1. Introduction

Phase modulation devices based on liquid crystals, including liquid crystal variable phase retarders (LCVR) [1], tunable photonic-crystal components [2,3], spatial light modulators (SLM) [4], beam steering devices [5] have been developed and extensively used in various optical systems due to its low cost, no moving parts, low driving voltage and small size [6–8]. Most of these electrically controlled phase modulation devices that employ nematic LCs confront one of the fundamental limits of slow response (several tens of milliseconds) [9]. Although the rise time can be effectively minimized by increasing the voltage amplitude, the fall time cannot be controlled since it depends strongly on the LC material parameters, such as viscosity and elastic constants, as well as cell gap and anchoring strength [10]. Many researchers proposed different approaches to overcome the limit, such as creating a polymer network within the bulk to let the strong polymer network anchoring make LC to relax quickly [11]. Yun-Hsing Fan proposed polymer network liquid crystal light modulator with response 250 times faster compared with pure E44 cell, but at the same time the threshold voltage also increases by 25 times [12]. John L. West demonstrated a stressed liquid crystal with rise and decay time of 75μs and 793μs driven under 160V_rms voltage [13]. Nevertheless, there are some problems associated with polymer network nematic liquid crystals, such as light scattering and a relatively high driving voltage.

Moreover, most LC based phase modulation devices are polarization-dependent, which sacrifices half of the optical efficiency. Thus, it is necessary to develop a fast response, polarization-independent phase modulator. In addition to the above fast response light modulator, several methods were proposed to obtain polarization-independent LC devices, including polymer-dispersed homeotropic [14–17], 90° twisted dual-frequency liquid crystal [18,19] and etc. However, they all need complicated fabrication process and the phase change is relatively small. Moreover, due to the presence of the pretilt angle, the phase delay cannot reach to zero without compensators. Now, it challenging to realize a phase modulator that possesses simultaneous properties of fast response, low driving voltage, zero phase modulation and polarization-independence.

In this paper, we first theoretically analyze the optical compensation of double layer LC plates, and then the polarization independence of the proposed phase modulator is simulated using Jones vector and verified in experiment. The presented design would be a promising approach for several advantages. First, the residual phase due to the surface anchored LC molecules is compensated and zero phase retardation is obtained. Second, the fast response is realized by means of converting the rise and fall time of phase retardation to the switching-on time of the nematic plates, and simultaneous relaxation of the two plates remains optically hidden during the synchronized voltages falling. Third, the phase modulator is polarization independence with low driving voltage. This fast response phase modulator has great potential for applications in optical communications and light field imaging systems.

2. Basic principles

The parallel-aligned nematic LC plate appears optically as a uniaxially birefringent plate whose optical axis coincides with the aligning direction of LC molecules. When external voltage is applied cross the sample, the long axis of LC molecules tends to tilt towards the electrical field., Based on the long-range order of molecules, the averaged tilt angle θ determines the effective refractive index of the sample cell:

Here, n_eff is the effective refractive index, n_e/n_o are the extraordinary/ordinary refractive index of LC material. Averaged tilt angle of the cell is a function of amplitude of electrical field, written as θ(E), which varies from θ(E = 0)≈0 to θ(E>E_sat) = θ_sat, where θ_sat represents the saturation of tilt angle with electrical field amplitude increases. Normally, initial value of averaged tilt angle is a small value determined by the pretilt angle of alignment. Also, the saturated average value θ_sat can never goes to 90° and is determined by anchoring condition. Phase delay δ induced by the LC layer with birefringence Δn = n_eff - n_o is δ = 2πdΔn/λ, where λ is the wavelength of the incident light, d is the thickness of LC layer. The electrically tunable range of phase delay is from δ_res to 2πd(n_o-n_e)/λ, where δ_res is the residual delay due to the mismatch of θ_sat and 90°. To solve this problem, a passive compensator is always employed to provide a phase bias to correct it to 0° or even <0°. When there are two identical LC cells, with birefringence Δn₁/Δn₂ and thickness d₁ = d₂ = d can be used to form a double cell phase modulator as shown in Fig. 1. As it is illustrated that the two nematic cells are arranged in series parallel to each other with optical axis of Cell 1 and Cell 2 perpendicular to each other.

 

Fig. 1 Schematic representation of the double cell nematic LC phase modulator.

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Let the incident light be arbitrary polarization and decomposed to two orthogonal of x-/y-polarization:

The Jones vector of the exiting light, which pass through double cell nematic modulator is

Given cell gaps of the samples are determined, only birefringence is controlled by the application of an electrical field. Moreover, the phase delay is linearly related to birefringence, therefore, birefringence is used to analyze the phase delay. With no voltage applied or same voltage amplitude applied, the birefringence of Cell 1 and Cell 2 are $\Delta n_1(E)$and $\Delta n_2(E)$ respectively. Based on the optical compensation, total retardation${\delta }_{tot}=2\pi d(\Delta n_1\text{-}\Delta n_2)/\lambda$equals zero, since Cell 1 and Cell 2 are aligned perpendicular to each other leading to $\Delta n_1=\Delta n_2$. Because of the anchoring from alignment, the phase delay of single nematic LC cell can never reach zero. However, due to the role of optical compensation, the phase delay of double cell modulator can reach zero after careful adjustment and proper driving.

As illustrated by Eqs. (3) and (4), if we synchronize the driving signals to make the phase delay of Cell 1 and Cell 2 identical, the two orthogonal components of exit light E_out carries the same phase. Thus, the polarization state of the exit light is the same as the incident light, a pure phase modulator without polarization modulation is obtained. For validation purpose, the Poincare sphere is plotted and simulated when polarization of incidence changes. As shown in Fig. 2(a), for linearly-polarized incidence represented by blue stars, the exit light is also linear represented by red dots when the fixed amplitude is applied to the cells. Similarly, when incident light changes from linear to circular as modulated by rotating quarter wave plate, the exit light follows the same trace on the sphere. Additionally, since no polarizer or analyzer is required, the transmittance of the double cell modulator is at least two times higher than the traditional device.

 

Fig. 2 Polarization states of incident and exit light on Poincare sphere when incidence changes (a) linear-polarization direction and (b) between linear and circular polarization.

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For the transient response of total phase retardation, the compensation occurs when the driving signals of two cells change simultaneously:

With electrical field above Fredericks threshold applied, the LC molecules tend to tilt from planar to vertical. First, for total retardation increases from zero value, the driving voltage of one cell is set to be increasing, with the other cell maintains fixed. In this case, the total birefringence is tuned electrically. Similarly, by keeping the driving voltage of one cell at a high amplitude, the retardation drop is achieved by increasing the voltage of the other cell. Thus, the fast response is realized by means of converting the rise and fall time of phase retardation to the switching-on time of the nematic plates. Moreover, the simultaneous relaxation of the two plates remains optically hidden during the synchronized voltages falling. The proposed design is considerably easier and cheaper than the previous techniques and thus is more compatible for various applications.

3. Experiment

The experimental setup for measuring of the electro-optical characteristics is shown in Fig. 3. The solid-state laser (CNI, 532 nm) and He-Ne laser (632.8 nm) were used as light source, and a pair of crossed polarizer and analyzer was used to measure the transmittance and then get the derived phase retardation according to the relationship $I=I_0{\sin }^2(\delta /2)$. Two identically parallel-aligned nematic cells, which are with cell gap of 5μm and filled with E7 nematic mixture (with n_e of 1.741 and n_o of 1.517), were arranged in series with the optical axis perpendicular to each other. The directors are placed at an angle of 45° and −45° with respect to the polarization direction of the polarizer, respectively. The analyzer is placed behind to facilitate the detection of changes in light intensity. Finally, the electrically modulated optical signals were recorded by a photo-detector. Among them, the driving signals of square wave at 4k Hz frequency are generated by NI-USB-6343 data acquisition card. The signal received by the photo-detector is acquired by the card and analyzed by LabVIEW software.

 

Fig. 3 Experimental setup for measuring transmittance and response waveforms.

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4. Results and discussion

Figure 4(a) shows the curves of transmittance changing with voltage measured in the experiment using a solid-state laser (532nm) as light source at room temperature. The cell gaps of the Cell 1 and Cell 2 are 5 µm. The green line represents the transmittance versus voltage curve (TVC) of Cell 2 under the condition that voltage for Cell 1 is fixed at V₁ = 0V. In this case, the total transmittance starts from zero due to the compensation, then increases after the voltage exceeds threshold voltage, bounces back and forth and finally saturates at the value determined by the total phase delay due to Cell 2. The corresponding phase retardation illustrated in Fig. 4(b) green line also starts from zero. The total phase retardation decreases when the driving voltage of Cell 2 is fixed at V₂ = 10 V and the V₁ increase from 0V to 10V as shown by black line in Fig. 4(b).

 

Fig. 4 (a) Transmittance versus voltage curves at 532 nm and (b) phase retardation of double cell nematic LC phase modulator at 532nm.

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The driving scheme and corresponding response are shown in Fig. 5. Voltage signals with frequency of 4k Hz, which are generated by data acquisition card with the rise and fall time of 10ns, are loaded on Cell 1 and 2. The rising edge of V₂ induces the increasing of total phase delay, and the rising edge of V₁ is responsible for the decreasing of total phase delay. The synchronized falling edges of V₁ and V₂ show optical compensation.

 

Fig. 5 Waveforms of 4k Hz driving signals and electro-optical responses at 532 nm.

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The phase delay δ₁ /δ₂ of Cell 1 and Cell 2 are measured with the absence of the other cell. The responses in Figs. 5(c) and 5(d) show residual phase delay, which is δ₁(from t₂ to T) and δ₂(from t₁ to T), of around 1.3rad respectively, and they are compensated shown in Fig. 5(e) that residual phase delay of exit light, which is δ(from t₂ to T), is less than 0.1rad. At time t₁, to let the total retardation increases from lower to higher value, V₁ is set to be a fixed level above threshold, which is 1.5V in Fig. 5, and V₂ increases. Different from traditional LCVR, the drop of total phase retardation is realized by increasing V₁ with V₂ maintained at a high level as illustrated at time t₂. In this case, the switching time corresponds to the rise time of the Cell 1 as shown by Fig. 5(e). At time T, the simultaneous falling of V₁ and V₂ makes the total phase retardation unchanged, since the two orthogonal aligned LCs are optically compensated by each other during the relaxation process. But in reality, a bump in the waveform was observed in experiment as shown in Fig. 5(e) marked in red circle. The appearance of the fluctuation is due to the mismatch of transient distribution of directors in the two LC cells. For minimizing the fluctuation experimentally, the calibration can be done in several manners. Firstly, the cell gaps of two orthogonal aligned cells need to be kept the same for the full compensation in phase delay, which corresponds to the residual phase of the double layer modulator. Moreover, the strong anchoring or high order parameter is preferred for the transient matching of relaxation. The minimized bump shows 0.6rad, which is less than 10% of the retardation, is obtained using our hand-made cell. The precisely control of cell gap and anchoring condition can potentially eliminate the fluctuation.

Table 1 shows the response time of double cell phase modulator and LC cells. For comparison, the maximum retardation change is obtained by a square wave with frequency of 4k Hz and amplitude changing from 1.5V to 10V. The rise time of LC cells is in the range above 1ms, and the decay time is in the range above 10ms. The switching-on and -off time of double cell modulator is 0.86ms and 1.83ms, and can be further minimized by thinner cell gap.

Table 1. Response time of double cell modulator and LC cells

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Using the experimental setup shown in Fig. 3, with a He-Ne laser (632.8 nm) as the light source, we tested the polarization independence of the double layer phase modulator. The polarization-independence of proposed double-cell modulator is tested by two orthogonal linear polarizations of 0° and 90°. Figure 6 illustrates the voltage dependent behavior of the total retardation for two input polarizations, one parallel to x axis and the other perpendicular. The result indicates that the tested design has very good polarization independence which is in accordance with theoretical results.

 

Fig. 6 TVC of double cell modulator under different polarized incident light, (a) at V₁ = 0 V, (b) at V₂ = 10 V by a He-Ne laser at room temperature. The cell gaps of the Cell 1 and Cell 2 are 6.8 µm.

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5. Conclusion

In summary, this work demonstrated a fast response and polarization-independent phase modulator based on double cell nematic liquid crystal. The response time total phase delay changes from zero to maximum value is 0.86ms, and that from maximum value back to zero is 1.83ms, which can be further minimized by thinner cell gap. Based on the optical compensation, residual phase delay of nematic cells, which is due to the surfaced anchored layer even under high voltage, are cancelled and total phase delay can actually be zero. Phase modulation depth is over 3π at 532nm with double 5 μm cell filled with E7. Such a phase modulator has several advantages, including a low driving voltage, a fast response, polarization-independent, and zero phase retardation. The potential applications are optical communication, polarization imaging and etc.