Electrically controllable Fresnel lens in 90&#x00B0; twisted nematic liquid crystals

Chie-Tong Kuo; Chien-Yu Li; Shih-Hung Lin; Hui-Chen Yeh

doi:10.1364/OE.23.026041

1. Introduction

Fresnel lenses have attracted considerable attention in various fields since the first lens was fabricated in 1871. Because of the large birefringence and high flexibility, Fresnel lenses based on liquid crystal (LC) materials have been extensively developed [1–7]. The key features of the 90° twisted nematic (TN) LC configuration include a low operating voltage and a fast response, and hence, this configuration has been widely used in displays. However, the twisted director configuration renders their optical properties sensitive to the polarization of light. Polarization-independent devices using 90° TN-LCs can be achieved by operating at higher voltages [8,9]. In the high-electric-field regime, the LC director in the bulk tends to be oriented along the electric field, whereas the molecules near the substrates are aligned parallel to the rubbing directions. Two orthogonal birefringent layers with an electrically controllable thickness are formed on the substrates. For normally incident light, the phase retardation through one of the two boundary layers compensates that through the other. The total phase retardation of the light polarized parallel to the front LC director through the whole cell is identical to that of the light polarized perpendicular to the front director. Consequently, the polarization state of incident linearly polarized light is not changed after passing through the cell, and the optical properties of the cell are polarization independent. For surface-mode switching Fresnel lenses, two high electric fields were used to control the thickness of the surface LC layers in the odd and even zones [9]. Because of the self-phase compensation of the boundary layers, the phase shifts between the odd and even zones, and therefore the diffraction efficiency is independent of the polarization of incident light.

This study theoretically and experimentally developed an electrically controllable Fresnel lens based on 90° TN-LCs. An electric field was applied only to the odd zones. At low voltages, the device was polarization sensitive. The maximum diffraction efficiency was close to the 41% theoretical limit. By choosing a suitable cell gap according to the Gooch-Tarry conditions for a 90° TN cell, polarization-independent Fresnel lenses can be achieved at higher voltages. The Jones matrix method was used to analyze the optical fields of the light emerging from the odd and even zones. The diffraction efficiency at various voltages could be calculated using the Fresnel diffraction theory. The simulation effectively explained the experimental results.

2. Theoretical analysis

2.1 Optical properties of 90° TN-LC

In a 90° TN cell, the rear LC alignment direction is twisted by 90° relative to the front LC alignment direction. In the absence of an electric field, the LC molecules are all aligned parallel to the substrates and uniformly twisted because of the boundary conditions. The Jones matrix method is used to analyze the optical properties of this medium. In Fig. 1, a 90° TN cell is sandwiched between a pair of crossed polarizers. The x and y axes were chosen to be parallel to the LC director orientations at the entrance and exit, respectively. The transmission axis of the entrance polarizer was at an angle β with respect to the x axis, and the transmission axis of the exit polarizer was perpendicular to the entrance polarization for various β. According to Fig. 1, the normalized transmittance $T_{⊥}$ derived using the Jones matrix method is given by [10]

T_{⊥} = \cos^{2} X + {(\frac{Γ}{2 X} \cos 2 β)}^{2} \sin^{2} X,

where

X = {[{(π / 2)}^{2} + {(Γ / 2)}^{2}]}^{1 / 2}

,

Γ = 2 π d Δ n / λ

, d is the cell gap, Δn is the birefringence of the LC, and λ is the wavelength of the incident light. In the limit of slow twist with

Γ ≫ π / 2

, known as the Mauguin condition,

T_{⊥}

approaches unity for incident beams with β = 0° (E-mode) or 90° (O-mode). In a 90° TN cell satisfying the Mauguin condition, the polarization of the incident E-mode and O-mode beams appears to follow the twist of the local LC directors as light propagates in the cell. Therefore, the output polarization is rotated by 90°. This is called the polarization rotation effect.

Fig. 1 Schematic of the orientations of the polarizer axes and LC directors of a 90° TN cell in the xy plane.

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Figure 2 depicts plots of the normalized transmittance in Eq. (1) as a function of $d Δ n / λ$ and β for a 90° TN cell. The maximum $T_{⊥}$ value of 1 occurs when X is an integral multiple of π, independent of β. These integral multiples correspond to $d Δ n / λ = \sqrt{3} / 2, \sqrt{15} / 2, \sqrt{35} / 2, \dots$ , which are known as the Gooch-Tarry conditions for a 90° TN cell and can be formulated as

\frac{d Δ n}{λ} = {(q^{2} - \frac{1}{4})}^{1 / 2},

where q is an integer. When the Gooch-Tarry conditions are satisfied, a linearly polarized beam remains linearly polarized after passing through a 90° TN cell, except for the plane of polarization being rotated by 90°, regardless of β.

Fig. 2 Normalized transmittance $T_{⊥}$ of a 90° TN cell as a function of $d Δ n / λ$ and β.

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When a voltage is applied along the twist axis, the LC molecules with positive dielectric anisotropy are tilted in the direction of the electric field. The final distribution of the director, which is described using tilt angle and twist angle, can be obtained by minimizing the free energy of the LC cell. A commercial nematic LC material E7 was used as a sample to calculate the distribution of the director. E7 had an extraordinary refractive index n_e of 1.713, an ordinary refractive index n_o of 1.5, a dielectric anisotropy Δε of 14.3, a splay elastic constant k₁₁ of 11.7 × 10⁻¹² N, a twist elastic constant k₂₂ of 9.0 × 10⁻¹² N, and a bend elastic constant k₃₃ of 19.5 × 10⁻¹² N. Figure 3 shows the calculated tilt and twist angles of the director in a 90° TN cell for several applied voltages. The Freedericksz transition voltage V_th is approximately 1 V, and it is obtained from the expression [11]

V_{t h} = π \sqrt{\frac{k_{11} + (k_{33} - 2 k_{22}) / 4}{ε_{0} Δ ε}} .

Above this voltage, the LC directors start to tilt. The twist distribution remains uniform until the applied voltage reaches the optical threshold V_op, which is approximately twice of V_th. Substantial changes of the LC director distribution occur above the optical threshold, disrupting the polarization rotation effect [12].

Fig. 3 Distribution of the (a) tilt angle and (b) twist angle of the LC director in a 90° TN cell along the z axis for different applied voltages.

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2.2 Electrically controllable Fresnel lens in 90° TN-LCs

A Fresnel lens consists of a series of concentric zones. The radius r_m of the mth zone is determined by

r_{m}^{2} = m r_{1}^{2},

where r₁ is the radius of the innermost zone. In a binary phase Fresnel lens, all zones are transparent, and a phase retardation exists between adjacent zones. A plane wave passing through a Fresnel lens is diffracted and forms multiple diffraction orders. The first-order focal length f is given by [13]

f = \frac{r_{1}^{2}}{λ},

where λ is the wavelength of the incident beam. The structural relation in Eq. (4) indicates that the light field emerging from a binary Fresnel lens is periodic in r² with period

2 r_{1}^{2}

, and therefore, it can be represented as a Fourier series expansion. The nth Fourier coefficient A_n of the emerging light field E_out is given by [13]

\begin{array}{l} A_{n} = \frac{1}{2 r_{1}^{2}} \int_{0}^{2 r_{1}^{2}} E_{o u t} \exp (- i π n \frac{r^{2}}{r_{1}^{2}}) d (r^{2}) \\ = \frac{1}{2 r_{1}^{2}} [\int_{0}^{r_{1}^{2}} E_{o u t - o d d} \exp (- i π n \frac{r^{2}}{r_{1}^{2}}) d (r^{2}) + \int_{r_{1}^{2}}^{2 r_{1}^{2}} E_{o u t - e v e n} \exp (- i π n \frac{r^{2}}{r_{1}^{2}}) d (r^{2})], \end{array}

where E_out-odd and E_out-even represent the light fields emerging through the odd and even zones, respectively. The first-order diffraction efficiency η can be obtained from Eq. (6) as

η = \frac{{| A_{1} |}^{2}}{{| E_{i n} |}^{2}} = \frac{{| E_{o u t - o d d} - E_{o u t - e v e n} |}^{2}}{π^{2} {| E_{i n} |}^{2}},

where E_in is the incident light field.

In this study, a 90° TN-LC cell was used to fabricate an electrically controllable Fresnel lens. In theoretical calculations, a uniform electric field was applied to the odd zones, causing the redistribution of the LC director. No electric field was assumed to exist in the even zones, and the LC molecules were assumed to maintain the perfect twist structure. According to the distribution of the LC director in Fig. 3, E_out-odd at an applied voltage and E_out-even at zero voltage can be determined using the Jones matrix method. Substitution of E_out-odd and E_out-even in Eq. (7) yields the diffraction efficiency at various voltages. Figure 4 shows the calculated diffraction efficiency of 90° TN Fresnel lenses as a function of the voltage applied to the odd zones for different β of the incident linearly polarized beam with a wavelength of 633 nm. The cell gap in Fig. 4(a) corresponds to the third Gooch-Tarry condition, namely, m = 3 in Eq. (2). Both cell gaps in Figs. 4(a) and 4(b) satisfy the Mauguin condition, indicating that the polarization rotation effect is valid for the E-mode and O-mode beams passing through the two cells in the absence of an applied voltage.

Fig. 4 Calculated diffraction efficiency of 90° TN Fresnel lenses with (a) d = 8.8 μm and (b) d = 10 μm as a function of the voltage applied to the odd zones and polarization direction β for the wavelength of 633 nm.

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For an applied voltage below V_th, the LC director distributions in the odd and even zones remain unchanged, and the diffraction efficiency is zero. When the applied voltage is between V_th and V_op, the LC directors in the odd zones tilt but retain a uniform twist. Under the polarization rotation effect, the effective refractive index of the E-mode beam in each layer of the odd zones is less than the extraordinary refractive index n_e in the even zones. The light fields E_out-odd and E_out-even are parallel to each other but have a phase shift, resulting in a focusing effect because of diffraction. The diffraction efficiency can reach the theoretical limit of 41% when the phase shift between E_out-odd and E_out-even equals π. However, the tilt of the LC molecules in the odd zones does not alter the effective refractive index of the O-mode beam. The light has the same refractive index n_o in the odd and even zones, causing very little diffraction.

When the applied voltage is between V_op and 4V_th, the twist in the odd zones is broken, and therefore, the polarization rotation effect no longer holds. The light fields E_out-odd and E_out-even are elliptically polarized and linearly polarized, respectively. The diffraction efficiency is dependent on the applied voltage and β. At voltages greater than 4V_th, the LC molecules in the bulk of the odd zones are aligned nearly parallel to the applied electric field, and therefore, they do not change the polarization of light. For the E-mode and O-mode beams, E_out-even is rotated by 90° with respect to E_out-odd because of the polarization rotation effect, yielding the same diffraction efficiency. This result is applicable to cells satisfying the Mauguin condition (Figs. 4(a) and 4(b)). However, the situation is different for an incident beam with β = 45°. Under the Gooch-Tarry conditions, the polarization rotation effect is valid at any β. This indicates that the diffraction efficiency is independent of polarization, as shown in Fig. 4(a). When the Gooch-Tarry conditions are not satisfied, the incident linearly polarized beam with β = 45° does not show the polarization rotation effect. Therefore, E_out-even is elliptically polarized, and the diffraction efficiencies are different from those for the E-mode and O-mode beams, as shown in Fig. 4(b).

3. Experiments

Two indium tin oxide (ITO)-glass substrates were used to assemble a TN-LC Fresnel zone plate. The ITO film on a glass substrate was patterned by applying the photolithography technique. The odd-zone electrodes were retained using a Fresnel-zone-patterned photomask. The innermost radius of the photomask was 0.5 mm. Both substrates were spin coated with an alignment film of polyvinyl alcohol (PVA) and rubbed orthogonally. To prevent reverse-twist disclinations, a chiral dopant S811 was added to the nematic LC E7 at a concentration of 0.4%. The mixture was stirred thoroughly and injected into the empty cell via the capillary effect to enable diffusion throughout the entire cell, resulting in the formation of a 90° TN structure. The cell gap was chosen as 8.8 μm to satisfy the third Gooch-Tarry condition. Figure 5 depicts the experimental setup for examining the focusing characteristics of the formed 90° TN Fresnel zone plate. A linearly polarized He-Ne laser beam with a wavelength of 633 nm was expanded and collimated to cover the aperture of the zone patterns. The quarter-wave plate converted the linearly polarized beam into a circularly polarized beam. A polarizer was used to select the polarization of the beam incident on the sample. A detector was placed at a distance of 39.5 cm from the sample to measure the transmitted beam intensity.

Fig. 5 Experimental setup for examining the focusing characteristics of the 90° TN Fresnel zone plate. L, lens; I, iris diaphragm; P, polarizer; λ/4 WP, quarter-wave plate for 633 nm; S, sample; AC, alternating-current power supply; D, detector.

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4. Results and discussion

Figure 6 shows microscope images of the 90° TN Fresnel lens under crossed polarizers in the presence of an applied voltage. Below an optical threshold of 2 V, the polarization rotation effect exists for light passing through the odd and even zones. The E-mode beam is rotated by 90° after passing through the cell, and hence, the microscope images are almost uniformly bright. The LC molecules in the even zones retain a twisted structure because of the absence of an electric field, and the molecules in the odd zones are reoriented toward the electric field with an increase in the applied voltage. Therefore, the even zones remain bright, and the odd zones gradually become black. At high applied voltages, a leakage fringe field extended into the even zones [4,14]. The orientation of the LC molecules in the even zones in the direction of the applied electric field resulted in irregular dark lines when observed under crossed polarizers, as shown in the magnified pictures in Fig. 6.

Fig. 6 Microscope images of the 90° TN Fresnel lens under crossed polarizers at (a) 0, (b) 1.4, (c) 2, (d) 3, (e) 4, and (f) 10 V, and a magnified view of images obtained at (g) 3, (h) 4, and (i) 10 V. P and A denote the transmission axes of the polarizer and analyzer, respectively, and R denotes the rubbing direction.

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Figure 7(a) shows the diffraction efficiency of the 90° TN Fresnel lens as a function of the applied voltage and β. Under the Gooch-Tarry conditions, the focusing characteristics are in agreement with the theoretical calculation, as shown in Fig. 4(a), except at high applied voltages. This disagreement can be attributed to the orientation of the LC in the even zones due to the leakage fringe field, which decreased the difference between the LC configurations in the odd and even zones resulting in a reduction in diffraction efficiency. A theoretical model is currently being developed to explain this disagreement. Figure 7(b) shows the intensity profile of the outgoing beam in the focal plane, and the inset shows the focusing pattern. Most of the incident light diffracts into the first-order [13].

Fig. 7 (a) Diffraction efficiency of the 90° TN Fresnel lens as a function of the applied voltage and β. (b) The intensity profile and focusing pattern in the focal plane at β = 0° and an applied voltage of 1.4 V.

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5. Conclusion

This study presents a theoretical analysis and experiments on an electrically controllable Fresnel lens in a 90° TN-LC cell. One substrate was completely coated with ITO. A Fresnel-zone-patterned electrode was formed on the other substrate. The cell gap was selected to satisfy the Gooch-Tarry conditions, and therefore, the polarization rotation effect existed regardless of the polarization direction of the incident beam. At low applied voltages, the diffraction efficiency was sensitively dependent on the applied voltage and polarization. In the high-voltage regime, the diffraction efficiency was polarization insensitive. Theoretical calculations effectively explained the experimental measurements.

Acknowledgments

The authors would like to thank the Ministry of Science and Technology, Taiwan, for financially supporting this research (grant no. MOST 103-2221-E-327-016).

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Electrically controllable Fresnel lens in 90° twisted nematic liquid crystals

Abstract

1. Introduction

2. Theoretical analysis

2.1 Optical properties of 90° TN-LC

2.2 Electrically controllable Fresnel lens in 90° TN-LCs

3. Experiments

4. Results and discussion

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (7)

Equations (7)

Optics Express