Non-periodic photon sieve

The proposed wavefront modulator confines the active area of each LCD pixel through pinholes of the photon sieve (Fig. 2a–c). The pinholes diffract the light fields at wide angles20,21, thus increasing the viewing angle of the holographic images. We designed the photon sieve such that its pinholes show a one-to-one correspondence with the pixels; only one pinhole is positioned on the active area of each LCD pixel, and the total number of pinholes are same as the pixel numbers in LCD panel (Fig. 2a). The probability distribution of the lateral (x–y plane) displacement of the pinhole position from the centre position of the LCD pixel is a continuous uniform distribution. The optical field scattered from each pinhole is independently modulated by each LCD pixel and render dynamic holographic images.

Fig. 2 Wide-angle large-area holographic display using non-periodic photon sieve. a Scheme of the system. The non-periodic photon sieve increases the diffraction angles of transmitting light. A one-to-one correspondence between SLM pixels and pinholes allows independent modulation of the optical field transmitted from each pinhole. b Photograph of the system. The pinholes are placed close to the transmissive SLM to maintain the small form factor. c Micrographs from a scanning electron microscope of the non-periodic photon sieve fabricated using conventional photolithography. The diffraction angles of the transmitting light field vary according to the pinhole size. d Intensity profile of light transmitted through the non-periodic photon sieve. The size of the pinholes in the photon sieve is 2.2 μm. The distance between the image plane and the photon sieve is 10 mm. e Spatial frequency map corresponding to the intensity pattern of the image in d. The dashed circle indicates the numerical aperture corresponding to the viewing angle of 30 degrees. f Scheme of dynamic optical focusing over a wide volume. Optical focus is generated at arbitrary positions by displaying optimal patterns at the LCD panel. A 4-f telescopic imaging system (NA = 0.75) is mounted on a movable stage to capture images. g, h The size and shape of the focus vary according to its position because the addressable transversal wave vector depends on the displaying geometry. Effective viewing angles of the foci were examined from the size of the foci Full size image

The operation of the proposed system relies on the randomness of the linear optical transformation induced by the non-periodicity of the pinholes. In practice, if pinholes are periodically aligned, reconstructed holograms can only carry optical information within the Nyquist frequency defined by the pixel pitches. As a result, spatial aliasing artefacts occur; unwanted multiple cloned images are generated. In contrast, pseudo-randomly oriented pinholes suppress the duplication of the holographic images within the diffraction angle defined by the pinhole size, but at the expense of background noises (see Supplementary Figure 1 and Supplementary Note 1).

Importantly, the positions and shapes of the pinholes in the photon sieve are completely known and do not change over time, and the light transport through the LCD panel and the photon sieve occurs in a deterministic manner. Furthermore, because each pinhole is located within the area of each corresponding LCD pixel, the present method does not require complex time-consuming calibration and calculation associated with an optical transmission matrix22,23.

We should note that our methodology is fundamentally different from previously reported photon sieve based optical holograms. The previous methods have tailored the sizes and distributions of the pinholes for generating static holograms24,25 with given design principles. In contrast, we tailor the light field incident on the pinholes to generate dynamic holograms while the lateral positions of the pinholes are designed in a pseudo-random manner. This pseudo-random optical transformation by the pinholes enables to manipulate holograms with a large degree of freedom.

Experimental verification

To verify the generation of holographic images with a wide viewing angle, we first measured a light field scattered from the photon sieve, which was fabricated using a conventional photolithography process (see Methods). The size of the pinholes in the photon sieve is 2.2 μm (Fig. 2c). When the photon sieve is close to the LCD panel (1.8 inches diagonal, 1024 × 768 pixels) and illuminated by a collimated green laser beam (λ = 532 nm), a random speckle pattern is generated (Fig. 2d). As shown in Fig. 2e, the speckle pattern contains high-spatial-frequency components capable of providing wide viewing angles to the holographic images.

Next, we verified the dynamic focusing at arbitrary positions, as illustrated in Fig. 2f. An optimal pattern is displayed on the LCD panel to produce the optical focus at the desired position. We determined the phase values, which match the optical path lengths of the light fields scattered from the pinholes to the desired focal position, by simple algebraic calculations (see Methods). The resulting bias values corresponding to the phase values of the patterns were displayed on the LCD panel. In practice, the LCD panel is neither amplitude- nor phase-only modulator. The randomness of the photon sieve scrambles the optical properties of the propagating light fields and allows the conventional LCD panel to be used as an efficient wavefront modulator upon precise calibration (Supplementary Figure 2). By using a 4-f telescopic imaging system with a high numerical aperture (NA = 0.75), we observed the tight optical focus as shown in Fig. 2g and Fig. 2h.

To examine an effective viewing angle of the proposed system, we measured the size of the optical focus. For a given diffraction-limited imaging system with a circular aperture, a shape of an optical focus follows an Airy pattern. A full-width at half-maximum (FWHM) value of the pattern is directly related to the NA (or viewing angle) of the imaging system as FWHM = 0.51λ/NA. Along the axial direction, FWHM values of the foci at 1, 2, 3 cm from the screen were measured as 1.0 μm. The corresponding viewing angle is approximately 30° (Fig. 2g). However, when the distance between the screen and the focus increases over a certain distance (Z 1,2 in the Fig. 1), the effective viewing angle decreases inversely proportional to the distance since the supported spatial frequency range is limited by the geometry rather than diffraction angles of the pinholes. Similarly, the radial position of the focus imposes asymmetry in the addressable spatial frequency ranges for the radial directions, thus changes the shape and size of the focus6. Also, the non-uniform diffraction of the laser beam through the pinholes also may decrease the effective viewing angles.

It is worth noting that unlike the previous research on wavefront shaping techniques that consider disordered systems22,26,27,28,29,30, we used a deterministically generated pseudo-random mask to exploit randomness without requiring a time-consuming calibration process31. Furthermore, the absence of relaying optics between the pinholes and the LCD panel results in minor aberrations, induced by the LCD panel and the glass substrate of the metal mask, and should be calibrated only once for optimal performance. Although the holographic images were formed in a large-optical-mode space (>109), the complexity of the linear optical transformation, i.e., the number of non-zero entries of the transmission matrix of the photon sieve, is equal to the number of pixels on the LCD panel (~106). Thus, the calculation of the optimal pattern for focusing requires only O(~106) rather than O(>109) computations.

Holographic image generation

To confirm the 3D nature of the holographic images, we generated and captured a helix hologram consisting of 75 points at different viewing angles (Fig. 3). The helix was positioned 4.2 cm behind the photon sieve, which consists of 2.2 μm sized pinholes. For this experiment, we mounted the 4-f telescopic imaging system (NA = 0.16) on a movable stage to observe the motion parallax of the hologram. Figure 3b clearly shows that the images of the hologram vary according to the observation angle, and the experimental results are consistent with those from simulations. Although the actual viewing angle was measured as 30°, the holographic images can be observed at higher angles because our imaging system has a non-zero numerical aperture. It is worth noting that we achieved a wide viewing angle of about 30° using a single flat-panel wavefront modulator without additional optics. By using the smaller sized pinholes, the viewing angle can be further enhanced (Supplementary Figure 3; Supplementary Figure 4).

Fig. 3 Holographic images of 3D helixes. To demonstrate the motion parallax of holographic images, a helix consisting of 75 foci was generated and placed within a depth of focus of the proposed imaging system. a Geometry of the holographic helix. b 2D intensity images captured at different observation angles using the 4-f telescopic imaging system with a numerical aperture of 0.16 and corresponding simulation results. c Holographic image for projection in a large volume. The diameter of the helix varies from 0.25 cm to 0.5 cm with a length of 5 cm. d 2D intensity maps of the helical trajectory captured at each depth without lenses mounted on the camera Full size image

To demonstrate the projection of holographic images on a large volume, we increased the size of the helix to the centimetre scale (Fig. 3c). The diameter of the helix varies along the axial direction, and the helix has maximum diameter and length of 0.5 cm and 5 cm, respectively (Fig. 3d). We captured the 2D projections of the hologram by placing a camera without lenses at each depth plane and observed the corresponding images. In Fig. 3d, the blurred images of out-of-plane foci are also shown. In practice, although the large screen size and the wide diffraction angles were confirmed, the actual viewing angles of holographic images are restricted by its displaying geometries. The holographic images must be lie between the observers’ eyes and the scattering surface (the photon sieve) since the light is emitted from the displaying screen rather than the virtual objects32.

A contrast factor, defined as the ratio between the intensity of the holographic image and the average value of the background noise, is directly proportional to the number of controlled optical modes in our system. By using all the pixels in the XGA resolution (1024 × 768 pixels) of the LCD panel, we achieved a factor of approximately 2.0 × 105 (Supplementary Figure 5), thus notably outperforming previous works on wavefront shaping through disordered systems33,34. Although this value, to the best of our knowledge, is the highest ever reported, still insufficient to generate complex holographic images. When a large number of optical modes (or optical foci) are addressed simultaneously, the signal is redistributed into the optical modes. In that case, the signal level decreases in proportion to the number of the optical modes while the background noise remains the same. In order to maintain the high visibility of the holographic images, either the signal must be sparse or to utilise a large number of controlled optical modes (pixel numbers).

In Table 1, the performance of the LCD panel integrated with the photon sieve for displaying holographic images in terms of the screen size and diffraction angle is compared with commercially available SLMs and our previous research on a 3D display26. The values of the proposed system in Table 1 are achieved using a non-periodic photon sieve of 2.2 μm sized pinholes. In our demonstration, the product value of the screen size and diffraction angle was enhanced over a factor of ~1200 compared to the use of only a wavefront modulator with XGA resolution while the holographic image generated from our display attain the same SBP (or complexity) value (see Discussion).

Table 1 Characteristics of the holographic displays Full size table

We also verified the generation of dynamic colour holograms using a single wavefront modulator by presenting a rotating cube with three colours, as shown in Fig. 4. Red (λ = 639 nm), green (λ = 532 nm), and blue (λ = 473 nm) laser beams illuminated the LCD panel simultaneously, and the optimal patterns were displayed to generate coloured foci at different spatial positions. The optical transformations of the photon sieve are spectrally uncorrelated for the given wavelengths. Therefore, the colour images were projected simultaneously using the modulator resembling space-division multiplexing such that no colour filter was required35 (see Methods).

Fig. 4 Dynamic colour holograms of a rotating cube. Frames of a 3D holographic rotating cube with red, green, and blue colours. See Supplementary Movie 1 for the full set of frames Full size image

Varying pinhole size and number of pinholes per LCD pixel

Because the active area of the LCD pixel is confined by the size of the pinholes, the photon sieves are suffered from low light efficiency. In our demonstration, the light transmittance through the photon sieve, which consists of 2.2 μm sized pinholes, was measured as 0.16%. To mitigate the issue, we demonstrated the proof of concept experiments with the photon sieves consisting of either large-sized pinholes or larger number of pinholes.

As shown in Fig. 5a, the transmittance of the proposed method is readily adjustable upon demand by compromising the viewing angles. We should note that although the increased transmittance was achieved in trade relations between diffraction angles. Still, the photon sieves notably increase the viewing angle of the transmissive LCD, which was originally 0.8 degree. Compared to the use of reflective SLMs, it is more challenging to narrow down the pixel pitch of the transmissive SLMs because of the embedded electrodes and required optical pathlengths of rotating birefringent molecules. Until now these technical hurdles obstruct the realization of flat-panel holographic displays with large viewing angles. The larger the number of pinholes, the higher the transmittance (Fig. 5b). However, the calculation burden also increases linearly with the number of pinholes. Nevertheless, we successfully demonstrated the generation of holographic images, a tetrahedron consisting of 60 foci, with a total of 7.8 million pinholes (Fig. 5d). The holographic image converges and diverges rapidly. The shape highly depends on the observing planes, which reflect the nature of the large diffraction angle of the holographic display.