Reconstruction of weak near-infrared images in methyl red-doped nematic liquid crystals via stochastic resonance

Wentong Ji; Wentong Ji; Zhaolu Wang; Nan Huang; Hongjun Liu; Hongjun Liu

doi:10.1364/OE.462740

1. Introduction

When a image is scattered by smoke, turbid water, or atmospheric turbulence before reaching the detector, the image quality will be degraded due to the existence of multiplicative or additive noise. For general weak signals, the computer can directly restore and optimize the image through digital image processing technology. When the signal is scattered by strong noise so that the information directly captured by the detector cannot be effectively processed, the intervention of physical methods becomes a necessary means. Aiming at the problem of optical imaging under the influence of strong noise, image reconstruction techniques such as polarization imaging, range gating, wavefront shaping and speckle correlation have been proposed and developed continuously [1–4]. The above technologies consider noise to be harmful. In some nonlinear systems, which are called stochastic resonance, the role of noise is complex, a certain level of noise can enhance the signal. In 2010, Dylov et al. proposed a method for image reconstruction using nonlinear optics, which showed that optical images annihilated by the additive noise can be reconstructed through a photorefractive crystal without any post-processing [5]. This method took use of the nonlinear coupling between coherent signal and incoherent noise based on modulation instability (MI) [6].

NIR light has been widely used in optical imaging, such as biomedical imaging and night vision, due to its excellent penetration, safety to human eyes, and location in the atmospheric window. In previous studies on image reconstruction via stochastic resonance, researchers often used photorefractive crystals [5], such as Sr_0.6Ba_0.4Nb₂O₂(SBN). SBN crystals require a strong DC electric field and are limited by the spectral response range. The highest photorefractive sensitivity is limited to the visible light band, the photorefractive sensitivity in NIR band is low, and the MI gain in the NIR band is small [7]. NLCs are widely used to study physical phenomena such as optical spatial solitons, MI, and beam bending [8–11]. Due to the unique structure and molecular arrangement, NLCs are good nonlinear chromophores, showing great optical nonlinearity. In addition, because of the Freédericks effect of NLC molecules, the external field (electric or magnetic field) required for nonlinear effects is low, and the response time is fast. The absorption of NLC molecules in the NIR band is weak, and the nonlinear effect can be obtained in a wide range [12]. Therefore, it is possible to reconstruct NIR weak images in NLCs. In 2019, Feng. et al. theoretically demonstrated the nonlinear image reconstruction based on the reorientation response of NLCs molecules. In the additive noise system, they applied 3.6 V voltage across the flat LC. When the input signal intensity was $1.4 \times {10^4}W/\textrm{c}{\textrm{m}^2}$, the noise-to-signal intensity ratio was 30:1, and the spatial coherence length was $170\mu m$, the hidden image information can be extracted effectively. Whereas, the boundary conditions of the flat LC have a great influence on the imaging effect, the narrow light passage surface leads to unsatisfactory reconstruction [13]. In 2021, Zhang. et al. proposed to use bulk NLCs instead of flat NLCs, and the magnetic field was used to replace the electric field. In the multiplicative noise system, when the input signal intensity was $1.1 \times {10^4}W/\textrm{c}{\textrm{m}^2}$, the spatial coherence length was $120\mu m$, and the pretilt angle of the magnetic field was $50^\circ $, the image reconstruction can be realized based on magnetic-optical molecular reorientation effects. However, this work also has the thermal effect problem caused by the strong incident light [14]. In this paper, MR doped-NLCs are selected in order to reduce the light intensity threshold required for reorientation and realize the NIR image reconstruction via stochastic resonance in magnetic field.

The passage of the laser beam through the transparent NLCs causes reorientation of the nematic director. Due to their anisotropic optical polarizability, reorientation effect can be explained by the optical torque acting on NLC molecules [15]. A small amount (less than 1%) of MR dye doped in NLCs can result in a strong enhancement of nonlinear optical response [16] and 2 orders of magnitude enhancement of reorientation [17]. The explanation proposed by Janossy is associated with the metastable state of electron excitation in dye molecules. The interaction of excited-state dye molecules with NLC molecules is different from that of ground-state dye molecules [17]. One of the side effects of dye doping is the enhancement of absorption on the green part to the blue part of the spectrum, leading to other nonlinear effects such as thermal effects, density changes and photo-induced phase transition. It is necessary to use a laser with wavelengths that lie outside the peak dye absorption [18], such as a NIR laser with the wavelength of 1064 nm. By reasonably optimizing magnetic field direction, dye doping concentration, coherence length and magnetic field intensity, the scattered image is effectively recovered when the input signal intensity was $0.9W/\textrm{c}{\textrm{m}^2}$, which is 3-4 orders of magnitude lower than the incident light intensity required in undoped NLCs [13,14], the cross-correlation coefficient between the reconstructed image and the original image is increased from 0.26 to 0.54, and the maximum cross-correlation gain is 2.25.

2. Method

We used a sample of 6CHBT NLC doped with MR as a nonlinear medium. The nonlinear propagation of light in NLCs requires initial preorientation of the molecular director [19]. Since 6CHBT exhibits positive diamagnetic anisotropy, its molecules follow the direction of the magnetic field. It can not only avoid the space and boundary restrictions of flat NLCs, but also can realize the reorientation response of NLCs molecules with a low light intensity threshold [20]. The signal is reinforced at the expense of the noise via stochastic resonance, due to the change of the spatial refractive index in NLC [21].

MR doped-NLCs are sensitive to e-polarized light. Assuming that E is the optical field envelope and the beam propagates along the z-axis. ${B_0}$ is the strength of the external magnetic field, the direction of the magnetic field is limited in the y-z plane. The alignment direction of LC molecules is the same as that of the magnetic field without the beam incidence [22]. NLC molecules pre-deflection angle and magnetic field direction are both ${\theta _m}$ (angle with the z-axis). $\psi $ is the small deflection angle of NLC molecules under optical stimuli. We can use the coupled equation to describe the propagation form of E in the inhomogeneous light-induced refractive index distribution, which is [23]:

(1)$$2i{k_0}n({\theta _0},{T_0})(\frac{{\partial E}}{{\partial z}} + \frac{{\partial E}}{{\partial y}}) + \frac{{{\partial ^2}E}}{{\partial {x^2}}} + {D_y}\frac{{{\partial ^2}E}}{{\partial {y^2}}} + k_0^2({n^2}(\theta ,T) - {n^2}({\theta _0},{T_0}))E = 0, $$

where ${k_0} = 2\pi /{\lambda _0}$ is the wave number. Neglecting walk-off effect, ${\theta _0} = {\theta _m}$ is the pre-deflection angle of the NLC molecule, $\theta = {\theta _0} + \psi $. ${D_y}$ is the diffraction coefficient across y, $n({\theta ,T} )= {({cos ^2}\theta /n_o^2(T )+ {sin ^2}\theta /n_e^2(T ))^{ - 1/2}}$ is the effective refractive index [24]. ${n_o}(T )$ and ${n_e}(T )$ are the ordinary and extraordinary refractive indices at the temperature of T, respectively. T₀ is the initial temperature, and we use T₀= 22°C during the simulation.

According to the theory of liquid crystal elastomers, the free energy of NLCs can be expressed as [25,26]:

(2)$$F = \frac{1}{2}{K_{11}}{(\nabla \cdot \hat{n})^2} + \frac{1}{2}{K_{22}}{(\hat{n} \cdot \nabla \times \hat{n})^2} + \frac{1}{2}{K_{33}}{(\hat{n} \times \nabla \times \hat{n})^2} + {F_{field}}. $$

The first three parts of Eq. (2) represent the spreading, twisting and bending free energy density, and the fourth part represents the field-matter interaction. ${K_{11}},{K_{22}}$ and ${K_{33}}$ are the elastic constants under spreading, twisting, and bending deformations, respectively. Therefore, the total free energy of the NLC can be written as [25,27]:

(3)$$\begin{array}{l} F = {F_d} + {F_{field}}\\ = \frac{1}{2}{K_{11}}{(\cos \theta \frac{{\partial \theta }}{{\partial x}} - \sin \theta \frac{{\partial \theta }}{{\partial z}})^2} + \frac{1}{2}{K_{22}}{(\frac{{\partial \theta }}{{\partial y}})^2} + \frac{1}{2}{K_{33}}{(\cos \theta \frac{{\partial \theta }}{{\partial z}} + \sin \theta \frac{{\partial \theta }}{{\partial x}})^2}\\ - \frac{1}{2}\Delta \varepsilon {\varepsilon _0}{\sin ^2}\theta - \frac{A}{{2{\mu _0}}}B_0^2\Delta \chi ^{\prime}{\sin ^2}\theta - \frac{1}{{2{\mu _0}}}B_0^2\Delta \chi {\sin ^2}(\theta - {\theta _0}) \end{array}, $$

where $\mathrm{\Delta }\chi $ and $\mathrm{\Delta }\chi ^{\prime}$ are the diamagnetic susceptibility anisotropy of NLC molecules and MR molecules, respectively. A is the concentration of MR molecules. ${B_0}$ is the magnitude of the magnetic field, and $\mathrm{\Delta }\varepsilon $ is the dielectric anisotropy of NLC.

The Euler Lagrange equation with the minimum free energy density satisfied by the formula is [26]:

(4)$$\frac{{\partial F(\theta )}}{{\partial \theta }} - \frac{d}{{d{x_i}}}(\frac{{\partial F(\theta )}}{{\partial (\frac{{d\theta }}{{d{x_i}}})}}) = 0. $$

When ${\; }{K_{11}} = {K_{22}} = {K_{33}} = K$, substituting Eq. (3) into Eq. (4) can obtain the relational expression satisfied by the deflection angle of NLC molecules:

(5)$$\frac{{{\partial ^2}\theta }}{{\partial {x^2}}} + \frac{{{\partial ^2}\theta }}{{\partial {y^2}}} + \frac{{\Delta \varepsilon \cdot {\varepsilon _0}}}{{2K}}{|E |^2}\sin (2\theta ) + \frac{{A\Delta \chi ^{\prime}B_0^2}}{{2K{\mu _0}}}\sin (2\theta ) + \frac{{\Delta \chi B_0^2}}{{2K{\mu _0}}}\sin 2(\theta - {\theta _0}) = 0. $$

The absorption is weak in NIR wavelengths, but it also leads to heating of the medium. Its temperature equation satisfies the Poisson distribution [28,29]:

(6)$${\nabla ^2}T ={-} \frac{{c{\varepsilon _0}\alpha }}{{2\kappa }}{|E |^2}, $$

where α is the absorption coefficient, κ is the thermal conductivity, and c is the speed of light. It is solved by the fourth order Runge-Kutta algorithm and multigrid finite-difference method.

The Foch-Leontovich equation is reduced to one dimension and used to represent incoherent beams based on the Wigner transform [29–31]:

(7)$$f({\boldsymbol r},{\boldsymbol k},z) = {(2\pi )^{ - 3}}\int_{ - \infty }^{ + \infty } {{d^3}} {\boldsymbol \xi } \cdot \textrm{exp} (i{\boldsymbol k\xi })\left\langle {{E^\ast }({\boldsymbol r} + \frac{{\boldsymbol \xi }}{2})E({\boldsymbol r} - \frac{{\boldsymbol \xi }}{2})} \right\rangle , $$

where ${\; }{E^\mathrm{\ast }}({r,z} )E({r,z} )= \mathop \smallint \nolimits_{ - \infty }^{ + \infty } {d^3}kf({r,k,z} )$.

The Wigner-Moyal equation can be obtained as:

(8)$$\frac{{\partial f}}{{\partial z}} + \frac{{{k_x}}}{{{k_0}n({\theta _0},{T_0})}}\frac{{\partial f}}{{\partial x}} + \frac{{{k_0}}}{{n({\theta _0},{T_0})}}({n^2}(\theta ,T) - {n^2}({\theta _0},{T_0}))\sin (\frac{1}{2}\frac{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftarrow$}} \over \partial } }}{{\partial x}}\frac{{\vec{\partial }}}{{\partial {k_x}}})f = 0. $$

The Vlasov-like radiative transfer equation is obtained:

(9)$$\frac{{\partial f}}{{\partial z}} + \frac{{{k_x}}}{{{k_0}n({\theta _0},{T_0})}}\frac{{\partial f}}{{\partial x}} + \frac{{k_0^2\beta }}{2}\frac{{\partial \Delta n}}{{\partial x}}\frac{{\partial f}}{{\partial {k_x}}} = 0, $$

for $\varDelta n = {n^2}({\theta ,T} )- {n^2}({{\theta_0},{T_0}} )$, $\beta = 1/{k_0}n({{\theta_0},{T_0}} )$.

Incoherent MI can be studied with standard perturbation theory. Considering the perturbation around the spatially uniform distribution ${f_0}(k )$, the solution of the equation can be written in the form of the following uniform background and perturbation superposition:

(10)$$f(x,{k_x},z) = {f_0}({k_x}) + {f_1}\textrm{exp} [i(\alpha x - gz)], $$

${f_0}$ represents the uniform and balanced distribution of the background light, while $\; {f_1}$ represents the small disturbance caused by the light wave, and ${f_0} \gg {f_1}$. $g\; $ represents the incoherent MI gain.

Substitute Eq. (10) into Eq. (9) to get the dispersion equation:

(11)$$1 + \frac{\kappa }{\beta }\int\nolimits_{ - \infty }^{ + \infty } {\frac{{{f_0}({k_x} + \frac{\alpha }{2}) - {f_0}({k_x} - \frac{\alpha }{2})}}{{\alpha ({k_x} - \frac{g}{{\alpha \beta }})}}} d{k_x} = 0, $$

where

\kappa = \frac{{{k_0}}}{{2n({\theta _0},{T_0})}}({(\frac{{{{\cos }^2}{\theta _0}}}{{n_o^2(T)}} + \frac{{{{\sin }^2}{\theta _0}}}{{n_e^2(T)}})^{ - 2}}(\frac{{\sin 2{\theta _0}}}{{n_o^2(T)}} - \frac{{\sin 2{\theta _0}}}{{n_e^2(T)}})\gamma + {n^2}({\theta _0},T) - {n^2}({\theta _0},{T_0})),

\gamma = \frac{{\frac{{\Delta \varepsilon \cdot {\varepsilon _0}}}{{2K}}{{|E |}^2}\sin (2{\theta _0}) + \frac{{A\Delta \chi ^{\prime}B_0^2}}{{2K{\mu _0}}}\sin (2{\theta _0})}}{{{\alpha ^2} - \frac{{\Delta \varepsilon \cdot {\varepsilon _0}}}{K}{{|E |}^2}\cos (2{\theta _0}) -{-} \frac{{\Delta \chi B_0^2}}{{K{\mu _0}}} - \frac{{A\Delta \chi ^{\prime}B_0^2}}{{K{\mu _0}}}\cos (2{\theta _0})}}.

For the angular spectrum of the scattered image with Gaussian distribution:

(12)$${f_0}({k_x}) = \frac{1}{{\sqrt {2\pi \Delta {k^2}} }}\textrm{exp} ( - \frac{{k_x^2}}{{2\Delta {k^2}}}), $$

where $\varDelta k = 2\pi /{l_c}$ is the spectral spread of the beam with coherence length ${l_c}$. The effective gain of intensity perturbation is given by:

(13)$$g = {(\kappa {\alpha ^2}\beta )^{ - 1/2}}(1 + i\frac{\pi }{2}\frac{\kappa }{\beta }\frac{{\partial {f_o}}}{{\partial {k_x}}}). $$

Substituting Eq. (12) into Eq. (13), we get:

(14)$$\begin{array}{l} {g_{eff}} = {\left. {\frac{\pi }{2}{{(\kappa {\alpha^2}\beta )}^{ - 1/2}}\frac{\kappa }{\beta }\frac{{\partial {f_o}}}{{\partial {k_x}}}} \right|_{{k_x} = k_\varphi ^{}}}\\ \begin{array}{*{20}{c}} {}&{}& = \end{array}\textrm{exp}( - 3/2)\sqrt {\frac{\pi }{8}} \frac{{{\kappa ^2}}}{\beta }\frac{\alpha }{{\Delta {\textrm{k}^\textrm{3}}}}\textrm{exp}( - \frac{{{\kappa ^2}}}{{2\Delta {\textrm{k}^\textrm{2}}}}) \end{array}. $$

According to Eq. (14), we can obtain the relationship between MI gain coefficient and different variables. Figure 1 shows the gain of MI with different input light intensities. The curves in Fig. 1(a) and 1(b) are the gains obtained in MR doped-NLC and undoped NLC, respectively. As shown in Fig. 1(a), the gain tends to increase and then decrease with the increase of incident light intensity. When the input light intensity is small, the self-focusing nonlinearity is weak, part of the energy of the noise is coupled to the signal. The signal is reinforced via stochastic resonance. The MI gain achieves its maximum value when the light intensity is $0.9W/c{m^2}$, and decreases with the continued increase of the light intensity because of incoherent MI originated from the strong self-focusing nonlinearity. The peak of the curve in Fig. 1(b) is obtained at $I = 9000W/c{m^2}$. The reason is that the nonlinear effect of the medium is greatly enhanced by doping, and the presence of MR molecules reduces the light intensity required for the reorientation response by about 3-4 orders of magnitude.

Fig. 1. MI gain coefficient with different input light intensities. (a) MI gain in MR doped-NLC. (b) MI gain in undoped NLC. Other parameters are fixed at ${\theta _0} = {50^\circ },{l_c} = 90\mu m,A = 0.8{\%}wt,{B_0} = 0.12T.$

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Then we present MI gain coefficient for different values of variables in Fig. 2. The curve in Fig. 2(a) demonstrates the gain at different perturbation modes. When the light passes through the MR doped-NLC, the low frequencies are hardly amplified. High frequencies are selectively amplified as the MI gain rate curve changes. At a specific light intensity, the optimal amplification frequency of the MI is optimally matched to the mode of the signal frequency component, at which time the system is in resonance and the gain reaches its peak value. The curve in Fig. 2(b) shows the gain with different magnetic field directions. As the angle between the magnetic field direction and the z-axis increases, the gain first increases and then decreases, and the maximum gain is obtained when the magnetic field direction is 50°. As shown in Fig. 2(c), the gain increases and then decreases when the concentration of MR increases in NLC. When the concentration is too large, the absorption of light by the dye molecules causes a great thermal effect, leading to a decrease of the gain. The optimal concentration is 0.8%wt. The liquid crystal used above is MR doped-6CHBT with background temperature of 22°.

Fig. 2. MI gain coefficient for different values of variables. (a) MI gain at different perturbation modes. (b) MI gain with different magnetic field directions. (c) MI gain with different dye doping concentrations. (f) MI gain with different magnetic field intensities. Other parameters are fixed at (a)$I = 0.9W/c{m^2},\; {\theta _0} = {50^\circ },{l_c} = 90\mu m,A = 0.8{\; \%}wt,{B_0} = 0.12T.$ (b) $I = 0.9W/c{m^2},\; {l_c} = 90{\; }\mu m,A = 0.8{\%}wt,{B_0} = 0.12T.$ (c) $I = 0.9W/c{m^2},\; {\theta _0} = {50^\circ },{l_c} = 90\mu m,{B_0} = 0.12T.$

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3. Results and analysis

In practical applications, multiplicative noise is more common than additive noise [32]. The schematic diagram of the weak optical image reconstruction via stochastic resonance is shown in Fig. 3(a). The rotating diffuser scatters the 1064 nm laser after passing the resolution chart so that the signal image is completely overwhelmed by noise. With the rotating diffuser, the beam is expanded, then the beam passes through the imaging lens. We control the spatial coherence length of the signal beam by adjusting the focal length of the lens. After that, it enters the MR doped-NLC ($20mm \times 20mm \times 1mm$). The pre-deflection angle of the NLC molecules is the same as the direction ${\theta _m}$ of the applied magnetic field. The applied magnetic field intensity is ${B_0}$. The output image is captured by a charge-coupled device (CCD) camera. Figure 3(b) and 3(c) show the original image and the scattering images of the simulation. The image characteristic length is about 90 $\mu m$.

Fig. 3. (a) Schematic diagram. The beam passing through the RC is scattered by the RD, and then enters into the CCD for imaging after MR doped-NLC. The pre-deflection angle of LC molecules is the same as the direction of the applied magnetic field. RC: Resolution Chart, RD: Rotating Diffuser. (b) The simulated unscattered original image. (c) The simulated noise annihilation image. Yellow scale bar: 2 $0\; um$.

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In order to quantitatively measure the reconstruction effect of the output image, the cross-correlation coefficient is used to evaluate the similarity between the reconstructed image and the original image. The higher the similarity between the reconstructed image and the original image, the closer the cross-correlation coefficient is to 1, which is defined as [33]:

(15)$${C_{I,{I_0}}} = \frac{{\left\langle {\left( {I - \left\langle I \right\rangle } \right)\left( {{I_0} - \left\langle {{I_0}} \right\rangle } \right)} \right\rangle }}{{{{\left[ {\left\langle {\left. {{{\left( {I - \left\langle I \right\rangle } \right)}^2}} \right\rangle \left\langle {{{\left( {{I_0} - \left\langle {{I_0}} \right\rangle } \right)}^2}} \right.} \right\rangle } \right]}^{1/2}}}}$$

where I denotes the reconstructed image and I₀ denotes the original image. Then the cross-correlation gain can be defined as ${C_g} = {C_{{I_{\textrm{out}}},{I_0}}}/{C_{{I_{\textrm{in}}},{I_0}}}$.

Our simulations are based on the particle model [33]. By treating light as a particle beam composed of a large number of optical quasi-particles, tracking the propagation trajectory of the optical quasi-particle under nonlinear refractive index modulation. By counting the intensity distribution and angular spectrum distribution of the optical quasi-particle, the reconstruction of weak near-infrared images in MR-doped NLCs via stochastic resonance can be clearly demonstrated. To describe the motion of particles, we use the motion equations [33]:

(16)$${\textbf F} = \frac{{\partial \kappa }}{{\partial {\textbf r}}}, $$

(17)$$d{\textbf k} = \frac{{\textbf F}}{m} \cdot \frac{{dz}}{{{k_0}}}, $$

(18)$$d{\textbf r} = {\textbf k} \cdot \frac{{dz}}{{{k_0}}}, $$

(19)$$\begin{array}{l} \kappa = \frac{{{k_0}}}{{2n({\theta _0},{T_0})}}({(\frac{{{{\cos }^2}{\theta _0}}}{{n_o^2(T)}} + \frac{{{{\sin }^2}{\theta _0}}}{{n_e^2(T)}})^{ - 2}}(\frac{{\sin 2{\theta _0}}}{{n_o^2(T)}} - \frac{{\sin 2{\theta _0}}}{{n_e^2(T)}})\gamma + {n^2}({\theta _0},T) - {n^2}({\theta _0},{T_0})),\\ \gamma = \frac{{\frac{{\Delta \varepsilon \cdot {\varepsilon _0}}}{{2K}}{{|E |}^2}\sin (2{\theta _0}) + \frac{{A\Delta \chi ^{\prime}B_0^2}}{{2K{\mu _0}}}\sin (2{\theta _0})}}{{{\alpha ^2} - \frac{{\Delta \varepsilon \cdot {\varepsilon _0}}}{K}{{|E |}^2}\cos (2{\theta _0}) -{-} \frac{{\Delta \chi B_0^2}}{{K{\mu _0}}} - \frac{{A\Delta \chi ^{\prime}B_0^2}}{{K{\mu _0}}}\cos (2{\theta _0})}} \end{array}$$

where ${\boldsymbol F}$ represents the self-consistent driving force, ${\boldsymbol r}$ and ${\boldsymbol k}$ represent the position and momentum vectors, respectively. $m = \lambda /({2\pi n({{\theta_0},{T_0}} )} )$ is the mass of each particle, and ${k_0}$ is the wave number, z is the propagation direction.

In Fig. 4, the functional relationship between the cross-correlation coefficient and different light intensities is given. The overall correlation numbers show a trend of increasing and then decreasing with the increase of light intensity. When the input light intensity is weak, the noise energy coupled to the signal is small, and the cross-correlation coefficient does not increase much. It reaches the maximum when the light intensity increases to $0.9W/c{m^2}$ gradually. When the light intensity continues to increase, the NLC molecules are over-deflected, and the cross-correlation coefficient gradually decreases, due to the generation of incoherent MI and the enhancement of the absorption of the MR on light, which affects the image reconstruction effect.

Fig. 4. Cross-correlation coefficient of the recovered images versus the input light intensity. Other parameters are fixed at ${\; }{\theta _0} = {50^\circ },{l_c} = 90\mu m,A = 0.8{\%}wt,{B_0} = 0.12T.$

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The simulation reconstruction images with different input light intensities are shown in Fig. 5. In MR doped-NLC, the impurities, vacancies, defects and a small amount of MR molecules inside the medium can be excited with free charges under the light irradiation, and the movement of free charges is related to the light intensity distribution. In this way, the spatial charge distribution corresponding to the spatial distribution of light intensity is formed in the medium. These charges will generate the spatial-charge field in the medium, and then the space charge field will cause the refractive index of the medium to change accordingly through the reorientation [16,17,34,35]. Therefore, the light intensity required to generate reorientation is much smaller than that of undoped NLC. When the input light intensity is $0.1W/c{m^2}$, it is not enough to induce the movement of space charges, so that the NLC molecules cannot be reorientated. At this time, the output image is completely hidden by noise, the cross-correlation coefficient is 0.26. When the light intensity increases slightly, as shown in Fig. 5(b) and 5(c), the noise energy is gradually transferred to the signal, at which time the NLC molecules are deflected to an angle, the refractive index gradient changes inside the LC, and the output image is reconstructed to some extent. The cross-correlation coefficients are 0.33 and 0.45, respectively. At the input light intensity of $0.9W/c{m^2}$, as shown in Fig. 5(d), the reconstructed image quality is the best, and the cross-correlation coefficient reaches 0.54, the cross-correlation gain is 2.25. The stochastic resonance occurs between the signal and the noise, and the energy of the noise coupled to the signal is up to the maximum, so that the output weak light image is effectively enhanced [36]. Further increasing the light intensity, as shown in Fig. 5(e) and 5(f), the reconstructed image quality is deterioration and the cross-correlation coefficients are 0.42 and 0.34, respectively. One of the reasons is that the strong self-focusing nonlinearity induces incoherent MI. Another reason is that the absorption of dye molecules on light interferes with the nonlinear coupling process of signal and noise, resulting in the output image becoming blurred and dark.

Fig. 5. Simulation results of the output images at 1064 nm with different values of the input light intensity. (a)-(f) The input light intensities are $I = 0.1W/c{m^2},{\; \; }0.5W/c{m^2},0.7W/c{m^2},0.9W/c{m^2},1.4W/c{m^2},1.6W/c{m^2}.$ Other parameters are fixed at ${\; }{\theta _0} = {50^\circ },{l_c} = 90\mu m,A = 0.8{\%}wt,{B_0} = 0.12T.$ Cross-correlation coefficients (a) 0.26, (b) 0.33, (c) 0.45,(d) 0.54, (e) 0.42 and (f) 0.34. Cross-correlation gains (a)1.08, (b)1.38, (c) 1.88, (d)2.25, (e) 1.75 and (f) 1.42.

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In Fig. 6, the simulation results of the reconstruction images with different dye doping concentrations are shown. For Fig. 6(a)–6(c), the reconstruction images are obtained in bulk samples of NLC doped with a too small concentration of MR. The space charges distribution inside the NLC does not change much and the reorientation of the molecules is weak. So that the refractive index gradient changes less, resulting in an unsatisfactory reconstruction effect. For too large doping concentration, the reconstruction images are shown in Fig. 6(e) and 6(f), the thermal effect caused by the enhanced absorption of light by the dye molecules leads to poor reconstruction quality and darker reconstructed image brightness. The reconstructed image quality is the best at the doping concentration of $A = 0.8{\%}wt$. The low-light image reconstruction can be better realized by choosing the appropriate doping concentration in NLCs.

Fig. 6. Simulation results of the output images at 1064 nm with different values of the doping concentrations. (a)-(f) Dye doping concentrations are $A = 0.2{\%}wt,{\; \; }0.4{\%}wt,{\; \; }0.7{\%}wt,{\; \; }0.8{\%}wt,\; \textrm{1}\%wt,{\; \; }2{\%}wt$. Other parameters are fixed at ${\theta _0} = {50^\circ },{l_c} = 90\mu m,I = 0.9W/c{m^2},{B_0} = 0.12T$. Cross-correlation coefficients (a) 0.24, (b) 0.25, (c) 0.50, (d) 0.54, (e) 0.49 and (f) 0.29. Cross-correlation gains (a)1, (b)1.04, (c) 2.08, (d)2.25, (e) 2.04 and (f) 1.21.

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The simulation results of the reconstruction images with different magnetic field intensities are shown in Fig. 7. MR doped-NLCs have greater diamagnetic anisotropy and will reorient the NLC molecules under weaker magnetic fields. The reconstructed image qualities improve with the increase of magnetic field intensities until ${B_0} = 0.12T$. The correlation coefficient is improved from 0.29 to 0.54. When the magnetic field intensity continues to increase, the non-local nonlinear effect is too strong and causes a slight decrease in the reconstruction. Then we show the simulation of the reconstruction images with different magnetic field directions in Fig. 8. Since 6CHBT has positive diamagnetic anisotropy, its molecules tend to align along the direction of the magnetic field. Therefore, the pre-deflection angle of the NLC molecules can be controlled by controlling the distribution of the angle of the magnetic field. When the magnetic field angle is ${\theta _\textrm{m}} = {40^\circ },{50^\circ },{60^\circ }$, the pre-deflection angle of the NLC molecules is the same as the magnetic field deflection angle, which is ${\theta _0} = {40^\circ },{50^\circ },{60^\circ }$. The best image reconstruction image quality is achieved when ${\theta _0} = {50^\circ }$. Therefore, the optimal nonlinear response of MR doped-NLC molecules can be achieved by reasonably setting the magnetic field intensity and direction.

Fig. 7. Simulation results of the output images at 1064 nm with different values of the magnetic field intensities. (a)-(f) Magnetic field intensities are ${B_0} = 0.04T,{\; \; }0.08T,{\; \; }0.1T,{\; \; }0.12T,{\; \; }0.16T,{\; \; }0.20T$ .Other parameters are fixed at ${\theta _0} = {50^\circ },{l_c} = 90\mu m,I = 0.9W/c{m^2},A = 0.8{\%}wt.$ Cross-correlation coefficients (a) 0.29, (b) 0.35, (c) 0.45, (d) 0.54, (e) 0.49 and (f) 0.42. Cross-correlation gains (a)1.21, (b)1.46, (c) 1.88, (d)2.25, (e) 2.04 and (f) 1.75.

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Fig. 8. Simulation results of the output images at 1064 nm with different magnetic field direction. (a)-(c) The magnetic field direction (angle with z-axis) is ${\theta _0} = {40^\circ },{50^\circ },{60^\circ }$, respectively. Other parameters are fixed at ${B_0} = 0.12T,{l_c} = 90\mu m,I = 0.9W/c{m^2},A = 0.8{\%}wt.$ Cross-correlation coefficients (a) 0.46, (b) 0.54 and (c) 0.47. Cross-correlation gains (a)1.92, (b)2.25 and (c) 1.96.

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Then we show the simulation results of the recovered images with different coherence lengths in Fig. 9. The coherence length is the numerical measurement of beam spatial coherence. For partially coherent light, the coherence length is a finite value between infinite (coherent light) and 0 (incoherent light). When the coherence length is less than the characteristic length of the signal image, the signal can be amplified with the same magnitude of the coherence length. Figures 9(a) and 9(d) are the original and reconstructed images when ${l_c} = 80\mu m$, respectively. The qualities of reconstructed images cannot achieve the best result. At this point, although the background noise is eliminated, the maximum capacity that stochastic resonance can handle is not yet reached. This kind of circumstance can be called “under-resonance”, the cross-correlation coefficient is improved from 0.11 to 0.41. When the coherence length matches the image characteristic length, ${l_c} = 90\mu m$, as shown in Fig. 9(b) and 9(e), the noise can be maximally transferred to the spatial position of the initial signal distribution, thus achieving signal enhancement and realizing the optimal state of stochastic resonance. The cross-correlation coefficient is improved from 0.26 to 0.54. As shown in Fig. 9(c) and 9(f), when the coherence length is $120\mu m$, larger than the characteristic length, only the part of the noise, which coherence length is the same as the image characteristic length, can be coupled to the signal via stochastic resonance. Although the signal is enhanced at the expense of noise, the excess noise still exists in the system, and this part of noise only plays the role of reducing the signal-to-noise ratio, that is, it exceeds the capacity of stochastic resonance system. This situation can be called “over-resonance" [37], and the cross-correlation coefficient is improved from 0.15 to 0.46.

Fig. 9. Simulation images at 1064 nm with different coherence lengths. (a)-(c) The scattering images with coherence lengths of ${l_c} = 80\mu m,{\; \; }90\mu m,{\; \; }120\mu m$ and (d)-(f) the recovered images with coherence lengths of ${l_c} = 80\mu m,{\; \; }90\mu m,{\; \; }120\mu m$. Other parameters are fixed at $\; {B_0} = 0.12T,\; {\theta _0} = {50^\circ },\; I = 0.9W/c{m^2},\; A = 0.8{\%}wt.$

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4. Conclusion

In summary, we propose a NIR image reconstruction method based on the molecules reorientation in MR doped-NLC, and deduce the intensity perturbation gain equation of MI. The noise energy is coupled to the signal energy, inducing the NIR weak optical image be reconstructed. The recovered image qualities are influenced by the parameters such as dye doping concentration, magnetic field intensity, magnetic field direction and coherence length. We find that the molecules reorientation response in MR doped-NLC can be optimally achieved when input light intensity is $\; 0.9W/c{m^2}$ . The light intensity required for image reconstruction in the MR doped-NLC is 3-4 orders of magnitude lower than that in the undoped NLC. The work suggests a potential method for NIR image reconstruction technologies and has important implications for the application of dye-doped NLCs in image processing.

Funding

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

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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Reconstruction of weak near-infrared images in methyl red-doped nematic liquid crystals via stochastic resonance

Abstract

1. Introduction

2. Method

3. Results and analysis

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Equations (21)

Optics Express