Modulation of shape and polarization of beam using a liquid crystal q-plate that is fabricated via photo-alignment

Yao-Han Huang; Shih-Wei Ko; Ming-Shian Li; Shu-Chun Chu; Andy Y.-G. Fuh

doi:10.1364/OE.21.010954

Introduction

Gaussian beams (in transverse electromagnetic wave (TEM₀₀) mode) are the most common used laser beam-profiles. Their particular distribution is preserved as the beams propagate, and such beams can be focused into a diffraction-limited spot. In recent decades, the manipulation of the shapes of laser beams has rapidly developed because of technological improvements in beam-shaping devices and the ever-increasing demand for their applications [1–5]. The profile intensity of a laser-beam can be shaped by modulating the amplitude [6,7] or the phase [8,9] of the beam. Although amplitude modulation is effective, it usually wastes much of the energy of the laser beam. When properly performed, phase modulation can almost losslessly reproduce the intensity of a beam. Hence, phase modulation methods are more efficient in practice. A novel device, called a q-plate (QP), which is based on the phase modulation of liquid crystal (LC), has been developed [10]. It can be patterned with a specific transverse topology and contains a well-defined integer or semi-integer topological charge at its center [11–13]. This device has been applied to convert a Gaussian into a beam with orbital angular momentum [14,15] and it has been used in various areas in the field of optics [16–18]. Vector vortex beams that are produced by the modulating circularly polarized Gaussian beams are commonly coincident with corresponding singular points in the optical phase. Vortex beams are strongly correlated to singular optics with the optical phase singular point and can be used in optical tweezers [19,20], imaging [21,22], atomic trapping [23], and quantum information [24].

The work presents LC QPs with positive or negative integer and semi-integer q values, using the axially symmetric photo-alignment method. The shift in the phase-retardation that is caused by fabricated LC QP devices when a voltage is investigated and the fabricated QPs are shown to be useful for the electrical tuning the beam shapes. Moreover, the polarization distribution of a linearly Gaussian beam after it has been modulated by an LC QP is analyzed. Numerical simulation was performed to confirm the LC alignment structures on the substrates and for comparison of the results with observations made under a crossed-polarized optical microscope (POM). The results of the simulation are consistent with the experimental results. The LC QPs were utilized to modulate a linearly polarized Gaussian laser beam and the distributions of linear polarization in three modes upon the application of a suitable voltage were studied. The LC QP device can be utilized as a beam shape modulator and a spatial polarization converter in diffractive optics and imaging systems.

Experiments

The dye-doped LC (DDLC) consisted of 99 wt% LC (E7 from Merck) and 1 wt% azo-dye (Methyl Red from Aldrich) [25]. An empty cell was fabricated using two indium-tin-oxide-coated glass slides without surface treatment, separated by 12 μm ball spacers. A homogeneous mixture was injected into this empty cell to produce a DDLC sample. Figure 1 shows the setup for fabricating the sample. A diode-pump solid state laser (DPSS; λ = 532 nm) with a power intensity of 50 mW/cm² was used as a pumping beam source. The illumination duration was ~30 minutes. The light beam was polarized by the combination of a quarter-wave plate and a rotatable linear polarizer. The linearly polarized laser beam was then passed through a beam expander, focused into a line-shaped beam with a width of approximately 200 μm by a cylindrical lens, and passed through an aperture onto the rotating DDLC cell. The rotatable polarizer and the sample were attached to rotating step-motors that were controlled by a computer. The polarizer and sample were rotated at angular velocities of ω_p and ω_s, respectively. Photo-alignment was performed in the present study following the procedure in our previous report [26]. Varying the ratio of the angular velocities of the two motors, ω_p to ω_s and yields various q values. The q value is given by $q = 1 \pm \frac{ω_{p}}{ω_{s}}$ , where the “+” and “−” signs indicate the same and opposing directions of rotation of the two motors, respectively.

Fig. 1 Sample fabrication setup.

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Q-plate structures

Equation (1) gives the orientation profile of the director axis of the LC molecules on the substrates of a QP

α_{(x, y)} = α_{(ϕ)} = q ϕ + α_{(0)},

where

ϕ

= arctan (y/x) is the transverse azimuthal coordinate;

α_{(x, y)}

is the orientation of the fast axis at a given azimuthal angle

ϕ

on the component; q is the rate of change in

α_{(x, y)}

with respect to

ϕ

;

α_{(0)}

is the orientation of the fast axis at

ϕ

= 0, and q is an integer or a half integer that is determined by the relative rotational velocities of the two motors. Figures 2(a)–2(c) present images of the LC QPs observed under the cross-POM, the results of the numerical simulation, and the schematic LC director profiles for q values −2, −1.5, 1.5, and 2, respectively. Figure 2 clearly reveals that the POM images are consistent with the simulated images. Notably, the number of modulations at each q value is six for q =

\pm

1.5 and eight for q =

\pm

2. The reason for the dependence of the modulation number on the q-value, but its lack of dependence on the sign can be determined from the LC director distribution-profile in Fig. 2(c). The LC QPs operate as half-wave plates [9]; thus, a linearly polarized input field will be converted into another linearly polarized output field but being rotated an azimuthal angle. Notably, no significant aging effect is found for a six-month old sample after being exposed by ambient light. The temporal stability of these devices is good.

Fig. 2 (a) POM images of LC QPs with q = 2, 1.5, −1.5, and −2; (b) numerically simulated POM images with q = 2, 1.5, −1.5, and −2; (c) schematics of LC director.

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Figure 3 shows the setup to analyze the optical modulation properties of these LC QPs. The transmittance of the DDLC cell after being pumped by a DPSS laser with a power intensity of 50 mW/cm² for 30 mins were measured to be 17.6%, 95.7% at 532 nm and 632.8 nm, respectively. In order to avoid the absorption, the measurement was performed using a y-polarized Gaussian He–Ne laser (λ = 632.8 nm) with an intensity of 20 mW/cm² to pass through the center of an LC QP, to which was applied a suitable AC voltage (1 kHz), and the diffracted pattern was projected on a screen. The diameter of the laser beam was ~1.5 mm.

Fig. 3 Experimental setup for analyzing liquid crystal QP devices.

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Figure 4 shows the images of the beams that were modulated by LC QPs with q values of $\pm$ 1.5 and $\pm$ 2 under suitable applied voltages. LC QPs with the same absolute q values identically modulate a linearly polarized TEM₀₀ Gaussian beam, as revealed in the POM images. Equation (1) also describes the alignment structures in the LC QPs. The LC molecules at a particular azimuthal angle on the substrate with the same absolute q values equally retard the phase of the linearly polarized TEM₀₀ Gaussian beam. As the applied voltage is varied, each LC QP has three modes, which are a donut-shaped mode and two beam structures with 2q dark spots. The dark spots are resulting from the phase singularities that caused by far-field beam interference, and can be calculated using the Fraunhofer diffraction [27] for the propagation of a linearly polarized Gaussian beam through the center of LC QP. The shift in the phase-retardation of the beam through the LC cell is described as $Δ ϕ = 2 π d (n_{e f f (V)} - n_{o}) / λ$ , where d is the cell gap; $n_{e f f (V)}$ is the effective LC refractive index, and λ is the wavelength of the incident beam. When a suitable voltage that yield $Δ ϕ = π$ was applied to the LC, the LC QP device can be considered to be as a single birefringent plate with a homogeneous phase retardation of $π$ (half-wave plate) for light that propagates in the longitudinal z direction, but with a transversely inhomogeneous optical axis in the x–y plane, as described in Eq. (1) and presented in Fig. 2(c). Under such conditions, the LC QP functions as a half-wave LC QP, and a donut-shaped beam is produced. Changing the applied voltage shifts the value of $Δ ϕ$ from $π$ , and results in the formation of the other two beam shapes, which have 2|q| dark spots. When the applied voltage is high enough (~15 V), the LC molecules are reoriented almost perpendicularly to the substrates, and the beam shape reverts to Gaussian.

Fig. 4 Various beam shapes were obtained by modulation by LC QP with q values of $\pm$ 1.5 and $\pm$ 2 when suitable voltages were applied.

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The theoretical optical–field distributions, E_x and E_y, are numerically simulated in 1D-DIMOS software and MATLAB [28] for comparison with the experimental results. The Jones matrix for an LC QP is described by Eq. (2), where $ϕ$ is the azimuthal angle of the component; d is the thickness of the LC layer; $n_{e f f (V)}$ is the effective refractive index, which can be modulated by applying the voltage V; $λ$ is the wavelength of the Gaussian beam.

M_{(ϕ, V)} = [\begin{matrix} \cos (ϕ) & - \sin (ϕ) \\ \sin (ϕ) & \cos (ϕ) \end{matrix}] [\begin{matrix} 2 π d \cdot n_{e f f (V)} / λ & 0 \\ 0 & 2 π d \cdot n_{o} / λ \end{matrix}] [\begin{matrix} \cos (ϕ) & \sin (ϕ) \\ - \sin (ϕ) & \cos (ϕ) \end{matrix}],

The optical field of the incident Gaussian beam

{\vec{E}}_{i n}

is defined by Eq. (3):

{\vec{E}}_{i n} = E_{1 y} \vec{j} = E_{1 y} [\begin{matrix} 0 \\ 1 \end{matrix}], and E_{1 y} = E_{0} \frac{ω_{0}}{ω (z)} exp [(- \frac{r^{2}}{ω^{2} (z)} - \frac{j k r^{2}}{2 R (z)} - j k z)],

where j is the unit imaginary number; k is the wave-number of the electromagnetic field, and

r = \sqrt{x^{2} + y^{2} + z^{2}}

is the distance over which the beam has propagated.

E_{0}

is the maximum amplitude of the electric field;

ω^{2} (z) = ω_{0}^{2} (1 + z^{2} / z_{0}^{2})

;

ω_{0}

is the beam waist; z is the Rayleigh range;

z_{0} \equiv π ω_{0}^{2} n / λ

, and

R (z) = z (1 + z_{0}^{2} / z^{2})

is the radius of curvature of the Gaussian beam. The modulated optical field,

{\vec{E}}_{o u t}

is re-decomposed using two orthogonal electric components,

E_{2 x}

and

E_{2 y}

, which can be expressed as Eq. (4):

{\vec{E}}_{o u t} = M_{(ϕ, V)} E_{1 y} [\begin{matrix} 0 \\ 1 \end{matrix}] = [\begin{matrix} E_{2 x} \\ E_{2 y} \end{matrix}] = E_{2 x} [\begin{matrix} 1 \\ 0 \end{matrix}] + E_{2 y} [\begin{matrix} 0 \\ 1 \end{matrix}],

The far-field optical field that is projected on the screen can be calculated using Fraunhofer diffraction [27], and is given by Eq. (5):

U (ε, η) = \frac{e^{j k z} e^{j k (ε^{2} + η^{2}) / 2 z}}{j λ z} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} U (x, y) \exp [- j \frac{2 π}{λ z} (ε x + η y)] d x d y,

U (x, y)

and

U (ε, η)

are defined as the optical fields before and after the propagation of a beam by a distance z, respectively. The two orthogonal electric fields

E_{2 x}

and

E_{2 y}

after modulation by LC QP as

U (x, y)

are substituted into Eq. (5). The far-field electric field can be obtained as shown in Fig. 5. Notably,

n_{e f f (V)}

is the voltage-dependent effective refraction index of LC, which ranges from the ordinary refractive index (1.5216) to the extra-ordinary refractive index (1.7462) of E7. The applied voltage for modulation was verified;

n_{e f f (V)}

from 1.7462 to 1.59 was applied and the shift in phase-retardation shift was from 1.6

π

to 0.8

π

. Only

E_{2 y}

was affected. The

E_{2 x}

intensity patterns retained a number of 4q bright radial petals. For the DDLC q-plate with q =

\pm

1.5, comparing the simulated patterns (first column and from top to bottom in Fig. 5) with the experimental ones (top row and from left to right in Fig. 4), it is seen that the simulated patterns of modulated beam shape under various phase shifts agree well with the experimental ones of the sample under the applied voltage. The three dark spots merge into one, and then separate again into three dark spots as the applied voltage is increased. Similar results were obtained for the q-plate with q =

\pm

2 (comparing fourth column and from top to bottom in Fig. 5 with second row and from left to right in Fig. 4). Notably, the switching of the modulated beam patterns was repeatable, and reversible.

Fig. 5 Simulated intensity distribution, obtained by summing |(E)_2x|² and |(E)_2y|², when a y-polarized Gaussian beam was incident onto the LC QPs for various phase-retardation shifts at various applied voltages.

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Figure 6 shows the simulated polarization distribution profiles of an incident y-polarized Gaussian beam that were modulated by LC QPs with q values of $\pm$ 1.5 and $\pm$ 2 under an applied voltage that yields a phase-retardation shift, as determined. The cell gap was set to 12.7 μm. The LC QPs with the same absolute q values yielded equally retarded the phase of the linearly polarized incident beams. When the LC QP shifted the phase-retardation by $π$ , the polarized director varied continuously at the center of the profile of the modulated Gaussian beam; the dark spots in the profile are attributed to phase singularities. When the voltage had been applied, and phase-retardation had therefore been shifted, the number of phase singularities was 2|q|.

Fig. 6 Numerically simulated polarization distributions in transverse plane, following modulation by an LC QP with q = $\pm$ 1.5 and q = $\pm$ 2.

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Conclusion

This work demonstrates that LC QPs with the same absolute q values equally shift the phase retardation of a linearly polarized beam even when if their alignment structures vary. The shape of a linearly polarized TEM₀₀ Gaussian beam can be modulated by an LC QP to one of three different shapes, donut-shaped and two shapes with a number of 2|q| dark spots. The dark spots can be explained using Fraunhofer diffraction, which occurs when a linearly polarized Gaussian beam propagates through the center of an LC QP. Varying the applied voltage shifts the phase retardation $Δ ϕ$ , and leads to the formation of various beam shapes. The beam shape and polarization distribution profiles of an incident linearly polarized Gaussian beam that is modulated by an LC QP were simulated. The experimental far-field patterns are consistent with the simulated. These devices can be used as beam-shaped modulators and spatial polarization converters in diffractive optics and imaging systems. Also, the proposed beam-shape manipulation can be used in an advanced optical tweezers system.

Acknowledgments

The authors would like to thank the National Science Council of Taiwan for financially supporting this research under Grant No. NSC 101-2112-M-006-011-MY3. Additionally, this work is partially supported by the Top University Program of the National Cheng Kung University as well.

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Modulation of shape and polarization of beam using a liquid crystal q-plate that is fabricated via photo-alignment

Abstract

Introduction

Experiments

Q-plate structures

Conclusion

Acknowledgments

References and links

Cited By

Figures (6)

Equations (5)

Optics Express