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Self-diffraction can be induced using a biased photorefractive crystal in the Fourier plane of an imaging system where the light beam intensity is naturally high due to the concentration effect of an optical lens. The spatial frequency spectrum of the output image is proportional to the optical power density distribution in the Fourier plane. A photorefractive crystal with small size can be used and hence an reduced amount of biased voltage is needed to obtain significant diffraction effect in the image plane. When the input image is an overlay of a signal and a noise pattern, theoretic model reveals that the induced diffraction in the Fourier plane may be preferably applied on the noise pattern. In order to illustrate the effect experimentally, a signal from a weakly illuminated object is coupled with an overwhelming noise pattern and then the hidden signal is successfully recovered using a SBN61 crystal with an applied voltage of 800 V in the Fourier plane. Such technology can be employed in encrypted spatial communication systems for security purposes.
© 2015 Optical Society of America
1. Introduction
Biased photorefractive crystals have been extensively explored in order to achieve novel diffracted patterns such as spatial solitons. Studies on the important properties such as dimensionality, size and spatial charge field for bright, dark and gray solitons have been conducted in great detail [1–4]. Experiments have also confirmed that nonlinear self-focusing and self-defocusing can be achieved using partially coherent beams and incoherent light beams [5–11]. Modulation instability has been analyzed to reveal the source traced to the noninstantaneous response properties of the photorefractive crystals [12, 13]. Pattern formations including 2D clustered and discrete solitons have been generated with spatially incoherent light beams [14–16]. Information usually not directly observable can then be retrieved from noisy images using biased photorefractive crystals [17–20]. Meanwhile, efforts have been continuously devoted to coherent beam propagation through biased photorefractive crystals for the phase encoding mechanism on an image input associated with the intensity distribution [21–23].
Phase contrast can be obtained on optical images when a photorefractive thin film is applied at the focal point of an optical lens [24, 25]. Self-focusing and self-defocusing can also be obtained on the optical images using biased photorefractive crystals applied in the Fourier plane of an imaging system [26]. The refractive index change Δne,o induced by partially coherent light beam inside the crystal can be described using the following equation [6]:
where r33 and r13 are the electro-optic coefficients, Eo is the effective electric field across the crystal, I(x,z) the light beam power density inside the crystal, and Id the so-called dark irradiance. The dark irradiance Id is an inherent property of each specific photorefractive crystal and equal to the light beam power density where the biased induced refractive index change is half of the maximum amount, which is a useful parameter to indicate the refractive index modulation sensitivity of the photorefractive crystal to the light beam power density. In this work a typical photorefractive crystal SBN61 doped with 0.002% CeO2 is used. The crystal has a size of 5 × 10 × 5 mm3 with the distance along the c-axis for poling and bias voltage of 5 mm and the light propagation distance inside the crystal of 10 mm. The electro-optic coefficients r33 and r13 are 250 pm/V and 47 pm/V, respectively; the refractive indices ne and no at 633nm are 2.2817 and 2.3103, respectively. The refractive index change distribution can be derived after Eq. (1) and written as follows:From Eq. (2) we can see two features of the light beam may have important impact on the diffraction effect: one is the light beam power density profile ∂I(x,z)/∂x,z and the other is the relative value of the absolute light intensity compared to the dark irradiance I(x,z)/Id. Typical self-diffraction experiments are conveniently conducted based on high power density light beams, such as high power or pulsed lasers, in order to achieve both effective power density profile and high I(x,z)/Id ratio .
In contrast, significantly lower light beam power density is needed to achieve self diffraction results in the Fourier plane of an optical imaging system, simply due to the remarkably high light power density concentration ratio at the focal point of an optical lens. For each coherent component of the light beam, according to Gaussian optics principles, the self-diffraction in the Fourier plane will be rendered to the output images in the image plane, as the power density distribution in both planes are Fourier Transform conjugates. For illustration, a weak collimated laser beam is employed to illuminate a binary object and the light beam is then focused by a lens with a focal length of 150mm. When a SBN61 photorefractive crystal is placed at the focal point and certain bias voltage is applied across the c-axis, self-focusing and self-defocusing can be observed in the image plane. The self-focused and self-defocused images obtained from a CCD camera are illustrated in Fig. 1, where Di is the self-defocused output image with a bias voltage of -i × 100V, FFT(Di) the calculated spatial frequency spectrum of Di, Fi the self-focused output image with a bias voltage of i × 100V, FFT(Fi) the calculated spatial frequency spectrum of Fi. The calculated spatial frequency spectrum is proportional to the power density distribution in the Fourier plane and it can be clearly noticed that the low spatial frequency zones are affected more than the high frequency zones by the bias voltage on the photorefractive crystal. Also please notice that a photorefractive crystal of smaller size is needed for a highly concentrated light beam in the Fourier plane and hence significantly reduced applied voltage is required for desirable diffraction observations.
Fig. 1 Self-focusing and self-defocusing experimental results of a weakly illuminated object with a biased photorefractive crystal applied in the Fourier plane of the imaging system. Di: the self-defocused output image with a bias voltage of -i × 100V; FFT(Di): the calculated spatial frequency spectrum of Di; Fi: the self-focused output image with a bias voltage of i × 100V; FFT(Fi): the calculated spatial frequency spectrum of Fi; i = 0, 1, ... 5. The spatial frequency spectra are in log scale and zero-centered.
2. Theoretical analysis
Previous results explain that: (1) self-diffraction can be induced in the Fourier plane even for an input image with weak power density; (2) self-diffraction is preferably induced at the center of the light beam in the Fourier plane with high ∂I(x,z)/∂x,z and I(x,z)/Id values; and (3) the center of the Fourier plane corresponds to the low spatial frequency zones of the output image so that the self-diffraction causes low spatial frequency shifting effect in the output images. We can then consider combining a signal from a binary object with an overwhelming optical noise pattern consisting of a plurality of grain sizes and analyze the possibility of recovering the signal from the noisy image by applying the biased photorefractive crystal SBN61 in the Fourier plane of an optical imaging system. For instance, as shown in Fig. 2, modeling results demonstrate that an image hidden in a noise pattern with grain sizes of average 1.4 × 1.4 pixels can be revealed with applied voltage to reducing the average noise grain sizes down to 0.9 × 1.2 pixels and 0.5 × 1.1 pixels, respectively. Next we will achieve the results experimentally.
Fig. 2 Modeling results illustrate the hidden signal recovery process: (a) is the input image combining a weak signal and an overwhelming noise pattern; (b) and (c) are output images with increased bias voltage. The average noise grain sizes in (a-c) are 1.4 × 1.4, 0.9 × 1.2, and 0.5 × 1.1 pixels on the CCD camera, respectively.
3. Experiment and results
A 532nm laser beam (with ~200mW output power) is expanded to be a Φ20mm collimated beam and the center relative uniform region is used for the experiment (Fig. 3). The light beam is split into two: the first in the “noise arm” is focused onto a stationary diffuser and then the diffused light is collimated to provide the noise pattern; and the second in the “signal arm” is directed on a binary object with an adjustable neutral density filter to tune the signal intensity so that the signal is properly hidden in the noise pattern when both arms are combined. The combined light beam is then focused by an optical lens with a biased photorefractive crystal (SBN61) at the focal point and the output images are recorded by a CCD camera.
Fig. 3 Experimental scheme for hidden image recovery using a photorefractive crystal in the Fourier plane of the imaging system. BS: beam splitter; L: lens; D: diffuser; A: aperture; M: mirror; AF: adjustable ND filter; BD: beam dump.
Three different inputs, representing the cases for (1) noise only, (2) signal only, and (3) noise plus signal, respectively, are designed for the imaging system. Ni, Si, and Ti are the recorded output images with bias voltages of 0V, 200V, 400V, 600V, and 800V, respectively (Fig. 4). The spatial frequency spectra of the output image are calculated and highlighted in Fig. 4 as FFT(Ni), FFT(Si), and FFT(Ti), respectively. As the optical power density distribution in the Fourier plane of the imaging system for each output image is proportional to its calculated spatial frequency spectrum, we can trace the evolution of the optical power density distribution through the biased photorefractive crystal and examine the impact from the induced self-diffraction. When no bias voltage is applied on the photorefractive crystal, N0, S0, and T0 are the linear output of the imaging system, respectively, and T0 is the linear superimposition of N0 and S0 (Fig. 5). The power density of the signal is properly tuned to submerge the signal in the noise pattern and FFT(S0) is also very difficult to be recognized from FFT(T0). With increased bias voltage, the signal image gets more blurred and the noise pattern looks much darker since quite amount of the optical power may be diffracted off the effective image plane. For the noise plus signal output images, the noise pattern in Ti evolves very similar to that in Ni, however the signal seems nearly intact compared to S0. The image system is no longer linear with biased photorefractive crystal (Fig. 5). FFT(Ti) results reveal that the self-diffraction impact is preferably applied on the low spatial frequency components of the noise pattern with high ∂I(x,z)/∂x,z and I(x,z)/Id values and the signal image is rarely affected instead.
Fig. 4 The output images of the optical imaging system with a biased photorefractive crystal in the Fourier plane. Ni, Si, and Ti are output images for noise, signal, and noise plus signal, respectively. The bias voltage is i × 200V for i = 0, 1, ...4, respectively and N0, S0, and T0 are the linear output images with zero bias.
Fig. 5 Hidden image recovery with bias voltage applied on photorefractive crystal. The system is nonlinear with applied voltage.
The results obtained in the experiment indicate that image signal can be encoded in an overwhelming noise pattern for secure communication purposes. The encoded image can be simply decoded using an optical receiver with a biased photorefractive crystal applied at the Fourier plane of the imaging system of the receiver. This can be very useful for laser based secure spatial communication systems.
4. Conclusion
We experimentally demonstrate the feasibility of recovering a hidden signal from an overwhelming noise pattern using a photorefractive crystal based receiver. A SBN61 crystal with bias voltage is employed to induce self-diffraction in the Fourier plane of the imaging system. The hidden signal can be successfully recovered with a bias voltage of 800V. The technology is promising to be employed in encrypted spatial communication systems for security purposes.
Acknowledgment
Authors would like to thank support from AQSIQ (AJG1301 & AJG1411) and National Science Foundation of China (No. 11104257).
References and links
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