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According to Malus' law, when a linearly polarized light beam passes through an analyzer (linear polarizer), the intensity of the light transmitted by the analyzer is directly proportional to the square of the cosine of angle θ between the transmission axes of the analyzer and the polarizer (see Figure 2a), that is, I=I0cos2θ, where I0 is the intensity of incident light. A structured beam with inhomogeneous polarization distribution can generate a spatial intensity distribution when passing through a polarizer, which provides a new degree of freedom to encode an image. Based on Malus' law, an arbitrary grayscale image can be hidden in the linear polarization profile of a light beam. Recently, metasurfaces5-10 have enabled us to engineer the spatial distribution of the amplitude, phase and polarization response at subwavelength resolution, thus enabling us to develop a plethora of ultrathin devices with unusual properties6, 11-27.
Fig. 2
Malus' law and image-hidden mechanism. (a) According to Malus' law, when a linearly polarized light beam passes through an analyzer (linear polarizer), the intensity of light transmitted by the analyzer is I=I0cos2θ, where I0 is the intensity of incident light, and θ is the angle between the transmission axes of the analyzer and the polarizer. A grayscale image is hidden in the linear polarization profile of a light beam. (b) The target image of James Clerk Maxwell's grayscale portrait. (c) The details of the selected area from the eyebrow area with 10×10 pixels. The left side shows the grayscale profile, and the right side shows the required polarization distribution for the analyzer with a transmission axis along the vertical direction. (d) A linear polarization is generated by a coherent superposition of two planar circularly polarized beams with opposite handedness, which propagates along the same direction. (e) As the sign of the geometric phase generated at the interface of metasurface only depends on the handedness of the incident light, under the illumination of a linearly polarized beam, the off-axis reflected beams with opposite handedness will meet, interfere with each other, and generate the desired polarization profile for the hidden image on both sides.Figure 2b shows a high-resolution grayscale image with 1300 × 1300 pixels, which is to be hidden in the optical beam. The resultant beam has a dimension of 390 × 390 μm because each pixel has a size of 300 × 300 nm, which exhibits subwavelength resolution. To explain our approach, we select an area from the eyebrow region with 10×10 pixels (Figure 2c). The enlarged intensity profile and corresponding polarization distribution are shown in the left and right sides of Figure 2c, respectively. In our design, the transmission axes of the polarizer and the analyzer (polarizer) are along the horizontal and vertical directions, respectively.
The required light beam with an inhomogeneous linear polarization profile can be decomposed into the superposition of two circularly polarized beams with equal components and opposite handedness (Figure 2d and 2e), which can be described as:
$$ \begin{array}{l} {\rm{\vec E}}\left( {x,y} \right) = {E_0}\left[ {\hat x\cos \varphi \left( {x,y} \right) + \hat y\sin \varphi \left( {x,y} \right)} \right]\\ \;\;\;\;\;\;\;\;\;\;\;\; = \frac{{{E_0}}}{{\sqrt 2 }}\left[ {\exp \left( {i\varphi \left( {x,y} \right)} \right){{\hat e}_{\rm{R}}} + \exp \left( { - i\varphi \left( {x,y} \right)} \right){{\hat e}_{\rm{L}}}} \right] \end{array} $$ where φ(x, y) is the relative phase difference between two orthogonal circular polarization states; ${{\hat e}_{\rm{L}}} = \left( {\hat x + i\hat y} \right)/\sqrt 2 $ and ${{\hat e}_{\rm{R}}} = \left( {\hat x + i\hat y} \right)/\sqrt 2 $ are unit vectors of the left circular polarization (LCP) and the right circular polarization (RCP). A geometric metasurface is used to realize the handedness-dependent phase profile while maintaining a constant amplitude10, 16, 18, 28. Here, a single reflective metasurface is designed to generate the desired structured beams by manipulating the superposition of two beams with opposite circular polarization states, which emerge from the same metasurface (Figure 2e). The key point here is to generate a phase profile that can simultaneously generate a pair of centro-symmetrically distributed off-axis beams with the identical phase profile φ(x, y) upon the illumination of RCP light. As the sign of the geometric phase generated at the interface of the metasurface depends on the handedness of the incident light, when the incident beam is changed from RCP to LCP, a pair of off-axis beams with the phase profile −φ(x, y) is generated. Obviously, under the illumination of the linearly polarized light beam, which is the superposition of LCP and RCP components, the reflected beams with opposite handedness will combine and generate the desired polarization profile for the hidden images on both sides. A detailed explanation is provided in Supplementary Information Section 1. The combination of two sets of phase profiles for the two opposite incident handedness values will generate the desired polarization profile for the hidden image. To eliminate the effect of the non-converted beam, the off-axis configuration is used for the metasurface design. A detailed explanation of the off-axis design is also given in Supplementary Information Section 1.