Since the early days of holography, computer-generated holograms (CGHs) have been used for creating wavefronts corresponding to arbitrary, even non-existing objects [1]. The capabilities of such holograms have been vastly improved due to the progress in computational speed and miniaturization of phase-modulating devices. As a result, CGH has found a wide range of applications, including optical testing [2], beam shaping [3,4] (including forming optical tweezers [5]), and visual displays [6], the latter being our main interest in this field. However, CGH is still hindered by its numerous intrinsic drawbacks. In this work, we address the problem of visible higher diffractive orders.

The most common devices used for displaying CGHs are liquid crystal on silicon (LCoS) spatial light modulators (SLMs). The holograms are reconstructed in the far field as a result of light diffraction on the phase patterns displayed on the regularly pixelated surface of the SLM. Inevitably, this periodicity leads to the unfavorable presence of additional images in the far-field light distribution, referred to as higher-order images. Such an effect is especially undesired in the case of animated holographic projections, in which the multiplied images severely distract the viewer. Removing the higher-order images is possible with spatial filtering [7], which, however, is not elegant and significantly increases the complexity and volume of the optical setup. Here we propose an alternative solution, which is the effective randomization of the positions of CGH pixels (samples) in the plane of phase modulation.

Due to the fact that commonly used SLMs comprise a regular, fixed pattern of relatively large pixels, the implementation of small, random displacements of CGH pixels is troublesome. Therefore in this work, we additionally propose a new, photo-magnetic medium, better suited for recording of randomized patterns.

The idea of applying a photorefractive or a photochromic material for light modulation has been previously presented, as many researchers showed the possibility of employing doped liquid crystals [8–10], polymer-based materials [11–17], or metamaterials [18–21] for such a purpose. Most of the proposed solutions, however, possess disadvantages that exclude them from the field of translucent holographic displays. A number of them employ visible light for recording the diffractive patterns [8–15,22,23], or are designed for reconstructing the wavefront in the invisible part of the spectrum [19]. Other drawbacks include, e.g., long recording times measured in milliseconds [10,17], neccessity of applying high temperatures for erasing the patterns [11,12,19], and temporal instability of the diffractive patterns that can be affected by external conditions, i.e., voltage or temperature [11,12,19]. Notably, there have been reports of metamaterials in which ultrafast switching of their states [21], and even recording of grayscale pixel values [20] are possible. In this Letter, we propose a photo-magnetic cobalt-doped yttrium iron garnet (YIG:Co) [24] as a new promising material for ultrafast recording of holographic patterns.

The theoretical approach of finding the reconstructed field from a CGH displayed on a pixelated modulator is rather straightforward, as the CGH plane and projection (playback) planes are Fourier related. In Eq. (1), $g$ corresponds to the computer hologram, while $G$ is its Fourier transform (FT), constituting the played-back intensity pattern. The comb component on the right side of Eq. (1) represents the problematic copies of the playback pattern, and its origin is obviously the comb function on the left side of the equation, corresponding to the regular sampling on regular pixels of the modulator:

It can be concluded that a diffraction pattern with deliberately disturbed periodicity of the pixels in the modulation plane would create a far-field light distribution with reduced visibility of such undesired additional images. In order to confirm this, numerical simulations of pixel randomization were carried out. The area of the pre-calculated binary CGH of a test image of a triangle was divided into smaller subsections. From each of those sections, a single, randomly selected pixel was preserved, while the rest of the pixels were set to an intensity of zero. The size of the sub-areas is referred to as a degree of randomization (none, medium, high) and was equal to $1 \times 1$ sample, $3 \times 3$ samples, and $4 \times 4$ samples, respectively (Fig. 1). Simulations of hologram reconstructions were then numerically computed by fast FT (FFT) and analyzed, with regard to contrast and signal-to-noise ratio (SNR) (Table 1). The SNR was defined as the ratio of average intensity of an image to the standard deviation of intensity of the background. The obtained results support the presented thesis, showing an almost 80% decrease in SNR in the areas corresponding to the second diffraction order, with a simultaneous drop of only 19% of the SNR in the main (central) image area. This results in a clearly observable reduction of the visibility of ghost images.

 

Fig. 1. Magnified central areas of the holograms (top) and their numerical reconstructions (bottom) for increasing degree of randomization: (left column) none; (middle column) medium; (right column) high.

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Table 1. Intensity SNR of the Images Reconstructed from Randomized Holograms—Simulation Results

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The initial proof-of-concept experiment utilized an LCoS SLM (Holoeye Pluto 2 VIS) as the phase-modulating device with a fixed, regular array of 8 µm-sized pixels. Analogous binary CGHs of a test triangle image with randomly displaced pixels were displayed on the chosen SLM, which was illuminated by a beam from a He–Ne laser, convergent at the matrix of a camera. The experimental reconstructions (Fig. 2) confirmed the simulations, showing significant suppression of higher-order fields, hence proving the merit of pixel randomization. Notably, there was little change in the contrast, brightness, and speckle noise of the useful images in the lowest diffractive orders.

 

Fig. 2. Experimental reconstructions of CGH with second-diffraction order area marked and enlarged: (left) no pixel randomization; (right) high pixel randomization.

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Inevitably, the presented demonstration required the use of oversampling, as there is no physical possibility of applying small displacements in the fixed and relatively large pixels of the SLM. For this reason, as the next step we attempted the writing of diffractive patterns in a non-pixelated photo-magnetic medium. There the recording positions of the CGH spots can be freely chosen, and moreover, they are intrinsically randomized by the stochastic nature of internal magnetic domains.

While the initial experiment proved the possibility of implementing the pixel randomization with the use of the SLM, it simultaneously showed intensity losses, as only a small fraction of SLM pixels was used to create the observed image. To increase the efficiency of the proposed method, an alternative recording medium must be employed that would allow small random displacements of CGH samples. From the point of view of potential applications, we additionally assumed that such a material should allow optical contact-less recording of arbitrary patterns resulting in local changes of the refractive index with the use of invisible wavelengths. It is also desirable that the medium has no pre-existing internal pixel structure and allows fast and permanent rewriting of patterns. Additionally, the material must not require any auxiliary external fields or electrodes, work at room temperature, and be optically transparent in order to facilitate the direct see-through observation of the diffracted fields, i.e., in near-eye displays. Finding a material that meets all the above expectations is not a trivial task and to our best knowledge has not been previously demonstrated in CGH. As the answer to that problem, we propose here a photo-magnetic medium with a set of unique magneto-optical properties. YIG:Co in the form of a thin film was chosen from a range of possibilities due to its desirable properties and non-volatile magnetic memory [24].

In order to record a diffractive pattern with arbitrary localization of spots on the YIG:Co surface, we developed the magneto-optical bench comprising a micro-electro-mechanical systems (MEMS) mirror (see Fig. 3). This mirror was used to direct the sequence of linearly polarized laser pulses of 50 fs duration and 1 kHz repetition rate. The pump laser beam at 1300 nm was focused on the surface of the sample into a spot of 50 µm in diameter and moved across the material in a raster-scan manner. Each point of the pattern was recorded individually according to a script containing previously specified positions with the use of a single laser pulse. The exposure time of separate writing was fixed at 1 ms. This time was well above the characteristic time of photo-magnetic recording of a single domain, which is around 20 ps. It has been found that in YIG doped with Co-ions, femtosecond linearly polarized laser pulses can control magnetic anisotropy [25] and change the equilibrium orientation of the magnetization by changing the polarization of light. Upon excitation, there occurs the reversal of the magnetization component perpendicular to the sample plane in the magnetic domains. By changing the linear polarization, one can control the magnetization state at the domain for writing or erasing. The switched magnetic domains in a garnet film are formed at an ultrafast time scale after a single pump pulse and remain stable for a few days [24] without extrinsic forces until an intentional erasing. It is possible to record arbitrary patterns multiple times without notable heating or destruction of the sample. Recently it was also shown that laser pulses can write and rewrite magnetic bits in time intervals of down to 60 ps, hence such photo-magnetic recording is potentially feasible with repetition rates approaching 20 GHz [26].

 

Fig. 3. Scheme of the optical setup performing the raster-scanned recording of arbitrary patterns in the photo-magnetic medium. B, beam splitter; P, polarizer; A, analyzer; L, convex lenses; S, sample; O, objective lenses; D, diaphragm; M, reflective mirror; PUMP, 1300 nm pulsed laser for writing; PROBE, 650 nm CW laser for readout; CCD, cameras with bare photosensitive matrices; LED, white light illumination.

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For the experiment, two cameras were used. The first one was focused to infinity and allowed direct observation of the diffraction effects of the readout laser beam in the far field. The second camera was used for real-time investigation of the recording process accuracy by illumination of the sample with a polarized white light and the use of a properly crossed analyzer. The effects of recording and erasing are shown in Fig. 4. The erasing of the written domain with the brief external magnetic field or with the orthogonally polarized pulse returned the sample to its initial state [(a)–(c)]. The application of the MEMS mirror allowed fast and precise recording of the desired arbitrary patterns. The change of the distance between points on the sample hit by the writing beam allowed the creation of separated domains [(d)–(e)] or homogeneous, consolidated groups (f).

 

Fig. 4. Differential changes of magneto-optical images using Faraday geometry (a) before and (b) after single pump laser pulse excitation with linear polarization along [100] (solid line) in the YIG:Co, showing (b) writing and (c) erasing with linear polarization along [10] (dashed line). The magnetic domain images after MEMS scanning: (d) point-by-point at $3 \times 3$ pattern, (e) arbitrary with selective points or erasing previously recorded points of a $3 \times 3$ pattern, and (f) binary diffraction grating.

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For the proof of concept, a simple recording of a binary diffractive grating was chosen. It was expected that such a pattern should redirect the incident light beam into distinguishable diffraction orders, ${+}1$st and ${-}1$st in particular, as they are typically most visible, even without the suppression of higher orders. This was proven experimentally. The gratings of varying periods were recorded in the YIG:Co material, and the retrieved differential images showed high contrast of the obtained patterns (see Fig. 5). The visible irregularities result solely from the intrinsic randomization of the material structure, while the scanning during the writing process was done along straight lines. In other materials exhibiting lower intrinsic randomization, a higher degree of MEMS mirror control and deliberate programmatic randomization would be required in order to achieve similar results. The sample with written patterns was simultaneously probed with a 650 nm collimated CW laser beam, and the diffraction of this beam corresponded to the simulation results (see Fig. 5). The zeroth-order spot was removed from the diffraction photographs for better visibility, as it saturated the sensor. Due to small phase modulation in the sample, the resulting diffractive efficiency was estimated below 1%. While in most potential applications, this would be problematic, in the case of translucent near-eye diffractive displays, this constitutes an advantage of the see-through perception unobstructed by chromatic diffractive effects. If necessary, one of the possible methods of increasing the magneto-optic contrast by as much as 10 times is using Bi-substituted iron garnets (BIG) exhibiting giant Faraday rotation [27].

 

Fig. 5. Diffraction gratings of period $ d $ recorded in the photo-magnetic medium: (left) recorded magnetic structures; (center) experimental diffracted fields; (right) simulation results.

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The measured positions of diffracted spots for variable grating periods are gathered in Fig. 6. The obtained diffraction images contain easily recognized maxima of the first and ${-}1$st orders of diffraction. Notably, the maxima in spurious higher diffractive orders were not observed in the experimental and numerical reconstructions. This can be attributed to their low intensities, potentially being the positive effect of the intrinsic randomization of the written patterns on the stochastic magnetic domains. The confirmation of this hypothesis on more complex holographic recorded patterns will be our future work. Reducing the size of the unit domain in the pattern will lead to an increase in feasible density of recorded patterns and the total count of written spots. This will be possible by precise control over the intensity and focus of the writing laser spot. However, the demagnetization field in this particular garnet prevented stable domains of sizes below 20 µm.

 

Fig. 6. First-order diffraction in the Fourier plane as a function of the grating period for a 650 nm CW probe beam.

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The proposed method of reduction of visibility of higher-order images by randomization of pixel patterns was proved to be effective in both simulations and experiment. Both numerical and visual analyses of the holographic reconstructions showed a decline in the contrast and SNR in the areas of spurious images almost four times higher than the resulting decrease in the main image. The experiment was carried out with the use of the LCoS light modulator, limited by its periodic pixel structure, thus leading to intensity losses in the hologram reconstruction when the randomization was applied. In order to avoid such costs of randomization, a new photo-magnetic medium was proposed as a new diffractive material. In the preliminary experiments, diffraction gratings of various periodicity were recorded in the YIG:Co thin film. The analysis of the obtained results proved that the diffraction of the visible light on the said gratings matched the theoretical and simulational predictions. Further research into the application of such photo-magnetic media may include both the improvement in diffractive efficiency and complexity of holographic patterns recorded in the garnet. Finding a suitable photo-magnetic medium with a high rotation of polarization for high-density recording is still an open problem.