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Flat magic window

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Abstract

Magic windows (or mirrors) consist of optical devices with a surface deformation or thickness distribution devised in such a way to form a desired image. The associated image intensity distribution has been shown in previous works to be related to the Laplacian of the height of the surface relief. Exploiting the Laplacian theory to calculate the needed phase pattern, we experimentally realize such devices with flat optics employing optical polarization-wavefront coupling, which represent a new paradigm for light manipulation. The desired pattern and experimental specifications for designing the flat optics was implemented with a reconfigurable spatial light modulator, which acted as the magic mirror. The flat plate, an optical polarization-wavefront coupler, is then fabricated by spatially structuring nematic liquid crystals. The plate is used to demonstrate the concept of a polarization-switchable magic window, where, depending on the input circular polarization handedness, one can display either the desired image or the image resulting from the negative of the window’s phase.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

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]

$${I_{{\rm{Laplacian\, Window}}}}({\textbf{R}},z) \simeq 1 - z (n - 1) {\nabla ^2}h({\textbf{R}}).$$

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.

 figure: Fig. 1.

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 [1114]. 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.

 figure: Fig. 2.

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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 figure: Fig. 3.

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

$${{\textbf{e}}_{\pm }} {\mathop{\rm MP}\limits_{\longrightarrow}}{\cos}\left( \frac{\delta }{2} \right){{\textbf{e}}_{\pm }}+{i}\ {\sin}\left( \frac{\delta }{2} \right){{\textbf{e}}_{\mp }}{{e}^{\pm i\chi (\boldsymbol{r})}},$$
where ${{\textbf{e}}_ +}$ and ${{\textbf{e}}_ -}$ stand for the left and right circular polarization unit vectors, respectively. The sample optical retardation, $\delta$, can be tuned by applying an AC voltage to the plate. Here, we can see that a perfectly tuned PBOE with $\delta = \pi$ results into the complete conversion of the input circular polarization to the opposite handedness with the addition of the desired phase ${\pm}\chi ({\textbf{r}})$, where $\chi ({\textbf{r}})$ is the inverse Laplacian of the image. Therefore, by flipping the incident polarization state from left- to right-handed, one can gain a ${+}\chi ({\textbf{r}})$ and ${-}\chi ({\textbf{r}})$ phase at the output, respectively. Due to the different signs in the phase, the two circular polarization components propagate in different ways, as dictated by the Fresnel diffraction integral, ${E_ \pm}({\textbf{r}},z) = \int K({\textbf{r}},{\textbf{r}^\prime},z){E_ \pm}({\textbf{r}^\prime},0){d^2}{\textbf{r}^\prime}$, where $K({\textbf{r}},{\textbf{r}^\prime},z) = \exp [ik{| {{\textbf{r}} - {\textbf{r}^\prime}} |^2}/2z]{e^{\textit{ikz}}}/i\lambda z$. From $K({\textbf{r}},{\textbf{r}^\prime},z) = {K^*}({\textbf{r}},{\textbf{r}^\prime}, - z)$, one can see that ${E_ +}({\textbf{r}},z) = E_ - ^*({\textbf{r}},z)$. As a consequence, if one circular polarization experiences the formation of an image, which is being focused due to the radial variation in $\chi ({\textbf{r}})$, the opposite circular polarization will be defocused, and the image [from Eq. (1)] will be the negative. The University of Ottawa logo was chosen to be used for the spin–orbit magic plate. The phase pattern written on the plate for the magic window is shown in Fig. 2 with the experimental setup. The plate is fabricated in our own LC facility [8]. A pair of ITO glass plates are spin-coated with a polyamide. The ITO plates then are kept at 4 µm distance using spacers and are glued to each other with epoxy glue. The chosen polyamide can be photoaligned through illumination from linearly polarized UV light. We are able to control the orientation of the polyamide by changing the polarization of an incident UV-beam on a pixel-by-pixel basis by using a digital micromirror device (DMD). The pattern written on the polyamide dictates the orientation of the LC molecules, which are added between the plates in the successive stage. The pattern was written with 32 phase steps, thus 32 polarization settings illuminating different parts of the plate. We characterized the action of the fabricated plate illuminating it with both spatially incoherent and coherent light. The magic plate optical retardation was set to $\delta = \pi$ by applying an AC voltage. As a source with low transverse spatial coherence, we used a LED, followed by a polarizing beam splitter, to illuminate the magic plate. The light was filtered with a bandpass filter centered at 633 nm (which increases the longitudinal coherence but does not affect the transverse spatial coherence). We choose the input polarization to be either linear or right/left circular by using a quarter-wave plate before illuminating the magic plate. The transmitted intensity was recorded on the CMOS camera. With the linear polarized input, we observe the simultaneous formation of the University of Ottawa logo and its negative image (with an imperfect overlap due to polarization dependent lensing), as shown in Fig. 4(a). The appearance of the negative image is due to the input right circular polarization component, which gains the phase ${-}\chi ({\textbf{r}})$ [this has the effect of flipping the relative sign in Eq. (1)]. It is possible to isolate the image or its negative by choosing input left or right circular polarization, respectively [Fig. 4(a)]. In Fig. 4(b), we show theoretical simulation of the intensity evolution from a source with low transverse coherence. The source was simulated by sampling the transverse plane in regions where all the pixels are in phase and imposing random phase noise between the different regions. The transverse coherence length was, thus, proportional to the region width, which we fixed at 15 pixels. The simulations show the results averaging over 200 realizations of the random noise (uniformly distributed between 0 and $2\pi$). Similar effects are observed in the case of illumination with a coherent laser beam. We used a He–Ne laser ($\lambda = 633\,\,\rm nm$) prepared with left/right circular or linear polarization. The resulting intensity in the case of input left circular polarization corresponds to the desired pattern [Figs. 4(c) and 4(d)]. We also observe fringes due to the transverse coherence of the source. As in the incoherent illumination case, an input right circular polarization gives rise, within the magic window theory approximations, to the negative of the desired image. When a beam with linear polarization is sent onto the magic plate, as opposed to one of the circular polarizations, the resultant image is a coherent linear combination of the images one would achieve from a left and right circular input. Moreover, the magic plate optical retardation $\delta$ can be altered to not be $\pi$, but any other values. In Figs. 4(e) and 4(f), we show how, by tuning the optical retardation $\delta$, we can switch, at a given plane, between the input beam intensity distribution and the image encoded in the plate. The interplay between source coherence and polarization-conditioned action of the device leads to the formation of polarization singularities during the beam propagation. When an electric field has a non-uniform polarization pattern, an interesting phenomenon can arise whereby the polarization azimuth is undefined [1517]. These singularities of
 figure: Fig. 4.

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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the complex scalar field are called C-points. C-points are loci of
 figure: Fig. 5.

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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exactly circular polarization; thus, the orientation of the major axis of the polarization ellipse cannot be defined. When the magic plate is illuminated by linearly polarized light, the outgoing beam has both the right and left circular component, whose propagation is dictated by, respectively, the plate phase and the negative of the plate phase. The intensity patterns of these two components evolve differently: if one of them experiences a focusing effect, the other circular component has a divergent intensity pattern. As a consequence of the different propagations, in particular the formation of diffraction fringes with different distributions for the two circular polarizations, C-points will arise in regions where only one of the two components has a zero intensity. This is a general feature that can be observed in PBOE elements with a radially varying phase. The interaction of these two co-propagating beams gives rise to C-points with different topological charges. In the plane of the LC magic plate, the polarization remains uniformly linear, since we still have a uniform intensity distribution, albeit with a different phase for the left and right components, i.e., $({e^{+ {\rm{i}}\chi ({\textbf{r}})}} {{\textbf{e}}_ -} + {e^{- {\rm{i}}\chi ({\textbf{r}})}} {{\textbf{e}}_ +})/\sqrt 2$. Here $\chi ({\textbf{r}})$ is the spatially dependent phase imparted on the beam by the magic plate. The phase given to the right component is, thus, the negative of that gained by the left component. It is not until propagation to the image plane where differences in the diffraction pattern of the right and left components result in the appearance of the polarization C-points, as shown in Figs. 5(a) and 5(b). Due to conservation of total topological charge, C-points appear at given planes in pairs with opposite topological charges. The free-space dynamics of the C-points generated here is rich and requires individual investigation.

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.

Funding

Ontario Early Research Award; Natural Sciences and Engineering Research Council of Canada; Canada First Research Excellence Fund; Canada Research Chairs.

Acknowledgment

E. K. acknowledges the fruitful conversation with Sir Michael Berry. This work was supported by Canada Research Chairs, Ontario Early Research Award (ERA), Canada First Research Excellence Fund (CFREF), and Natural Sciences and Engineering Research Council of Canada (NSERC).

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.

REFERENCES

1. M. V. Berry, “Oriental magic mirrors and the Laplacian image,” Eur. J. Phys. 27, 109–118 (2005). [CrossRef]  

2. W. E. Ayrton and J. Perry, “II. The magic mirror of Japan. Part I,” Proc. R. Soc. London 28, 127–148 (1879). [CrossRef]  

3. K. Kugimiya, “Characterization of polished surfaces by ‘Makyoh,’” J. Cryst. Growth 103, 461–468 (1990). [CrossRef]  

4. Z. J. Laczik, “Quantitative Makyoh topography,” Opt. Eng. 39, 2562–2567 (2000). [CrossRef]  

5. K. Kugimiya, “Characterization of polished mirror surfaces by the ‘Makyoh’ principle,” Mater. Lett. 7, 229–233 (1988). [CrossRef]  

6. M. V. Berry, “Laplacian magic windows,” J. Opt. 19, 06LT01 (2017). [CrossRef]  

7. M. Brand and D. A. Birch, “Freeform irradiance tailoring for light fields,” Opt. Express 27, A611–619 (2019). [CrossRef]  

8. H. Larocque, J. Gagnon-Bischoff, F. Bouchard, R. Fickler, J. Upham, R. W. Boyd, and E. Karimi, “Arbitrary optical wavefront shaping via spin-to-orbit coupling,” J. Opt. 18, 124002 (2016). [CrossRef]  

9. Y. Schwartzburg, R. Testuz, A. Tagliasacchi, and M. Pauly, “High-contrast computational caustic design,” ACM Trans. Graph. 33, 74 (2014). [CrossRef]  

10. E. Cohen, H. Larocque, F. Bouchard, F. Nejadsattari, Y. Gefen, and E. Karimi, “Geometric phase from Aharonov–Bohm to Pancharatnam–Berry and beyond,” Nat. Rev. Phys. 1, 437–449 (2019). [CrossRef]  

11. F. Flossmann, O. Kevin, M. R. Dennis, and M. J. Padgett, “Polarization singularities in 2D and 3D speckle fields,” Phys. Rev. Lett. 100, 203902 (2008). [CrossRef]  

12. H. Larocque, D. Sugic, D. Mortimer, A. J. Taylor, R. Fickler, R. W. Boyd, M. R. Dennis, and E. Karimi, “Reconstructing the topology of optical polarization knots,” Nat. Phys. 14, 1079–1082 (2018). [CrossRef]  

13. M. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213, 201–221 (2002). [CrossRef]  

14. F. Cardano, E. Karimi, L. Marrucci, C. de Lisio, and E. Santamato, “Generation and dynamics of optical beams with polarization singularities,” Opt. Express 21, 8815–8820 (2013). [CrossRef]  

15. J. F. Nye and J. Hajnal, “The wave structure of monochromatic electromagnetic radiation,” Proc. R. Soc. London. A 409, 21–36 (1987). [CrossRef]  

16. J. F. Nye, Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations (CRC Press, 1999).

17. M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009). [CrossRef]  

18. E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light Sci. Appl. 3, e167 (2014). [CrossRef]  

19. R. C. Devlin, A. Ambrosio, N. A. Rubin, J. B. Mueller, and F. Capasso, “Arbitrary spin-to–orbital angular momentum conversion of light,” Science 358, 896–901 (2017). [CrossRef]  

20. W. T. Chen, A. Y. Zhu, J. Sisler, Z. Bharwani, and F. Capasso, “A broadband achromatic polarization-insensitive metalens consisting of anisotropic nanostructures,” Nat. Commun. 10, 355 (2019). [CrossRef]  

References

  • View by:

  1. M. V. Berry, “Oriental magic mirrors and the Laplacian image,” Eur. J. Phys. 27, 109–118 (2005).
    [Crossref]
  2. W. E. Ayrton and J. Perry, “II. The magic mirror of Japan. Part I,” Proc. R. Soc. London 28, 127–148 (1879).
    [Crossref]
  3. K. Kugimiya, “Characterization of polished surfaces by ‘Makyoh,’” J. Cryst. Growth 103, 461–468 (1990).
    [Crossref]
  4. Z. J. Laczik, “Quantitative Makyoh topography,” Opt. Eng. 39, 2562–2567 (2000).
    [Crossref]
  5. K. Kugimiya, “Characterization of polished mirror surfaces by the ‘Makyoh’ principle,” Mater. Lett. 7, 229–233 (1988).
    [Crossref]
  6. M. V. Berry, “Laplacian magic windows,” J. Opt. 19, 06LT01 (2017).
    [Crossref]
  7. M. Brand and D. A. Birch, “Freeform irradiance tailoring for light fields,” Opt. Express 27, A611–619 (2019).
    [Crossref]
  8. H. Larocque, J. Gagnon-Bischoff, F. Bouchard, R. Fickler, J. Upham, R. W. Boyd, and E. Karimi, “Arbitrary optical wavefront shaping via spin-to-orbit coupling,” J. Opt. 18, 124002 (2016).
    [Crossref]
  9. Y. Schwartzburg, R. Testuz, A. Tagliasacchi, and M. Pauly, “High-contrast computational caustic design,” ACM Trans. Graph. 33, 74 (2014).
    [Crossref]
  10. E. Cohen, H. Larocque, F. Bouchard, F. Nejadsattari, Y. Gefen, and E. Karimi, “Geometric phase from Aharonov–Bohm to Pancharatnam–Berry and beyond,” Nat. Rev. Phys. 1, 437–449 (2019).
    [Crossref]
  11. F. Flossmann, O. Kevin, M. R. Dennis, and M. J. Padgett, “Polarization singularities in 2D and 3D speckle fields,” Phys. Rev. Lett. 100, 203902 (2008).
    [Crossref]
  12. H. Larocque, D. Sugic, D. Mortimer, A. J. Taylor, R. Fickler, R. W. Boyd, M. R. Dennis, and E. Karimi, “Reconstructing the topology of optical polarization knots,” Nat. Phys. 14, 1079–1082 (2018).
    [Crossref]
  13. M. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213, 201–221 (2002).
    [Crossref]
  14. F. Cardano, E. Karimi, L. Marrucci, C. de Lisio, and E. Santamato, “Generation and dynamics of optical beams with polarization singularities,” Opt. Express 21, 8815–8820 (2013).
    [Crossref]
  15. J. F. Nye and J. Hajnal, “The wave structure of monochromatic electromagnetic radiation,” Proc. R. Soc. London. A 409, 21–36 (1987).
    [Crossref]
  16. J. F. Nye, Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations (CRC Press, 1999).
  17. M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
    [Crossref]
  18. E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light Sci. Appl. 3, e167 (2014).
    [Crossref]
  19. R. C. Devlin, A. Ambrosio, N. A. Rubin, J. B. Mueller, and F. Capasso, “Arbitrary spin-to–orbital angular momentum conversion of light,” Science 358, 896–901 (2017).
    [Crossref]
  20. W. T. Chen, A. Y. Zhu, J. Sisler, Z. Bharwani, and F. Capasso, “A broadband achromatic polarization-insensitive metalens consisting of anisotropic nanostructures,” Nat. Commun. 10, 355 (2019).
    [Crossref]

2019 (3)

M. Brand and D. A. Birch, “Freeform irradiance tailoring for light fields,” Opt. Express 27, A611–619 (2019).
[Crossref]

E. Cohen, H. Larocque, F. Bouchard, F. Nejadsattari, Y. Gefen, and E. Karimi, “Geometric phase from Aharonov–Bohm to Pancharatnam–Berry and beyond,” Nat. Rev. Phys. 1, 437–449 (2019).
[Crossref]

W. T. Chen, A. Y. Zhu, J. Sisler, Z. Bharwani, and F. Capasso, “A broadband achromatic polarization-insensitive metalens consisting of anisotropic nanostructures,” Nat. Commun. 10, 355 (2019).
[Crossref]

2018 (1)

H. Larocque, D. Sugic, D. Mortimer, A. J. Taylor, R. Fickler, R. W. Boyd, M. R. Dennis, and E. Karimi, “Reconstructing the topology of optical polarization knots,” Nat. Phys. 14, 1079–1082 (2018).
[Crossref]

2017 (2)

M. V. Berry, “Laplacian magic windows,” J. Opt. 19, 06LT01 (2017).
[Crossref]

R. C. Devlin, A. Ambrosio, N. A. Rubin, J. B. Mueller, and F. Capasso, “Arbitrary spin-to–orbital angular momentum conversion of light,” Science 358, 896–901 (2017).
[Crossref]

2016 (1)

H. Larocque, J. Gagnon-Bischoff, F. Bouchard, R. Fickler, J. Upham, R. W. Boyd, and E. Karimi, “Arbitrary optical wavefront shaping via spin-to-orbit coupling,” J. Opt. 18, 124002 (2016).
[Crossref]

2014 (2)

Y. Schwartzburg, R. Testuz, A. Tagliasacchi, and M. Pauly, “High-contrast computational caustic design,” ACM Trans. Graph. 33, 74 (2014).
[Crossref]

E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light Sci. Appl. 3, e167 (2014).
[Crossref]

2013 (1)

2009 (1)

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[Crossref]

2008 (1)

F. Flossmann, O. Kevin, M. R. Dennis, and M. J. Padgett, “Polarization singularities in 2D and 3D speckle fields,” Phys. Rev. Lett. 100, 203902 (2008).
[Crossref]

2005 (1)

M. V. Berry, “Oriental magic mirrors and the Laplacian image,” Eur. J. Phys. 27, 109–118 (2005).
[Crossref]

2002 (1)

M. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213, 201–221 (2002).
[Crossref]

2000 (1)

Z. J. Laczik, “Quantitative Makyoh topography,” Opt. Eng. 39, 2562–2567 (2000).
[Crossref]

1990 (1)

K. Kugimiya, “Characterization of polished surfaces by ‘Makyoh,’” J. Cryst. Growth 103, 461–468 (1990).
[Crossref]

1988 (1)

K. Kugimiya, “Characterization of polished mirror surfaces by the ‘Makyoh’ principle,” Mater. Lett. 7, 229–233 (1988).
[Crossref]

1987 (1)

J. F. Nye and J. Hajnal, “The wave structure of monochromatic electromagnetic radiation,” Proc. R. Soc. London. A 409, 21–36 (1987).
[Crossref]

1879 (1)

W. E. Ayrton and J. Perry, “II. The magic mirror of Japan. Part I,” Proc. R. Soc. London 28, 127–148 (1879).
[Crossref]

Ambrosio, A.

R. C. Devlin, A. Ambrosio, N. A. Rubin, J. B. Mueller, and F. Capasso, “Arbitrary spin-to–orbital angular momentum conversion of light,” Science 358, 896–901 (2017).
[Crossref]

Ayrton, W. E.

W. E. Ayrton and J. Perry, “II. The magic mirror of Japan. Part I,” Proc. R. Soc. London 28, 127–148 (1879).
[Crossref]

Berry, M. V.

M. V. Berry, “Laplacian magic windows,” J. Opt. 19, 06LT01 (2017).
[Crossref]

M. V. Berry, “Oriental magic mirrors and the Laplacian image,” Eur. J. Phys. 27, 109–118 (2005).
[Crossref]

Bharwani, Z.

W. T. Chen, A. Y. Zhu, J. Sisler, Z. Bharwani, and F. Capasso, “A broadband achromatic polarization-insensitive metalens consisting of anisotropic nanostructures,” Nat. Commun. 10, 355 (2019).
[Crossref]

Birch, D. A.

M. Brand and D. A. Birch, “Freeform irradiance tailoring for light fields,” Opt. Express 27, A611–619 (2019).
[Crossref]

Bouchard, F.

E. Cohen, H. Larocque, F. Bouchard, F. Nejadsattari, Y. Gefen, and E. Karimi, “Geometric phase from Aharonov–Bohm to Pancharatnam–Berry and beyond,” Nat. Rev. Phys. 1, 437–449 (2019).
[Crossref]

H. Larocque, J. Gagnon-Bischoff, F. Bouchard, R. Fickler, J. Upham, R. W. Boyd, and E. Karimi, “Arbitrary optical wavefront shaping via spin-to-orbit coupling,” J. Opt. 18, 124002 (2016).
[Crossref]

Boyd, R. W.

H. Larocque, D. Sugic, D. Mortimer, A. J. Taylor, R. Fickler, R. W. Boyd, M. R. Dennis, and E. Karimi, “Reconstructing the topology of optical polarization knots,” Nat. Phys. 14, 1079–1082 (2018).
[Crossref]

H. Larocque, J. Gagnon-Bischoff, F. Bouchard, R. Fickler, J. Upham, R. W. Boyd, and E. Karimi, “Arbitrary optical wavefront shaping via spin-to-orbit coupling,” J. Opt. 18, 124002 (2016).
[Crossref]

E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light Sci. Appl. 3, e167 (2014).
[Crossref]

Brand, M.

M. Brand and D. A. Birch, “Freeform irradiance tailoring for light fields,” Opt. Express 27, A611–619 (2019).
[Crossref]

Capasso, F.

W. T. Chen, A. Y. Zhu, J. Sisler, Z. Bharwani, and F. Capasso, “A broadband achromatic polarization-insensitive metalens consisting of anisotropic nanostructures,” Nat. Commun. 10, 355 (2019).
[Crossref]

R. C. Devlin, A. Ambrosio, N. A. Rubin, J. B. Mueller, and F. Capasso, “Arbitrary spin-to–orbital angular momentum conversion of light,” Science 358, 896–901 (2017).
[Crossref]

Cardano, F.

Chen, W. T.

W. T. Chen, A. Y. Zhu, J. Sisler, Z. Bharwani, and F. Capasso, “A broadband achromatic polarization-insensitive metalens consisting of anisotropic nanostructures,” Nat. Commun. 10, 355 (2019).
[Crossref]

Cohen, E.

E. Cohen, H. Larocque, F. Bouchard, F. Nejadsattari, Y. Gefen, and E. Karimi, “Geometric phase from Aharonov–Bohm to Pancharatnam–Berry and beyond,” Nat. Rev. Phys. 1, 437–449 (2019).
[Crossref]

De Leon, I.

E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light Sci. Appl. 3, e167 (2014).
[Crossref]

de Lisio, C.

Dennis, M.

M. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213, 201–221 (2002).
[Crossref]

Dennis, M. R.

H. Larocque, D. Sugic, D. Mortimer, A. J. Taylor, R. Fickler, R. W. Boyd, M. R. Dennis, and E. Karimi, “Reconstructing the topology of optical polarization knots,” Nat. Phys. 14, 1079–1082 (2018).
[Crossref]

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[Crossref]

F. Flossmann, O. Kevin, M. R. Dennis, and M. J. Padgett, “Polarization singularities in 2D and 3D speckle fields,” Phys. Rev. Lett. 100, 203902 (2008).
[Crossref]

Devlin, R. C.

R. C. Devlin, A. Ambrosio, N. A. Rubin, J. B. Mueller, and F. Capasso, “Arbitrary spin-to–orbital angular momentum conversion of light,” Science 358, 896–901 (2017).
[Crossref]

Fickler, R.

H. Larocque, D. Sugic, D. Mortimer, A. J. Taylor, R. Fickler, R. W. Boyd, M. R. Dennis, and E. Karimi, “Reconstructing the topology of optical polarization knots,” Nat. Phys. 14, 1079–1082 (2018).
[Crossref]

H. Larocque, J. Gagnon-Bischoff, F. Bouchard, R. Fickler, J. Upham, R. W. Boyd, and E. Karimi, “Arbitrary optical wavefront shaping via spin-to-orbit coupling,” J. Opt. 18, 124002 (2016).
[Crossref]

Flossmann, F.

F. Flossmann, O. Kevin, M. R. Dennis, and M. J. Padgett, “Polarization singularities in 2D and 3D speckle fields,” Phys. Rev. Lett. 100, 203902 (2008).
[Crossref]

Gagnon-Bischoff, J.

H. Larocque, J. Gagnon-Bischoff, F. Bouchard, R. Fickler, J. Upham, R. W. Boyd, and E. Karimi, “Arbitrary optical wavefront shaping via spin-to-orbit coupling,” J. Opt. 18, 124002 (2016).
[Crossref]

Gefen, Y.

E. Cohen, H. Larocque, F. Bouchard, F. Nejadsattari, Y. Gefen, and E. Karimi, “Geometric phase from Aharonov–Bohm to Pancharatnam–Berry and beyond,” Nat. Rev. Phys. 1, 437–449 (2019).
[Crossref]

Hajnal, J.

J. F. Nye and J. Hajnal, “The wave structure of monochromatic electromagnetic radiation,” Proc. R. Soc. London. A 409, 21–36 (1987).
[Crossref]

Karimi, E.

E. Cohen, H. Larocque, F. Bouchard, F. Nejadsattari, Y. Gefen, and E. Karimi, “Geometric phase from Aharonov–Bohm to Pancharatnam–Berry and beyond,” Nat. Rev. Phys. 1, 437–449 (2019).
[Crossref]

H. Larocque, D. Sugic, D. Mortimer, A. J. Taylor, R. Fickler, R. W. Boyd, M. R. Dennis, and E. Karimi, “Reconstructing the topology of optical polarization knots,” Nat. Phys. 14, 1079–1082 (2018).
[Crossref]

H. Larocque, J. Gagnon-Bischoff, F. Bouchard, R. Fickler, J. Upham, R. W. Boyd, and E. Karimi, “Arbitrary optical wavefront shaping via spin-to-orbit coupling,” J. Opt. 18, 124002 (2016).
[Crossref]

E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light Sci. Appl. 3, e167 (2014).
[Crossref]

F. Cardano, E. Karimi, L. Marrucci, C. de Lisio, and E. Santamato, “Generation and dynamics of optical beams with polarization singularities,” Opt. Express 21, 8815–8820 (2013).
[Crossref]

Kevin, O.

F. Flossmann, O. Kevin, M. R. Dennis, and M. J. Padgett, “Polarization singularities in 2D and 3D speckle fields,” Phys. Rev. Lett. 100, 203902 (2008).
[Crossref]

Kugimiya, K.

K. Kugimiya, “Characterization of polished surfaces by ‘Makyoh,’” J. Cryst. Growth 103, 461–468 (1990).
[Crossref]

K. Kugimiya, “Characterization of polished mirror surfaces by the ‘Makyoh’ principle,” Mater. Lett. 7, 229–233 (1988).
[Crossref]

Laczik, Z. J.

Z. J. Laczik, “Quantitative Makyoh topography,” Opt. Eng. 39, 2562–2567 (2000).
[Crossref]

Larocque, H.

E. Cohen, H. Larocque, F. Bouchard, F. Nejadsattari, Y. Gefen, and E. Karimi, “Geometric phase from Aharonov–Bohm to Pancharatnam–Berry and beyond,” Nat. Rev. Phys. 1, 437–449 (2019).
[Crossref]

H. Larocque, D. Sugic, D. Mortimer, A. J. Taylor, R. Fickler, R. W. Boyd, M. R. Dennis, and E. Karimi, “Reconstructing the topology of optical polarization knots,” Nat. Phys. 14, 1079–1082 (2018).
[Crossref]

H. Larocque, J. Gagnon-Bischoff, F. Bouchard, R. Fickler, J. Upham, R. W. Boyd, and E. Karimi, “Arbitrary optical wavefront shaping via spin-to-orbit coupling,” J. Opt. 18, 124002 (2016).
[Crossref]

Marrucci, L.

Mortimer, D.

H. Larocque, D. Sugic, D. Mortimer, A. J. Taylor, R. Fickler, R. W. Boyd, M. R. Dennis, and E. Karimi, “Reconstructing the topology of optical polarization knots,” Nat. Phys. 14, 1079–1082 (2018).
[Crossref]

Mueller, J. B.

R. C. Devlin, A. Ambrosio, N. A. Rubin, J. B. Mueller, and F. Capasso, “Arbitrary spin-to–orbital angular momentum conversion of light,” Science 358, 896–901 (2017).
[Crossref]

Nejadsattari, F.

E. Cohen, H. Larocque, F. Bouchard, F. Nejadsattari, Y. Gefen, and E. Karimi, “Geometric phase from Aharonov–Bohm to Pancharatnam–Berry and beyond,” Nat. Rev. Phys. 1, 437–449 (2019).
[Crossref]

Nye, J. F.

J. F. Nye and J. Hajnal, “The wave structure of monochromatic electromagnetic radiation,” Proc. R. Soc. London. A 409, 21–36 (1987).
[Crossref]

J. F. Nye, Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations (CRC Press, 1999).

O’Holleran, K.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[Crossref]

Padgett, M. J.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[Crossref]

F. Flossmann, O. Kevin, M. R. Dennis, and M. J. Padgett, “Polarization singularities in 2D and 3D speckle fields,” Phys. Rev. Lett. 100, 203902 (2008).
[Crossref]

Pauly, M.

Y. Schwartzburg, R. Testuz, A. Tagliasacchi, and M. Pauly, “High-contrast computational caustic design,” ACM Trans. Graph. 33, 74 (2014).
[Crossref]

Perry, J.

W. E. Ayrton and J. Perry, “II. The magic mirror of Japan. Part I,” Proc. R. Soc. London 28, 127–148 (1879).
[Crossref]

Qassim, H.

E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light Sci. Appl. 3, e167 (2014).
[Crossref]

Rubin, N. A.

R. C. Devlin, A. Ambrosio, N. A. Rubin, J. B. Mueller, and F. Capasso, “Arbitrary spin-to–orbital angular momentum conversion of light,” Science 358, 896–901 (2017).
[Crossref]

Santamato, E.

Schulz, S. A.

E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light Sci. Appl. 3, e167 (2014).
[Crossref]

Schwartzburg, Y.

Y. Schwartzburg, R. Testuz, A. Tagliasacchi, and M. Pauly, “High-contrast computational caustic design,” ACM Trans. Graph. 33, 74 (2014).
[Crossref]

Sisler, J.

W. T. Chen, A. Y. Zhu, J. Sisler, Z. Bharwani, and F. Capasso, “A broadband achromatic polarization-insensitive metalens consisting of anisotropic nanostructures,” Nat. Commun. 10, 355 (2019).
[Crossref]

Sugic, D.

H. Larocque, D. Sugic, D. Mortimer, A. J. Taylor, R. Fickler, R. W. Boyd, M. R. Dennis, and E. Karimi, “Reconstructing the topology of optical polarization knots,” Nat. Phys. 14, 1079–1082 (2018).
[Crossref]

Tagliasacchi, A.

Y. Schwartzburg, R. Testuz, A. Tagliasacchi, and M. Pauly, “High-contrast computational caustic design,” ACM Trans. Graph. 33, 74 (2014).
[Crossref]

Taylor, A. J.

H. Larocque, D. Sugic, D. Mortimer, A. J. Taylor, R. Fickler, R. W. Boyd, M. R. Dennis, and E. Karimi, “Reconstructing the topology of optical polarization knots,” Nat. Phys. 14, 1079–1082 (2018).
[Crossref]

Testuz, R.

Y. Schwartzburg, R. Testuz, A. Tagliasacchi, and M. Pauly, “High-contrast computational caustic design,” ACM Trans. Graph. 33, 74 (2014).
[Crossref]

Upham, J.

H. Larocque, J. Gagnon-Bischoff, F. Bouchard, R. Fickler, J. Upham, R. W. Boyd, and E. Karimi, “Arbitrary optical wavefront shaping via spin-to-orbit coupling,” J. Opt. 18, 124002 (2016).
[Crossref]

E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light Sci. Appl. 3, e167 (2014).
[Crossref]

Zhu, A. Y.

W. T. Chen, A. Y. Zhu, J. Sisler, Z. Bharwani, and F. Capasso, “A broadband achromatic polarization-insensitive metalens consisting of anisotropic nanostructures,” Nat. Commun. 10, 355 (2019).
[Crossref]

ACM Trans. Graph. (1)

Y. Schwartzburg, R. Testuz, A. Tagliasacchi, and M. Pauly, “High-contrast computational caustic design,” ACM Trans. Graph. 33, 74 (2014).
[Crossref]

Eur. J. Phys. (1)

M. V. Berry, “Oriental magic mirrors and the Laplacian image,” Eur. J. Phys. 27, 109–118 (2005).
[Crossref]

J. Cryst. Growth (1)

K. Kugimiya, “Characterization of polished surfaces by ‘Makyoh,’” J. Cryst. Growth 103, 461–468 (1990).
[Crossref]

J. Opt. (2)

M. V. Berry, “Laplacian magic windows,” J. Opt. 19, 06LT01 (2017).
[Crossref]

H. Larocque, J. Gagnon-Bischoff, F. Bouchard, R. Fickler, J. Upham, R. W. Boyd, and E. Karimi, “Arbitrary optical wavefront shaping via spin-to-orbit coupling,” J. Opt. 18, 124002 (2016).
[Crossref]

Light Sci. Appl. (1)

E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light Sci. Appl. 3, e167 (2014).
[Crossref]

Mater. Lett. (1)

K. Kugimiya, “Characterization of polished mirror surfaces by the ‘Makyoh’ principle,” Mater. Lett. 7, 229–233 (1988).
[Crossref]

Nat. Commun. (1)

W. T. Chen, A. Y. Zhu, J. Sisler, Z. Bharwani, and F. Capasso, “A broadband achromatic polarization-insensitive metalens consisting of anisotropic nanostructures,” Nat. Commun. 10, 355 (2019).
[Crossref]

Nat. Phys. (1)

H. Larocque, D. Sugic, D. Mortimer, A. J. Taylor, R. Fickler, R. W. Boyd, M. R. Dennis, and E. Karimi, “Reconstructing the topology of optical polarization knots,” Nat. Phys. 14, 1079–1082 (2018).
[Crossref]

Nat. Rev. Phys. (1)

E. Cohen, H. Larocque, F. Bouchard, F. Nejadsattari, Y. Gefen, and E. Karimi, “Geometric phase from Aharonov–Bohm to Pancharatnam–Berry and beyond,” Nat. Rev. Phys. 1, 437–449 (2019).
[Crossref]

Opt. Commun. (1)

M. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213, 201–221 (2002).
[Crossref]

Opt. Eng. (1)

Z. J. Laczik, “Quantitative Makyoh topography,” Opt. Eng. 39, 2562–2567 (2000).
[Crossref]

Opt. Express (2)

Phys. Rev. Lett. (1)

F. Flossmann, O. Kevin, M. R. Dennis, and M. J. Padgett, “Polarization singularities in 2D and 3D speckle fields,” Phys. Rev. Lett. 100, 203902 (2008).
[Crossref]

Proc. R. Soc. London (1)

W. E. Ayrton and J. Perry, “II. The magic mirror of Japan. Part I,” Proc. R. Soc. London 28, 127–148 (1879).
[Crossref]

Proc. R. Soc. London. A (1)

J. F. Nye and J. Hajnal, “The wave structure of monochromatic electromagnetic radiation,” Proc. R. Soc. London. A 409, 21–36 (1987).
[Crossref]

Prog. Opt. (1)

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[Crossref]

Science (1)

R. C. Devlin, A. Ambrosio, N. A. Rubin, J. B. Mueller, and F. Capasso, “Arbitrary spin-to–orbital angular momentum conversion of light,” Science 358, 896–901 (2017).
[Crossref]

Other (1)

J. F. Nye, Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations (CRC Press, 1999).

Supplementary Material (4)

NameDescription
Visualization 1       The intensity propagation of the Gee-Gees logo produced by the liquid crystal magic mirror is shown over a distance of 5 centimetres.
Visualization 2       The intensity propagation of the Structured Quantum Optics group logo produced by the liquid crystal magic mirror is shown over a propagation distance of 5 centimetres.
Visualization 3       The intensity propagation of the University of Ottawa logo produced by the liquid crystal magic mirror is shown over a distance of 5 centimetres.
Visualization 4       The polarization handedness of the image produced by the magic window upon propagation is shown. The left circular component is shown in red while the right circular is shown in green.

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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Figures (5)

Fig. 1.
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.
Fig. 2.
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.
Fig. 3.
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.
Fig. 4.
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.
Fig. 5.
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.

Equations (2)

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I L a p l a c i a n W i n d o w ( R , z ) 1 z ( n 1 ) 2 h ( R ) .
e ± MP cos ( δ 2 ) e ± + i   sin ( δ 2 ) e e ± i χ ( r ) ,
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