the material substrate. On the other hand, at lower energies, it becomes challenging to achieve the retardance required for polarization measurement. Balancing these trade-offs to find the optimum values will be a future challenge.

Finally, we performed simultaneous polarization and wavefront measurements using our unique optical elements. We evaluated our imaging sensor using structured light, which had spatially controlled polarization and wavefronts. We measured the ellipticity and azimuth for the polarization measurement, and Δz and its direction (see Fig. 1(c2)) for the wavefront measurement. Figures 7(a) and 7(b) show that the radial and azimuthal polarization is linearly polarized with ε = 0 and the azimuth is axisymmetric. The wavefronts showed a saddle-shaped distribution. When right- and left-hand optical vortices with left- and right-hand circular polarizations were evaluated, as shown in Figs. 7(c) and 7(d), the polarization distributions were elliptically polarized at −0.83 and 0.86, respectively. The detection of perfect circular polarizations was limited by the error in the polarization measurement and a slight error in the adjustment of the optical system that generated the polarization. The vector map, representing the difference Δz between the wavefront and the reference and its angle shows left- and right-hand vortices, rather than radial and azimuthal polarization. Comparison with theoretical results clearly shows the effectiveness of this measurement. Our results demonstrate that we have achieved simultaneous detection of the polarization states and wavefront, in contrast to conventional techniques, where the detection of structured light with controlled polarization and wavefront requires two separate measurement devices, such as a polarimeter and a wavefront sensor.

5. Summary and outlook

We have demonstrated the simultaneous detection of the polarization states and wavefront of light using a micro-retarder-lens array fabricated by laser processing of a polymethyl methacrylate substrate dyed sky blue. The retardance was concentrically distributed and the azimuthal angle was axial-symmetrically oriented around the lens. Using this unique optical element, we have proposed a calibration method for polarization measurement. The detection accuracy of the polarization states, such as ellipticity ε and azimuth φ, were ε = ± 0.01 and φ = ± 1.0°, respectively. Moreover, the detection resolution of the wavefront was 25 nm (= 0.05λ). We achieved simultaneous detection of both polarization states and wavefront for four types of structured light with radial and azimuthal polarization, as well as right- and left-hand vortex wavefronts. Our current image sensor is still only 7 × 7 pixels. However, further integration of micro-retarder-lenses is feasible. These micro-retarder-lenses can form images just like conventional lenses. We expect that our sensor can realize 3D light field measurement, as well as the simultaneous measurement of polarization and wavefronts. Our concept can be applied to various applications in the fields of laser physics, astronomy, and even ophthalmology.

Funding

Terumo Foundation for Life Sciences and Arts (2023); Sumitomo Foundation (2200578); Murata Science Foundation (2022); Amada Foundation (AF-2019204-B2, AF-2022217-B3); Casio Science Promotion Foundation (202134); Uehara Memorial Foundation (2019, 2022); Japan Society for the Promotion of Science (20H02157, 21H03842).

Disclosures

The authors declare no conflicts of interest.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

References

1. S. P. Bos, “The polarization-encoded self-coherent camera,” Astron. Astrophys. 646, A177 (2021). [CrossRef]

2. D. S. Kim, S. Čopar, U. Tkalec, et al., “Mosics of topological defects in micropatterned liquid crystal textures,” Sci. Adv. 4(11), eaau8064 (2018). [CrossRef]

3. M. Yoshida, Y. Kozawa, and S. Sato, “Subtraction imaging by the combination of higher-order vector beams for enhanced spatial resolution,” Opt. Lett. 44(4), 883–886 (2019). [CrossRef]

4. M. Li, S. Yan, Y. Zhang, et al., “Optical sorting of small chiral particles by tightly focused vector beams,” Phys. Rev. A 99(3), 033825 (2019). [CrossRef]

5. T. Ejima, T. Wakayama, N. Shinozaki, et al., “Demonstration of stimulated emission depletion phenomenon in luminescence of solid-state scintillator excited by soft X-rays,” Sci. Rep. 10(1), 5391 (2020). [CrossRef]

6. C. Hnatovsky, V. G. Shvedov, and W. Krolikowski, “The role of light-induced nanostructures in femtosecond laser micromachining with vector and scalar pulses,” Opt. Express 21(10), 12651–12656 (2013). [CrossRef]

7. X. Zhang, S. Yang, W. Yue, et al., “Direct polarization measurement using a multiplexed Pancharatnam-Berry metahologram,” Optica 6(9), 1190–1198 (2019). [CrossRef]

8. S. Nechayev and P. Banzer, “Mimicking chiral light-matter interaction,” Phys. Rev. B 99(24), 241101 (2019). [CrossRef]

9. X. Zeng, Y. Li, L. Feng, et al., “All-fiber orbital angular momentum mode multiplexer based on a mode-selective photonic lantern and a mode polarization controller,” Opt. Lett. 43(19), 4779–4782 (2018). [CrossRef]

10. M. Piccardo, M. de Oliveira, A. Toma, et al., “Vortex laser arrays with topological charge control and self-healing of defects,” Nat. Photonics 16(5), 359–365 (2022). [CrossRef]

11. A. Chong, C. Wan, J. Chen, et al., “Generation of spatiotemporal optical vortices with controllable transverse orbital angular momentum,” Nat. Photonics 14(6), 350–354 (2020). [CrossRef]

12. Y. Lang, Q. Xu, X. Chen, et al., “On-Chp plasmonic vortex interferometrs,” Laser Photonics Rev. 16(10), 2200242 (2022). [CrossRef]

13. N. A. Rubin, G. D’Aversa, P. Chevalier, et al., “Matrix Forurier optics enables a compact full-Stokes polarization camera,” Science 365(6448), 43 (2019). [CrossRef]

14. T. Wakayama, T. Higashiguchi, H. Oikawa, et al., “Determination of the polarization states of an arbitrary polarized terahertz beam: Vectorial vortex analysis,” Sci. Rep. 5(1), 9416 (2015). [CrossRef]

15. R. A. Chipman, Polarimetry. Chapter 22 in Handbook of Optics II (McGraw-Hill, New York, 1995).

16. E. Terasawa, D. Satoh, S. Maru, et al., “Ultrafast time-resolved single-shot birefringence microscopy for laser-induced anisotropy,” Opt. Lett. 47(15), 3728–3731 (2022). [CrossRef]

17. M. Garcia, C. Edmiston, R. Marinov, et al., “Bio-inspired color-polarization imager for real-time in situ imaging,” Optica 4(10), 1263–1271 (2017). [CrossRef]

18. N. C. Bruce, O. G. Rodriguez-Herrera, J. M. Lopez-Tellez, et al., “Experimental limits for eigenvalue calibration in liquid-crystal Mueller-matrix polarimeters,” Opt. Lett. 43(11), 2712–2715 (2018). [CrossRef]

19. A. W. Kruse, A. S. Alenin, I. J. Vaughn, et al., “Perceptually uniform color space for visualizing trivariate linear polarization imaging data,” Opt. Lett. 43(11), 2426–2429 (2018). [CrossRef]

20. M. Shoji, T. Wakayama, H. Ishida, et al., “Single-shot multispectral birefringence mapping by supercontinuum vector beams,” Appl. Opt. 59(23), 7131–7138 (2020). [CrossRef]

21. T. Onuma and Y. Otani, “A development of two-dimensional birefringence distribution measurement system with a sampling rate of 1.3 MHz,” Opt. Commun. 315, 69–73 (2014). [CrossRef]

22. Y. Yao, F. Zhang, B. Wang, et al., “Polarization imaging-based radomics approach for the staging of liver fibrosis,” Biomed. Opt. Express 13(3), 1564–1580 (2022). [CrossRef]

23. G. A. Parra-Escamilla, D. I. Serrano-Garcia, J. L. Flores, et al., “Pixelated polarizing system for dynamic interferometry events employing a temporal phase unwrapping approach,” Opt. Commun. 458, 124862 (2020). [CrossRef]

24. D. Wang and R. Liang, “Simultaneous polarization Mirau interferometer based on pixelated polarization camera,” Opt. Lett. 41(1), 41–44 (2016). [CrossRef]

25. J. Wu, Y. Guo, C. Deng, et al., “An integrated imaging sensor for aberration-corrected 3D photography,” Nature 612(7938), 62–71 (2022). [CrossRef]

26. V. Akondi and A. Dubra, “Accounting for focal shift in the Shack-Hartmann wavefront sensor,” Opt. Lett. 44(17), 4151–4154 (2019). [CrossRef]

27. T. Nirmaier, G. Pudasini, and J. Bille, “Very fast wave-front measurements at the human eye with a custom CMOS-based Hartmann-Shack sensor,” Opt. Express 11(21), 2704–2716 (2003). [CrossRef]

28. B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. 17(5), S573–S577 (2001). [CrossRef]

29. F. Pfeiffer, “X-ray ptychography,” Nat. Photonics 12(1), 9–17 (2018). [CrossRef]

30. Z. Yang, Z. Wang, Y. Wang, et al., “Generalized Hartmann-Shack array of dielectric metalens sub-arays for polarimetric beam profiling,” Nat. Commun. 9(1), 4607 (2018). [CrossRef]

Input	$(\begin{matrix} \begin{matrix} 1 \\ 1 \end{matrix} \\ \begin{matrix} 0 \\ 0 \end{matrix} \end{matrix})$	$(\begin{matrix} \begin{matrix} 1 \\ 0 \end{matrix} \\ \begin{matrix} 1 \\ 0 \end{matrix} \end{matrix})$	$(\begin{matrix} \begin{matrix} 1 \\ - 1 \end{matrix} \\ \begin{matrix} 0 \\ 0 \end{matrix} \end{matrix})$	$(\begin{matrix} \begin{matrix} 1 \\ 0 \end{matrix} \\ \begin{matrix} - 1 \\ 0 \end{matrix} \end{matrix})$	$(\begin{matrix} \begin{matrix} 1 \\ 0 \end{matrix} \\ \begin{matrix} 0 \\ 1 \end{matrix} \end{matrix})$	$(\begin{matrix} \begin{matrix} 1 \\ 0 \end{matrix} \\ \begin{matrix} 0 \\ - 1 \end{matrix} \end{matrix})$
Before	$(\begin{matrix} \begin{matrix} 1.89 \\ 3.85 \end{matrix} \\ \begin{matrix} - 0.54 \\ 0.20 \end{matrix} \end{matrix})$	$(\begin{matrix} \begin{matrix} 1.87 \\ 0.89 \end{matrix} \\ \begin{matrix} 3.28 \\ 0.07 \end{matrix} \end{matrix})$	$(\begin{matrix} \begin{matrix} 2.58 \\ - 3.04 \end{matrix} \\ \begin{matrix} 0.52 \\ - 0.01 \end{matrix} \end{matrix})$	$(\begin{matrix} \begin{matrix} 2.52 \\ 0.07 \end{matrix} \\ \begin{matrix} - 4.13 \\ 0.03 \end{matrix} \end{matrix})$	$(\begin{matrix} \begin{matrix} 2.56 \\ 0.34 \end{matrix} \\ \begin{matrix} - 0.38 \\ - 2.71 \end{matrix} \end{matrix})$	$(\begin{matrix} \begin{matrix} 2.57 \\ 0.27 \end{matrix} \\ \begin{matrix} 0.63 \\ 2.70 \end{matrix} \end{matrix})$
After	$(\begin{matrix} \begin{matrix} 0.96 \\ 1.01 \end{matrix} \\ \begin{matrix} 0.02 \\ - 0.02 \end{matrix} \end{matrix})$	$(\begin{matrix} \begin{matrix} 0.94 \\ 0.04 \end{matrix} \\ \begin{matrix} 0.92 \\ - 0.00 \end{matrix} \end{matrix})$	$(\begin{matrix} \begin{matrix} 0.96 \\ - 0.99 \end{matrix} \\ \begin{matrix} 0.02 \\ - 0.02 \end{matrix} \end{matrix})$	$(\begin{matrix} \begin{matrix} 0.94 \\ 0.04 \end{matrix} \\ \begin{matrix} - 1.08 \\ - 0.00 \end{matrix} \end{matrix})$	$(\begin{matrix} \begin{matrix} 1.10 \\ - 0.05 \end{matrix} \\ \begin{matrix} 0.06 \\ 1.02 \end{matrix} \end{matrix})$	$(\begin{matrix} \begin{matrix} 1.10 \\ - 0.05 \end{matrix} \\ \begin{matrix} 0.06 \\ - 0.98 \end{matrix} \end{matrix})$

Simultaneous detection of polarization states and wavefront by an angular variant micro-retarder-lens array

Abstract

1. Introduction

2. Principle of simultaneous detection of polarization states and wavefront

2.1 Polarization measurement

2.2 Calibration of polarization measurement

2.3 Measurement of wavefront

3. Experimental results

3.1 Birefringence properties on laser processing mark

3.2 Angular variant micro-retarder-lens array

3.3 Simultaneous detector for polarization states and wavefront of light

3.4 Structured beam evaluation

4. Discussion

5. Summary and outlook

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (7)

Tables (1)

Equations (20)

Optics Express

Toshitaka Wakayama	https://orcid.org/0000-0003-2491-5355
Takeshi Higashiguchi	https://orcid.org/0000-0003-3940-7126