## Abstract

We theoretically propose the magneto-optically reorientation-induced image reconstruction in bulk nematic liquid crystals (NLCs). The underlying signals are reinforced and recovered at the expense of scattering noise under reorientation-induced self-focusing nonlinearity. The intensity perturbation gain is derived and the numerical results are presented to show the response of NLC molecules to the diffusive images. The nonlinear image recovery is influenced by the input light intensity, the magnetic field direction, and the correlation length. The results suggest an alternative approach to detect noisy images and promote the application of NLCs in image processing.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Imaging through turbid media, such as fog, muddy water, and tissue has been widespread concerned in optical field. Absorption and scattering are two main factors that constrain the vision quality. Conventional imaging technologies, such as the spatial gating, the polarization gating and the time gating improve the image quality by filtering scattering noise. These noise-filtering methods sacrifice part of the signals and decrease the brightness of the image to eliminate the diffusive light. However, noise is not always harmful to signals. The relationship between signals and noise is much complex in nonlinear systems. The weak signals can be enhanced with the assistance of noise in the system of stochastic resonance (SR) [1]. SR was first proposed by Benzi et al. and has broad applications in pulse signal recovery, ranging from biology, to electricity to optics [2–5]. In 2010, Dylov et al. raised a new type of SR to recover the noise-hidden images. The underlying signals are reinforced at the expense of noise under self-focusing nonlinearity in a photorefractive crystal [6–8]. Subsequently, the nonlinear image recovery was extended to pulse image recovery, underwater image recovery, and white-light image recovery [9–12]. The SR-based method restores more detailed information by exploiting the nonlinear coupling between signals and noise.

So far, photorefractive crystals (PCs) are the primary functional material to implement the SR-based image reconstruction. PCs are rather expensive, require the control of high voltages and have long relaxation times. By comparison, NLCs have the similar functional properties with PCs to support the localization of light beams [13]. NLCs are affordable and have a large all-optical response. Thus, Feng et al. theoretically demonstrated the nonlinear image reconstruction based on the molecular reorientation response in NLCs [14]. Yet there is a problem with the NLCs they used. The NLCs with planar cells have the typical thickness of 0.1mm or less for ensuring the lateral anchoring and the appropriate molecular alignment [15,16]. The dimensional restriction of the planar structures destroys the symmetry of the spatially nonlocal nonlinearity of NLCs and the spatial-frequency composition of images. In order to solve the problem, the bulk NLCs are theoretically selected as the functional material to implement the SR-based image reconstruction in this paper [13,17,18]. Besides, the magnetic field rather than the electric field is adopted to control the initial alignment of NLC molecules.

In this paper, we theoretically demonstrate the SR-based image reconstruction via magneto-optical molecular reorientation in bulk NLCs. Selecting bulk NLCs as the functional material avoids the dimensional restriction. The gain of intensity perturbation and the numerical results are presented to analyze the nonlinear image reconstruction process. The underlying signals are reinforced by coupling with scattering noise under reorientation-induced self-focusing nonlinearity. The diffusive images are effectively recovered and the cross-correlation gain of 1.37 is obtained by reasonably optimizing the input light intensity, the magnetic filed direction, and the correlation length.

## 2. Model

The theoretical scheme of the nonlinear image reconstruction via magneto-optical molecular reorientation is shown in Fig. 1. The continuous uniform extraordinary-polarized light with the wavelength of 532nm passes through a resolution chart to generate a coherent binary image. The coherent image is scattered by a rotating diffuser, and then is imaged onto bulk NLCs. The diffuser rotates fast enough to ensure the response of NLCs to the time-averaged light intensity. As described in Refs. [13, 17 and 18], the used bulk NLCs are a cell and filled with the NLC mixture 6CHBT. The size of the NLC cell is 10mm×10mm and its thickness is 1mm along the beam propagation direction *z*. The applied uniform magnetic filed (*B*_{0}=0.2T) is used to preset the initial alignment of NLC molecules. The magnetic field is assumed to be sufficiently strong to reorient the molecular axes along the magnetic vector *B* in the absence of external optical stimuli. When the optical signals are injected into NLCs, the light electric field exerts an optical torque on the NLC molecules and drives their directors to align along the electric field vector. The local director reorientation will lead to the change of the spatial refractive index of NLCs and the occurrence of the intensity-dependent self-focusing nonlinearity, which allows the signals to be reinforced at the expense of scattering noise.

The enhancement of the noise-hidden signals is treated as the growth of small amplitude and phase perturbation from noise under reorientation-induced self-focusing nonlinearity. Here, the intensity perturbation gain of noise-hidden signals is derived to analyze the SR-based image reconstruction process. The propagation of the light beam in NLCs is described by Foch-Leontovich equation [18,19]:

*A*represents the light field, the light intensity

*I*=|

*A*|

^{2}

_{,}

*z*is the propagation direction,

*k*

_{0}=2

*π*/

*λ*is the wave number for the light of wavelength

*λ*, and

*n*(

*θ*,

*T*)=(cos

^{2}

*θ*/

*n*

_{o}

^{2}(

*T*)+sin

^{2}

*θ*/

*n*

_{e}^{2}(

*T*))

^{−1/2}is the refractive index for the extraordinary polarized light.

*n*(

_{o}*T*) and

*n*(

_{e}*T*) are ordinary and extraordinary refractive indices at the temperature of

*T*, respectively.

*θ*

_{0}=

*θ*-

_{m}*δ*, where

*θ*is the average tilting angle of NLC molecules with respect to the propagation direction

_{m}*z*and

*δ*is the walk-off angle.

*θ*=

*θ*

_{0}+

*ψ*, where

*ψ*represents the small tilting angles of NLC molecules under optical stimuli. Since

*ψ*<<

*θ*,

_{m}*δ*is practically determined by

*θ*[13]:

_{m}The light-induced reorientation angle *ψ* is governed by the following relation [20–23]:

*K*is an effective elastic constant, Δ

*ε*(Δ

*χ*) is dielectric (diamagnetic) anisotropy of NLCs, and

*B*

_{0}is the magnetic field strength. The temperature distribution in NLCs is described by [24,25]

*κ*is thermal conductivity,

*c*is the speed of light, and

*α*is the absorption coefficient.

The light scattered by a rotating diffuser is spatially incoherent. For describing the spatially incoherent properties, the Winger transform method is adopted [26–29]. Based on this transform, the radiation transfer equation is obtained by

*G*is the nonlinear response of NLCs to the input light field,

**(**

*r**r*,

_{x}*r*) and

_{y}**(**

*k**k*,

_{x}*k*) are the spatial position and momentum vectors, respectively. Reducing the system to one transverse dimension and linearizing Eq. (5) by writing

_{y}*f*=

*f*

_{0}+

*f*

_{1}exp[

*i*(

*αx*-

*gz*)] gives the dispersion relation:

*f*

_{0}is the base distribution,

*f*

_{1}is the perturbation distribution,

*g*=

*g*+i

_{R}*g*is the propagation constant for a perturbed mode with the wave number

_{I}*α*, and

*β*=1/[

*k*

_{0}

*n*(

*θ*

_{0},

*T*

_{0})]. By considering the long-wavelength perturbation with

*α*<<

*g*/(

*βk*), Eq. (6) is expanded as

_{x}By assuming that the growth rate |*g _{I}*|<<|

*g*|, the principal value and pole in Eq. (8) is given by

_{R}*g*≈

_{P}*g*is a mean propagation constant. The angular spectrum of the diffusive light is considered as a Gaussian distribution

_{R}*f*

_{0}(

*k*)=(2

_{x}*π*Δ

*k*

^{2})

^{−1/2}exp(-

*k*

_{x}^{2}/2Δ

*k*

^{2}), where Δ

*k*=2

*π*/

*l*represents the spectral spread for a beam with correlation length

_{c}*l*. The effective gain of intensity perturbation is given by

_{c}Obviously, the instability gain is determined by the input light intensity, the magnetic field direction, and the correlation length for a fixed perturbed mode.

Figure 2 shows the gain variation of intensity perturbation versus the input light intensity and the magnetic field direction under the reorientation-induced self-focusing nonlinearity, respectively. In simulation, we used the material parameters of 6CHBT: *K*=3.6×10^{−12}N, *B*=0.2T, *α*=5.769m^{−1}, *κ*=0.135W/(m°C), *T*_{0}=22°C [25,30]. As shown in Fig. 2(a), the gain curve versus the input light intensity first increases, and then decreases. The light intensity determines the strength of reorientation-induced self-focusing nonlinearity. The underlying signals are enhanced by coupling with scattering noise and the instability gain gradually increases under weak self-focusing nonlinearity. Incoherent modulation instability occurs and the enhancement process of signals is destroyed under strong self-focusing nonlinearity. The instability gain reaches its maximum value at the light intensity of 1.1×10^{4}W/cm^{2}. As shown in Fig. 2(b), the gain curve versus the magnetic field angle first increases, and then decreases. This is because the NLCs have the maximum reorientation response to the light field at the angle of about 50°. In simulation, the NLCs exhibit negligible thermal nonlinear effects at the used input light intensity, which indicates that the thermal effects have no influence on the nonlinear image reconstruction [17,25].

## 3. Results

In this section the numerical results are presented to study the SR-based image reconstruction under magnetic-optical molecular reorientation effects. The quasi-particle motion equations are derived from Eqs. (5)–(6) [12,31]:

*G*represents the reorientation-induced nonlinear response of NLCs to the input light field,

**represents the gradient-driven force,**

*F**m*=

*λ*/(2π

*n*(

*θ*

_{0},

*T*

_{0})) is the mass of quasi-particles,

**and**

*r***represent the position and momentum vectors of quasi-particles perpendicular to the propagation direction z, respectively. In simulation, Eqs. (3) and (4) were solved using the finite difference method. Equations (12)–(15) were solved using the quasi-particle model based on the particle-in-cell method [31].**

*k*We first investigate the angle distributions of the NLC molecules and the intensity distributions of the recovered images with different beam propagation lengths in bulk NLCs. When the extraordinary-polarized signals are injected into the bulk NLCs, the light field exerts an optical torque and aligns the director of the molecules along its own direction. The positive refractive index gradient is created under molecular reorientation effects in bulk NLCs. In turn, the gradient potential allows the signals to be reinforced by coupling with scattering noise. This is a positive feedback process. The underlying signals and the induced potential are continuously reinforced in the process of nonlinear beam propagation. As shown in Figs. 3(a)–3(f), the gradient potential gradually becomes steep and clear. As shown in Figs. 3(g)–3(l), the underlying signals are gradually enhanced at the expense of scattering noise. It is seen that the strips are clearly visible after the nonlinear image reconstruction, which indicates that the SR-based method is able to process the noisy image. In addition, the walk-off causes the image to shift downward. The offset of the image position is 95µm at the walk-angle of 5.4° in Fig. 3.

Then we presents the recovered images with different input light intensities under magnetic-optical molecular reorientation effects. The input light intensities impact the strength of the reorientation-induced self-focusing nonlinearity [17,26]. As shown in Figs. 4(a) and 4(b), the optical field is too weak to tilt the director of NLC molecules and induce the self-focusing nonlinearity. The strips are blurred after linear and weakly nonlinear propagation. As shown in Figs. 4(c) and 4(d), the enhancement effect of the signals gradually appears with the increase of the input light intensity in the condition of weak self-focusing nonlinearity. The underlying signals are obviously reinforced by coupling with scattering noise at the light intensity of 11000W/cm^{2} in Fig. 4(d). The corresponding power of the scattered light is 4W. It is seen that the strips are clearly seen and the scattering noise has no obvious intensity modulation. As shown in Figs. 4(e) and 4(f), the images look sharp in the condition of strong self-focusing nonlinearity. The reason is that incoherent modulation instability occurs and the scattering noise breaks into random-distributed light spots, which destroys the quality of the recovered images [26]. The better recovered image is obtained in Fig. 4(d). To quantitatively evaluate the image reconstruction effect, the cross-correlation coefficient between the reconstructed image and the pure image is calculated [31]. The value closer to 1 means that the image quality is better. As shown in Fig. 5, the evaluation curve first increases, and then decreases. The peak value is obtained at the light intensity of 11000W/cm^{2}. The cross-correlation coefficient is improved from 0.3 to 0.41 after the SR-based image reconstruction. The image quality gradually increases with the increase of the light intensity in the weak nonlinear region. The image quality gradually decreases due to the occurrence of incoherent modulation instability with the increase of the light intensity in the strong nonlinear region. It is necessary to reasonably adjust the input light intensity to obtain better recovered results.

Next Fig. 6 shows the recovered images with different magnetic field directions under magnetic-optical molecular reorientation effects. The initial alignment of NLC molecules is determined by the magnetic field direction. As shown in Fig. 6(b), the recovered image has the highest visibility at the tilting angle of 50°. The reason is that the nonlinear response of NLCs to the input light field is optimal and we maximize the image recovery effect at this angle. The self-focusing nonlinearity is weaker and the image quality is worse when the tilting angle is smaller or bigger than 50° at the fixed light intensity. The cross-correlation coefficient of the recovered images are 0.37, 0.41 and 0.38 in Fig. 6, respectively. Note that the walk-off angles are different in different initial alignment. The corresponding walk-off angles are 4.5°, 5.4° and 3.8°, respectively. The offset of the image positions is 79, 95 and 66µm, respectively.

Figure 7 shows the recovered images with different correlation lengths under magnetic-optical molecular reorientation effects. Figures 7(a)–7(c) are the diffusive images with the correlation lengths of 120, 115 and 110µm, respectively. Figures 7(d)–7(f) are the corresponding recovered images. The correlation length represents the divergence degree of scattered signals. The shorter correlation length means the larger divergence degree. It is seen that the quality of the scattered images decreases with the decrease of the correlation length. The quality of the corresponding recovered images also gradually decreases. This is because the less ballistic signals are remained and the weaker signals excite weaker seeded modulation instability under strong scattering conditions. The cross-correlation coefficient is improved from 0.3 to 0.41, from 0.25 to 0.34, and from 0.2 to 0.28 at the correlation lengths of 120, 115 and 110µm, respectively. Overall, the input light signals trigger the intensity-dependent self-focusing nonlinearity under magnetic-optical molecular reorientation effects in bulk NLCs. The underlying signals are effectively reinforced and recovered at the expense of scattering noise under reorientation-induced self-focusing nonlinearity. The dimension restrain is avoided by selecting the bulk NLCs as the nonlinear medium. The quality of the diffusive images is effectively improved by reasonably controlling the input light intensity, the magnetic field direction, and the correlation length.

It is seen that the magnetic field rather than the electric field is adopted to control the initial alignment of NLC molecules for maximizing the nonlinear response and making the light beam propagate perpendicularly through the NLC cell. In general, the electric field is applied using the ITO (indium tin oxide) transparent electrodes and its direction is perpendicular to the NLC cell. Thus, the electric field cannot be used to implement the pre-tilting of NLC molecules with respect to the beam propagation direction in our selected NLC configuration. Whereas combining the magnetic field and the electric field may help accelerate the molecular relaxation process based on the dual field effect [32,33]. The required magnetic field strength is 0.2T in the nonlinear process and it can be availably generated by a permanent neodymium magnet [13]. Note that the intense light is required to implement the light-induced molecule reorientation. Doping dye in NLCs is an alternative method to weaken the high power condition and improve the sensitivity of NLCs to the input light field in future work. This is because the intermolecular torque on NLC molecule exerted by dye molecules is stronger that the optical torque [34]. Although some problems exist in the proposed technology, providing a method of implementing SR and exploring the application of NLCs in image recovery is meaningful.

## 4. Conclusion

In summary, a nonlinear image reconstruction technology is proposed based on magnetic-optical molecular reorientation effects in bulk NLCs. The underlying signals are reinforced and recovered by coupling with scattering noise under reorientation-induced self-focusing nonlinearity. The gain of intensity perturbation is derived and the numerical results are presented to reveal the nonlinear response of NLC molecules to the input light signals. Our model clearly predicts the influence of the input light intensity, the magnetic filed direction, and the correlation length on the nonlinear image recovery. The work suggests a potential method to recover the noisy images and promotes the application of NLCs in the area of image processing.

## Funding

National Natural Science Foundation of China (61775234, 61975232).

## 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.

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