1. Introduction

Microscopic imaging is one of the important means to reveal the microstructure and reaction kinetics [1]. Owe to the applicability in quantitative analysis of molecular vibrational signatures, mid-infrared light are widely used in biomedical tissue and chemical imaging [2–4]. Traditional imaging methods by converting mid-infrared information to visible or near-infrared ranges inevitably lead to the loss of image features [5,6]. However, the direct imaging in mid-infrared band is also severely limited by the optical transfer function (OTF). According to Rayleigh criterion and linear Abbe theory, the maximum resolution obtained by classical imaging systems is δ = 0.61λ/NA, where λ is the imaging wavelength and NA is effective numerical aperture [7]. Therefore, in scalar diffraction, only the low-order wavenumbers can be detected. Minimizing the focusing spot can only improve the resolution to half of the imaging wavelength. However, the Richards-Wolf’s vector diffraction theory proves that, the diffraction limit can be broken by optimizing the distributions of wave polarization, phase, amplitude, and even modes interactions [8]. That means, a reasonable nonlinear reconstruction can effectively decouple the superimposed diffracted images and recover it clearly [9–11].

Since the principle of nonlinear structured-illumination microscopy is proposed, various optical effects, such as saturated excitation of fluorescent molecules, acoustic phonons scattering, optical Kerr effect and polymer photorefractive, et al, have been investigated to generate the suitable nonlinearity [12–16]. Although seemingly as the incremental versions of structured-illumination microscopy (SIM), their fundamentals are different. In SIM, the illumination frequency of structural grating is still limited by linear OTF, so the maximum resolution can only be doubled [17]. In contrast, the nonlinear resolutions are only affected by “soft” factors, such as the signal-to-noise ratio (SNR) and optical stability [18,19]. Of course, not all classical nonlinear effects can yield the expected improvement. For example, the two-photon absorption fluorescence, as a quadratic nonlinearity, has no obvious resolution enhancement due to the twice linear excitations of the same fluorophore [20]. That is, only the higher-order nonlinearity with serial harmonics can infinitely extend the observed spectrum.

In this paper, we demonstrate a far-field mid-infrared super-resolution imaging method that relies on the frequency shifting of evanescently decaying waves. The process is based on the orientational photorefractive grating established in nematic liquid crystal (NLC). Attributed by the rod-like structure and photoinduced reorientation of molecules, the photorefractive grating equivalently produces an expanded nonlinearity (mathematically expansion with ±1st, ± 2nd… Taylor series), leading to the theoretically infinite improvement of imaging resolution [21,22]. In addition, due to the Freedericksz Transition, the intensity required for molecular reorientation is much lower than general photorefractive materials [23,24]. The discussion of imaging process is divided into three sections: the establishment of photorefractive grating in NLC, the spatial shifting of evanescent waves, and the image reconstruction via phase decoupling. The results are in good agreement with theoretical simulation, which indicates the effectiveness of subwavelengths extraction.

2. Establishment of photorefractive grating in NLC

The designed imaging set-up is shown in Fig. 1. The nematic liquid crystal 5CB: 2%MR (4'-n-pentyl-4-cyanobiphenyl, doped with C₆₀) is used as the nonlinear medium. The long and short diameters of NLC molecules are 2.5 nm and 0.5 nm, respectively. Polarizabilities versus directions leads to the birefringence that benefits the regulation of refractive index. C₆₀ are excited as the photosensitizer to enhance the photoconductivity [25]. In this way, the medium has a strong nonlinearity and diffraction efficiency. As the measured absorption of 5CB is strongest around 500 nm and almost transparent around 2.94 µm, beams with the two wavelengths are selected to writing the grating and imaging, respectively [26]. NLC with a thickness d ≈ 20 µm is placed between two indium-tin oxide-coated glass slides. Original orientations of molecules are set parallel to x direction by coating, in order to maximize the nonlinear response. First, an s-polarized diode-pumped laser at wavelength λ₁= 514.5 nm is divided into two beams I₁ and I₂, and then symmetrically irradiated onto NLC. Intensities of I₁ and I₂ are about 85 mW/cm², and the angle between the two beams is θ ≈ 20°. They will be removed when the photorefractive grating is established. A p-polarization Er: YAG CW laser at wavelength λ₂= 2.94 µm and intensity 35 mW/cm² irradiates the object and then focused onto the liquid crystal in near field. A piezo-actuated mirror (PAM) phase-shifts the reference beam with a step of λ₂/4, and is used to calculate the absolute phase. A high-resolution cooled mid-infrared CCD captures the output images. A 3.5 V DC voltage is applied across the material to control the coupling gain [27]. Note that the voltage or room temperature should not be too high, otherwise the 5CB will become isotropic and decrease the nonlinearity [28].

 

Fig. 1. Schematic of proposed imaging set-up. The photorefractive grating in NLC is written by beams I₁ and I₂ at wavelength 514.5 nm. Once the steady grating is established, the writing beams are removed. Another beam at wavelength 2.94 µm illuminates the object and then transmits through the NLC. Reference beam is phase-stepped by PAM and used to calculate the phase. NLC: nematic liquid crystal. PBS: polarization beam splitter. PAM: piezo-actuated mirror.

Download Full Size | PDF

It should be emphasized that the measured absorption coefficients of material for writing beams and imaging beam are α₁ = 40 mm^-1 and α₂= 3 mm^-1, respectively. Thus, the influence of imaging beam on grating formation is negligible. Rod-shaped molecules will undergo a periodic quadratic polarization along the major axis under the combined modulation of electric field and internal charge field. Electro-optical effects and birefringence both contribute to the reorientation, forming a phase grating at molecular level. Since the intensities of writing beams are equal, the grating fringes have no bend, which is important for modes shifting without introducing any external distortion. With the slow-varying approximation, the period Λ and diffraction efficiency η of the grating can be given as [25]

Here, Δε is the anisotropic dielectric constant, θ_re is the reorientation angle of molecules, n₀ is the linear refractive index, and β is the angle between molecule and incident beams. Since the dimensionless parameter Q = 2πdλ₁/n₀Λ² > 1, the grating operates in Raman-Nath regime [29]. As the constant application of voltage makes the doped molecules fixed on the substrate surface, the writing time of grating is about nanosecond, while the duration can be several months. Even the writing beams are removed, the molecules still preserve the polarized states [30].

3. Spatial shifting of evanescent modes

Before the reorientation of NLC molecules, the output images are simply a diffracted version of input, in which evanescent waves disappear. When the photorefractive grating is established, the multistage diffraction will convert the evanescent waves into propagation region. This makes the maximum detectable frequency much higher than linear case. For the grating with a period Λ, the distribution of dielectric constant meets the relation ε = ε₀ + Δε·cos (2π/Λ·x). Raman-Nath diffraction theory indicates that the diffraction is in central symmetry. For simplicity, we can reasonably ignore the multiple reflections and backscattering, and only consider the forward scattering of object modes in z direction.

Since the wave interaction depends on the product of amplitudes, the weak coupling of high-frequencies are negligible. The approximation is valid, as k_g ≠ k₀ − k _± j, j = 1, 2, 3… can be always satisfied by controlling the incidence of writing beams, where k_g = 2π/Λ, k₀ = ω/c and k _± j are the vectors of grating, paraxial wave, and jth-order evanescent waves, respectively. Thus, there is no Bragg match between the imaging waves and grating. As shown in Fig. 2(a), in Raman-Nath diffraction regime, mode k_j is simultaneously diffracted into the adaxial component k_j-1 and the abaxial component k_j + 1, namely, the “upshift” and “downshift”. As a result, part of modes transfer towards the paraxial region. With the approximation of amplitudes A₀ >>A_j, the redistribution of A_j satisfies [31]

Then, the efficiencies of “downshift” component diffracted from jth mode can be derived as

Obviously, the severely off-axis modes are difficult to be incorporated back into the paraxial region. Increasing the effective dielectric constant is beneficial to frequency shift, while increasing k_g will squeeze the number of transferable modes. Series of evanescent waves will be coupled to the paraxial region and produce new modes like the beat-frequency. This process is depicted in Fig. 2(b). The observable spectrum expands in the form of k_expand = ∑k_j, and the resolvable region increases to k₀+ k_expand. Using the undepleted approximation of paraxial wave, the final detected intensity with averaging the background is [32]

By placing an array lens on the focal plane, the DC term of Eq. (6) can be separated by a low-pass filter, and the oscillating term can be decoupled by an appropriate algorithm.

 

Fig. 2. (a) Diffraction of high-order modes through the photorefractive grating. (b) Expansion of equivalent OTF. The black circle represents the original observation window. The dashed coils represent the extensions when the high-order modes are incorporated back into paraxial region.

Download Full Size | PDF

4. Image reconstruction via phase decoupling

Reconstruction algorithm is such important as an unfavorable approach will produce unpredictable artifacts [33]. Spatial intensity received on CCD is the superposition of multiple image components, whose displacements corresponds to the modal shifts in k-space [34]. Only when the scaling factors are precisely determined, can the components be decoupled and moved back to real positions. We can decompose the phase shift into two parts: one is the linear phase shift for multi-diffracted orders, and another is the additional phase shift caused by nonlinear coupling. The four-frame phase-shift method with the step length set at λ₂/4 is used to calculated the second part. In this way, the total phase shift of jth diffracted order can be given as

For j ≥ 2, γ_j≤ 1/2. Thus, except for ±1st modes, the diffracted image sizes of j ≥ 2 modes should be decreased for reconstruction. Although the theoretical shifts of modes are infinite, only the finite wavenumbers that exceed the noise level should be decoupled in practice. Thus, the observable modes mainly come from the paraxial waves and evanescent waves j = ±1, ± 2 orders.

5. Analysis of super-resolution imaging results

The theoretical resolution of reconstructed image can be closely expressed as

Taking the parameters n_||= 1.7, n_⊥= 1.5, NA = 1.4, the detectable resolution is 0.3213 µm, which is 3.268 times higher than the value of 1.05 µm in linear case. In order to verify the theory, we simulate the evolution of spectrum in k-space as Fig. 3 using finite-domain time-difference (FDTD) solution. Wave coupling is assumed to occur between the paraxial wave and evanescent waves of single beam. Meanwhile, the paraxial wave is undepleted in horizontal direction, and its amplitude component in vertical direction can be ignored. Optical response satisfies periodic boundary conditions in x direction and perfect matching layer (PML) condition in z direction. 5% Gaussian noise is added, and the data is averaged over several time periods. As shown in Fig. 3(a), only the paraxial wave is detected in linear imaging. When the ±1 modes are shifted to the paraxial region as Fig. 3(c), the observed frequency bound can reach 2k₀, resulting in a doubled observation bandwidth as Fig. 3(d). Furthermore, under an appropriate nonlinearity, the ±1 and ±2 modes are both re-incorporated into the propagation at the distance about 6λ₂. The transferred energy concentrates on the sidelobes near the axis. The results also show that the shifts of jth modes can improve the resolution by j times. Of course, considering the noise, we only simulate the j ≤ 2 modes here, so that the equivalent bandwidth is increased by about 3 times, which has a high consistency with the result of 3.268 times obtained by Eq. (9).

 

Fig. 3. (a), (c), (e) Simulation of power spectra in phase space under improved imaging conditions, and (b), (d), (f) the corresponding OTFs. The dashed regions indicate the spatial shift of high-order modes, which corresponds to the extension of space spectra circled by red rings.

Download Full Size | PDF

In order to further demonstrate the resolution ability for multi-objects, the sample is prepared by depositing 300-nm-diameter monodisperse silicon-dioxide microspheres onto the cover glass. Since the self-fluorescence of sample is weak, it is the transmitted light that mainly contributes to imaging. Figure 4 shows part of reconstructed result intercepted from a region of 50 × 50 µm². Gaussian noise is also added. In linear case, the resolution is about 1.05 µm, where the two spheres with a 600 nm center distance are imaged as a single bright spot. While under the nonlinear conditions, they can be distinguished. The improvement of resolution also can be read from the transect curves of intensity in Fig. 4(c), where the full width at half-maximum (FWHM) of a single sphere is compressed from 1.10 µm to 350 nm. The enhancement of resolution is about 3.143 times, which is in good agreement with the calculation.

 

Fig. 4. Comparison of (a) the linear imaging and (b) super-resolution imaging for multiple microspheres. Images of a single and two close-range balls are framed as the typical examples. (c) and (d) are the corresponding transverse intensities and spatial spectra (insets), respectively.

Download Full Size | PDF

In addition, the influences of system parameters on imaging results are briefly analyzed. A convenient controllable parameter is the writing angle of beams I₁ and I₂, because the grating period directly affects the imaging resolution. As seen from Fig. 5, increasing the angle significantly enhances the imaging resolution. However, this comes at the expensive of the reduction of diffraction efficiency. At the same time, the increase of diffraction cascade will lead to an excessive modal coupling, which inevitably weakens the intensity of paraxial wave, as depicted in Figs. 3(c) and (e). The accumulating system noise are also detrimental to the contrast of reconstructed images. Therefore, the writing angle is set to about 20°, corresponding the resolution enhancement about 3.2 times and diffraction coefficient about 8%. This is an acceptable choice after all consideration.

 

Fig. 5. Influences of writing angle on diffraction efficiency and resolution. Writing beams are injected with an angle about 20°. This is acceptable with considering the resolution and efficiency.

Download Full Size | PDF

6. Conclusion

In summary, we investigate a far-field mid-infrared microscopy method by utilizing the orientational photorefractive grating in NLC. Evanescent decaying waves can be redetected in the observation window through the spatial frequency shifting. This method provides a much higher resolution than conventional structured illumination microscopy. It is noteworthy that, for NLC, the resolution is on the order of several hundred nanometers due to the photorefractive response in Raman-Nath regime. Besides, the clearing point of 5CB, i.e., the nematic-to-isotropic phase transition temperature T_g, is about 35°, close to the imaging temperature of biological samples. When T_g is exceeded, 5CB will become isotropic. That means, the material should work at an ambient temperature between T_g and the melting point T_m (about 25°), which can benefit the imaging performance. We also note that, doped with some liquid crystal chemicals (with central bonds, such as fluorobenzene structure), may be expected to improve the clearing point and conducive to preserve the nematic state [36]. More or less, it provides a potential way for mid-infrared super-resolution imaging applications, such as highly sensitive thermal imaging, label-free chemical sensing, and non-invasive tissue microscopy.