Electrically tunable gradient-index lenses via liquid crystals: beyond the power law

Yi-Hsin Lin; Wei-Cheng Cheng; Victor Reshetnyak; Victor Reshetnyak; Hao-Hsin Huang; Ting-Wei Huang; Chang-Chiang Cheng; Yung-Hsun Wu; Chiu-Lien Yang

doi:10.1364/OE.504586

Optics Express
Vol. 31,
Issue 23,
pp. 37843-37860
(2023)
•https://doi.org/10.1364/OE.504586

Electrically tunable gradient-index lenses via liquid crystals: beyond the power law

Yi-Hsin Lin, Wei-Cheng Cheng, Victor Reshetnyak, Hao-Hsin Huang, Ting-Wei Huang, Chang-Chiang Cheng, Yung-Hsun Wu, and Chiu-Lien Yang

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Yi-Hsin Lin, Wei-Cheng Cheng, Victor Reshetnyak, Hao-Hsin Huang, Ting-Wei Huang, Chang-Chiang Cheng, Yung-Hsun Wu, and Chiu-Lien Yang, "Electrically tunable gradient-index lenses via liquid crystals: beyond the power law," Opt. Express 31, 37843-37860 (2023)

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Abstract

In this study we present an investigation of electrically tunable progressive lenses utilizing liquid crystals (LC). We introduce a polarized progressive LC lens capable of dynamically adjusting its focal length, functioning as either a positive or negative lens. Our findings reveal that the spatial distribution of lens power within the progressive LC lens, ranging from +4D to -3D, far surpassing the range of -0.87D to +0.87D which one may expect within the parabolic wavefront approximation. For a lens with a 30 mm aperture a total tunable range is 7.6 D (from +5.6D to -2D) which is 4.75 times larger than the traditional parabolic prediction∼1.6D (from +0.8D to -0.8D). This study not only challenges conventional limitations set by optical phase differences in gradient-index LC lenses (the power law) but also ushers in a new possibility for ophthalmic applications. The profound insights and outcomes presented in this paper redefine the landscape of LC lenses, paving the way for transformative advancements in optics and beyond.

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Supplementary Material (6)

Name	Description
Data File 1	Data File 1 for the coefficients of each polynomial fitting in Figs. 7(a) to 7(f).
Data File 2	Data File 2 for the coefficients of each parabolic fitting in Figs. 7(a) to 7(f).
Visualization 1	Visualization 1 corresponding to operation from Figs. 6(a) to 6(c).
Visualization 2	Visualization 2 corresponding to operation from Figs. 6(d) to 6(e).
Visualization 3	Visualization 3 corresponding to operation from Figs. 10(c) to 10(f).
Visualization 4	Visualization 4 corresponding to operation from Figs. 10(g) to 10(j).

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Fig. 1. General concept of wavefront modulation by a LC lens. Grey dotted arrows stand for oscillation of electric field (i.e. x-linearly polarization). Black straight lines are plane waves.

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Fig. 2. Structure of the spatially extended LC lens. (a) Without applied voltage, the LC molecules in LC layer are parallel to x-axis. The plane wave of entering x-linearly-polarized light remains unmodulated. (b)When applied inhomogeneous electric fields to each LC layer, the plane wave of entering x-linearly-polarized light is modulated by a gradient distribution of optical path difference contributed from all LC layers.

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Fig. 3. Phase profile of the sample. (a)-(d) are positive lenses and (e)-(h) are negative lenses. The corresponding conditions of applied voltages and frequencies are listed in Table 1. The aperture size is 20 mm. λ=543 nm.

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Fig. 4. The optical path difference (OPD) as a function of x-pupil coordinate. The curves of (a) to (h) are obtained from Figs. 3(a) to 3(h), respectively.

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Fig. 5. The optical path difference (OPD) as a function of x-pupil coordinate. (a) Change the applied voltage conditions of (V₁, V₂, f₁) in |r|<5 mm at fixed (V₃, V₄, V₅, f₂) = (20V_rms, 5V_rms, 20V_rms, 10 Hz) in 5 mm<|r|< 10 mm. (b) Change the applied voltage conditions of (V₃, V₄, V₅, f₂) in | 5 mm<|r|< 10 mm at fixed (V₁, V₂, f₁) = (20V_rms, 6V_rms, 2200 Hz) in |r|<5 mm. (c) Change the applied voltage conditions of (V₁, V₂, f₁) in |r|<5 mm at fixed (V₃, V₄, V₅, f₂) = (5V_rms, 30V_rms, 30V_rms, 10 Hz) in 5 mm<|r|< 10 mm.(d) Change the applied voltage conditions of (V₃, V₄, V₅, f₂) in | 5 mm<|r|< 10 mm at fixed (V₁, V₂, f₁) = (3 V_rms, 20 V_rms, 200 Hz) in |r|<5 mm.

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Fig. 6. Image performance of the LC sample as (V₁, V₂, f₁, V₃, V₄, V₅, f₂)= (a) (20 V_rms, 6 V_rms, 2200 Hz, 30 V_rms, 5 V_rms, 30 V_rms, 10 Hz), (b) (20 V_rms, 8 V_rms, 2200 Hz, 20 V_rms, 5 V_rms, 20 V_rms, 10 Hz), (c) (20 V_rms, 6 V_rms, 2200 Hz, 15 V_rms, 5 V_rms, 15 V_rms, 10 Hz), (d) (1 V_rms, 20 V_rms, 200 Hz, 5 V_rms, 30 V_rms, 30 V_rms, 10 Hz),(e) (3 V_rms, 20 V_rms, 200 Hz, 5 V_rms, 20 V_rms, 20 V_rms, 10 Hz), (f) (6 V_rms, 20 V_rms, 2000Hz, 5 V_rms, 30 V_rms, 30 V_rms, 10 Hz) and,(g) (40 V_rms, 40 V_rms, 1000 Hz, 40 V_rms, 40 V_rms, 40 V_rms, 1000 Hz). The lens powers calculated under parabolic approximation are (a) 1.01D, (b) 0.8D, (c) 0.8D, (d) -0.8D, (e)-0.66D, (f) -0.66D, and (g) 0D. Aperture size is 20 mm. (See

Visualization 1

for the operation from (a) to (c) and

Visualization 2

for the operation from (d) to (e).)

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Fig. 7. The OPD as a function of x-pupil coordinate. The voltage conditions of (a) to (f) are identical to the ones in Figs. 6(a) to 6(f), respectively. Assume the OPD function or the wavefront function W(r) is a parabolic function:

$W(r) = {a_2}{r^2} + {a_0}$

, then the lens power is

$- 2{a_2}$

. The lens powers of the parabolic fitting are (a) + 1D, (b) + 0.8 D, (c) + 0.8 D, (d) -0.8 D, (e) -0.66 D and (f) -0.66 D. See

Data File 1

and

Data File 2

for the coefficients of each polynomial fitting and each parabolic fitting.

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Fig. 8. The lens power as a function of x-pupil coordinate. The voltage conditions of (a) to (e) are identical to the ones in Figs. 6(a) to 6(f), respectively. The green lines are the lens power calculated from parabolic functions of OPD in Fig. 7(a) to 7(f).

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Fig. 9. Phase profiles of the sample with three layers for the aperture size of 30 mm. (a) The voltage condition was applied to the bottom LC layer for 10 mm<|r|< 15 mm. (b) The voltage condition was applied to the bottom LC layer and the middle LC layer for 5 mm<|r|< 15 mm. (c) The voltage condition was applied to three LC layers for |r|< 15 mm. (c) is a positive lens. Similarly, we turned on the voltages of the bottom LC layer to the top LC layer using different voltage condition in (d)(e) and (f). (f) is a negative lens. (g) is the phase profile as no applied voltages. The corresponding electric conditions: (V₁, V₂, f₁, V₃, V₄, V₅, f₂, V₆, V₇, V₈, f₃)= (a) (50 V_rms, 50 V_rms, 200 Hz, 50 V_rms, 50 V_rms, 50 V_rms, 200 Hz, 20 V_rms, 5 V_rms, 10 V_rms, 10 Hz), (b) (50 V_rms, 50 V_rms, 200 Hz, 15 V_rms, 5 V_rms, 5 V_rms, 40 Hz, 20 V_rms, 5 V_rms, 10 V_rms, 10 Hz), (c) (35 V_rms, 5 V_rms, 2000Hz, 15 V_rms, 5 V_rms, 5 V_rms, 40 Hz, 20 V_rms, 5 V_rms, 10 V_rms, 10 Hz), (d) (50 V_rms, 50 V_rms, 200 Hz, 50 V_rms, 50 V_rms, 50 V_rms, 200 Hz, 8 V_rms, 25 V_rms, 15 V_rms, 10 Hz), (e) (50 V_rms, 50 V_rms, 200 Hz, 6 V_rms, 13 V_rms, 6 V_rms, 10 Hz, 8 V_rms, 25 V_rms, 15 V_rms, 10 Hz), (f) (4 V_rms, 27 V_rms, 100 Hz, 6 V_rms, 13 V_rms, 6 V_rms, 10 Hz, 8 V_rms, 25 V_rms, 15 V_rms, 10 Hz). The aperture size is 30 mm. λ=543 nm.

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Fig. 10. (a) The corresponding OPD of Figs. 9(c) and 9(f). (b) is the corresponding lens power as a function of x-pupil coordinate in (a). (c)Image performance at voltage off. Images when we turned on (d) the bottom LC layer, (e) both of the bottom and middle LC layers, and (f) all three LC layers. (d)-(f) The sample is a positive lens. The corresponding OPD of (f) is blue line in (a). Similarly, (g)-(j) are the image performance when we turned off and on the voltage from the bottom LC layer, two LC layers, three LC layers. The corresponding OPD of (j) is red line in (a). (See

Visualization 3

for the operation from (c) to (f) and

Visualization 4

for the operation from (g) to (j).)

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Fig. 11. (a) The corresponding 2 dimensional OPD and (b) 2D map of the lens power of Figs. 6(c). (c)The corresponding 2 dimensional OPD and (d) 2D map of the lens power of Figs. 6(f).

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Tables (1)

Table 1. The list of corresponding conditions of applied voltages and frequencies for Figs. 3(a)-(h). The lens power is calculated from $P_{l e n s p o w e r} = 2 \cdot O P D^{'} / r_{0}^{2}$ . OPD’= maximum difference of OPD in Fig. 4.

View Table

Equations (17)

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t_{L C l e n s} = p (x, y) \cdot e^{j \cdot Φ_{1} (x, y)} \cdot e^{j \cdot Φ_{2} (x, y)} \cdot \dots \cdot e^{j \cdot Φ_{q} (x, y)} = p (x, y) \cdot \prod_{q} e^{j \cdot Φ_{q} (x, y)} = e^{j \cdot Σ_{}^{} Φ_{q} (x, y)}

Σ_{q = 1}^{N} Φ_{q} (x, y) = k_{0} \cdot W(x, y) .

W (ρ, θ) = \sum_{i} C_{i} \cdot Z_{i} (ρ, θ) = \sum_{i} C_{i} \cdot R_{\tilde{n}}^{| m |} (ρ) \cdot Θ^{m} (θ) .

R_{\tilde{n}}^{| m |} (ρ) = \sqrt{\tilde{n} + 1} \cdot \sum_{s = 0}^{(\tilde{n} - | m |) / 2} \frac{{(- 1)}^{s} \cdot (\tilde{n} - s)!}{s! \cdot [(\tilde{n} + m) / 2 - s]! \cdot [(\tilde{n} - m) / 2 - s]!} \cdot ρ^{\tilde{n} - 2 s},

Θ^{m} (θ) = {\begin{cases} \sqrt{2} \cdot \cos | m | θ & (m > 0) \\ 1 & (m = 0) \\ \sqrt{2} \cdot \sin | m | θ & (m < 0) \end{cases} .

P (r, θ) = \sum_{i} \frac{- 1}{r} \cdot \frac{\partial W(r, θ)}{\partial r} = \sum_{i} c_{i} \cdot \frac{- 1}{r} \cdot \frac{\partial Z_{i} (r, θ)}{\partial r}

W (r) = Σ_{m = 0}^{N} a_{m} \cdot r_{}_{}^{m},

P (r) = - Σ_{m = 1}^{N} a_{m} \cdot m \cdot r^{m - 2}

W (r)_{p a r a b o l a} = d \cdot (\frac{n_{b} - n_{c}}{{r_{o}}^{2}}) \cdot {r^{2}}_{}

P_{p a r a b o l a} = - 2 d (\frac{n_{b} - n_{c}}{{r_{o}}^{2}}) .

P_{s p a t i a l} = \frac{2 \cdot Δ n \cdot \sum_{i}^{N} d_{i}}{{r_{N}}^{2}},

W (r_{1}) - W (r = 0) = a_{2} \cdot {r_{1}}^{2} = - Δ n \cdot d_{1},

W (r_{2}) - W (r_{1}) = a_{2} \cdot {r_{2}}^{2} - a_{2} \cdot {r_{1}}^{2} = - Δ n \cdot d_{2},

W (r_{N}) - W (r_{N - 1}) = a_{2} \cdot {r_{N}}^{2} - a_{2} \cdot {r_{N - 1}}^{2} = - Δ n \cdot d_{N} .

\frac{r_{N}}{r_{1}} = \sqrt{\frac{\sum_{i = 1}^{N} d_{N}}{d_{1}}} = \sqrt{N}

Δ r \equiv r_{N} - r_{N - 1} = (\sqrt{N} - \sqrt{N - 1}) \cdot r_{1} .

P ={-} 2 \cdot a_{2} = \frac{- 2 \cdot Δ n \cdot \sum_{i = 1}^{N} d_{i}}{{r_{N}}^{2}} = \frac{- 2 \cdot Δ n \cdot N \cdot d_{1}}{{(\sqrt{N} \cdot r_{1})}^{2}} = \frac{- 2 \cdot Δ n \cdot d_{1}}{{r_{1}}^{2}} .

Voltage condition: (V₁, V₂, f₁, V₃, V₄, V₅, f₂)		Calculated lens power, Diopter
(a)	(20V_rms, 6V_rms, 2200 Hz, 30V_rms, 5V_rms, 30V_rms, 10 Hz)	1.01
(b)	(20V_rms, 6V_rms, 2200 Hz, 10V_rms, 5V_rms, 10V_rms, 10 Hz)	0.67
(c)	(20V_rms, 8V_rms, 2200 Hz, 20V_rms, 5V_rms, 20V_rms, 10 Hz)	0.80
(d)	(20V_rms, 18V_rms, 2200 Hz, 20V_rms, 5V_rms, 20V_rms, 10 Hz)	0.44
(e)	(1V_rms, 20V_rms, 200 Hz, 5V_rms, 30V_rms, 30V_rms, 10 Hz)	-0.81
(f)	(14V_rms, 20V_rms, 200 Hz, 5V_rms, 30V_rms, 30V_rms, 10 Hz)	-0.47
(g)	(3V_rms, 20V_rms, 200 Hz, 5V_rms, 20V_rms, 20V_rms, 10 Hz)	-0.66
(h)	(3V_rms, 20V_rms, 200 Hz, 5V_rms, 12.5V_rms, 12.5V_rms, 10 Hz)	-0.52

Abstract

Supplementary Material (6)

Data availability

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Figures (11)

Tables (1)

Equations (17)

Optics Express