## Abstract

A novel type of integrated refractive-diffractive varifocal membrane lens is designed and analyzed by wave-optical methods. In contrast to other hybrid devices, the diffractive microstructure is directly imprinted onto the soft deflecting membrane, allowing for a high level of integration. Elastic deformation is taken into account by mechanical simulations with the finite element method (FEM). We show, that the superimposed structure can considerably suppress chromatic and spherical aberration. Furthermore, our algorithm is successfully applied to design a confocal hyperspectral lens.

© 2014 Optical Society of America

## 1. Introduction

Varifocal membrane lenses have been subject of extensive research in the last years [1–5]. A variety of different applications such as beam shaping, imaging and optical interconnection were gaining increasing significance and led to commercial products [6]. However, in the case of tunable microlenses, the optical performance may considerably deteriorate due to non-idealities of the lens shape. Therefore, different approaches were investigated to correct for monochromatic and chromatic aberrations. Multi-chamber systems address both kinds by using liquids with different dispersion characteristics as well as appropriate lens curvatures [7, 8]. Compensation of chromatic aberration by using a diffractive optical element (DOE) is reported for a static diffractive structure [9] as well as for a dynamically tunable DOE [10].

In addition to the improvement of the optical performance, varifocal membrane lenses with extended functionality are reported as well. A new application has been presented by Cu-Nguyen *et al.* [11], who introduced a hyperchromatic lens system which enables confocal hyperspectral sensing by moving a chromatically dispersed focus along the optical axis. The dispersion was achieved by combining the tunable lens with a static DOE.

All described enhancements in optical performance or functionality necessitate additional optical components. We demonstrate here that the required change in wavefront can also be achieved by employing a microstructured membrane as optical interface of a tunable refractive lens, yielding an integrated refractive-diffractive tunable lens system of high compactness.

We present an approach to design and optimize such diffractive structures by physical optics calculations and a genetic algorithm. The method allows to tailor wavefront and dispersion in a broad focal range which is demonstrated for three different scenarios.

## 2. Simulation method

#### 2.1. Mechanical simulation

To our knowledge, there is no experimentally confirmed analytical description for the deformation of soft elastic membranes existing so far. Therefore, all resulting shapes have been gained by FEM simulations. In the presented approach, structured as well as unstructured membranes of polydimethylsiloxane (PDMS, *E* = 1.86MPa, *ν* = 0.48) have been investigated. Figure 1 shows the axially symmetric FEM model and corresponding results simulated in the commercial package COMSOL.

The model is set up accordingly to the MEMS fabrication process in [12] which is planned to be used in later experiments. It consists of a rigid silicon substrate, provided with a circular hole on which the membrane is spanned. The membrane defines the optical interface of a liquid-filled chamber and its curvature determines the refractive power due to the applied pressure. In the simulation, the temperature-induced shrinkage of the PDMS membrane during curing is considered. The maximum element size in the regions of the phase structure has been set to 400 nm, which corresponds roughly to one quarter of the relief height. Lens deflections are calculated for a pressure range of 0.2 – 8 kPa, for lenses with a membrane thickness of 22 μm and an optical aperture of 1.5 mm, resulting in focal lengths of 50 – 4.2 mm.

#### 2.2. Wave-optical simulation

In order to evaluate the optical properties of the deformed structure, the Huygens-Fresnel diffraction integral has been solved numerically in 2D. In this procedure, the radial cross-section of the deformed lens structure is sampled along the radial coordinate into *n* samples with a sampling distance of Δ*r* < *λ*/10. These samples serve as a launch set for spherical waves which start with an individual phase, resulting from their optical path difference (OPD) through the lens. By assuming an axial symmetry and plane wave illumination, this approach corresponds to the 2D superposition of circular waves on the optical axis. The complex electric field *U* at a point *P*_{0} on the optical axis can be expressed by a Riemann sum over *n* sample points

*χ*denotes the angle difference between the normal to the wavefront at (

_{k}*r*,

_{k}*z*(

*r*)) and the vector pointing from this source point to

_{k}*P*

_{0}.

Figure 2 shows the corresponding geometric representation and compares the method to 3D Gaussian beam calculations performed by a commercial package.

In comparison to common numerical wave propagation methods, such as Gaussian beam propagation or Fresnel propagation, the method benefits from fewer approximations, fast computation time and an easy-to-use implementation in optimization algorithms. Furthermore, in compliance with the symmetry, the approach allows to extend to three dimensions by using only a quarter section, albeit at a cost of increased computing time.

## 3. DOE design

As a starting point, a set of deflected membrane geometries has been gained from FEM simulations. The resulting shapes have been fitted by 6^{th} order even polynomials. Thus, the smooth surface shape *h _{s}*(

*r*) is represented as a function of pressure

*p*by

For the design procedure, the local distortion of the soft microstructure resulting from the membrane deflection as well as radial strain components are assumed to be negligible. This first order assumption is verified by a comparison with FEM simulations of structured membrane lenses, which include such effects (see Section 4.1). Hence, the combined refractive-diffractive surface is expressed as the superposition of the smooth membrane surface function *h _{s}*(

*r*) and a diffractive relief function

*h*(

_{r}*r*). This relief function can be described by a modulated n

^{th}order even polynomial as

*λ*

_{0}and the refractive indices

*n*

_{1}(

*λ*

_{0}) and

*n*

_{2}(

*λ*

_{0}). Figure 3 shows an example of a relief and a membrane structure.

By inserting *z*(*r*) = *h _{s}*(

*r*) +

*h*(

_{r}*r*) into Eq. (1), the refracted/diffracted electric field on the optical axis can be calculated for plane wave incidence. Depending on the design goal, different merit functions can be constructed and optimized as described in Section 4.

To find optimum DOE coefficients *b _{n}*, a shuffled complex evolution (SCE) algorithm has been used which was introduced by Duan

*et al.*[13]. Compared to a DOE design based on ray tracing, our wave-optical method includes effects such as efficiency and multiple order diffraction. All simulations and optimizations are performed with water as filling liquid and an aperture stop of 1 mm in diameter.

## 4. Results

#### 4.1. Effects of local distortion and radial strain

For the design, the diffractive structure of the membrane lens has been assumed to be static over a limited pressure range, while local distortion and radial strain are neglected. To test the validity of this assumption, a structured membrane has been accurately modeled with the finite element analysis as described in Section 2.1. Figure 4 compares the static DOE with a pressure-deformed structure. For Fig. 4(a), the underlying membrane has been subtracted to allow direct comparison of the reliefs. Although the pressure leads to a significant radial elongation, the phase function after demodulation is not greatly affected leading to very similar optical results.

The optical results show, that the DOE significantly improves the focal properties. We see a slight shift in the peak between the static case and the realistic FEM model and an increased amount of stray light in the FEM simulation. Nonetheless, modeling a static DOE is a fully justifiable and reasonable assumption in the design process.

#### 4.2. Suppression of spherical aberration

Since spherical aberration automatically decreases if the peak irradiance at the focal position is increased, the optimization algorithm was adapted to maximize this value. Figure 5 shows longitudinal 2D sections through the focal spot at different focal lengths *f* with and without the optimized phase structure for a mechanical model as described in Section 2.1. The effect of the structured membrane becomes clearly visible. In particular, at higher pressure values (shorter focal lengths), the superimposed diffractive structure leads to a higher peak irradiance and a better defined focal spot. Figure 6 depicts the corresponding irradiance distributions in the focal plane, scaled to their individual maximum. Particularly at high pressures, the phase relief on the membrane helps to remove unwanted side lobes in the point spread function and brings it close to the diffraction limit.

#### 4.3. Suppression of chromatic aberration

For suppressing longitudinal chromatic aberration, the DOE has been optimized to match the peak positions on the optical axis at the two wavelengths 420 and 650 nm at a pressure of 0.75 kPa. Figure 7 shows the relative focal shift as a function of wavelength at three different settings of the lens pressure, together with the wavelength-dependent diffraction efficiency for setting two.

At this medium pressure of 0.75 kPa (*f* = 14 mm), the corrected lens is achromatic with a remaining secondary spectrum of 0.5 %. For higher and lower pressures, chromatic aberration increases but is still considerably lower compared to the unstructured membrane.

At the design wavelength of 535 nm, diffraction efficiency reaches almost 100 % due to the continuous phase profile and a weak diffractive contribution of the DOE. For higher and lower wavelengths, diffraction efficiency decreases since the relief height is optimized for the design wavelength of 535 nm.

#### 4.4. Hyperchromatic system

The diffractive structure on the membrane can be used to realize a strongly dispersed focus, which can be moved by the applied pressure. This allows to design a chromatic confocal detection method, thus realizing a hyperspectral lens [11]. To maximize the spectral resolution, the dispersion of the lens has to be be maximized while spherical aberrations shall remain small. In the present case, high dispersion was achieved by applying a negative pressure to the lens, yielding a concave lens, and designing a DOE with high positive diffractive power.

Figure 8 shows the results of the hyperchromatic lens design. For a spectral investigation, the optical power at a circular detector (pinhole) with a diameter of 5 μm was calculated as a function of wavelength. The detector is placed at *z* = 21 mm and the membrane pressure is varied in discrete steps between −0.5 kPa and −1.4 kPa. The average full width at half maximum (FWHM) of the peaks is 15.7 nm.

## 5. Conclusion

Diffractive structures imprinted on the membrane of liquid-filled varifocal lenses have been designed and simulated. Using wave-optical calculations and a genetic algorithm, the optical performance of such devices could be improved over a broad focal range. In contrast to analytical or ray tracing-based designs, our fast method includes arbitrary lens geometries and considers effects such as efficiency and stray light. Local distortions and radial strain of the surface relief due to the membrane deformation are shown to be negligible in the design process.

The design variants presented here were optimized to demonstrate correction of spherical as well as chromatic aberration of a smooth, purely refractive membrane lens. In case of spherical aberration, the peak irradiance at the focal position could be increased by more than 130 % at short focal lengths, while staying constant at longer focal lengths. Longitudinal chromatic aberration was shown to be corrected for a medium focal length around 15 mm and considerably suppressed at higher and lower focal lengths. In a third example, we successfully applied our algorithm to the design of a continuously tunable filter with an average linewidth of 15.7 nm to form a confocal hyperspectral lens.

## Acknowledgments

The authors gratefully acknowledge the Baden-Württemberg Stiftung (project ‘HYAZINT’) for financial support.

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