1. INTRODUCTION

The mechanism behind ancient magic mirrors from China and Japan was not understood until the 20th century, despite the earliest creations of these artistic pieces dating back to 2000 BC [1]. The cast bronze mirrors presented as normal mirrors while viewing one’s reflection. However, when sunlight was shone directly on the mirror, it acted as a subtly parabolic mirror forming an image—corresponding to patterning on the back side of the mirror—presented on the floor or a screen [2]. A similar phenomenon can be observed in the reflection of the sun off large windows and onto a street below. Though the window appears flat and does not significantly distort an image while we look through it, the slight deformations from tension around the edges result in a non-uniform reflection onto the ground in the shape of an “${{X}}$”-pattern. Despite deriving from a millennia old tradition, magic mirrors inspired a measurement technique (called Makyoh topography after the Japanese word for “wonder mirror”) for detecting surface deformities in silicon wafers in the late 20th century [3,4]. This approach has the advantage of being very simple and practical for industry-based applications, in comparison with other measurement techniques such as interferometry or atomic force microscopy [5].

The magic mirror effect can be quantitatively explained through standard diffraction theory [3]. However, the final understanding of how images are formed from the magic mirrors was derived in 2005 by Sir Michael Berry [1]. Here, it was shown that the intensity of the image is given to the first-order approximation by the Laplacian of the height of surface reliefs on the mirror. The principle of the magic mirror can be applied to devices working in transmission, the so-called “magic windows,” which can produce a similar effect forming the Laplacian image through very slight thickness deformations [6]. Specifically, the surface should be “smooth” enough, with gentle variations, such that caustics are not formed before the image appears. It is shown that the intensity of the Laplacian image is given in terms of the height of the surface relief, $h$, by ${I_{{\rm{Laplacian\,Mirror}}}}({\textbf{r}},Z) \simeq 1 + Z {\nabla ^2}h({\textbf{r}})$ [1]. Here, $Z = 2D/M$ and ${\textbf{r}} = R/M$ are the rescaled distance along the propagation direction and transverse position from the center of the mirror, respectively, normalized to the magnification of the convex mirror $M$. $D$ and $R$ are the distances from the mirror and transverse position of the image, respectively. The Laplacian image produced from a magic window, however, depends on the relative refractive index of the window, $n$, in addition to the height of the surface relief, $h$, which is given by [6]

Here $z$ is the distance of the image plane from that of the window, and $R$ is the transverse distance from the center of the window. Given any image, we can, thus, find the necessary surface of the magic window or mirror by solving the Poisson equation in the transverse plane. A remarkable property of magic mirrors or windows is that, in contrast with conventional windows and lenses, the image can be observed in several observational planes without a substantial change in sharpness.

 

Fig. 1. Magic optics with liquid crystal displays. (a) Operation principle of SLMs and liquid crystal PBOEs. Both types of displays rely on rotated liquid crystals to impart a transverse phase profile onto an incident light beam. The SLM relies on out-of-plane rotations to impart a phase onto linearly polarized light whereas the PBOE relies on in-plane rotations to add a phase onto circularly polarization light at the expense of flipping its handedness. (b) Corresponding surfaces that are classically considered in magic optics.

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There has been recent interest in the problem of shaping light intensity often referred to as freeform optics [7]. In this work, we show how magic mirrors/windows can be implemented with flat optical devices. We use liquid crystal (LC) based devices, a reflective spatial light modulator (SLM), and a Pancharatnam–Berry optical phase element (PBOE) [8] to construct our magic mirror and magic window, respectively. We also exploit the Laplacian theory, which gives a simpler and more direct approach to calculate the desired phase patterns in contrast with more elaborate techniques, e.g., caustic design [9]. Figure 1(a) illustrates how these two types of LC devices can be programmed or fabricated to act as the reflective or transmissive surfaces shown in Fig. 1(b). Both types of devices rely on the uniaxial birefringence of its constituent nematic LCs to implement the magic mirror/window effect. On one hand, standard LC on silicon SLMs rely on a reflective back-plane that rotates LCs about an axis perpendicular to the direction of propagation of an incident optical beam. The resulting optical medium is, thus, defined by two linear polarization eigenstates. The first is along the rotation axis and is defined by a refractive index of ${n_o}$, the ordinary refractive index of the LCs. The second is orthogonal to the propagation direction of the beam and to the rotation axis, and it has a refractive index of $n(\chi) = ((\mathop {\cos}\nolimits^2 \chi)/n_o^2 + (\mathop {\sin}\nolimits^2 \chi)/n_e^2{)^{- 1/2}}$, where ${n_e}$ and $\chi$ are the extraordinary index and the rotation angle of the LCs, respectively. Therefore, by exposing an SLM to an optical beam polarized along this second direction, we can impart a transverse phase profile onto the beam controlled by the relative angle of the LC in the display. An SLM can, thus, operate like a magic mirror if its programmed transverse phase profile replicates that attributed to the height profile of the mirror. On the other hand, LCs in PBOEs are rotated about an axis parallel to the direction of propagation of an optical beam. The resulting polarization eigenstates are, thus, either aligned or orthogonal to the orientation axis of the LCs and have indices of ${n_e}$ and ${n_o}$, respectively. When the thickness of this system imparts a $\pi$ phase shift between these two linear polarizations, then, as prescribed by Jones calculus, an incident circularly polarized beam experience a phase shift of ${\pm}2\chi$ accompanied by a flip in handedness upon propagating such a LC cell, where $\chi$ is the rotation angle of the LCs. PBOEs leverage this phenomenon by rotating the orientation angle of the LCs across a transmissive optical display such as to impart a desired phase profile onto a circularly polarized beam. Thus, a LC PBOE acts like a magic window if it has a LC orientation pattern that produces the same phase profile as that induced by the thickness variations of the window. PBOEs, through an effect known as light’s spin-to-orbital angular momentum coupling [10], also allow us to implement a polarization dependent phase distribution. Thus, one can observe the image resulting from the phase or its negative by switching the input polarization from left to right circular. In addition, the introduced topic of spin-to-orbit coupling allows one to explore the complex patterns of polarization singularities when the LC magic plate is illuminated with a linear superposition of left- and right-handed circular polarization. There have been many investigations into singular optics, arising from such polarization singularities, introduced in polarization system such as Stokes singularities, polarization knots, and links [11–14]. It is, thus, interesting to reconstruct the polarization topology of the images formed by a LC magic plate. In particular, we track the trajectory in the three-dimensional space of C-point singularities, i.e., loci of circular polarization. We show that C-points accumulate in points of the transverse plane where the image is forming.

2. LIQUID CRYSTAL MAGIC MIRROR

The magic mirror is realized using a Hamamatsu SLM with a screen resolution of 800 by 600 pixels. Figure 2(b) shows the detail of the experimental setup for both magic mirror and magic plate. Given the desired image, the required phase pattern for the mirror is computed. There is a freedom to increase the steepness of the pattern, i.e., increasing the number of times the phase pattern goes from 0 to $2\pi$. This can be seen as altering the concavity of the mirror, which results in changing how quickly the image is formed. In practical settings, care must be taken in choosing the window size and phase steepness. Due to beam divergence, an image that forms too slowly will lose its sharpness. At the same time, the pattern must be smooth enough such that caustics are not formed before the image plane [6]. The phase pattern was uploaded to the SLM with the addition of a vertical grating. The SLM is shined with a laser beam with an enlarged Gaussian beam shape to mimic an input, coherent, plane wave. The first order of diffraction is selected, filtering out the rest using an iris in the center of a ${\rm{4 - f}}$ lens system, to remove unconverted light resulting from inefficiencies in the SLM. In addition to selecting the first order of diffraction, the ${\rm{4 - f}}$ lens system is also used to image the SLM plane and probe the intensity at different propagation distances. The evolution of the intensity distribution is recorded on a CMOS camera from the plane of the SLM down to the image formation planes. Additional planes were also imaged beyond the latter to capture the formation of caustics. The implementation of three SLM magic mirrors is shown in Fig. 3, where the intensity distribution smoothly evolves from the input beam profile to the desired image pattern.

 

Fig. 2. Calculation principle of phase patterns and experimental setup. In (a) we illustrate the steps used to calculate the phase pattern needed for generating a desired image. The BMP image of the desired intensity is used to calculate the mirror height using the discrete sine transform to solve Eq. (1). The maximum height for the window corresponding to the Gee-Gees logo is $6\,\,\unicode{x00B5}{\rm m}$ as shown. The mirror height is then used to calculate the mirror phase by taking the modulus for the specific wavelength, i.e., ${\rm{Phase}} = {\rm{Mod}}({\rm{Height}}/(2\pi \lambda),2\pi)$. In (b) we show the experimental setup. A 633 nm He–Ne laser was used for characterizing the magic mirror and magic plate. The setup for the magic mirror ${\textbf{b}}$ consists of a SLM with a resolution of 600 by 800 pixels. After the SLM, a ${\rm{4 - f}}$ system is used to image the pattern displayed on the SLM, allowing for us to both make measurements starting precisely from the SLM plane and also filter out the first diffraction order using a pinhole placed at the focus of the ${\rm{4 - f}}$ system. Following the ${\rm{4 - f}}$ lens system, two mirrors are placed on a translation stage to construct a trombone, which is followed by a CMOS camera. In the magic window setup, the PBOE is placed in the same plane as the SLM. The addition of a QWP before and a QWP, HWP, and PBS after is required to perform polarization tomography on the output beam of the magic window. The phase pattern used for the University of Ottawa logo is shown for the magic window and mirror.

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Fig. 3. Intensity images recorded at different propagation distances after reflection from an SLM-based magic mirror. The first image at the back shows the SLM plane, where there is a uniform intensity pattern. The propagation shows the contrast of the image improving upon propagation away from the SLM. The insets show the phase distributions, for the three different examples, encoded on the SLM in hue colors, which encode a phase range from 0 to $2\pi$. Videos showing the free space evolution of the above images can be found in Visualization 1, Visualization 2, and Visualization 3.

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The goal of the experiment is to show a magic mirror and magic plate, encoded using the Laplacian theory, using LC technology in an SLM and PBOE. To achieve this, we must generate the phase pattern corresponding to the chosen intensity image. This phase pattern is then displayed on the SLM and PBOE. The image patterns that we use are converted to a bitmap form such that the pixels contain only a 1 or 0 for the intensity [see Fig. 2(a)]. Based on Eq. (1), the discretized intensity function $I({\textbf{R}},z)$ is then used to find the height of the surface relief $h({\textbf{r}})$, where ${\textbf{r}} = {\textbf{r}}({\textbf{R}})$ at the image plane. The inverse of the Laplacian is solved numerically with Dirichlet boundary conditions using the two-dimensional discrete sine transform [1]. In the case of flat optics, we are not varying the height of the window, but rather the index of refraction $n({\textbf{r}})$. Thus, Eq. (1) becomes ${I_{{\rm{Magic\,Plate}}}}({\textbf{R}},z) = 1 + h z {\nabla ^2} n({\textbf{R}})$, where $n({\textbf{r}})$ is the transverse spatially dependent index of refraction of the plate, $h$ is again the height of the plate though it is now a constant, and $z$ is the distance from the image plane to the window. The resulting window phase pattern, as shown in Fig. 2(a), is plotted in radians.

3. LIQUID CRYSTAL MAGIC PLATE

As the next step, we bring together the concepts of magic window imaging and photonic polarization-wavefront coupling. Such a device, which we call the spin–orbit magic plate, is based on PBOEs, i.e., slabs of uniaxial anisotropic materials (LCs, in our case) with an extraordinary axis orientation that is spatially varying in the plate’s plane. The action of a PBOE element with extraordinary axis orientation $\chi ({\textbf{r}})/2$ and (spatially uniform) retardation $\delta$ is given by

 

Fig. 4. Working principle of a flat spin–orbit magic plate. In (a) and (b) we show the intensity distributions of light transmitted by the plate with different polarization inputs from a LED source with low transverse coherence. The resulting image from the linear input is the sum of the left and right circular contributions. The image resulting from the right circular input is not an exact negative of the left circular because it experiences a slight defocusing from the window while the left component is slightly focused. In (c) and (d) we show the intensity distributions of light transmitted by the plate with different polarization inputs: left circular, right circular, and linear polarization from and a 633 nm He–Ne laser. See also Visualization 4 for the free space evolution. Panels (e) and (f) show the images resulting from different $\delta$ across the plate as given by Eq. (2), with the left circular polarization input. When $\delta = 0$, there is no conversion from left- to right-handed polarization; thus, the beam does not acquire the phase of the magic window. When $\delta = \pi$, the input polarization is fully converted; thus, we see the desired image. Tuning parameters between 0 and $\pi$ result in partial conversion, and we see interference between the intensity profile of the converted beam and the input beam.

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Fig. 5. In (a), we show simulation of the propagation of C-points from the spin–orbit magic plate. The red points in the three-dimensional graphics show the C-point trajectory along the propagation of the beam. The intensity profiles are shown at different propagation planes. The two-dimensional plot above represents the polarization azimuth, where C-points are labeled with black circles. The inset shows a magnified image of the polarization azimuth where the singularities are clearly visible at the location where the polarization azimuth is undefined. Upon close observation, we can see that these singularities have opposite charge. The hue color coding corresponds to polarization azimuth values ranging from 0 to $\pi$. In (b), we show the experimentally reconstructed polarization azimuth obtained through polarization tomography. The tracking of C-points here is very sensitive to the camera resolution as well as the exposure time, particularly in the regions of low intensity. We are, however, still able to see singularities form in the predicted areas, particularly along the bottom of the logo.

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4. CONCLUSION

In summary, we have realized a LC-based magic plate exploiting the principle of light manipulation with flat optics, where the impinging light wavefront is modulated by an inhomogeneous refractive index distribution. By exploiting the physics of patterned anisotropic media, we fabricated a flat spin–orbit magic plate. This device, depending on whether the input polarization is circularly left- or right-handed, creates a desired pattern or its negative, respectively. The flat magic plate can be tuned for operation at different wavelengths since its optical retardation can be adjusted by applying an external electric field to the plate. While we used this device under monochromatic illumination, in principle it can work as well under broadband illumination, if one carefully selects only the converted contribution by means of achromatic wave plates and polarizing beam splitters, even though the conversion efficiency will not be uniform at the different wavelengths. The working principle was demonstrated for both incoherent and coherent sources. In the latter case, interference effects lead to the formation of polarization singularities (C-points). Our experiment was based on the use of LC devices with thickness of several wavelengths. However, as it has been shown in [18,19], PBOEs can also be realized with dielectric metasurfaces with thickness smaller than the wavelength. Furthermore, one could consider using achromatic and polarization-insensitive metasurfaces to form magic windows and further reduce the wavelength dependence of the devices [20]. Hence, our results also introduce the possibility of scaling down the thickness of these flat magic windows to sub-wavelength scales.