Generalized refractive tunable-focus lens and its imaging characteristics

Antonin Miks; Jiri Novak; Pavel Novak

doi:10.1364/OE.18.009034

1. Introduction

Recently the first types of tunable-focus lenses with variable optical parameters appeared on the market [1,2] that give a possibility to design optical systems, which have no analogy in classical systems. The advantage of these active lenses is their capability to change continuously the focal length within a certain range. Using several tunable-focus lenses one can build optical systems which change their parameters (focal length, magnification, etc.) in a continuous way without a need for changing their mutual position. Such lenses with a tunable focal length in a wide range and lens type convertibility make possible to design optical systems with functions that are difficult to combine using conventional approaches. A novel design of lens systems with tunable-focus lenses is promising for future, especially due to a possibility for size reduction, a lower complexity and costs, better robustness, and a faster adjustment of optical parameters of such systems.

Different types of either refractive or diffractive tunable lenses with variable focal lengths were developed in recent years [1–18] and some of them are offered commercially nowadays [1,2]. The technology of tunable-focus lenses is inspired with an active change of optical parameters of the human eye. Several different technical approaches were developed for controlling the focal length of lenses. Tunable-focus lenses can use the principle of electrowetting [1,3–8], the controlled injection of fluid into chambers with deformable membranes [9–11], thermooptical or electroactive polymers [2,12], or voltage-controlled liquid crystals as active optical elements [13–17]. The development of tunable-focus lenses is of great importance for a number of practical applications, ranging from adaptive eyeglasses for vision correction [18] to fast and miniaturized zooming devices in various cameras, camcorders, and mobile phones [19–21].

In this work we focused on analysis of refractive tunable-focus lenses that can be fabricated, for example, using two liquids and electrowetting phenomena, in which an electrically induced change in surface-tension changes the surface curvature of liquid [3,5]. Adjusting the shape of the surface between two immiscible liquids can be used for forming a positive or negative lens. Optical power, shape and material are fundamental optical parameters of the lens which affects its imaging properties [22–26]. Aberrations are essential factors which affect the image quality of the lens. Thus, it is important for designing optical systems composed of tunable-focus lenses to analyze paraxial imaging properties and aberrations of such lenses. Only a few papers address imaging properties and aberration analysis of tunable-focus lenses and their systems [27–29].

The purpose of this work is to show a possible approach for the calculation of fundamental paraxial properties and the third order aberration coefficients of refractive tunable-focus lenses and their combinations into more complex optical systems analogical to classical lenses. We perform a detailed theoretical analysis of different optical elements based on refractive tunable-focus lenses composed of two immiscible liquids with an interface of a variable curvature. The calculation of aberrations and parameters of these elements is presented on several examples. The provided analysis may serve for the initial design of non-conventional optical systems using refractive tunable-focus lenses.

2. Basic formulas for calculation of parameters of refractive tunable-focus lenses

From the optical and technological point of view a simple refractive tunable-focus lens can be most easily designed as an optical system consisting of three optical surfaces, whereas the first and the last surface is planar, and the inner surface has a spherical shape with an adjustable curvature. Such tunable-focus lenses can be fabricated, for example, using two immiscible liquids and electrowetting phenomena, in which an electrically induced change in surface-tension changes the surface curvature of liquid [3,5]. Adjusting the shape of the surface between two immiscible liquids can be used for forming an optical lens. A change in curvature of this inner surface between two liquids by electrowetting leads to a change in the focal distance of the lens. Further, we will not concern with a detail technical realization of tunable-focus lenses. Several variants of refractive tunable-focus lenses were described in literature [1–16] and some of them are being fabricated commercially [1,2]. Such lenses can be used in various interesting applications in practice. We will focus our analysis on a model of fluidic tunable-focus lenses. We do not consider a thickness and material of thin covering planparallel plates which are used in fluidic lenses for separation of liquids from the surrounding media. In the following analysis performed in this work we will deal mostly with an optical design using a thin lens approximation, where we can neglect the influence of thin covering plates and the thickness of lenses. The problem of replacing a thin lens by a thick lens is treated in Ref [30]. An optical scheme of the simple tunable-focus lens is shown in Fig. 1 .

Fig. 1 Simple refractive tunable-focus lens

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The following relations hold for raytracing the paraxial aperture ray through the optical system having K surfaces [22–26]

\begin{array}{l} n_{i}^{'} σ_{i}^{'} = n_{i} σ_{i} + h_{i} (n_{i}^{'} - n_{i}) / r_{i}, h_{i + 1} = h_{i} - d_{i} σ_{i}^{'}, σ_{i + 1} = σ_{i}^{'}, n_{i + 1} = n_{i}^{'}, \\ n_{i}^{'} / s_{i}^{'} - n_{i} / s_{i} = (n_{i}^{'} - n_{i}) / r_{i}, s_{i + 1} = s_{i}^{'} - d_{i}, n_{i + 1} = n_{i}^{'}, i = 1, 2, ..., K, \end{array}

where

σ_{i}

is the paraxial angle of the aperture ray incident at i-th surface of the optical system,

σ_{i}^{'}

is the paraxial angle of the aperture ray refracted at i-th surface,

n_{i}

is the index of refraction in front of the i-th surface,

n_{i}^{'}

is the index of refraction behind the i-th surface,

h_{i}

is the incident height of paraxial aperture ray at i-th surface,

r_{i}

is the radius of curvature of i-th surface,

d_{i}

is the axial distance of the vertex of the i-th surface and the vertex of (i+1)-st surface,

s_{i} = h_{i} / σ_{i}

is the distance of the axial point of the object, which is formed by the part of the optical system in front of the i-th surface, from i-th surface,

s_{i}^{'} = h_{i} / σ_{i}^{'} = h_{i} / σ_{i + 1}

is the image distance of the axial point of the object, which is formed by first i surfaces, from the i-th surface of the optical system. The transverse magnification m is given by the formula

m = \frac{y_{0}^{'}}{y_{0}} = \frac{n_{1} σ_{1}}{n_{K}^{'} σ_{K}^{'}} .

Now, consider imaging of the object at infinity (

σ_{1} = 0

). We obtain using Eq. (1)

φ = (n_{3} - n_{2}) / r_{2}, σ_{4} = h_{1} φ, h_{3} = h_{1} (1 - \frac{d_{2}}{n_{3}} φ),

where φ is the optical power of the tunable-focus lens. We can derive for the focal length

f^{'}

and positions

s_{F^{'}}^{'}

,

s_{F}

of the image and object focal points the following formulas

f^{'} = \frac{1}{φ} = \frac{h_{1}}{σ_{4}} = \frac{r_{2}}{(n_{3} - n_{2})}, s_{F^{'}}^{'} = \frac{h_{3}}{σ_{4}} = f^{'} - \frac{d_{2}}{n_{3}}, s_{F} = - f^{'} + \frac{d_{1}}{n_{2}} .

Equations (4) make possible to calculate fundamental paraxial parameters of the tunable-focus lens. Consider imaging of the point A in the distance

s = s_{1}

from the first surface of the tunable-focus lens, then the image A' is situated in the distance

s^{'} = s_{3}^{'}

from the last surface of the lens. We obtain using Eq. (1)

\frac{s}{f^{'}} = \frac{1}{m} - 1 + \frac{d_{1}}{n_{2} f^{'}}, \frac{s^{'}}{f^{'}} = 1 - m - \frac{d_{2}}{n_{3} f^{'}},

where m is the transverse magnification. Equations (5) enable to calculate s and s' for a given value of the transverse magnification m. Further, it holds the following image equation [22–26]

q q^{'} = - {f^{'}}^{2}, m = - q^{'} / f^{'} = f^{'} / q,

where q is the distance of point A from the object focal point F, and q' is the distance of point A' from the image focal point F'.

3. Third order aberrations of tunable-focus lenses

Aberrations are essential factors which affect the image quality of the lens. Thus, it is very important for designing optical systems composed of tunable-focus lenses to know aberrations of such lenses. Consider that the system of refractive tunable-focus lenses is a rotationally symmetric (Fig. 2 ) consisting of K spherical surfaces. In case we know radii of curvature of lenses, their thicknesses, indices of refraction and distances between individual lenses we can simply calculate aberration coefficients of the third order [22,24,25]. Firstly, we calculate two paraxial (auxiliary) rays through the optical system, namely a paraxial aperture ray and paraxial principal ray. The following relations are valid for raytracing the paraxial principal ray through the optical system having K surfaces

\begin{array}{l} {n^{'}}_{i} {\bar{σ}}^{'}_{i} = n_{i} {\bar{σ}}_{i} + {\bar{h}}_{i} (n_{i}^{'} - n_{i}) / r_{i}, {\bar{h}}_{i + 1} = {\bar{h}}_{i} - d_{i} {\bar{σ}}^{'}_{i}, {\bar{σ}}_{i + 1} = {\bar{σ}}^{'}, n_{i + 1} = n_{i}^{'}, \\ n_{i}^{'} / {\bar{s}}^{'}_{i} - n_{i} / {\bar{s}}_{i} = (n_{i}^{'} - n_{i}) / r_{i}, {\bar{s}}_{i + 1} = {\bar{s}}^{'}_{i} - d_{i}, n_{i + 1} = n_{i}^{'}, i = 1, 2, ..., K, \end{array}

where

{\bar{σ}}_{i}

is the paraxial angle of the principal ray incident at i-th surface,

{\bar{σ}}_{i}^{'}

is the paraxial angle of the principal ray refracted at i-th surface,

{\bar{h}}_{i}

is the incident height of paraxial principal ray at i-th surface,

{\bar{s}}_{i} = {\bar{h}}_{i} / {\bar{σ}}_{i}

is the distance of the image of the entrance pupil, which is formed by the part of the optical system in front of the i-th surface, from i-th surface,

{\bar{s}}_{i}^{'} = {\bar{h}}_{i} / {\bar{σ}}_{i}^{'} = {\bar{h}}_{i} / {\bar{σ}}_{i + 1}

is the distance of the image of the entrance pupil, which is formed by first i surfaces, from i-th surface of the optical system. The meaning of other symbols is the same as in the case of the paraxial aperture ray. The angular magnification in pupils of the optical system can be expressed as

\bar{γ} = {\bar{σ}}_{K}^{'} / {\bar{σ}}_{1}

.

Fig. 2 General optical system

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Denoting $(x_{P}, y_{P})$ as coordinates of the intersection of the ray with the entrance pupil plane, s₁ as the distance of the object plane from the first surface of the optical system, ${\bar{s}}_{1}$ as the distance of the entrance pupil from the first surface of the optical system, and $y_{0}$ as the object height, then transversal ray aberrations $δ x^{'}$ , $δ y^{'}$ of the third order of the rotationally symmetric optical system can be calculated from [22]

\begin{array}{l} δ x^{'} = - \frac{x_{P} (y_{P}^{2} + x_{P}^{2})}{2 {n^{'}}_{K} {σ^{'}}_{K} {(s_{1} - {\bar{s}}_{1})}^{3} σ_{1}^{3}} S_{I} + \frac{2 y_{0} y_{P}^{} x_{P}^{}}{2 {n^{'}}_{K} {σ^{'}}_{K} {(s_{1} - {\bar{s}}_{1})}^{3} σ_{1}^{2} {\bar{σ}}_{1}} S_{I I} - \\ - \frac{y_{0}^{2} y_{P}^{}}{2 {n^{'}}_{K} {σ^{'}}_{K} {(s_{1} - {\bar{s}}_{1})}^{3} σ_{1}^{} {\bar{σ}}_{1}^{2}} (S_{I I I} + H^{2} S_{I V}), \\ δ y^{'} = - \frac{y_{P} (y_{P}^{2} + x_{P}^{2})}{2 {n^{'}}_{K} {σ^{'}}_{K} {(s_{1} - {\bar{s}}_{1})}^{3} σ_{1}^{3}} S_{I} + \frac{y_{0} (3 y_{P}^{2} + x_{P}^{2})}{2 {n^{'}}_{K} {σ^{'}}_{K} {(s_{1} - {\bar{s}}_{1})}^{3} σ_{1}^{2} {\bar{σ}}_{1}} S_{I I} - \\ - \frac{y_{0}^{2} y_{P}^{}}{2 {n^{'}}_{K} {σ^{'}}_{K} {(s_{1} - {\bar{s}}_{1})}^{3} σ_{1}^{} {\bar{σ}}_{1}^{2}} (3 S_{I I I} + H^{2} S_{I V}) + \frac{y_{0}^{3}}{2 {n^{'}}_{K} {σ^{'}}_{K} {(s_{1} - {\bar{s}}_{1})}^{3} {\bar{σ}}_{1}^{3}} S_{V}, \end{array}

where aberration coefficients of the third order can be expressed for the centered optical system of spherical surfaces as [22,24,25]

\begin{array}{l} S_{I} = \sum_{i = 1}^{i = K} h_{i} U_{i}, S_{I I} = \sum_{i = 1}^{i = K} h_{i} U_{i} (\frac{Δ {\bar{σ}}_{i}}{Δ σ_{i}}), S_{I I I} = \sum_{i = 1}^{i = K} h_{i} U_{i} {(\frac{Δ {\bar{σ}}_{i}}{Δ σ_{i}})}^{2}, \\ S_{I V} = \sum_{i = 1}^{i = K} \frac{1}{h_{i}} \frac{Δ (n_{i} σ_{i})}{n_{i} {n^{'}}_{i}}, S_{V} = \sum_{i = 1}^{i = K} [h_{i} U_{i} {(\frac{Δ {\bar{σ}}_{i}}{Δ σ_{i}})}^{2} + \frac{H^{2}}{h_{i}} \frac{Δ (n_{i} σ_{i})}{n_{i} {n^{'}}_{i}}] (\frac{Δ {\bar{σ}}_{i}}{Δ σ_{i}}), \\ and \\ U_{i} = {(\frac{Δ σ_{i}}{Δ (1 / n_{i})})}^{2} Δ (\frac{σ_{i}}{n_{i}}), \\ H = n_{1} σ_{1} y_{0} = {n^{'}}_{K} {σ^{'}}_{K} {y^{'}}_{0} = n_{1} ({\bar{h}}_{1} σ_{1} - h_{1} {\bar{σ}}_{1}) = {n^{'}}_{K} ({\bar{h}}_{K} {σ^{'}}_{K} - h_{K} {\bar{σ}}^{'}_{K}) = konst
. \end{array}

In previous relations we denoted

Δ σ_{i} = {σ^{'}}_{i} - σ_{i} = σ_{i + 1} - σ_{i}

and similarly for other differences. Furthermore, it holds

{\bar{σ}}_{1} = \frac{{\bar{h}}_{1}}{{\bar{s}}_{1}} = \frac{y_{0}}{{\bar{s}}_{1} - s_{1}} .

Individual aberration coefficients of the third order have the following meaning: S _I is the coefficient of spherical aberration, S _II is the coefficient of coma, S _III is the coefficient of astigmatism, S _IV is Petzval coefficient, and S _V is the coefficient of distortion. The quantity H is the Lagrange-Helmholtz invariant. It is evident from previous equations that one can use an arbitrary choice of input parameters $(h_{1}, σ_{1} = h_{1} / s_{1}, {\bar{h}}_{1}, {\bar{σ}}_{1} = {\bar{h}}_{1} / {\bar{s}}_{1})$ for the calculation of the third order aberration coefficients for a given object distance $s_{1}$ and a position ${\bar{s}}_{1}$ of the entrance pupil.

We can apply above-mentioned formulas on an optical element (lens) with a variable focal length which consists of three surfaces. The outer surfaces are planar ( $r_{1} = r_{3} = \infty$ ) and the inner surface is a spherical surface with the radius $r_{2}$ which can be changed in a continuous way. Figure 1 presents an optical scheme of such lens. Using Eq. (9) we obtain after a time-consuming calculation for aberration coefficients of the third order of thin tunable-focus lens (d ₁ = 0, d ₂ = 0) in air (n ₁=1, n ₄=1) the following formulas

\begin{array}{l} S_{I} = h M, S_{I I} = \bar{h} M - H N, S_{I I I} = \frac{{\bar{h}}^{2}}{h} M - 2 H \frac{\bar{h}}{h} N + H^{2} φ, \\ S_{I V} = φ C, S_{V} = \frac{{\bar{h}}^{3}}{h^{2}} M - 3 H \frac{{\bar{h}}^{2}}{h^{2}} N + H^{2} \frac{\bar{h}}{h} φ (3 + C), \end{array}

where

M = {(h φ)}^{3} A + σ {(h φ)}^{2} (4 B - 1) + σ^{2} (h φ) (3 + 2 C),

N = {(h φ)}^{2} B + σ (h φ) (2 + C) .

Functions

A = A (P, Q)

,

B = B (P, Q)

, and

C = C (P, Q)

can be expressed as

A = 1 + \frac{2 P - Q}{Q {(P - Q)}^{2}}, B = 1 + \frac{1}{Q (P - Q)}, C = \frac{1}{P Q},

where

P = n_{2}

,

Q = n_{3}

. As one can see from Eqs. (10-13) we expressed the third order aberration coefficients using three parameters, which depend only on refractive indices of fluids forming the tunable-focus lens and do not depend on the optical power φ of the lens. The graph which presents the dependence of functions A(P,Q) and B(P,Q) on parameters P and Q (refractive indices) is shown in Fig. 3 for the value P = 1.38.

Fig. 3 Graph of functions A(P,Q) and B(P,Q) for value P = 1.38

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Assume now a refractive rotationally symmetric aspheric surface of the second order. The formula of the meridian of a general surface of the second order is within the scope of the accuracy of the third-order aberration theory given by [24]

x = \frac{y^{2}}{2 r_{}} + (1 + b) \frac{y^{4}}{8 r_{}^{3}},

where x and y are the coordinates of an arbitrary point of the lens surface meridian, r is the radius of curvature on the optical axis, and b is the aspheric coefficient that characterizes the shape of the aspheric surface. We can determine the type of the curve by the value of the coefficient b. The curve represents hyperbola, if

- \infty < b < - 1

, parabola, if

b = - 1

, ellipse, if

- 1 < b < \infty

∧ b ≠ 0 or circle, if b = 0. If the inner surface tunable-focus lens is aspheric then we must replace the variable M in Eq. (11) by the following expression

M_{a s f} = M + b {(\frac{h}{r_{2}})}^{3} (Q - P) .

We can determine aberration coefficients of the third order of a tunable-focus lens using Eqs. (10-13) for an arbitrary value of optical power φ and position of the object plane s. We can write

σ^{'} - σ = h φ

for the entering paraxial aperture angle

σ = σ_{1}

and the exiting paraxial aperture angle

σ^{'} = σ_{4}

(Fig. 2), where

φ = (n_{3} - n_{2}) / r_{2}

is the power of the lens. We obtain for a system of K tunable-focus lenses

S_{I} = \sum_{j = 1}^{K} (S_{I})_{j}, S_{I I} = \sum_{j = 1}^{K} (S_{I I})_{j}, S_{I I I} = \sum_{j = 1}^{K} (S_{I I I})_{j}, S_{I V} = \sum_{j = 1}^{K} (S_{I V})_{j}, S_{V} = \sum_{j = 1}^{K} (S_{V})_{j},

where

{(S_{p})}_{j}

(p = I, II, III, IV, V) denotes the aberration coefficient of the j-th element of the optical system. We can provide calculations of the third-order aberration coefficients of an arbitrary optical system of thin tunable-focus lenses using Eq. (8) and Eqs. (10-14). The provided analysis may serve for the initial design of optical systems, and the calculated parameters can be used for its further optimization using optical design software. Chromatic aberrations can be simply calculated by substitution of refractive indices for corresponding wavelengths into Eqs. (8) – (14). The error due to neglecting the finite thickness of lenses is relatively small because the change of aberration coefficients with respect to the lens thickness is approximately few percents.

4. Imaging properties of generalized tunable-focus lens

We derived formulas for a simple refractive tunable-focus lens in the previous text. Now we focus on optical systems designed using several tunable-focus lenses. Optical systems in practice are always composed of several lenses. Every spherical lens is characterized by its radii of curvature of optical surfaces and index of refraction of the optical material. We will deal with an analogy of a classical lens using tunable-focus lenses. One has to use two simple tunable-focus lenses in air (a generalized tunable-focus lens) to obtain the element similar to a classical simple spherical lens. As we can see form Fig. 4 it holds that $r_{1} = r_{3} = r_{4} = r_{6} = \infty$ and $n_{1} = n_{4} = n_{7} = 1$ . We obtain using Eq. (4) for powers $ϕ_{1}$ , $ϕ_{2}$ and the distance d of the object principal plane of the second lens from the image principal plane of the first lens

φ_{1} = 1 / {f^{'}}_{1} = (n_{3} - n_{2}) / r_{2}, φ_{2} = 1 / {f^{'}}_{2} = (n_{6} - n_{5}) / r_{5}, d = d_{2} / n_{3} + d_{3} + d_{4} / n_{5} .

We can derive for the optical power φ, the position of the image focal point

{s^{'}}_{F^{'}}

and the position of the object focal point

s_{F}

[22,24–26]

Fig. 4 Generalized tunable-focus lens

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φ = 1 / f^{'} = φ_{1} + φ_{2} + d φ_{1} φ_{2}, {s^{'}}_{F^{'}} = f^{'} (1 - d / {f^{'}}_{1}), s_{F} = - f^{'} (1 - d / {f^{'}}_{2}) .

It holds for imaging of the point A by a thin ( $d_{i} = 0, i = 1, 2, ..., 5$ ) generalized tunable-focus lens (Fig. 5 )

{σ^{'}}_{1} - σ_{1} = h_{1} φ_{1}, σ_{2} = σ^{'}, {σ^{'}}_{2} - σ_{2} = h_{2} φ_{2}, {σ^{'}}_{2} - σ_{1} = h_{1} φ_{1} + h_{2} φ_{2} = h φ, φ = φ_{1} + φ_{2},

where we set

h = h_{1} = h_{2}

. Generally, we can write for a system of K thin simple tunable-focus lenses in contact in air (

h_{1} = h_{2} = ... = h_{K} = h

)

{σ^{'}}_{K} - σ_{1} = \sum_{i = 1}^{K} h_{i} φ_{i} = h φ .

We will focus on aberration properties of a generalized thin tunable-focus lens. The first tunable-focus lens is described by parameters

A_{1} = A (P = n_{2}, Q = n_{3}), B_{1} = B (P = n_{2}, Q = n_{3}), C_{1} = C (P = n_{2}, Q = n_{3}) .

The second lens is described by parameters

A_{2} = A (P = n_{5}, Q = n_{6}), B_{2} = B (P = n_{5}, Q = n_{6}), C_{2} = C (P = n_{5}, Q = n_{6}) .

A generalized tunable-focus lens (Fig. 4) can be practically realized from two simple tunable-focus lenses (Fig. 1) which are appropriately mutually oriented. Two ways exist of orientation of the second simple tunable-focus lens with respect to the first tunable-focus lens:

\begin{array}{l} 1) n_{2} = n_{5}, n_{3} = n_{6}, A_{1} = A_{2}, B_{1} = B_{2}, C_{1} = C_{2}, \\ 2) n_{2} = n_{6}, n_{3} = n_{5}, A_{1} \neq A_{2}, B_{1} \neq B_{2}, C_{1} = C_{2} . \end{array}

As one can see the relations for calculation of aberration coefficients will be simpler in the first case. The second case is similar to “a classical lens”. We obtain for the object at infinity (

σ_{1} = 0

) using Eq. (11) and Eq. (12)

\begin{array}{l} M_{12} = M_{1} + M_{2} = h^{3} [φ_{1}^{3} A_{1} + (φ - φ_{1})^{3} A_{2} + φ_{1}^{2} (φ - φ_{1}) (3 + 2 C_{2}) + φ_{1} (φ - φ_{1})^{2} (4 B_{2} - 1)], \\ N_{12} = N_{1} + N_{2} = h^{2} [φ_{1}^{2} B_{1} + (φ - φ_{1})^{2} B_{2} + φ_{1} (φ - φ_{1}) (2 + C_{2})] . \end{array}

If we choose a position of the entrance pupil identical with a generalized thin tunable-focus lens (

\bar{h} = 0

), we can express aberration coefficients using Eq. (10) as

S_{I} = h M_{12}, S_{I I} = - H N_{12}, S_{I I I} = H^{2} φ, S_{I V} = φ C, S_{V} = 0.

Now, we require a minimization of spherical aberration of a generalized tunable-focus lens. From necessary condition for the extremum (

\partial S_{I} / \partial φ_{1} = 0

) we obtain using Eq. (19) for the power

ϕ_{1}

the following equation

3 (A_{1} - A_{2} + 4 B_{2} - 2 C_{2} - 4) φ_{1}^{2} + 2 φ (3 A_{2} + 2 C_{2} - 8 B_{2} + 4) φ_{1} + φ^{2} (4 B_{2} - 3 A_{2} - 1) = 0.

By the previous quadratic equation we can calculate the power

ϕ_{1}

of the first lens for the case where the generalized thin tunable-focus lens has minimum spherical aberration.

Fig. 5 Thin generalized tunable-focus lens

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6. Imaging properties of three thin tunable-focus lenses

A cemented doublet appears either as an individual optical system (telescopes, collimators, etc.) or as a part of complex optical systems very frequently in practice. The cemented doublet has three surfaces of different curvature. In case we want to construct its analogy using tunable-focus lenses we have to use three simple tunable-focus lenses (a tunable-focus doublet). We obtain for the object at infinity

σ_{1} = 0, σ_{2} = h φ_{1}, σ_{3} = h (φ_{1} + φ_{2}), φ_{3} = φ - (φ_{1} + φ_{2}) .

Using Eq. (11) and Eq. (12) we can write

M_{123} = M_{1} + M_{2} + M_{3}, N_{123} = N_{1} + N_{2} + N_{3},

M_{123} = h^{3} (k_{1} φ_{1}^{3} + k_{2} φ_{2}^{3} + k_{3} φ_{1}^{2} φ_{2} + k_{4} φ_{1} φ_{2}^{2} + k_{5} φ_{1}^{2} + k_{6} φ_{2}^{2} + k_{7} φ_{1} φ_{2} + k_{8} φ_{1} + k_{9} φ_{2} + k_{10}),

N_{123} = h^{2} (p_{1} φ_{1}^{2} + p_{2} φ_{2}^{2} + p_{3} φ_{1} φ_{2} + p_{4} φ_{1} + p_{5} φ_{2} + p_{6}) .

The coefficients in Eq. (23) and Eq. (24) are given by the following formulas

b_{2} = 4 B_{2} - 1, b_{3} = 4 B_{3} - 1, c_{2} = 3 + 2 C_{2}, c_{3} = 3 + 2 C_{3},

k_{1} = A_{1} - A_{3} + b_{3} - c_{3}, k_{2} = A_{2} - A_{3} + b_{3} - c_{3}, k_{3} = - 3 A_{3} + 3 b_{3} - 3 c_{3} + c_{2},

k_{4} = - 3 A_{3} + 3 b_{3} - 3 c_{3} + b_{2}, k_{5} = φ (3 A_{3} - 3 b_{3} + c_{3}), k_{6} = k_{5}, k_{7} = 2 k_{5},

k_{8} = φ^{2} (- 3 A_{3} + b_{3}), k_{9} = k_{8}, k_{10} = φ^{3} A_{3},

p_{1} = B_{1} + B_{3} - (2 + C_{3}), p_{2} = B_{2} + B_{3} - (2 + C_{3}), p_{3} = 2 B_{3} + (2 + C_{2}) - 2 (2 + C_{3}),

p_{4} = (2 + C_{3}) - 2 B_{3}, p_{5} = p_{4}, p_{6} = φ^{2} B_{3} .

If we choose a position of the entrance pupil identical with a generalized thin tunable-focus lens (

\bar{h} = 0

), we obtain using Eq. (10)

S_{I} = h M_{123}, S_{I I} = - H N_{123}, S_{I I I} = H^{2} φ, S_{I V} = φ C, S_{V} = 0.

If we want to remove spherical and coma aberration of the system of three thin tunable-focus lenses in contact, then conditions

S_{I} = 0

,

S_{I I} = 0

must be fulfilled. We obtain using Eq. (23) and Eq. (24) the following equations

α_{0} φ_{1}^{3} + α_{1} φ_{1}^{2} + α_{2} φ_{1} + α_{3} = 0,

β_{0} φ_{1}^{2} + β_{1} φ_{1} + β_{2} = 0,

where

α_{0} = k_{1}, α_{1} = k_{3} φ_{2} + k_{5}, α_{2} = k_{4} φ_{2}^{2} + k_{7} φ_{2} + k_{8}, α_{3} = k_{2} φ_{2}^{3} + k_{6} φ_{2}^{2} + k_{9} φ_{2} + k_{10},

β_{0} = p_{1}, β_{1} = p_{3} φ_{2} + p_{4}, β_{2} = p_{2} φ_{2}^{2} + p_{5} φ_{2} + p_{6} .

Equations (26) and (27) represent a system of two non-linear equations and values of power

φ_{1}

and

φ_{2}

are the solutions of these equations. If both Eq. (26) and Eq. (27) have the identical solution

φ_{1}

, then their resultant must be equal to zero [31]. We can derive one equation for the unknown value

ϕ_{2}

, which can be solved. The power

φ_{1}

can be calculated by the backward procedure. The resultant R of Eqs. (26) and (27) can be expressed as

R = | \begin{matrix} α_{0} & α_{1} & α_{2} & α_{3} & 0 \\ 0 & α_{0} & α_{1} & α_{2} & α_{3} \\ β_{0} & β_{1} & β_{2} & 0 & 0 \\ 0 & β_{0} & β_{1} & β_{2} & 0 \\ 0 & 0 & β_{0} & β_{1} & β_{2} \end{matrix} | = 0.

We can calculate the resultant (28) from the following formula

\begin{array}{l} R = α_{0} α_{3} (2 β_{0} β_{1} β_{2} - β_{1}^{3}) + α_{0} β_{2} (α_{0} β_{2}^{2} + α_{2} β_{1}^{2} - α_{2} β_{0} β_{2} - α_{1} β_{1} β_{2}) + \\ + α_{3} β_{0} (α_{1} β_{1}^{2} + α_{3} β_{0}^{2} - α_{2} β_{0} β_{1} - α_{1} β_{0} β_{2}) + \\ + β_{0} β_{2} (α_{1}^{2} β_{2} + α_{2}^{2} β_{0} + α_{0} α_{3} β_{1} - α_{1} α_{3} β_{0} - α_{0} α_{2} β_{2} - α_{1} α_{2} β_{1}) = 0. \end{array}

We obtain the power

φ_{2}

by solving Eq. (29), which can be substituted into Eqs. (26) and (27) and the power

φ_{1}

can be calculated. In case we require the optical system with specific values of aberration coefficients

S_{I}

and

S_{I I}

, then previous relations are still valid, only the coefficients k ₁₀ and p ₆ are changed in the following way

k_{10} \to (k_{10} - S_{I} / h^{4}), p_{6} \to (p_{6} + S_{I I} / H h^{2}) .

It can be noted that we can proceed in a similar way even in the case of more complicated optical systems which consist of a larger number of thin tunable-focus lenses. For example, an equivalent of a traditional non-cemented doublet must be composed of four tunable-focus lenses, a triplet must be composed of six tunable-focus lenses (i.e. three generalized tunable-focus lenses), the Petzval lens must be composed of six tunable-focus lenses (i.e. two tunable-focus doublets), etc. It is possible to combine tunable-focus lenses, traditional lenses and optical systems and design hybrid optical systems with variable optical characteristics (e.g. focal length, magnification). The fundamental advantage of optical systems with tunable-focus lenses over classical optical systems is the possibility to continuously change properties of these systems without the need for changing position of individual elements of the optical system.

7. Examples of tunable-focus lenses

We will show several examples of thin tunable-focus lenses in air and we provide a comparison of imaging properties to traditional lens systems. We have chosen two cases of values of refractive indices of fluids in tunable-focus lenses: $(n_{2} = 1.38, n_{3} = 1.55)$ and $(n_{2} = 1.38, n_{3} = 1.99)$ . Furthermore, we consider imaging of the object at infinity ( $σ_{1} = 0$ ) and the entrance pupil is identical with the plane of lenses ( ${\bar{s}}_{1} = 0$ ). Linear dimensions in the following examples are given in mm.

Example 1

We consider a simple thin tunable-focus lens with the optical scheme shown in Fig. 1. Parameters of the lens and the coefficient of spherical aberration S_I is given in Table 1 for both cases of refractive indices.We obtain for the transverse spherical aberration $δ y^{'}$ and longitudinal spherical aberration $δ s^{'}$ the following formulas

δ y^{'} = - \frac{y_{P}^{3}}{2 {f^{'}}^{3}} S_{I} = - \frac{1}{2} \sin^{3} U^{'} S_{I}, δ s^{'} = - \frac{1}{2} \sin^{2} U^{'} S_{I},

where

U^{'}

is the image aperture angle of the optical system. Now, we compare parameters of the thin simple tunable-focus lens with parameters of the classical thin lens in air, which is made from optical glass with the refractive index n and the minimum spherical aberration for the object at infinity. We have for the longitudinal spherical aberration

δ {s^{'}}_{\min}

of the classical lens

δ {s^{'}}_{\min} = - \frac{1}{2} \sin^{2} U^{'} {(S_{I})}_{\min},

where

{(S_{I})}_{\min} = \frac{n (4 n - 1)}{4 (n + 2) {(n - 1)}^{2}} f^{'} .

Using as optical glass Schott BK7 with n = 1.516 (λ = 589 nm) we can calculate for the focal length

f^{'} = 100 mm

the aberration coefficient

{(S_{I})}_{\min} = 205

. If we use optical glass Schott N-LASF46A with n = 1.904 (λ = 589 nm) we obtain

{(S_{I})}_{\min} = 98.7

for the value

f^{'} = 100 mm

. One can see by comparison to the thin tunable-focus lens with the same focal length

f^{'} = 100 mm

that the traditional thin lens has approximately fourteen times lower residual spherical aberration than the first case of the simple thin tunable-focus lens (n ₂ = 1.38 and n ₃ = 1.55). In the second case (n ₂ = 1.38 and n ₃ = 1.99) the tunable-focus lens has almost the same residual spherical aberration as the classical lens from the glass BK7. It is clear from the presented example that the difference (n ₃ – n ₂) between indices of refraction must be relatively large for achieving small residual aberration of the simple tunable-focus lens. It is also evident from Fig. 3.

Table 1. Simple thin tunable-focus lens

View Table | View all tables in this article

Example 2

We consider a generalized thin tunable-focus lens with minimum spherical aberration. The optical scheme of this lens is shown in Fig. 4 and Fig. 5. We can calculate parameters of the lens using Eq. (21). These parameters together with the coefficient of spherical aberration (S_I)_min are presented in Table 2 . As we can see from Table 2 the first solution gives always lower value S_I of spherical aberration.

Table 2. Generalized thin tunable-focus lens

View Table | View all tables in this article

By comparison with a classical lens we can see that the generalized tunable-focus lens with minimum spherical aberration has 2.2 times lower residual spherical aberration than the classical thin lens made from the glass BK7 and approximately the same residual aberration as the lens from the glass N-LASF46A.

Example 3

Now, we consider an optical system of three simple tunable-focus lenses and we choose the focal length equal to 1 mm ( $f^{'} = 1 / φ = 1 mm$ ). By solving Eq. (29), Eq. (27), and Eq. (26) we obtain parameters of the optical system which are shown in Table 3 and Table 4 ( $φ_{3} = φ - φ_{1} - φ_{2}$ ).We provided only four combinations of tunable-focus lenses. Other combinations can be obtained by different orientation of simple tunable-focus lenses, but it is not a goal of this work. As one can see from Table 3 and Table 4 it is possible to design an optical system, which has corrected spherical aberration and coma ( $S_{I} = S_{I I} = 0$ ), using three simple tunable-focus lenses. Such optical system is analogical to the classical cemented doublet which has also corrected spherical aberration and coma. We also provided a verification of presented calculations using the Zemax software that gives the same results of aberration coefficients. The high refractive index n = 1.99 in the previous examples was chosen intentionally in order to show from the theoretical point of view that we need the difference of refractive indices as large as possible for obtaining small values of Seidel coefficients.

Table 3. Three tunable-focus lenses (n₂=n₅=n₈, n₃=n₆=n₉, A₁=A₂=A₃, B₁=B₂=B₃)

View Table | View all tables in this article

Table 4. Three tunable-focus lenses (n₂=n₆=n₈, n₃=n₅=n₉, A₁=A₃, B₁=B₃)

View Table | View all tables in this article

8. Summary

The work presents a possible approach to a general solution of the problem of calculation of fundamental paraxial parameters and the third order aberration coefficients of thin tunable-focus lenses and their combinations into more complex optical systems. It is shown that aberration coefficients of the third order of the thin tunable-focus lens are completely characterized by three functions A, B and C that depend only on refractive indices of fluids forming the tunable-focus lens and do not depend on the position and size of the object and the position of the entrance pupil. These functions are constant for a given type of the tunable-focus lens. A detailed theoretical analysis was performed for a simple tunable-focus lens, a generalized tunable-focus lens, a generalized tunable-focus lens with minimum spherical aberration, and three-element tunable-focus lens (a tunable-focus doublet), which is the equivalent of the classical cemented doublet. The derived equations enable to carry out calculations of all parameters of above-mentioned optical systems and are also fundamental for solving of more complex optical systems using tunable-focus lenses. For example, an analogy of a traditional non-cemented doublet is composed of four tunable-focus lenses, a triplet must is composed of six tunable-focus lenses (i.e. three generalized tunable-focus lenses), the Petzval lens is composed of six tunable-focus lenses (i.e. two tunable-focus doublets), etc. The calculation of parameters of the optical systems with tunable-focus lenses was presented on several examples. The provided analysis may serve for better understanding aberration and imaging properties of the refractive tunable-focus lenses and for the initial design of optical systems using such non-conventional lens systems. Tunable-focus lenses start to be used in various practical applications and in near future these lenses will impact considerably the design of modern non-conventional optical systems, e.g. zoom lenses.

Acknowledgements

This work has been supported by Ministry of Education of Czech Republic by the grant MSM6840770022 and GA 202/09/P553 from Czech Science Foundation.

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$f^{'} = 100 mm$ , $h = f^{'}$ , $r_{1} = r_{3} = \infty$ , $n_{1} = n_{4} = 1$
n ₂	n ₃	A ₁	B ₁	r ₂	S_I
1.38	1.55	28.0119	−2.7951	17.000	2801
1.55	1.38	44.1272	5.2626	−17.000	4413

1.38	1.99	2.0399	0.1762	61.000	204
1.99	1.38	6.0633	2.1879	−61.000	606

$f^{'} = 100 mm$ , $h = f^{'}$ , $r_{1} = r_{3} = r_{4} = r_{6} = \infty$ , $n_{1} = n_{4} = n_{7} = 1$
n ₂	n ₃	n ₅	n ₆	A ₁	A ₂	B ₁	B ₂	r ₂	r ₅	(S_I )_min
1.38	1.55	1.38	1.55	28.0119	28.0119	-2.7951	-2.7951	35.2180	32.8634	592.9
1.55	1.38	1.55	1.38	44.1272	44.1272	5.2626	5.2626	-32.6313	-35.4885	1399.9

1.38	1.99	1.38	1.99	2.0399	2.0399	0.1762	0.1762	122.210	121.791	93.8
1.99	1.38	1.99	1.38	6.0633	6.0633	2.1879	2.1879	-104.473	-146.594	295.6

$f^{'} = 1 mm$ , $h = f^{'}$ , $r_{1} = r_{3} = r_{4} = r_{6} = r_{7} = r_{9} = \infty$ , $n_{1} = n_{4} = n_{7} = n_{10} = 1$ , $S_{I} = S_{I I} = 0$
$ϕ_{1}$	$ϕ_{2}$	n ₂	n ₃	A ₁	B ₁	r ₂	r ₅	r ₈
0.8620	0.5287	1.38	1.99	2.0399	0.1762	0.7076	1.1538	-1.5613
0.3681	0.9641	1.38	1.99	2.0399	0.1762	1.6571	0.6327	-1.8368

$f^{'} = 1 mm$ , $h = f^{'}$ , $r_{1} = r_{3} = r_{4} = r_{6} = r_{7} = r_{9} = \infty$ , $n_{1} = n_{4} = n_{7} = n_{10} = 1$ , $S_{I} = S_{I I} = 0$
$ϕ_{1}$	$ϕ_{2}$	n ₂	n ₃	A ₁	A ₂	B ₁	B ₂	r ₂	r ₅	r ₈
1.0252	2.4505	1.38	1.99	2.0399	6.063	0.1762	2.188	0.5950	-0.249	-0.246
1.0250	-3.128	1.38	1.85	3.2268	8.610	0.1762	2.542	0.4487	0.1502	0.1515

$f^{'} = 100 mm$ , $h = f^{'}$ , $r_{1} = r_{3} = \infty$ , $n_{1} = n_{4} = 1$
n ₂	n ₃	A ₁	B ₁	r ₂	S_I
1.38	1.55	28.0119	−2.7951	17.000	2801
1.55	1.38	44.1272	5.2626	−17.000	4413

1.38	1.99	2.0399	0.1762	61.000	204
1.99	1.38	6.0633	2.1879	−61.000	606

Generalized refractive tunable-focus lens and its imaging characteristics

Abstract

1. Introduction

2. Basic formulas for calculation of parameters of refractive tunable-focus lenses

3. Third order aberrations of tunable-focus lenses

4. Imaging properties of generalized tunable-focus lens

6. Imaging properties of three thin tunable-focus lenses

7. Examples of tunable-focus lenses

Example 1

Example 2

Example 3

8. Summary

Acknowledgements

References and links

Cited By

Figures (5)

Tables (4)

Equations (48)

Optics Express