Tunable liquid crystal (LC) lenses are vital for a variety of applications in optics due to large birefringence and flexible control by external fields [1, 2, 3]. Because of simple fabrication processes and tunable optical properties, switchable LC Fresnel lenses have attracted considerable attention in research [4, 5, 6, 7, 8]. The proposed methods for fabricating LC Fresnel lenses have mostly relied on a patterned photomask or electrode in the form of concentric rings. Because the focal length of a Fresnel lens depends on the radius of the first ring, the produced photomask or electrode can be used only to fabricate lenses with a fixed focal length, limiting the practical applications of such LC Fresnel lenses. Nemati et al. proposed a simple holographic technique for fabricating an electrically switchable Fresnel lens manufactured from an LC– polymer composite [9]. The unique advantage is the ability to fabricate different focal lengths with the same arrangement. However, the requirement for high optical power for the curing process limits the application of this technique.

The present study demonstrates an LC Fresnel lens based on an azo-dye-doped LC (ADDLC) film using a Sagnac interferometer. Two identical but oppositely directed paths taken by the beams of the Sagnac interferometer produce a stable interference pattern. The pattern consists of a series of concentric circular bright and dark bands, which the Fresnel lens exhibits. UV light and a Fresnel-patterned green beam irradiate the ADDLC film simultaneously. The UV light induces trans–cis isomerization of azo-dye molecules. The bent cis isomers disturb the LC orientational order and transform the nematic LCs into an isotropic phase [10]. The green beam suppresses phase transitions by cis–trans back-isomerization, retaining the LCs in a nematic phase. The difference in structure induces an effective phase shift between the incident light through the bright and dark regions in the Fresnel pattern, causing a focal effect. The proposed ADDLC Fresnel lens exhibits several unique features. First, a regular homeotropic cell and a compact setup were used. The structure of the Fresnel lens in the ADDLC film is dynamic. Second, the focal effect can be switched on and off by turning the green beam on and off, accordingly, with UV irradiation. The focusing efficiency is tunable by varying the intensity of the green beam. Third, the homeotropic and isotropic configurations in the bright and dark regions of the green Fresnel pattern lead the focal effect toward being irrespective of the polarization of the incident beam. Fourth, no photomask is required and the focal length is tunable with the same arrangement. There is a limitation of working wavelength due to the absorption of the sample below $550\mathrm{nm}$. The problem of the proposed system involves slow switching speed with rise and decay times of $10\min$, which can be overcome by means of azobenzene LCs [11].

The host LC used in the present study was nematic E7 ($n_e=1.7371$ and $n_o=1.5183$ at wavelength $\lambda =633\mathrm{nm}$, $n_i=(n_e+2n_o)/3=1.5912$ in the isotropic phase, from Merck). The photoresponsive material was azo dye 4-Methoxyazobenzene (4MAB, from Fluka). The mixing ratio of $\mathrm{E}7:4\mathrm{MAB}$ was $85:15$ by weight. The homogeneous mixture was injected into a glass cell with a gap of $23\mathrm{\mu m}$. Both indium-tin-oxide (ITO)-glass substrates were coated with a homeotropically aligned film of N, N-dimethyl-N-octadecyl-3-aminopropyltrimethoxysilyl chloride (DMOAP). Figure 1 illustrates the experimental setup for forming the Fresnel pattern in ADDLC film using the Sagnac interferometer and analyzing the focusing properties. One s-polarized green beam, derived from an ${\mathrm{Ar}}^{+}$ laser (wavelength $514\mathrm{nm}$) and collimated using an expander, was divided by a beam splitter into two with equal intensities. The two beams traveled identical but oppositely directed paths in a closed loop and were then united to produce interference. The lens in the loop caused the interfering rays to diverge. By appropriate adjustment of the orientation of the mirrors, the Fresnel pattern could be produced. Figures 2a, 2b dis play the interference pattern created by the Sagnac interferometer and the intensity profile of the pattern, respectively. The radius of the central ring increased with increasing distance from the beam splitter due to the divergent interfering rays. The ADDLC cell was placed at a position such that the diameter of the Fresnel pattern was $7\mathrm{mm}$. One unpolarized UV beam with a fixed intensity of $5.09\mathrm{mW}/{\mathrm{cm}}^2$ illuminated the Fresnel pattern on the cell simultaneously. The focusing properties of the ADDLC Fresnel lens were probed using the collimated He–Ne laser beam (wavelength $633\mathrm{nm}$), which was expanded to cover the entire Fresnel pattern. The polarization state of the probe beam was altered by rotating the analyzer behind the quarter-wave plate, which converted the linearly polarized beam transmitting through the polarizer to the circularly polarized beam. The focusing properties were measured using a CCD camera and a powermeter, which were set $86.7\mathrm{cm}$ behind the ADDLC cell.

Figure 3 displays the absorption spectra of the azo dye based on a homeotropically aligned $15\text{\hspace{0.17em} wt.}\%$ 4MAB-doped E7 cell. The azo dyes are generally in a rodlike trans form when kept in the dark. The absorption bands of the trans-4MAB lie in the UV and blue–green regions. Following UV irradiation, the trans molecules absorbed the UV light and isomerized to the bent cis form. The absorbance in the UV region was reduced with an increase in the blue–green region due to the change in the trans and cis isomer populations. Under irradiation of a green beam, cis-4MAB dyes could convert back to the trans isomers. Consequently, the absorption curve of the dye tended to recover to the initial state.

In the homeotropic ADDLC cell, the rodlike trans-4MAB dyes were aligned with LC molecules via the guest–host effect. With UV and the Fresnel-patterned green beam irradiation, the dyes in the dark regions of the Fresnel pattern could absorb UV light and transform to the cis form. The bent cis isomers tended to destabilize the phase structure of the LC host such that the LCs gradually changed from a nematic to an isotropic phase. However, the green beam suppressed phase transitions of the LCs in the bright regions by cis–trans back-isomerization, maintaining the LCs in a homeotropically aligned nematic phase. With normal incidence, the probe beam experienced the ordinary refractive index $n_{om}$ and the average refractive index $n_{im}$ of the LC mixture through the bright and dark regions of the Fresnel pattern, respectively [12]. Because of the refractive index mismatch, the Fresnel pattern acted as a focusing lens through diffraction. Figure 4 displays the optically controllable focusing properties of the ADDLC Fresnel lens. Without irradiation of the green Fresnel pattern, no focal effect occurs and the image shows a uniform intensity due to the collimated probe beam [Fig. 4a]. Once the green beam is turned on, an image containing a sharp focus and circular noises due to diffraction is observed [Fig. 4b]. The focus intensity increases and then declines with the increase of the intensity of the green beam [Fig. 4c]. When the lower intensities were applied, the green beam could not suppress phase transitions of the LCs induced by the UV light. Therefore, the refractive index mismatch between the bright and dark regions of the Fresnel pattern was small. The phase shift induced by the refractive index mismatch gradually increased and achieved a maximal $2\pi (n_{im}-n_{om})d/\lambda$, where d and λ denote the cell gap and the wavelength of the probe beam, respectively. Meanwhile, the area of the bright regions of the Fresnel pattern gradually expanded with the increase of the intensity of the green beam, differing from a regular binary phase Fresnel zone plate, in which the areas of each zone are almost equal. The focusing efficiency of the lens reaches a maximal value when the sum of the optical disturbances from all zones at focus is maximal. As the intensity of the green beam continues to increase, the area in the nematic phase induced by the strong green beam gradually expands and the area in the isotropic phase shrinks. As a result, the focusing efficiency continuously declines. Compared with the holographic LC/polymer Fresnel lens proposed by Nemati et al. [9], the ADDLC Fresnel lens has a relatively sharp focus and large modulation of intensity at the focus but a slow switching speed.

According to the LC configuration of the ADDLC Fresnel lens, it could be expected that the focusing efficiency would be irrespective of the polarization state of the incident beam. Figure 5 displays the dependence of the focusing efficiency on the average intensity of the green Fresnel pattern with various linear polarizations of the probe beam. The focusing efficiency is defined as the ratio of the power in the focusing spot to the total incident power after passing through the sample. The variation in focusing efficiency is less than 1%.

In a regular binary phase Fresnel zone plate, the focal length f is related to the innermost zone radius $r_1$ as $f=r_1^2/\lambda$. Because of the divergent interference pattern of the proposed Fresnel lens, different focal lengths may be achieved by altering the position of the cell. Figure 6 displays the relationship between the focal length and the central ring radius of the Fresnel pattern. As a regular binary phase Fresnel zone plate, the focal length increased linearly with the increase of the square central ring radius of the Fresnel pattern.

To summarize, the present investigation developed a tunable Fresnel lens in an ADDLC film using an interference technique. One Fresnel-patterned green beam created using a Sagnac interferometer irradiated the UV-illuminated ADDLC cell. The UV-light-induced phase transition of the LCs from a nematic to isotropic phase through trans–cis isomerization. The green-beam- induced suppression of the phase transition by cis–trans back-isomerization. These mechanisms resulted in a concentric zone plate distribution with homeotropic and isotropic structures in bright and dark regions of the green interference pattern, which, in turn, induced the focusing effect through diffraction. The proposed Fresnel lens is polarization independent, focus tunable, and the focusing efficiency of the device can be optically controlled.

 

Fig. 1 Schematic experimental setup for forming the Fresnel pattern in an ADDLC cell and analyzing the focusing properties. BS, beam splitter; M, mirror; L, lens (focal length $20\mathrm{cm}$); DM, dichroic mirror, which reflects green light and transmits red light; $\lambda /4$ WP, quarter-wave plate for $633\mathrm{nm}$; P, polarizer; A, analyzer; F, filter, which absorbs green light and transmits red light. The inset presents the chemical structure of 4MAB.

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Fig. 2 (a) Interference pattern created by the Sagnac interferometer showing the Fresnel zone structure. (b) Intensity profile of the interference pattern along the horizontal line passing through the center of the rings at the position zero.

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Fig. 3 Absorption spectra of a $15\text{\hspace{0.17em} wt.}\%$ 4MAB-doped E7 cell in the dark (red curve), in the irradiation of the UV light for $10\min$ (blue curve), and in the irradiation of the UV light and the green beam with intensity of $43.9\mathrm{mW}/{\mathrm{cm}}^2$ for $10\min$ (green curve).

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Fig. 4 Optically controllable focusing properties of the ADDLC Fresnel lens. The probe beam image on the focal plane (a) before and (b) after irradiation of the green Fresnel pattern. (c) Beam intensity profiles on the focal plane at different average intensities of the green Fresnel pattern.

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Fig. 5 Focusing efficiency of the ADDLC Fresnel lens as a function of the average intensity of the green Fresnel pattern with various linear polarizations at angles of $0°$, $15°$, $30°$, $45°$, and $90°$ with respect to the horizon.

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Fig. 6 Focal length as a function of the square central ring radius of the Fresnel pattern.

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