3D Optical Wedge and Movable Optical Axis LC Lens

Abstract

Current liquid crystal (LC) lenses cannot achieve lossless arbitrary movement of the optical axis without mechanical movement. This article designs a novel bottom electrode through simulation and optimization, which forms a special LC lens with an Archimedean spiral electrode, realizing a 3D LC wedge and an arbitrarily movable LC lens. When only the bottom electrode is controlled, it achieves a maximum beam steering angle of 0.164°, which is nearly an order of magnitude larger than the current design. When the top and bottom electrodes are controlled jointly, a 0.164° movement of the lens optical axis is achieved. With focal length varies, the movement of the optical axis ranges from zero to infinity, and the lens surface remains unchanged during movement. The focus can move in a 3D conical area. When the thickness of the LC layer is 30 μm, the fastest response time reaches only 0.635 s, much faster than now.

1. Introduction

Nowadays, tunable optical wedges and optical axis motion (OAM) lenses are capable of achieving optical image stabilization (OIS) [1,2,3,4], auto-focus [5], super-resolution imaging [6,7,8], 3D reconstruction [9,10,11], and optical tweezers [12,13], etc. Meanwhile, the movement of traditional solid lenses needs mechanical movement, thereby increasing power consumption, occupying space, and presenting low tuning efficiency [14].

Traditional adaptive optics—a technology aimed at improving the performance of optical systems—can indeed achieve beam steering [15] and focusing [16,17,18], but it requires the control of multiple mirrors to construct a deformable mirror surface, resulting in high costs and complex structure. A comprehensive system is also indispensable for real-time feedback, including a wavefront detector, wavefront controller, and wavefront corrector [19], which leads to a large size. Therefore, it is usually used in fields such as astronomy, which requires high precision and multi-discipline. However, LC lenses have significant advantages in terms of miniaturization, are lightweight, and have a low cost when applied voltage of less than 10 V [20]. The millimeter-level dimensions and micron-level thickness render the LC lens portable and suitable for mobile phones and other small devices, making it perfectly convenient in life.

LC devices are the typical representative of novel adaptive optics and function as a phase modulation element. The researches encompass LC beam deflectors [21,22] and LC zoom lenses [23,24]. The beam deflectors are classified into refractive wedges and diffraction gratings [22]. The former generates a small refractive angle yet incurs a relatively small loss of light energy, while the latter exhibits a large angle but has multiple diffraction orders, resulting in a more significant light loss. The research on refractive LC wedges is gradually becoming more comprehensive. Love [25] and Hands [26] have achieved beam steering in 1D with a 0.01° steering angle and in 2D with a 0.024° steering angle. Kotova [27] employed a modal four-electrode device to observe the 3D continuous optical deflection experimentally. Nevertheless, the deflection theory was not improved until Xu [28], accomplishing 0.048° beam deflection.

The LC zoom lens, initially proposed by Sato and Ye [20,23,29,30,31], is capable of adjusting the focal length and optical axis without mechanical movement. Nevertheless, the OAM lens exhibits poor performance and significant aberrations, and it has difficulty maintaining shape during movement. Meanwhile, the modal electrodes produce an uneven and unstable voltage distribution. With the advancement of LC pattern electrodes with micro-structured [32,33,34,35,36], the voltage across the aperture can be precisely characterized by the expression. However, independent research on LC devices has resulted in low utilization efficiency of layer electrodes.

Therefore, through multi-simulations and optimizations, this paper proposes a novel LC lens with a movable optical axis that can achieve the movement of focus in 3D space without mechanical movement. It possesses the merits of adjustable focal length, lightweight, compact structure, no mechanical movement, and low power consumption. This paper elaborates on the design principle, working principle, and driving method of the electrodes and fabricates a liquid crystal lens with a liquid crystal layer of 30 μm. The interference fringes and positions during the movement process were recorded, and they were compared with the theoretical values.

2. Principle

2.1. Electrode Structure and Process

For the OAM-LC lens, the device structure is presented in Figure 1a, comprising five parts: glass, top electrode, LC layer, bottom electrode, and glass. The top electrode, as shown in Figure 1b, is an Archimedean spiral with equal spacing, which is designed based on the patterned electrode theory to achieve low aberration focusing [37]. The bottom electrode is designed based on vector synthesis and modal electrode theory [38], generating a planar phase difference to achieve beam steering and focus movement. The bottom electrode structure is shown in Figure 1c. When the top and bottom electrodes are jointly controlled, as depicted in Figure 1d, the focus can move within the 3D space, and its movement range lies within the conical area presented in Figure 1e. The dioptre power is within ±6D, the vibration of the l optical axis ranges from 0.164°, and the movement of the optical axis can reach infinity.

It is worth noting that the bottom electrode is composed of copper (Copper) and zinc oxide (Zinc Oxide); hence, the size and brief manufacturing process of the electrode must be explained. The size of the glass is 25 mm × 20 mm. Since there is an extremely negligible voltage drop across the copper wires, it is sufficient to ensure that they can be connected. The current design of copper wires features a width of 0.5 mm, with the central Zinc Oxide region being 6 mm×6 mm. The total radius of the top electrode is 1 mm, as depicted in Figure 1b, with electrode widths and gaps both 10 μm.

We simulated the voltage distribution by COMSOL, and a perfect oblique planar phase difference was generated in the central 2 mm × 2 mm region within the ZnO area, as shown in Figure 2a,b. This implies that the top electrode generates a parabolic voltage distribution while the bottom electrode generates a planar voltage distribution, as shown in Figure 2c. The voltage difference within the LC cell is determined by the top and bottom electrodes.

The bottom electrode is developed using two photolithographic steps for the deposition of high-resistance copper and zinc oxide films. The rotating speed of the spin-coated photoresist is 6000 RPM. The deposition duration for copper and zinc oxide high-resistance films is approximately 30 min and 35 min, respectively. The thickness of the zinc oxide film is approximately 25 nm. The resistance εr_cu of the copper electrode is approximately 300 Ω, and the ε_rZnO of the zinc oxide region is approximately 1 MΩ. The resistance ratio between the two is approximately 10³. In accordance with the simulation in Figure 2d, when the sheet resistance ratio (ε_r) between the two exceeds 10³, the copper will scarcely generate a voltage drop.

2.2. Function and Working Principle

This structure can realize an optical-axis-movable LC lens (OAM LC lens), which is a highly significant function. OAM lenses can improve the imaging performance of the optical system through adaptive zooming and simultaneously eliminate image shifts induced by the optical axis shift and tilt of the camera. Hence, the system holds potential applications in optical image stabilization (OIS). Thus, the system can realize multiple functions, and the specific functions and theoretical derivations will be expounded in detail in the subsequent sections.

3D Optical wedge: According to the simulation in Figure 2b, the center of the bottom electrode can generate a planar phase profile, which means an optical wedge can be made. Moreover, the steering angle θ_w and azimuth angle θ_a are controlled by two parameters, V_a and p. The θ_w and θ_a are expressed as:

$${\theta }_a=\arctan ({\displaystyle \frac{N\lambda }{Diameter}})=\arctan ({\displaystyle \frac{\Delta n\cdot h_{LC}}{Diameter}})$$

(1)

$${\theta }_a=\arctan ({\displaystyle \frac{\partial \Delta n}{\partial x}}/{\displaystyle \frac{\partial \Delta n}{\partial y}})$$

(2)

N: the number of stripes, λ: laser wavelength; Δn: refractive index difference; h_LC: LC layer thickness.

Zoom imaging: The voltage distribution of the top electrode and bottom electrode are parabolic and planar, respectively, as we can see in Figure 2c.

The voltage difference within the aperture can be expressed as Equation (2):

$$v=V_{max}-m\cdot x^2-t(x+1)$$

(3)

m: the parabolic coefficient (V·mm⁻²); t: the linear coefficient (V·mm⁻¹); V_max: the maximum voltage on the top electrode; V_min: the minimum voltage on the top electrode; x (mm) is the position of a point within the aperture. To generate parabolic and planar phase profiles, all voltage difference values within the aperture must fall within the linear response region of the LC material [39]. If t = 0, the voltage difference within the aperture is a parabolic distribution to be a lens with a small wavefront RMS error (<0.05 λ).

OAM lens: by transforming the Equation (3), the voltage difference can be expressed as Equation (4):

$$v=(V_{max}-t+{\displaystyle \frac{t^2}{4m}})-m(x+{\displaystyle \frac{t}{2m}}{)}^2$$

(4)

From Equation (4), the axis of the parabolic function has shifted, but its shape has not changed with parabolic coefficient m remains. This implies that when the top and bottom electrodes are jointly controlled, they can be OAM LC lenses and remain unchanged during the movement of the optical axis. The movement of the optical axis is obtained.

$$s={\displaystyle \frac{t}{2m}}$$

(5)

s: the movement of the optical axis of the OAM lens. From Equation (5), the optical axis of the lens can move to any position, but if it is too far away, it is meaningless for the 2 mm × 2 mm lens. Therefore, achieving movement within the 2 mm × 2 mm aperture is sufficient.

Offset image shifts caused by optical axis vibration: When photographing images of distant objects, minor shifts in the imaging system induced by external forces will give rise to the vibration of the lens. If the conventional solid lens is still employed, the imaging position will undergo displacement. The prevailing OIS technology is cropping the image. Nevertheless, by utilizing the aforementioned OAM lens, optical image stabilization can be accomplished without any information loss and mechanical movement through the lens moving in the opposite direction. Its working principle is shown in Figure 3a.

The offset of the image can also be explained by the following expression. The image movement is equal to the steering angle multiplied by the focal length. The focal length of the LC lens is expressed as:

$$f={\displaystyle \frac{r^2}{2d\times \Delta n}}$$

(6)

The offset of the image is:

$$\Delta h=f\tan \theta ={\displaystyle \frac{t}{2m}}=s$$

(7)

By combining Equations (5) and (7), the offset of the image is exactly equal to the shift of the lens optical axis, and they compensate for each other.

Overall, this is why the lens can be used for OIS. It can eliminate parallel movement of the optical axis and compensate for image offset caused by optical axis vibration. This shows the multifunctionality of the OAM lens.

2.3. Driving Voltage Condition

Within the aperture, the voltage difference must fall within the linear response region (1.6~2.5 V) of the LC material depicted in Figure 3b. Therefore, it is necessary to satisfy the condition that δV_max − δV_min≤ 5 V–3.2 V. Moreover, the driving voltages require rigorous calculations.

$$\delta V_{max}=V_{max}-m\cdot {(-{\displaystyle \frac{t}{2m}})}^2-t(-{\displaystyle \frac{t}{2m}}+1)=V_{\max }+{\displaystyle \frac{t^2}{4m}}-t\le 5$$

(8)

$$\delta V_{min}=V_{max}-m{r}^2-(r+1)t\ge 3.2$$

(9)

δV_max: the maximum voltage difference in the 2 mm × 2 mm aperture; δV_min: the minimum voltage difference in the 2 mm × 2 mm aperture; r is the radius of the LC lens. Combining Equations (8) and (9), we can obtain the trade-off between m and t as shown in Equation (10):

$$4m^2+(4tr-7.2)m+{(tr)}^2\le 0$$

(10)

Thus, the driving voltage range is expressed as follows:

$$V_{min}\ge 3.2+2tr$$

(11)

$$m\le {\displaystyle \frac{-(4tr-7.2)+\sqrt{7.2\times (7.2-8tr)}}{8}}$$

(12)

In this article, the radius of the OAM LC lens is 1 mm. The trade-off between m and t is depicted in Figure 3c.

3. Experiment and Result

3.1. Experimental Setup

We designed two interference optical paths in Figure 4 and Figure 5 for the experiment. The device in Figure 4 was used to record the fringes to verify the feasibility of the OAM lens. The laser wavelength is 532 nm, and the modulator is used to adjust the intensity of the laser. When the laser passes through the polarizer I, it is decomposed into two orthogonal components. The first one sees the LC ordinary refractive index, which is uniform. However, the second one shows the LC’s extraordinary refractive index, which is affected by the applied voltages. The two components have phase differences; as they pass through the polarizer II, the interference fringes are captured by the CMOS camera.

The setup in Figure 5 was used to capture the image shifts caused by the OAM lens in the imaging system. We place the OAM lens in the imaging system, and the optical wedge will deflect the incident light, causing an image shift.

The prepared LC lens is shown in Figure 5. The left electrodes control the bottom voltage to achieve the OAM. The two left electrodes control the upper voltage and the focusing effect of the LC. The positive nematic LC material used in the experiment is HTW 148700-100, produced by Hecheng Display Technology Co., Ltd. from Yangzhong, Jiangsu China, whose optical birefringence Δn = 0.259. The linear response range of this LC material is measured using the method reported in [40], which is 1.6~2.5 V_rms. Then, the driving voltage difference falls within this range to generate a specific phase profile. The driving voltages are square waves. Therefore, 1.6–2.5 V_rms is equivalent to 3.2–5.0 V_pp, and the voltage mentioned below in the article is the V_pp. The frequency of the driving voltage is all 1 kHz in this work.

3.2. Analysis of Light Deflection Experiment

The driving voltage is related to two parameters, V_a and p. V_a determines the azimuth angle θ_a, and p determines the steering angle θ_w. The interference fringes of the 3D optical wedge are recorded in Figure 6a–f with different driving voltages listed in

Table 1. The driving voltage and parameter of the different azimuth angle.

	(a)	(b)	(c)	(d)	(e)	(f)
Va	0.8 V	0.8 V	0.8 V	0.8 V	0.8 V	0.8 V
p	1.1 V	2.2	3.3	4.4	5.5	6.6
θa	77.6	61.3	43.8	26.3	10.6	0
θw	0.17°	0.17°	0.17°	0.17°	0.17°	0.17°

.

The results shown in Figure 6 demonstrated that the LC optical wedge adjusts from 0–90° as the fringes with different tilt angles. There are approximately 5.5 λ variations within the aperture, which means the wedge angle is 0.0838° from Equation (1).

According to the voltage from Table 1, the ratio of voltage drops across the X (ΔV_x) and Y (ΔV_y) axis is related to the slope of the fringes. Due to the slope of the tilt fringes and the fact that the ratio may be infinite, they were processed by means of the inverse tangent function. A relationship between the azimuth angle and the ratio is plotted in Figure 6g, demonstrating a linear relationship in the form of a proportional function.

The proportional relationship means that the structure has the advantage of controlling the azimuth angle of the beam steering by applying voltage ΔV_y and ΔV_x across the electrode. The linear fit is as high as 0.999, nearly to 1.

The LC optical wedge is placed into the imaging system in Figure 5, and the practicability of the LC optical wedge is verified by the imaging resolution plate in the experiment. Figure 7e is the original state; the surrounding images are shifted by the wedge. The movement in Figure 7a–i is depicted in Figure 7j.

In the optical imaging system shown in Figure 5, the object distance is 700 mm, with a 25 mm lens employed, giving rise to a maximum image shift of 1 mm. According to geometric calculations, the maximum angular deviation in one direction can reach 0.164°.

3.3. Analysis of Off-Axis Focusing Experiment

It should be noted that when applying the voltage to the OAM lens, the trade-off between the m and t needs to be considered according to Equation (12), as shown in Figure 3c.

According to Equations (4) and (5), m determines the dioptre power, and m and t jointly determine the movement of the lens s, which means that the trade-off between the m and t leads a trade-off between the movement s and the dioptre power as depicted in Figure 8. The maximum dioptre power of the lens is ±6D. Once the OAM lens moves to infinity, the dioptre power is nearly 0, the movement can be calculated by Equation (6), and the fringes are parallel.

Four experiments were proposed with different values of m and t to demonstrate the OAM lens. Here, we define S_figure as the real movement of the OAM lens in figures.

3.3.1. m and t at the Critical Condition

When the voltage differences within the aperture entirely fall within the linear response range of the LC material (1.6–2.5 V_rms), the LC lens can achieve the specific function. Hence, the critical conditions of m and n addressed in this article occur when δV_max = 2.5 V_rms and δV_min = 1.6 V_rms, as Equation (12) met the condition of equality. The values of m and t within the critical conditions are a must to guarantee that the voltage difference lies within the linear response range of the LC material. The critical condition of m and t is listed in

Table 2. m and t at the critical condition.

m (V·mm−2)	t (V·mm−1)	s (mm)	Sfigure (mm)	Va (V)
1.70	0.10	0.029	0.03	3.73
1.59	0.20	0.063	0.07	3.37
1.48	0.30	0.101	0.11	3.00
1.37	0.40	0.146	0.16	2.63
1.25	0.50	0.200	0.22	2.27
1.12	0.60	0.268	0.27	1.90
0.97	0.70	0.361	0.38	1.53
0.45	0.80	0.500	0.52	1.17
0.45	0.90	1.000	0.98	0.80

, and the linear response region of the LC material is fully used (δV_max = 5.0 V, δV_min = 3.2 V), ensuring that at least five fringes can be observed in Figure 9.

From Figure 9a–i, the lens shifts from the aperture center. The movement of the OAM lens S_figure is recorded by pixels, compared with the theoretical value s. The aberrations of the OAM lens were recorded to verify its feasibility as a lens in Figure 9j, and the aberrations were all less than 0.1 λ, which demonstrates the good imaging performance of the proposed OAM LC lens. Therefore, when the OAM lens moves unidirectionally at critical conditions, the RMS error is no longer recorded.

3.3.2. Fixed m, Gradual t, Fixed Focal Length OAM Lens with Equal Movement

Fixed m at 0.8, the maximum value of t was 0.8 from the Table 1. According to Equations (4) and (5), fixed m means fixed focal length and a gradual change in t means a gradual change in the movement of the OAM lens. s. t is gradually changed in increments of 0.2. These m and t are listed in

Table 3. The driving voltage and parameter of the OAM.

	(a)	(b)	(c)	(d)	(e)
Vmin (V)	7.3	7.5	7.7	7.9	8.1
Vmax (V)	8.1	8.3	8.5	8.7	8.9
m (V·mm−2)	0.8	0.8	0.8	0.8	0.8
t (V·mm−1)	0	0.2	0.4	0.17°	0.17°
s (mm)	0	0.125	0.250	0.375	0.500
Sfigure (mm)	0	0.126	0.251	3.780	0.495

. The interference fringes of the lens are measured by the setup in Figure 4. Accordingly, the interference fringes of the lens only exhibit movement in Figure 10a–e.

3.3.3. Changing m and t, Fixed OAM Lens with Changed Focal Length

According to Equations (4) and (5), if the optical axis of the OAM lens is to be fixed, both m and t should be adjusted simultaneously to ensure that the position of the optical axis s remains constant. In this part, three fixed optical axis OAMs were recorded in Figure 11. The movements of the OAM lens are s = 0.25 mm, s = 0.5 mm, and s = 1.00 mm. The concrete values of m and t are listed in

Table 4. The driving voltage and parameter of the fixed optical axis.

s (mm)	m (V·mm−2)	t (V·mm−1)	Va (V)
0.25	0.40	0.20	3.37
	0.60	0.30	3.00
	0.80	0.40	2.63
	1.00	0.50	2.27
0.50	0.40	0.40	2.63
	0.50	0.50	2.27
	0.60	0.60	1.90
	0.70	0.70	1.53
1.00	0.30	0.60	1.90
	0.35	0.70	1.53
	0.40	0.80	1.17
	0.45	0.90	0.80

When the optical axis of the OAM lens is fixed, we can observe that the m and t increase simultaneously, which means the number of fringes within the aperture increases. The red dashed line refers to the position of the center of the lens. A series of fringes demonstrated that once the optical axis of the OAM lens is fixed away from the center, the changes in the m and t can only influence the focal length of the lens from Equation (5); it can achieve zoom at a position away from the center of the lens.

3.3.4. Fixed m and t, the OMA Lens Moves within the Aperture in Multiple Directions

Finally, we demonstrated the multi-directional movement of the OAM lens within the aperture. In order to record fringes clearly, the m and t are fixed at 1.0 and 0.6, respectively. The constant movement of the optical axis is s = 0.3 mm. The interference fringes are captured during the optical axis shift in Figure 12a–e.

Unlike the previous unidirectional movement, the applying voltage V_a and p of the eight electrodes remained, but the applying positions needed to be changed. Assuming the driving voltage at the top left corner position is V_leftup, V_leftup was listed in

Table 5. The driving voltage and parameter of optical axis multi-directional movement.

m (V·mm−2)	t (V·mm−1)	s (mm)	Vleftup (V)	p (V)	Azimuth Angle
1.0	0.60	0.30	1.90	2.20	45° (left up)
1.0	0.60	0.30	4.10	2.20	45° (left down)
1.0	0.60	0.30	4.10	0	0°
1.0	0.60	0.30	4.10	2.20	45° (right up)
1.0	0.60	0.30	1.90	2.20	45° (right down)

. The total root-mean-square (RMS) errors of the wavefront are then extracted and are shown in Figure 12f. It can be seen that OAM lenses have a small RMS error during multi-directional movement.

In summary, through the selection of various m and t, we have verified the feasibility of equidistant OAM LC lenses, constant optical axis offset LC lenses, and arbitrary range OAM LC lenses. The theoretical movement amount was compared with the actual OAM, and the consistency of the two datasets verified the effectiveness of the simulation for the actual voltage application situation. The lens can be moved to infinity, which is the horizontal stripe. The maximum optical power is ±6D, and the minimum focal length is 16.7 cm.

3.4. Phase Delay Response Time

When no phase difference is generated, the eight electrodes of the OAM lens can be applied with the same voltages at 0 V and 4.1 V, but under the condition of 0 V, the response time of the OAM lens is too long to take 2–3 s, which will limit its application. However, when applying the voltage at 4.1 V, the response time of the OAM lens is significantly shortened for the LC, which is driven by pre-voltage. In this paper, the thickness of the LC is 30 μm, and response times can reach 0.635 s in Figure 13, which is regularly 2–3 s [21].

The response time is measured based on the intensity changes during the variations of the fringes. The response time of the nematic LC lens is expressed as [41]

$${\tau }_r={\displaystyle \frac{\gamma d^2}{\Delta \epsilon (V^2-V_{th}^2)}}$$

(13)

γ: the visco-elastic coefficient, d: the thickness of the LC layer, V: driving voltage, V_th: threshold voltage. In practical applications, it is possible to use a low-viscosity LC material or reduce the thickness of the LC layer to increase the response speed. Moreover, the overdrive voltage method and undershoot effect can be used to reduce the response time [42].

4. Conclusions and Discussion

In conclusion, an LC lens with a movable optical axis is proposed and experimentally demonstrated firstly without mechanical movement. It can also function as a 3D wedge with a 0.164° steering angle. Experimentally, the vibration of the optical axis ranges from 0.164°, and the movement of the optical axis can reach infinity and remain unchanged during the movement, but there is a trade-off between movement and dioptre power of the OAM LC lens. In addition to the extremely shortened response time, it can be used for OIS. The comparison is listed in

Table 6. Works compared with other typical papers.

LC Devices	Beam Deflectors			OAM Lens			Response Time
	Angle	Dimension	Thickness	Movement	Dimension	Aberration
Hands [26]	0.024°	3D	50 μm	—	—	—	2–3 s
Ye [13]	0.1°	2D	130 μm	—	—	—	—
J.F.Algorri [35]	0.008°	2D	50 μm	—	—	—	—
Feng [37]	—	—	50 μm	1.0 mm	1D	$<0.1 \lambda$	2–3 s
This paper	0.164°	3D	30 μm	0~∞ (firstly)	2D	$<0.1 \lambda$	0.635–3 s

.

For an LC with normal dispersion, the refractive index difference Δn is a function of wavelength.

$$\Delta n=n_e-n_o=a+{\displaystyle \frac{b}{{\lambda }^2}}+{\displaystyle \frac{c}{{\lambda }^4}}$$

(14)

a, b, and c are Cauchy dispersion coefficients related to the liquid crystal material. The maximum dioptre power (OP_max) of the LC lens can be expressed as

$$O{P}_{max}={\displaystyle \frac{2d}{r^2}}(a+{\displaystyle \frac{b}{{\lambda }^2}}+{\displaystyle \frac{c}{{\lambda }^4}})$$

(15)

It can be seen that even if the thickness of the LC layer, the size of the aperture, and the birefringence difference of the LC lens are determined, its dioptre power will still be affected by wavelength. When polychromatic light passes through the LC lens and becomes a converging/diverging spherical wave, it will still be affected by the chromatic dispersion of the LC layer. This causes the focal point of different wavelengths to be separated in different image planes, resulting in the difference in retardation at different wavelengths of the liquid crystal lens. This effect will become more severe as the thickness of the liquid crystal layer and the difference in the birefringence of the liquid crystal lens increase.