1. Introduction

Recently, the Fibonacci grating (FbG), with sophisticated Quasi-Periodic designed, is obtaining more and more attractions in super-resolution imaging and super-structure-grating technology [1]. Due to their ability of producing quasi-continuous frequency shifts, FbGs are used in superstructure fiber Bragg grating applications [2], far-field super-resolution [3,4], and diffractive focal lens [5]. Losing features for far object detection is a serious issue for several decades, primarily because of two reasons [6]. First, the phase shift for grating with fixed periodic structure. In other words, the corresponding phase shift in the frequency domain is not continuous, leading to the feature loss during the reconstruction in the far-field region. Secondly, some technologies, ranging from near-field scanning optical microscopy (NSOM) [7], structured illumination [8], and super-lens [9] are making efforts to acquire high-frequency information for the reconstruction of super-resolution images directly from the evanescent wave in the near-field. However, the limitations are inevitable for them. Firstly, all the measurement is constricted in the near-field region. Secondly, the point-by-point scanning for NOSM is time-consuming. Thirdly, hyper-fine information still experiences high loss because of the periodical structure of the super-lens. For the existing imaging system, the minimum resolution, which can be recovered is 1.22$\mathrm{\lambda }$. To acquire the information in the near-field region, the capability of translating the information in the evanescent wave at high-frequency into the propagation waves in the far-field for FbG is a key. This indirect reconstruction method not only extends the boundary to far-field but also reconstructs minor details from the evanescent wave generated and projected on the target by the FbG itself. The structure of Fibonacci grating follows Fibonacci optical super-lattice [10]. The recursive relationship for all terms of the sequence is ${F_{r + 1}} = {F_r} + {F_{r - 1}}$, where $\{{{F_r}} \}$ represents Fibonacci sequence. Table 1 shows the sequence of the FbG for F=1 &F=2 for the binary grating. When ${F_r}$ is approximating positive infinity, the ratio of the latter term to the former term is exactly the golden ratio ($\mathrm{\kappa } = \frac{{\sqrt 5 + 1}}{2} \approx 1.618$). This sequence is not periodic but is strictly long-range ordered. Interestingly, the quasi-periodicity of the Fibonacci array corresponding to the projection for a high-dimensional periodic structure. Due to this feature, FbG produces a set of diffraction spots with unequal energy, which generates a frequency shift with quasi-continuity. The location to the neighbour point is derived from the golden ratio.

Table 1. The sequence of FbG

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Despite the fixed FbGs mentioned above, a switchable liquid crystal (LC) FbG is recently disclosed, which can tune multiple diffraction efficiency at low voltage. However, the response time is very slow ∼26 ms and diffraction efficiency (first-order diffraction: 3.7% at 5 V) [11]. The LC grating mainly utilized on three fabrication approaches: 1) Use localized polymer network, 2) control the LC director by the external stimuli like electric or magnetic fields, and 3) use the photo-alignment to generate domains with distinct or continuous easy axis patterns. [12,13]. Among these methods, the photo-alignment approach can generate sub-wavelength grating pitches, which offers well-designed accurate grating domains with easy fabrication without any surface contacts [14]. For the speed, LC modes such as blue phase liquid crystals, cholesteric liquid crystals, thin nematic liquid crystals, and ferroelectric LC (FLC) are heavily explored. [15] The FLC provides the fastest response time at the lowest applied voltage. However, most of the FLC show intrinsic diffraction due to the ferroelectric domains, elastic domains, and helical periodic domains, and therefore, they are not suitable for the diffraction elements. Recently, we showed that the electrically suppressed FLC (ESHFLC) shows a response time of ∼100µs at the driving voltage of 3 V/µm, without any intrinsic diffraction and hysteresis [16].

In this work, we present a low voltage fast switchable ESHFLC-FbG with $\frac{\lambda }{{2.25}}$ spatial resolution based on the photoaligned ESHFLC [17]. The photo-alignment approach is used to provide patterned alignment for the ESHFLC that offers a phase difference of 0 and $\mathrm{\pi }$ for the impinging light in the adjacent alignment domains. The high-quality patterned alignment and binary electro-optical operations of the ESHFLC offer high-diffraction efficiency (DE) and polarization-independent diffraction [18]. The proposed FbG gratings are characterized by the fast response time ($\mathrm{\tau } = 100\mathrm{\mu }\textrm{s}$), low driving voltage 3 V/µm, high resolving power of $\frac{\lambda }{{2.25}}$ for the far-field objects, and DE of ∼97%. The DE for the 1st order and 2nd order is 6.6% and 21.2%, respectively. The response time drops to $55\mathrm{\mu }\textrm{s}$ for the driving voltage of 10 V. Thus, the proposed FbGs can find application in the super-resolution imaging system, superstructure fibre sensor, and other photonic applications.

2. Theoretical background

To understand light propagation after the diffractive optical system (FLCFbG), scalar diffraction theory and Rayleigh-Sommerfeld (RS) integral [19] are used. Derived from the physical definition, the far-field diffraction with function of diffraction angle T(f) can be determined from the spatial transmission coefficient t(x), and the input spatial light distribution E(x) [20]:

Where E0(x) is the amplitude distribution, and phase distribution φ(x). Moreover, t(x) is the spatial phase and amplitude distribution of the grating. k is the wavevector of the incident light and L is the aperture of the grating. Using the Fourier Transform, the far field distribution can be derived through the convolution of the Fourier Transform of t(x). Particularly, when parallel light incident on to the target, the Fourier transform of the aperture is also called point spread function (PSF).

However, if a diffractive aperture can translate the information contained in the near-field into the propagating waves at the far-field, then this problem could be solved. The quasi-periodicity of FbG enables the imaging system to shift the phase quasi-continuously, which overcomes the problem of fixed phase shift of imaging system and reconstruct hyperfine details of targets. Assuming a light of wavelength=$\mathrm{\lambda }$ incident on the FbG, ${K_0}$ is wavenumber of incident light ($K = \frac{{2\pi }}{\lambda } = 2\pi f$). The diffraction order is $\textrm{m} \in \{{ \ldots - 2, - 1,0,1,2 \ldots } \}$, since the DE for FbG is very low for the orders higher than 4, the transformation of information in evanescent wave can only be completed when $\textrm{m} ={-} 2, - 1,0,1,2$. The condition for phase matching (${K_1}$) can be expressed as:

Regarding the relationship between spatial frequency, wavenumber, and period of grating structure, this equation can be further derived into [4,5]:

3. Experiment

The ESHFLC mode and photoalignment technology based on the photo-reorientation mechanism of azo dyes SD1 are chosen. The ESHFLC is characterized by the fast response at a low applied voltage, no hysteresis, and no intrinsic diffraction, and thus, offer a high alignment quality for multi-domain structure. Furthermore, the sensitivity to the electric field polarity of the applied electrical field, the ESHFLC-FbG can switch between non-diffractive and diffractive state. Usually, the FLC shows intrinsic diffraction as stated above, but the ESHFLCs, because of the proper balance between the anchoring energy and elastic energy of the material, do not show intrinsic diffraction and show high optical quality [21]. In the absence of the electric field (E=0), the ESHFLC cell shows the degenerated optical state characterized by two optical domains that can be suppressed by the application of the electric field > critical field of the helix unwinding (Ec). Thus, in the present case, the electric field higher than 0.5 V/µm (i.e. > Ec), suppresses the helix, and the optical state becomes uniform. The ESHFLC mode requires proper balancing between the anchoring energy and elastic energy of the helix such that $Kq_0^2 \ge \frac{{2W_Q^0}}{{{d_{FLC}}}}$, where K is an average elastic constant and ${q_0}$ is the helix wave vector(${q_0} = \frac{{2\pi }}{{{R_0}}},\; {R_0}$: helix pitch), $W_Q^0$ is a coefficient of the anchoring energy,$\; {d_{FLC}}$ is the thickness of FLC cell [22]. Hence, for the E > Ec, FLC molecules switch uniformly between the two-switching position around the FLC cone, offering bright and dark states. The FLC selected for present grating was FD4004N (Dainippon Ink & Chemicals Ltd., Japan) that is characterized by helical pitch ρ ≈ 350 nm (at 25°C), spontaneous polarization Ps = 61 nC/cm2, and the tilt angle $\theta = {22.1^\circ }$. Selected FLC remains in its smectic C* phase up to 73°C and becomes isotropic at 105°C. Besides to maximize the phase difference of the two domains (0, $\mathrm{\pi }$), the tilt angle is $\theta \; {22.1^\circ }$. For one polarity electric field, the FLC molecules align parallel or perpendicular to the polarizer or analyzer and block all the light. Whereas for the other polarity of the electric field, the FLC molecules make an angle of 45° or 135° from the polarizer, and thus, work as half wave-plate (HWP) condition. The ${90^\circ }$ difference in the polarization azimuth between the output light from the two domains produce π phase difference that maximizes the diffraction efficiency. The details of the molecular dynamics are discussed later in the article. In another aspect of the alignment technology, an approach based upon the double exposure procedure on the same substrate to rewrite the alignment layer selectively with the FbG mask is applied. The fabrication details are illustrated in Fig. 1. Furthermore, as the ESHFLC is very sensitive to the anchoring energy, and therefore, the alignment layer is required to have tunable anchoring energy and also should have an ability to rewrite the alignment direction and have the same anchoring energy in the two alignment domains. Thus, the sulphonic azo dye (SD1) is used because of its considerably large anchoring energy and the re-align ability of the alignment direction. Moreover, for better orthogonal direction between each domain, the irradiation energy for the second exposure, to rewrite the alignment direction in the exposed area of the alignment layer through the FbG mask, is fixed to be twice larger than that of the first exposure. The doubled irradiation energy in the second exposure provides similar anchoring energy in the two alignment domains.

Figure 1(a) shows the fabrication process and detailed illustration. The design pattern as shown in Fig. 1(b) and (c) can be described as:

 

Fig. 1. (a). The fabrication procedure of FLCFbG. (b). The schematic of the mask structure based on 1D-Fibonacci optical super-lattice. (c). The schematic of the mask structure based on 2D-Fibonacci optical super-lattice for 2D FLCFbG. (d). The structure of the FLCFbG. (e). The detailed schematic of FLC molecular rotation.

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4. Results

Figure 2(a) & (b) show the polarizing optical microscope (POM) images of 1D and 2D ESHFLC-FbG, respectively. The domain size for both 1D and 2D ESHFLC-FbG is 10 µm. We used the experimental arrangement shown in Fig. 2(c-i) to characterize the diffraction profile. Figure 2(d) & (e) show the diffraction patterns of the respective 1D and 2D FLCFbG, respectively. Figure 2(d-ii) shows the dark state for the 1D ESHFLC-FbG, which is primarily the same for the 2D ESHFLC-FbG grating. To understand the diffraction properties for the designed gratings, we have plotted the intensity profiles of these diffraction patterns in Fig. 2(f)-(g). The high intensities of the first and second diffraction orders confirm the high diffraction efficiency of the FbG and low on-axis intensity, which is consistent with the simulation results shown in Fig. 2(i) and previous publications. The intensity difference of the zero order is mainly due to the defects induced between the neighboring domains which is almost negligible in the simulation. Rooted in the Fibonacci sequence, the first-order diffraction of the proposed rectangular binary grating splits it into two sub-peaks of diffraction and the ratio of their location, from 0th order, follows the golden ratio. The diffraction pattern is the Fourier transform of the transmittance index including the amplitude and phase distribution. Simultaneously, the Fibonacci sequence follows the rule that the phase profile is the sum of the previous two domains. Additionally, the sum of the Fibonacci sequence is also a Fibonacci sequence, see the diffraction pattern. Thus, the diffraction point position follows the golden ratio with the ratio of 0.618 and 1.618. Experimentally, we measured the ratio of 0.603 and 1.66. The small deviation from the theoretical value can be due to the fabrication and experimental errors. The envelope of the diffraction profile depends on the domain size due to the diffraction of the domain. Being a binary grating, the maximum continuous domain size of the FbG is double the minimum domain size, which limits the number of the diffraction orders to below 3. Thus, the ratio of the positions of the adjacent diffraction orders is either 0.603 or 1.658. And the diffraction peak splits it into two sub-diffraction orders.

In the proposed switchable ESHFLC-FbG, the ability of fast switching between non-diffractive state and diffractive state is one of the critical features. Firstly, as for the phase difference, the cell gap of the FLC cell is designed to fit the HWP condition for the impinging light (l=632nm) to ensure that the propagating light experiences the same phase difference between neighbouring domains in the non-diffractive and diffractive state. Secondly, the relative locations of the polarizer and the rotation of the FLC molecules are very important to tune the optical states of the ESHFLC-FbGs. The schematic of the ESHFLC molecular dynamics in the two adjacent alignment domains is shown in Fig. 3. The ‘S’ stands for the state, and ‘D’ stands for the domain of the FbG, where SnDm defines the state n and domain m. State 1 is a non-diffractive state, state 2 is diffractive, and state 3 represents the molecular state when no E is applied. As for ESHFLC, the spontaneous polarization vector of the molecule follows the applied electric field. Thus, the FLC molecules in each domain rotate with the electric field polarity in the presence of the square wave electric field and show two switching positions around the cone surface. State 1 is non-diffractive, where the ESHFLC molecules are aligned either along with the polarizer or analyzer in the two adjacent alignment domains. The analyzer blocks the output light, and no diffraction shows up. In-state 2, the ESHFLC molecules in two domains are rotated by $2\theta = {45^\circ }$ on the changing the polarity of the applied E. In this state, due to the retardation induced by the ESHFLC-FbG, the polarization azimuth of the impinging light rotates by ${90^\circ }$, and diffraction evolves.$\; $ Due to this phase shift from 0 to ${\boldsymbol \pi }$, the proposed quasi-periodic structure is considered to be polarization independent. Moreover, the switching time between these states is around 100µs at 5V. The electro-optical (EO) performance is discussed in the following paragraph

 

Fig. 2. (a). The POM of 1D FLFbG. (b). The POM of 2D FLCFbG. The defect line size between the two domains is 1 µm. (c-i) and (c-ii) show the experimental setup used to click the diffraction pattern and measure the response time, respectively. (d-i)The diffraction pattern of 1D ESHFLC-FbG at diffractive state. (d-ii). The diffraction pattern of ESHFLC-FbG at non-diffractive state. (e). the diffraction pattern of 2D ESHFLC-FbG at diffractive state. (f). The intensity map of 2D ESHFLC-FbG at diffraction state. (g). The intensity map of 1D ESHFLC-FbG at diffraction state. (h). The diffraction angle for 1D and 2D FLCFbG. (i)The simulation results of1D ESHFLC-FbG at diffraction state that matches well with the experimental results.

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Fig. 3. The schematic of output polarization after diffraction of ESHFLC-FbG and corresponding molecular orientation at (a) State 1: non-diffractive state (driving voltage=-5 V). (b). State 2: diffractive state (driving voltage=5 V). (c). State 3: E=0.

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We studied the electro-optical characteristics of the ESHFLC-FbG grating using the experimental setup shown in Fig. 2(c-ii). We measured the transmittance-voltage-curve (TVC) and response time for the 0th, 1st, 2nd, and 3rd diffraction orders. The response time (τ), which is a material property and is the same for all the diffraction orders in the same cell, is plotted in Fig. 4(a). The τ decreases at higher voltages. At 5 V, the response time is 100$\mu $s, which further decreases to 50$\mu $s at 10 V. The τ of the ESHFLC-FbG can be written as τ=${\gamma _\varphi }$/Ps*E. Here ${\gamma _\varphi }$ is the rotational viscosity; Ps is the spontaneous polarization of the ESHFLC; E is the electric field. The experimental data is in good agreement with the theoretical values. The TVC for different orders is bright and dark states are plotted in Fig. 4(b). The bright state transmittance, for all of the diffraction orders, increases with the applied voltage and saturates after the 2 V of the electric field. Similarly, the dark state transmittance decreases at higher voltages and saturates after 2 V of the electric field, showing a high contrast ratio for all the orders. The contrast ratio for the 1^st and 2^ndorder are 33:1, 100:1, respectively. The DE, due to the binary operations of the FLCs, is unexpectedly high in comparison to previously reported works. The first and third order show similar diffraction efficiency of 8.5%, and the second-order DE measured for any FbGs to date (see Table 2). The binary operation of the ESHFLC without any geometrical defects and π phase difference in neighbouring domains can be attributed to such a high DE. The diffraction angle of these orders also follows the golden ratio, see Fig. 2(i).

 

Fig. 4. (a). The response time of ESHFLC-FbG. (b). The TVC curve for different diffraction orders at diffractive (bright) state and non-diffractive (dark) state. (c). Diffraction efficiency of different diffraction order.

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Table 2. The comparison of diffraction efficiency and response time of different grating

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5. Conclusion

In summary, we disclose a low-voltage, fast-switching, and polarization-independent ESHFLC-FbG based on photo-alignment technology. The response time of ESHFLC-FbG is 103µs at 5 V, which shows a high diffraction efficiency of 8.5% and 30% for the 1st order and the 2nd diffraction order, respectively. The ESHFLC-FbG, to the best of our knowledge, shows superior properties than other previous reports, see Table 2. The resolving power of the proposed system is λ/2.25. Thus, with the fast response time and high diffraction efficiency, the proposed ESHFLC-FbG can find application in the super-resolution imaging system, superstructure fibre sensor, and other photonic applications.