1. Introduction

Many modern optical imaging systems require elements with variable optical power for auto-focus and optical zoom applications. Particularly interesting are miniature cameras, with clear apertures ranging from 1 to 2 mm, used for cell phones and webcams, where the legacy mechanical approaches are very difficult to implement. The range of optical power variability required, for example for auto focus application in cell phones, should be 10 Diopter (1/m). The duration of such auto focus should be approximately 1 sec. Most importantly, the size of the tunable lens must be as small as possible, ideally having a thickness below 0.5 mm. The corresponding miniaturization of the traditional mechanical elements (such as voice coil motors/VCM) increases their cost and fragility making them unacceptable for low-cost consumer products. Several alternative approaches have been explored to replace those mechanical elements. A variable-focus liquid lens has been demonstrated using changing aperture [1]. Electro wetting effect in conductive immiscible liquids has been also used to obtain focus changes [2]. However in both approaches the variable voltage, delivered to the cell, still causes mechanical movement of liquids. This is highly undesirable since the operation of such systems is very sensitive to environmental conditions. In addition, their cost appears to be comparable or even higher than those for VCMs.

It is presently well known that liquid crystals (LCs) may provide huge electrically controlled refractive index changes without mechanical movements [3]. Multiple attempts have been done to use this phenomenon for building variable focus lenses (see the pioneering works by S. Sato et al. [4]). The focus changes here require the creation of spatially non uniform and dynamically variable refractive index distribution across the LC layer. Such a modulation usually requires either a spatially non uniform LC layer (for example, by using an LC cell with non uniform cell gap [5]) or a spatially non uniform electric field that is applied to a uniform LC layer (see later). The LC orientation being typically obtained by mechanical rubbing, the non uniform LC layer method is very difficult to apply in industrial scale. An example of an “intermediate” type solution is the use of a gradient polymer stabilized LC material system [6]. This approach being relatively simpler is however suffering from light scattering. Those are the reasons why we shall not analyze further such solutions. In the alternative approach of uniform LC layers: the “simplest” method of obtaining a spatially varying electric field is the use of multiple transparent electrodes (such as pixelated Indium Tin Oxide/ITO) distributed on the LC cell substrates [7,8]. However, the fabrication of such structures requires high precision, their electrical driving requires rather complex electronic processing and their operation is degraded by the profile “granularity”, light diffraction and scattering. The optical loss and device cost being among the most critical parameters, this approach cannot be industrially adopted.

The required spatially non uniform electric field was also obtained without pixelization of electrodes by means of using a hole patterned electrode [9]. The main drawback of this structure is the necessity to use very thick intermediate substrates to be able to obtain the desired “soft” spatial profile of the electric field in the LC layer and to maintain good optical quality of the lens with low optical aberrations. This increases the lens thickness and control voltages, making such an approach unacceptable.

An alternative approach, by using a high resistivity (or weakly conductive) layer combined with a hole patterned electrode, was also proposed to obtain the required spatial profile of the electric field. This, so-called “modal-controlled”, LC lens was using a hole shaped electrode with its central part being coated by a high resistivity material [10–12]. The principle of operation of this lens was based on the gradual attenuation of the electric potential when approaching the center of the hole patterned electrode thanks to the high sheet resistance of the central part. The same high resistivity layer was used also in combination of inter-digitized (in the same plane) disc & hole shaped electrodes [13].

There are several problems in these approaches too, including the light absorption by the high resistivity layer and the complexity of the electrical control (necessity of simultaneous adjustment of voltage and frequency). However, the most important problem here is related to the fundamental limit of the electrical conductivity mechanism; the resistivity values required for the high resistivity layer are in the critical transition (“percolation”) zone, which makes impossible the fabrication of two environmentally stable layers with the same (or at least similar) sheet resistance value, discarding thus this approach as being not practical.

Combination of one planar and one external curved electrode has been described in Ref [14], which still allows the use of uniform LC layers. The non uniform electric field here is obtained thanks to the geometrical lens-like form of the external curved surface, which is coated, e.g., by ITO, to form the upper (curved) transparent electrode. In fact, such a structure has a focusing property at 0 volt (what we could call “fixed” or “residual” optical power), which causes problems of integration with various cameras (“base lenses”). This residual optical power may be eliminated (“hidden”) by using an additional polymer layer (with the same refractive index) that is placed over the curved (and ITO-coated) surface and which has flat upper surface [15]. This approach, in fact, allows eliminating the residual optical power, but the reproducible fabrication of a transparent electrode on a lens like structure and orientation defects (disclinations) remain very complicated problems for this approach.

In contrast with all previous examples (where the non uniform electric field is obtained by the use of geometrical form of curved or hole patterned electrodes) an elegant solution was proposed by Sato and his collaborators, which uses flat uniform LC layers and flat electrodes [16]. The mechanism of field modulation here is based on the use of an intermediate non uniform material layer with a given dielectric constant ε_DC at low (typically at 10³Hz) frequency of LC driving electric fields (here the “DC” is used only to distinguish the low frequency character of driving electric field versus the high frequency of optical fields, which are at the order of 10¹⁴Hz). Namely, the proposed intermediate layer is simply one glass lens (with ε^(g)_DC) of negative optical power, noted as “outer lens” in the Fig. 1 .

 

Fig. 1 Schematic representation of the hidden layer structure that is composed of an additional “mid” substrate, “inner” and “outer” lenses.

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The remaining part (“inner lens”) of the intermediate layer is “occupied” by air with ε^(a)_DC = 1. The application of the low frequency electric voltage to uniform ITO electrodes generates a spatially non uniform electric field inside the LC cell thanks to the spatial non-uniformity (when going from the center to the periphery of the structure) of the dielectric constant of the intermediate “combined” layer since ε^(g)_DC > ε^(a)_DC . The electric field in the central part of the LC cell will thus be different (weaker) from the electric field near to the periphery. The advantage of this approach is that the desired spatial form (gradient) of the field may be obtained by the use of the intermediate material of appropriate form (such as a lens) and its dielectric constant.

However, several major problems remain to be solved in this approach too. One of them is the inherent fixed residual optical power of this lens, which creates integration problems for different base-lenses. The second problem is related to the necessity of having additional and costly antireflection coatings to avoid high optical losses of this device due to the Fresnel reflections on glass-air surfaces (since the refractive index is quite different for glass n_g≈1.5 and air n_a≈1). Also, the achievable contrast of the electric field (and thus of the contrast of effective refractive index of the LC layer) is limited because of the maximum achievable contrast between ε^(g)_DC (which can be typically varied from 3.8 to 14.5 depending on the type of glass) and ε^(a)_DC = 1. The last point imposes also higher thicknesses of the intermediate layer and correspondingly higher driving voltages. In fact, one could use also higher ε^(g)_DC materials (e.g., amorphous semiconductor chalcogenide glasses) to build the outer lens, but it would impose higher refractive indexes (generating thus more Fresnel losses), sever environmental reliability problems and higher cost. The alternative is the use of more curved surfaces, which would further increase the thickness, voltages and will further complicate the problem of management of static and dynamic aberrations (another serious problem of base-lens adaptation). Thus, it is clear that the optical functions of the additional layer (here, defocusing) are inherently “coupled” with its dielectric functions (generation of the driving electric field’s spatial profile), which is limiting the performance of the device from several points of view.

2. The proposed solution

The solution that we propose in the present work, at first sight, looks like an illogical modification of the previous approach: that is, we propose to fill the gap (the “inner lens” area) by another dielectric. However with further explanation (see later) of the choice of the specific material to be used, it will be clear that this modification will not only eliminate the above-mentioned drawbacks, but also will provide better performance and functionality of the device. This will be achieved by exploiting the fundamental property of dispersion: the dependence of material’s complex dielectric constant ε upon the frequency ν of the electric field ranging from 10³ Hz to 10¹⁴ Hz (see below).

Namely, we propose to fill the inner lens area by a material that would have similar optical (at 10¹⁴Hz) properties as the outer lens material, but different dielectric properties at lower driving frequencies (at 10³Hz). Thus, we obtain an “optically hidden doublet lens” structure composed of inner and outer lenses (Fig. 1), which is optically uniform since n_{inner lens} = n_{outer lens}, but is strongly non uniform for low frequency electric field that is used to drive the LC orientation since ε_{inner lens} ≠ ε_{outer lens}. An example of such a material may be a water based solution with appropriate values of low-frequency dielectric constant ε^(w)_DC and high (optical) frequency dielectric constant ε^(w)_opt (let us recall that the optical refractive index n_w of this material is defined as n_w = [ε^(w)_opt]^0.5). Due to the specific dipolar molecular character, the water solutions may have huge ε^(w)_DC (at the order of 78) and, in the meantime, very low n_w (at the order of 1.3). Thus, we can resolve all above mentioned problems by using a combination of a water based solution (with ε^(w)_DC ≈78 and n_w ≈1.3) with an outer lens material having similar refractive index n^(out) and quite different ε^(out)_DC. An example of such material may be the fluorinated polymer (typically having very low values of n_pol ≈1.3 and ε^(pol)_DC ≈3). Thus, at optical frequencies, the combination of such inner and outer lenses will be a planar uniform plate without residual optical power (n_pol ≈n_w). In addition there will be no additional Fresnel reflections and optical losses. In the meantime, there will be an extra-ordinarily large contrast of dielectric constants at low frequency fields ε^(w)_DC - ε^(pol)_DC ≈75 (as opposed to the previous case of ε^(g)_DC - ε^(a)_DC ≈5.5). This contrast will allow the generation of spatially non-uniform electric field that will reorient the LC’s director (average orientation of long molecular axes of LC molecules) in a way that is required to generate lens-like refractive index modulation and to focus light.

Before discussing the theoretical and experimental details, let us note that there is a wide variety of materials and their combinations that could provide not only various contrasts of dielectric constants, but also different polarities. That is, the above mentioned example will generate stronger reorientation of the director in the central part of the lens, generating thus a “negative” tuning of optical power. However, the dielectric constant of the inner lens material may also be chosen to be lower than the one for the outer lens material. In this case, in contrast to the previous example, the central field will be weaker than the peripheral field. Thus we can obtain more reorientation of the director in the periphery of the structure (compared to the center) and thus a “positively” tunable lens.

Let us now shortly introduce the theory of the “hidden layer” concept, which is based on the principle of the “dielectric deviser” (in analogy with the “voltage deviser” in electronics). It describes the electrical potential distribution (in space) depending upon the dielectric constants of materials in that space. We analyze this phenomenon by using the simple example of two uniform layers with dielectric constants ${\epsilon }_1,{\epsilon }_2$ and thicknesses $d_1,d_2$ (see Fig. 2a ). Namely, if we apply an “external” voltage $V_0$ to ITO coatings then it would create voltage drops on each layer (along the z axes) that may be presented as

 

Fig. 2 a. Schematic demonstration of the principle of “dielectric devisor” by the example of two uniform dielectric layers (Mediums 1 and 2) with different dielectric constants ε and thicknesses d.

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From the continuity condition of inductions ${(D_1)}_{z=0}={(D_2)}_{z=0}$ we can obtain the relation of electric fields:

Let us note that this approach may be generalized for a multi layer dielectric structure (with n layers); then the drop of voltage on each i-th layer may be presented as

It is clear that the achievable optical power of the tunable lens depends upon two groups of experimental parameters; those of the LC and of the hidden layer. For the purpose of a qualitative demonstration only, corresponding theoretical calculations have been done, based on above presented theory and using the following simulation parameters: the spherical shaped (plano-convex) inner lens’s base diameter = 2.25 mm, height d = 200 μm and dielectric constant ε = 2.3, the glass-made mid substrate’s dielectric constant was ε = 6.9 and its thickness was = 70 μm, the outer lens was chosen as made of the same glass, embracing the inner ring. The parameters of the 30 μm thick LC used for simulation (only for a qualitative demonstration of the effect, not optimized) were chosen to be Δn = 0.27 @ 0.633 μm, ${\epsilon }_{\perp }$ = 4.9, Δε = 10, K₁₁ = 15.6 pN.

 

Fig. 2 b. Theoretical demonstration of the maximum achievable optical power (using simulation parameters described in the main text) depending upon the dielectric constant of the outer lens material.

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As it can be seen, the quite reasonable increase of the contrast between the dielectric constants of the inner and outer lens materials should provide huge increase of optical power. In addition, those values of optical power should be achievable for a rather thin hidden layer structure (with smaller separation of electrodes) and correspondingly with low voltages.

3. Experimental results and discussions

Again, just for a qualitative experimental demonstration, we used a standard planar aligned 30 μm thick home-made high birefringence nematic LC mixture with an optical birefringence Δn = 0.3991 @ 546 nm and Δn = 0.3655 @ 633 nm, both measured at room temperature (23 °C). The ratio of rotational viscosity over the elastic constant was measured (at room temperature) to be at the order of 6 msec/μm². This provided a typical relaxation time (do not confuse with the auto-focus convergence time) at the order of 0.5 sec. The inner lens was made of silica glass of plano-convex form. Its diameter was 2.25 mm, while its height was 227 μm. The inner lens was glued onto mid substrate by use of a very thin epoxy glue of the same refractive index as the silica glass. The outer lens was formed as a chamber (surrounding the inner lens) of height 250 μm between the mid substrate and bottom substrate (Fig. 1). The chamber was filled with glycerol/water solution. The refractive index of the solution was adjusted to match the refractive index of the silica glass (inner lens). This provides the geometry where the effective electric field (in the layer of the LC) will be stronger at the periphery of the cell, providing thus more reorientation in the periphery compared to the central part. We have thus a positively tunable lens.

Square shaped electrical signal of 1 kHz was applied to the ITO electrodes to drive the lens. First of all, a polarizing microscope was used to inspect the interference structure induced in the LC cell under applied voltage by positioning the lens director at 45° between two cross oriented polarizers and illuminating it by a lamp through a narrow bandpass (546nm) interference filter. As it can be seen from the micro photographies (Fig. 3 ), the obtained structures are initially uniform and there is no residual optical power due to the non uniformity of the LC gap before the application of the voltage (left photo). Typical “interferential rings” appear upon the application of the voltage (right photo), which confirm the smooth profile and high optical power obtained (see later).

 

Fig. 3 Typical microphotography of the tunable lens (placed between two crossed polarizers and illuminated by a lamp through a narrow bandpass interference filter) for low (left photo) and high (right photo) optical powers.

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The further increase of the electrical field’s amplitude reorients the LC director and tends to uniformize it across the cell up to a homeotropic alignment (not shown here).

Shack-Hartmann wavefront sensor (from Imagine Optics) was used to characterize the optical power and aberration dependence upon the driving conditions. Linearly polarized (extra ordinary mode in the LC layer) CW He-Ne laser (operating at 632.8 nm) was used as probe for this purpose. The driving RMS voltage dependence of the optical power and optical RMS aberrations (including spherical aberrations) are presented in Fig. 4 and Fig. 5 , respectively.

 

Fig. 4 Optical Power (in Diopters) versus driving RMS voltage (at 1 kHz square shaped signal) obtained by a Shack Hartman wavefront sensor. The experimental error is estimated to be ± 0.1 Diopters. The line is used for eye-guide only.

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Fig. 5 Optical RMS aberrations (in μm) versus driving RMS voltage (at 1 kHz square shaped signal) obtained by a Shack Hartman wavefront sensor. The experimental error is estimated to be ± 0.005μm. The line is used for eye-guide only.

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As it can be seen, the obtained optical powers are very good, at the order of magnitude predicted by our simulations. Note also that optical power and aberrations were severely transformed (not shown here) if, for the given geometry, the water solution was removed: then the electric field’s profile change becomes very drastic (non spherical), reducing thus the optical power and increasing unacceptably the optical aberrations. It must be noticed that the LC being initially homogeneously (“planar”) aligned, there is a threshold of reorientation by voltage (on the left side of the curve with respect to the maximum of optical power, Fig. 4) that must be over passed. The threshold value here is at the order of 10 V. It is not visible in the Fig. 4 since the first excitation voltage (15 V) used is chosen deliberately to be above that threshold (line is used only for eye-guiding). In fact, while it would be preferable to work by using as low as possible voltages (left side of the curve), the performance of the lens in the left side of the curve is rather compromised by the neighborhood of the threshold and corresponding optical aberrations. In contrast, the performance of the lens is rather good on the right side of the curve. Different approaches may be explored to optimize the lens operation, taking into account all parameters including, for example, the driving voltage and scattering phenomena, etc. The preliminary results of an interesting approach are described below.

Namely, in addition to above mentioned advantages, the proposed combination of two different dielectrics in a “hidden layer” provides the unique capacity of controlling the field’s spatial profile and strength by the choice of the driving frequency thanks to the difference of dispersion mechanisms of those two materials (difference of dielectric constant’s dependence upon the driving frequency for inner and outer lens materials). It may allow us to control the optical power of our lens by the adjustment of the driving frequency alone. The preliminary data we present below (Fig. 6 ) confirms this hypothesis. As it can be seen, a rather small change in driving frequency changes quite noticeably the optical power with still quite acceptable optical aberrations (Fig. 7 ).

 

Fig. 6 Optical Power (in Diopters) versus driving RMS voltage (square shaped drive signal) at three different driving frequencies; spheres: 1 kHz (lower curve), squares: 0.5 kHz (mid curve) and crosses: 0.1 kHz (top curve), obtained by a Shack Hartman wavefront sensor. The experimental error is estimated to be ± 0.1 Diopters. Lines are used for eye-guide only.

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Fig. 7 Optical RMS aberrations (in μm) versus driving RMS voltage (square shaped drive signal) at three different driving frequencies; spheres: 1 kHz (lower curve), squares: 0.5 kHz (mid curve) and crosses: 0.1 kHz (top curve), obtained by a Shack Hartman wavefront sensor. The experimental error is estimated to be ± 0.005μm. Lines are used for eye-guide only.

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In this case, we can use rather small driving voltages (e.g., 20 V) to change the optical power in the low-aberration range (right side of the curves in Figs. 6-7) by the simple frequency change. Thus, if we consider the tendency observed in the Fig. 6 (between 0.1 and 1 kHz), then we can expect to control the optical power almost 100% with a further increase of the driving frequency in the range of tens of kHz. Corresponding work is under way and the detailed results will be reported soon.

4. Conclusions

In conclusion, we believe that the obtained lens is very promising both from technical performance and manufacturing cost points of view. Obviously, the described lens is polarization dependent since the molecular reorientation occurs mainly in the plane defined by the initial director alignment and the applied electric field. Given that natural light’s polarization may be decomposed into two perpendicular components, then two such lenses must be used (with cross oriented optical axes) to handle the unpolarized light, for example, generated by sun. In this case, those lenses must perform synchronously and must be positioned very close to each-other, which implies the use of thin substrates. Namely, the polarization independent version of our lens is thin (~0.5 mm), very efficient (achievable optical powers and >10 D and less than 1 sec is required for auto focus convergence time) and is providing more functionality, such as frequency control (in addition to the voltage control possibility). The proposed architecture is compatible with low-cost wafer scale manufacturing, using for example, standard injection molding fabrication techniques. The hidden character of the doublet structure relaxes requirements on the surface quality further simplifying the manufacturing process. Finally, it uses uniform planar ITO electrodes and uniform planar LC cells, both being well mustered in the LC industry.