Abstract

We present an iterative design method for liquid-tunable aspherical lenses capable of diffraction-limited performance over a wide focal length range. The lenses are formed by a thin elastomer meniscus with a variable thickness profile engineered to deform into an ideal asphere under uniform pressure load. Compared to their more conventional counterparts, the proposed lenses significantly reduce spherical aberration over a larger portion of the aperture. The design procedure begins with the semi-analytical calculation of the meniscus thickness profile using large-deflection thin plate theory. This initial profile is then further optimized using coupled finite element analysis and ray-tracing simulations iteratively. We apply the developed method to design a tunable aspherical lens with 3 mm clear aperture and 8 mm optimum focal length, and numerically demonstrate the improvement in optical performance over conventional tunable-lenses over a focal length range from 6 mm to 12 mm. Using 80% of the clear aperture, the lens has better than λ/4 RMS surface error over the focal length range from 7.7 mm to 8.5 mm, corresponding to 10% tuning of focal length with diffraction-limited performance. The sources of potential fabrication errors in a practical implementation of such a lens are also analyzed in detail in terms of their influence on optical performance.

© 2015 Optical Society of America

1. Introduction

Tunable lenses are crucial elements for a wide range of applications including wavefront sensors [1, 2], optical communication systems [3], miniaturized imaging systems such endomicroscopes [4, 5]. Compared to the axial translation of lenses, tunable-lenses can attain similar zooming/focusing performance within significantly simpler optical arrangements. They exhibit a vast variety in size, tuning range, tuning mechanism, material and fabrication methods. Regardless of their tuning mechanism, the profile of tunable-lenses are predominantly spherical, rendering them inherently prone to spherical aberrations, particularly for high NA lenses [6]. The optical performance might degrade further if the lens profile deviates from sphericity, due to any parasitic effects related to the actuation mechanism or the mechanical suspension of the flexible structures. The practical implication of such effects is the narrowing of the usable optical aperture. For space-limited imaging applications such as endoscopic imaging probes, this is a significant factor limiting the imaging performance.

Various notable approaches have been proposed to minimize the spherical aberration in tunable-focus lenses. Mishra et al. [7] introduced a hybrid tunable lens combining fluidic and electrostatic tuning methods. Suspended in between two optical liquids of different refractive indices, a flexible elastomer membrane deformed under hydrostatic pressure defines the focal length of the lens. The quasi-spherical optical interface is then reshaped into an aspherical one by electrostatic forces acting between an electrode placed at a certain distance above the membrane and the conductive optical liquid. Despite its complex structure and non-trivial problem of determining the electrical and fluidic driving, this device is a versatile and flexible tunable lens. A more straightforward but less versatile solution came from Yu et al. [8], who developed a tunable liquid-filled lens integrated with a fixed aspherical surface that corrects for a fixed amount of spherical aberration. Once fabricated with enough fidelity to the design geometry, the lens provides good aberration correction at the focal length of choice. However, fabrication and integration of the device is cumbersome and prone to alignment errors. Furthermore, the spherical aberration grows rapidly as the focal length changes. Fuh et al. [9], on the other hand, adopted a biconvex lens employing two thin polyvinyl chloride membranes of different thicknesses to compensate for the spherical aberration. A large tuning range was demonstrated with this device with limited success in spherical aberration correction.

The basic idea of using a variable meniscus thickness for a liquid-tunable lens, which constitutes the basic approach of the work presented here, was proposed by Alvarado et al. [10]. However, rather than engineering the meniscus thickness profile to control the deformation profile, they used it as a static aberration corrector, and did not attempt to provide a systematic approach to determine the meniscus thickness profile necessary for a given lens profile.

In this paper, we present a generalized method to design liquid-tunable aspherical lenses based on a meniscus-like membrane with variable thickness, suspended over liquid-filled cavities. The essential design question for such a device is this: for a given aperture size and focal length, what should be the thickness profile of the meniscus, such that under uniform pressure load the meniscus deflection converges to the desired optical surface within certain tolerances. A first estimate for the meniscus thickness profile can be obtained by numerically solving the differential equations of the large-deflection thin plate theory. This first estimate is not sufficient due to two effects: for very fast lenses, the meniscus deflection is so large that the analytical model does not work accurately enough; the meniscus thickness profile calculated analytically has to have zero thickness at the edges due to the boundary conditions; however this is not practical in a real device and leads to an increasing deviation from the ideal profile towards the edge of the lens.

Therefore, an iterative numerical method is devised to optimize the meniscus thickness profile to ensure practical limits on the minimum thickness and avoidance of the edge effects for a slightly reduced aperture size. Using an FEA(finite element analysis) model, a reduced order model depicting the pressure dependent meniscus deflection is established, and then used for optical performance analysis using a ray-tracing software. At this stage, an optical merit function (integral of the MTF, cut-off frequency, RMS wavefront error etc.) allows for the optimization of the initial meniscus profile estimate through iterative FEA and ray-tracing simulations. Since the asphericity depends heavily on the focal length, the optimized meniscus performs perfectly only for a single focal length. We demonstrate that the deviation from the ideal profile can be kept below λ/4 for a significant range, within which the lens is practically diffraction-limited. Depending on the requirements of the particular applications, the tuning range can be much larger. Through simulations covering 67% of the optimum focal length, we also demonstrated that compared to its spherical counterparts, spherical aberration of the tunable aspherical lens is strongly reduced.

In Section 2, we first summarize our basic design approach, and discuss every step of the optimization process in detail. To illustrate the capabilities and the limitations of the method in practice, we apply the developed method to design an aspherical tunable lens of 3 mm aperture size and with an optimum focal length of 8 mm. Various types of fabrication errors and their effect on the lens performance of the developed lens are discussed in Section 3. Finally, we discuss the potential applications and practical limitations of the tunable aspherical lenses.

2. Lens structure and the design method

A liquid-filled cavity sealed by a flexible membrane from one or both sides is a common tunable lens architecture [Fig. 1(a)]. By varying the external pressure via the inlet, the membrane can be inflated or deflated, resulting in a shift in the lens focal length. The majority of the devices sharing this basic structure feature elastomer membranes (including earlier works from our group [11–13]), but different groups have explored different membrane materials [14–16]. Even though they need external pressure controllers, liquid-tunable lenses are capable of combining the optical aperture and the actuator in the same structure, leading to very compact devices. This is a major advantage for adaptive imaging systems limited in aperture size, such as endomicroscopes. The deflection profile converges to that of a perfect spherical lens at the center of the aperture, but deviates significantly at the edges due to the inflection points and significantly limits the usable aperture [17–19]. As the membrane thickness reduces the inflection points are pushed towards the edges, but eventually one hits the limit in practical concerns such as mechanical robustness and manufacturability. Making high-NA tunable lenses using uniform membranes are particularly difficult, due to the strong spherical aberration introduced even by the central parts of the lenses.

 

Fig. 1 (a) Schematic depiction of a liquid-filled tunable lens consisting of a rigid frame as the lens chamber sealed by an elastomer membrane from the top and rigid substrate from the bottom. The chamber is filled with a liquid. By applying external pressure through pressure inlet, the membrane can be inflated (deflated) to have a convex (concave) profile. (b) Schematic diagram of the tunable lens developed in this work, where a meniscus of variable thickness replaces the uniform membrane. The meniscus has a plano-convex shape with a base thickness tb and sag thickness of ts.

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Figure 1(b) depicts the cross-section view of the tunable aspherical lens concept developed in this work. Compared to the conventional form, the key structural difference here is the radially symmetric variation in the membrane thickness that forms a meniscus-like structure. This structural modification manifests itself in two main improvements: the possibility to deform the membrane into an asphere leading to an aberration-free optical surface, and shifting of the inflection points towards the edge as much as possible without reducing the membrane thickness into impractical limits. The main refractive interface is the initially flat meniscus/air interface. If the refractive indices of the membrane and the liquid are selected to be the same, no refraction occurs at the bottom surface and the lens behaves as a single monolithic component. This also ensures that any roughness on the bottom surface arising from the manufacturing process can effectively be rendered optically transparent.

The idea of using variable-thickness membrane as high-quality spherical mirrors was first proposed by Lemaitre et al. [20] and applied successfully for a cat’s eye system employed in the European Very Large Telescope [22,23]. They demonstrated that when the thickness profile of the membrane is properly designed and engineered, the deformation of the membrane can converge to the desired shape with very high fidelity, and the geometrical aberrations can be significantly reduced over a certain focal length range. Using a similar approach, Greger et al. developed a 20 mm size mirror, demonstrating the viability of the approach for miniaturized components as well [24]. If the membrane deflection is limited to a fraction of its thickness, analytical methods can be used to calculate the meniscus profile with sufficient accuracy [20]. For high-NA lenses however, the deflection range is much larger and a more involved design procedure is necessary.

2.1. The iterative lens design method

The main goal of the design process is the determination of the meniscus thickness profile for accurate reproduction of a specific optical profile under uniform pressure load. The deflection of the meniscus is expected to be significantly larger than the maximum meniscus thickness and the desired deflection profile is aspherical [26]. The flow-chart in Fig. 2 summarizes the iterative design procedure that combines analytical analysis, mechanical finite element analysis, and ray-tracing simulations.

 

Fig. 2 Flow chart of the design procedure to obtain the meniscus thickness distribution for a given profile.

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In the first step, the desired aspherical profile is defined. Then an analytical model based on large-deflection thin plate theory is developed to estimate the initial meniscus profile under articulated boundary conditions. The resulting profile acts as the starting point for the iterative optimization step. Using FEA simulations, the deformation of the initial profile is simulated to obtain a realistic profile, which differs from the desired one due to the edge effects and material nonlinearities. The resulting deflection profile is fitted with a polynomial model and imported into a ray tracing software to evaluate its optical performance. Depending on the optical performance, the same process is repeated with a slightly modified meniscus profile, until the target optical performance level is attained. This optimization procedure is performed automatically using MATLAB. In the following, we discuss each point in this process separately.

2.1.1. Lens profile

The design procedure begins with the definition of the ideal optical profile. An aspherical lens profile can be described by the standard equation:

Z(r)=Cr21+1(1+K)C2r2+k=2αkr2k,
where C is the spherical curvature of the surface, K is the conic constant, and the αk are the high-order apsherical coefficients. When αk are all zero, the conic constant K defines the shape of the meniscus surface, e.g. a spherical shape when (K = 0), a paraboloid (K = −1), a hyperboloid (K < −1), or an ellipsoid (−1 < K < 0 or K > 0). To demonstrate its potential for high-power lenses, we chose to apply the design procedure for a 3 mm aperture lens of 8 mm optimum focal length.

2.1.2. Evaluation of the starting meniscus profile

The first step towards the design procedure is to obtain a starting profile for the meniscus thickness based on the lens profile given in Fig. 1, which will be further optimized through numerical means. We hereby use the analytical formulation developed by Lemaitre to analyze the large deformation of thin plates with a variable thickness distribution under uniform loading and reaction at the edge (e.g. articulated boundaries) [20]. This formulation builds upon Timoshenko and Woinowsky-Krieger’s analysis [21] of large deformation of uniform thickness plates under the same boundary and loading conditions. Unlike the small-deflection analysis that is only valid if the maximum deflection is small compared to the mean meniscus thickness, the large deflection theory also accounts for the radial and tangential stresses in the middle surface of the meniscus. As shown below, however, the resulting equilibrium equations are nonlinear, and cease to have close-form analytical solutions. Therefore, it is necessary to employ numerical integration methods.

With t(r) denoting the radially symmetric thickness distribution, the structural rigidity of the meniscus is defined as

D(r)=Et3(r)12(1ν2),
where E and ν are Young’s modulus and Poisson ratio, respectively. Under static equilibrium, with u(r) and z(r) respectively denoting the radial and axial deflections of the meniscus, the relationship between the meniscus thickness distribution, the deflection, and the uniform pressure P acting on the articulated edge meniscus is given by the following pair of nonlinear differential equations [20]:
d2udr2+(1tdtdr+1r)dudr+(νtdtdr1r)ur+12(1tdtdr+1νr)(dudr)2+d2udr2dudr=0,
d(2z)dr+(d2zdr2+νrdzdr)1DdDdr1t2(dudr+νur+12(dzdr)2)dzdrPr2D=0.

The first of these equations is derived from the equilibrium of tensile forces acting on the middle surface of a meniscus segment defined by and dr; the second one arises from the equilibrium of the bending moments acting on the same segment. An analytical solution for t(r) cannot be achieved directly from this system of equations due to their nonlinear nature. With proper boundary conditions, however, a numerical solution can easily be found. For simplicity, the numerical integration can be carried out by the use of dimensionless variables ρ = r/a and τ = t/t0, where a is the meniscus radius, and t0 is the meniscus thickness at the center. It should be noted that t0 and the pressure P are free parameters that can be chosen within the limits defined by the pumps to be used to derive the hydraulically-actuated lens and the fabrication process of the meniscus. For the design presented in this paper, we first fix the pressure, and then calculate t0 using the small deformation theory for a uniform thickness meniscus [20].

In addition to the axial displacement, z(ρ), and its radial derivative, dz(ρ)/dρ, that are completely defined by the desired lens profile, we also know that the meniscus thickness τ(ρ) has to be null at the edge (ρ = 1) and to have a non-zero slope there. At the center, the meniscus thickness is equal to t0 and we have no radial displacement. On the other hand, the derivative of the radial displacement at the center has to have a unknown finite value of ε (e.g. the meniscus will be elongated by a certain amount in static equilibrium under pressure P). In the beginning of the numerical integration, ε will be set to an arbitrary small value, and be progressively increased in successive iterations until the boundary condition on meniscus thickness at ρ = 1 is satisfied. The complete set of boundary conditions are listed below:

t(ρ)ρ=0=1,t(ρ)ρ=1=0,u(ρ)ρ=0=0,(dudr)r=0=ε.

The numerical integration is initiated at a point δρ, which is incrementally far away from the meniscus center. Inserting the boundary conditions at ρ = 0 into Eq. 3 and Eq. 4 provides the values for d2u/dρ2 and dτ/dρ, the former of which yields the value of radial displacement at the next integration point ρ = 2δρ. Therefore, all elements necessary to do the same computation at the subsequent point ρ = 2δρ are known. This integration in the radial direction continues with successive increases by δρ until it reaches the meniscus edge at ρ = 1, where the meniscus thickness should diminish to satisfy the boundary conditions. If not, this entire process is repeated with successively larger values of ε, until the edge boundary condition on the meniscus thickness is satisfied.

Figure 3 depicts the meniscus thickness profiles calculated for a Polydimethylsiloxane (PDMS) meniscus of various initial thicknesses t0 and different pressures P. PDMS is a common material used for tunable lenses with mechanical parameters depending heavily on the preparation, deposition and curing conditions of the material. We used a Young’s modulus of 1.37 MPa, and a Poisson ratio of 0.499 for the PDMS meniscus in this work. These values are estimated to accurately reproduce the experimental results obtained in a prior investigation done by our group for similar lenses with uniform meniscus thickness profiles [26]. Correct modeling of the material properties are critical for an effective lens design.

 

Fig. 3 (a) Normalized meniscus thickness profiles calculated at different pressures and identical t0 of 38 µm. All three configurations produce the same lens profile z(r). As the pressure decreases, the meniscus decreases more gradually towards the edge, lowering the effective rigidity. (b) Normalized meniscus thickness profiles calculated for different t0 and an identical pressure of 1 KPa. In this case, the meniscus thickness at the center exhibits a more gradual decrease towards the edge. In all cases, the meniscus material is PDMS (E = 1.37 MPa, ν = 0.499).

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2.1.3. Optimization of the meniscus profile

Since we target high-NA lenses with aperture sizes in the range of several millimeters, the expected deflection of the meniscus is on the order of hundreds of micrometers. For diffraction-limited optical performance, however, the RMS surface error over the usable aperture should be limited to a fraction of the wavelength. This means the tolerance on the relative error in meniscus deflection is extremely tight and any deviation from the ideal case treated above would rapidly lead to significant degradation in optical performance.

There are two sources of such errors for the meniscus design obtained above. In a practical lens implementation, the meniscus thickness at the edges must have a non-zero minimum value, which would be limited by the capabilities of the fabrication process. The actual meniscus deflection, in this case, would deviate from the desired profile particularly towards the edges, and reduce the usable lens aperture. Material nonlinearity would also have a similar effect. If maximum stress in the meniscus is large, the effective Young’s modulus will have a radial dependence, which is equivalent to the modification of the thickness profile. To compensate for these factors and modify the ideal meniscus profile to ensure minimum optical error, we devised an iterative meniscus optimization procedure that rely on finite element simulations for meniscus deflection and ray-tracing simulations for the assessment of the optical performance. This procedure is summarized below.

In the first step, the initial profile is modified to have a base thickness tb, without changing the maximum thickness at the center. This base thickness should be chosen as small as possible without compromising manufacturability. This is done by fitting the profile with an exponential function of the form:

to(r)=m1exp(rn1)+m2exp(rn2)+tb,
where m1, m2, n1, n2 are coefficients. Depending on the shape of the initial profile, alternative forms can also be used for the fit function. To keep the maximum thickness at the center, we fix the values of m1 and m2, and only let n1 and n2 to be modified during the optimization procedure. Then, this profile is imported to ANSYS and the deflection of the menisus under the design pressure is simulated using a viscoelastic material model for PDMS represented with Prony series coefficients. The resulting deflection profile is fitted with an 8th order polynomial and imported into Code V ray-tracing software for optical performance analysis, which can be evaluated by any criteria that meets the needs of particular application. If the performance does not meet the criteria, the meniscus profile to (r) is modified by tweaking n1 and n2, and the whole process repeated until the criteria is met.

Figure 4 depicts the initial and optimized meniscus thickness profiles for the aspherical lens profile. The initial profile was calculated with a center thickness t0 of 38 µm at a pressure of 1 KPa. The optimized profile has a an additional base thickness tb of 20 µm. We set PDMS refractive index to 1.412, and assumed that the lens is filled with propylene carbonate with a refractive index of 1.416. The used merit function was the RMS error of the MTF measured at the focal point, with the criteria defined as a maximum error of λ/8. It should be noted that to avoid the edge effects, only the central 2.9 mm of the total meniscus deflection was selected from the ANSYS results and used for the optical analysis.

 

Fig. 4 Analytically calculated and optimized initial meniscus thickness distribution for the tunable aspherical lens. With the diameter of 3 mm, the normalized initial meniscus profile has a base thickness of 20 µm, and a maximum sag height of 38 µm.

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Figure 5(a) shows the ANSYS simulation results for the deflection of the optimized meniscus under pressure values ranging from 0.2 KPa to 3 KPa with a step of 0.2 KPa. The inflection points at the edges are still noticeable. However, compared to the uniform thickness membrane, they are located much closer to the edge. The nonlinear dependency of maximum deflection on the pressure is also depicted in this figure as the inset. This nonlinearity arises from the large meniscus deflection that leads to axial and radial stresses that modify the effective meniscus stiffness. These results are in good agreement with other PDMS based tunable lenses found in the literature [13].

 

Fig. 5 (a) FEA results depicting the meniscus deflection as a function of pressure ranging from 0.2 KPa to 3 KPa within steps of 0.2 KPa. The inset plots the axial deflection at the center as a function of pressure, which is highly nonlinear due to the large deformation effects. (b) The deviation from the desired aspherical profile at different focal lengths. The deviation is minimum at the design focal length of 8 mm and grows significantly as the lens is tuned. Due to the base thickness, the pressure necessary for the optimum focal length is 1.2 KPa instead of 1 KPa (the value used to obtain the analytical profile).

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Figure 5(b) plots the difference between the simulated deflection profiles and the ideal aspherical lens profiles for different focal lengths. As expected, the deviation from the ideal profile is smallest for the optimized design focal length of 8 mm. The deviation from the ideal profiles grows exponentially at the edges due to the effect of the inflection points. However, compared to conventional tunable lenses with uniform membranes, the deviation from an ideal aspherical profile is drastically improved. In the next section, we analyze how such an accurate reproduction of ideal lens profiles manifests itself in the optical performance, and demonstrate the drastic improvement compared to conventional tunable lenses.

2.2. Optical performance of the aspherical tunable lens

Instead of importing individual lens profiles under different pressures into the ray-tracing software separately, we devised an analytical model for the tunable lens deflection profile that takes pressure as the sole parameter [14]. In addition to facilitating the optical analysis, this model also provides the means to perform optimization of any optical system that employs the aspherical tunable lens as an adaptive element. Below, we first describe how this model is established, and then discuss the lens performance in detail based on the simulation results obtained using this model.

2.2.1. Pressure dependent meniscus deflection model

As depicted by Fig. 5(a), the meniscus deflection profiles under different pressures can easily be obtained from the FEA results. At each pressure level, the deflection profile can be fit with a polynomial function of the form:

ZP(r)=Anr8+Bnr7+Cnr6+Dnr5+Enr4+Fnr3+Gnr2+Hnr+In,
where n denotes individual deflection profiles, and AnIn denotes fitted polynomial coefficients for that individual profile. Once these coefficients are determined for all the profiles within the focal length of interest, the coefficients multiplying the same power of the parameter (r) can be collectively represented as a function of pressure. For example, the series of coefficients multiplying the r8 terms for lens profiles at different pressures can be represented as:
An(P)=i=08aiPi,
where ai are the coefficients of the polynomial describing the change in An as a function of pressure. Performing same fitting for the other coefficients Bn, Cn, … In yields the pressure-dependent aspherical lens profile model that has the following form:
ZP(r)=i=08aiPir8+i=08biPir7+i=08ciPir6+i=08diPir5+i=08eiPir4+i=08fiPir3+i=08giPir2+i=08hiPir+i=08iiPi.

This analytical expression of the lens profile can easily be imported into a standard ray tracing software as a user-defined optical component. Optimization of an optical system employing the modeled lens can be then performed by tuning a single parameter P.

2.2.2. Modulation transfer function (MTF) and focal length tuning

An analytical model as in Fig. 9 for the lens of optimized meniscus profile was established and imported into the ray-tracing software. We used the modulation transfer function to analyze the optical performance. As in the optimization of the meniscus profile discussed in Section 2, the simulations are performed with a collimated beam of wavelength of 589 nm and radius of 2.4 mm, entering the lens from the flat side. The MTF curves are obtained for the focal point on the optical axis. Fig. 6 plots the MTF curves for the tunable lens for 4 different focal lengths, ranging from 6 mm to 12 mm. At the optimized focal length of 8 mm, the lens performance at the diffraction limit. As the focal length deviates more from the optimum point, the MTF curves drops gradually. However, compared to the conventional tunable lenses of 30 µm uniform membrane thicknesses, the improvement in the optical performance is drastic. The conventional lens models used for these simulations were established with the method identical to the one used for the aspherical lens.

 

Fig. 6 MTF curves of the tunable lens with variable thickness meniscus as our designed for different focal lengths, by comparison, the blue curves are the MTF of the tunable lens with 30 µm uniform thickness meniscus. (a) Focal length of 6 mm. (b) Focal length of 8 mm. (c) Focal length of 10 mm. (d) Focal length of 12 mm.

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A common figure of merit to evaluate the optical quality of a component is the root mean square (RMS) error between its surface profile and an ideal aberration-free surface. The RMS surface error is also directly related to the Strehl ratio, and provides a quantitative way to compare the quality of different optical surfaces. We assume that for diffraction-limited performance from a lens, the RMS surface error over the used aperture should not exceed λ/4. This corresponds to a Strehl ratio of 0.94. In Fig. 7, we plot the RMS surface error for the tunable aspherical lens over a focal length range of 6–12 mm and an aperture size of 2.4 mm. The λ/4 limit is denoted by the horizontal dashed-red line. Between 7.7 mm and 8.5 mm, the RMS error is small enough to provide diffraction-limited performance, which was confirmed via MTF simulations as well (not shown). This corresponds to a tuning range of 10% of the initial focal length without compromising optical performance.

 

Fig. 7 The RMS error in the MTF as a function of effective lens focal length. The λ/4 limit is denoted by the red line. Diffraction limited performance can be obtained between 7.7 mm and 8.5 mm focal lengths, which corresponds to an effective tuning range 10%.

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The improvement in the optical performance that we demonstrated in this section can only be achieved if the meniscus thickness profile can accurately be reproduced. Of course, fabrication of microsystem components is always subject to tolerances, which would lead to performance degradation. In the next section, we explore the effects of various types of potential fabrication errors on the optical performance of the designed lens.

3. Effects of fabrication tolerances on the MTF curves

The realization of the tunable aspherical lens is not within the scope of this paper. But it is safe to assume that advanced elastomer patterning techniques such as hot embossing or injection molding are the most likely candidates for the meniscus replication. Both of these methods require a mold engraved with the negative of the meniscus profile. Due to the radial symmetry, one can fabricate very accurate molds for the meniscus using techniques like diamond turning. Therefore, we will assume error-free shaping of the bottom meniscus surface, and consider two basic types of fabrication errors that are more likely to arise in the molding process: deviation from the target meniscus base thickness and a tilt error in the flat side of the meniscus surface.

3.1. Meniscus base thickness

As depicted in Fig. 3, the base thickness chosen for the meniscus optimization directly influences both the optimized shape and the operation pressure. Any deviation from the chosen base thickness, therefore, will lead to surface errors. In Fig. 8, the MTF curves are plotted for the designed lens of 20 µm base thickness at different focal lengths with different thickness errors. For every thickness error, a pressure-dependent lens model equivalent to the Fig. 9 was established based on ANSYS results. These models are then imported into ray-tracing software. The ray-tracing simulation parameters are identical to those of Fig. 6, except for the lens pressures that are adapted to compensate for the changes in meniscus stiffness. In general, the drop in the MTF curves compared to the optimum device is negligible for ±0.5 µm thickness error, but quickly becomes substantial as the error grows. Interestingly, for the shorter focal length of 6 mm, the MTF curve slightly improves with increasing base thickness, but the lens is far from being diffraction-limited.

 

Fig. 8 MTF curves for tunable lenses with base thickness errors of ±0.5 µm, ±1 µm, ±2 µm. (a) Focal length of 6 mm. (b) Focal length of 8 mm. (c) Focal length of 10 mm (d) Focal length of 12 mm.

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3.2. Meniscus base tilt error

Another likely fabrication error is a tilt of the meniscus base thickness (e.g. base thickness linearly decreasing from one side to the other). Such an error destroys the radial symmetry of the device and is expected to cause coma and astigmatism. Following the same steps as for the base thickness error, the effects of various tilt errors are analyzed through ray-tracing simulations as well. The resulting MTF curves are depicted in Fig. 9 for tilt errors of 0.01°, 0.02°, 0.04° respectively. The drastic effect of the tilt error on the MTF is evident. Even for 0.02° error, the MTF curves at the optimum wavelength drop suddenly, particularly for the mid-frequency range. The overall form of the curves follows the expectations and exhibits strong coma and astigmatism.

 

Fig. 9 MTF curves for tunable lenses with meniscus tilt errors of 0.01°, 0.02° and 0.04°. (a) Focal length of 6 mm. (b) Focal length of 8 mm. (c) Focal length of 10 mm. (d) Focal length of 12 mm.

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Figure 10 depicts the RMS surface error between the ideal aspherical profiles and the simulated profiles for lenses with meniscus thickness and tilt errors as a function of lens focal length. The surface error is calculated over 80% of the full aperture. The effect of meniscus thickness error is mostly observed as a shift in the optimum focal length (where the minimum RMS surface error occurs). The effect of the tilt error is more drastic in terms of optical performance degradation, where the shift in the optimum focal length is limited. Even for 0.01° tilt error, the RMS surface error exceeds λ/4, and the lens ceases have diffraction-limited performance at any focal length. Therefore, in a practical implementation, special attention is required to eliminate any tilt error on the meniscus surface.

 

Fig. 10 The RMS error in the MTF as a function of effective lens focal length. The λ/4 limit is denoted by the black line. (a) shows the RMS due to the meniscus base thickness error. (b) shows the RMS due to the meniscus tilt.

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4. Discussion

As demonstrated by the results of the previous section, the fabrication tolerances for the tunable aspherical lenses designed this way are rather tight. This of course raises a question on the conditions in which such a development effort is justifiable. For lenses smaller than 1 mm in diameter and with NA smaller than 0.1, conventional tunable lenses with a thin uniform membrane can provide diffraction-limited performance if the aperture is limited to avoid the edge effects. As the lens diameter and/or the NA grows, both the spherical aberration and impact of edge effects increase significantly. For lenses larger than roughly 10 mm in diameter, however, gravitational sag becomes a problem and tunable liquid lenses cease to be a viable option [25]. Therefore, the aperture size range within which the presented design methodology is relevant can be estimated to be between 1–10 mm. Particularly few tunable aspherical optical components were developed for aperture sizes in the order a few mm, mostly due to the lack of adequate fabrication techniques. Modern micromachining methods provide the means to attain the precision necessary to engineer thin films at a reasonable cost. The main challenge would be the accurate replication of the meniscus profile, mostly through means of molding.

Apart from acting as a case study for the method in question, the device design presented here has have significant potential applications areas such as laser-to-fiber coupling, focal plane tuning for endoscopic confocal microscopes and miniaturized zoom cameras. From the optical performance perspective, the extension of the focal length range within which the lens performance remains diffraction-limited is crucial. The 10% tuning range is relatively large compared to the tunable lenses found in the literature, however it is likely to be reduced by even minute fabrication errors as demonstrated above. The meniscus optimization is based on a base thickness and operation pressure chosen according to the fabrication methods and pressure drivers we have available in the lab (a study on the realization and testing of the lens design provided here is under way). However, a more thorough study should optimize the choice of these free parameters to expand the diffraction-limited tuning range. We are currently working on such a study as well, and present the results in a future publication.

The potential use of the method presented here is not only limited to the design of tunable aspherical lenses. Once the aberration behavior of an optical system is well characterized, custom tunable correction elements for complex optical systems can also be developed. Instead of the device presented here, which is optimized for stand-alone use, tunable-lenses optimized to perform perfectly within a zoom/auto-focus systems can also be implemented. We have previously presented a similar device optimized for a miniaturized zoom microscope [26].

5. Conclusion

We presented a detailed account of a generalized method to design aspherical liquid-filled tunable lenses based on variable thickness elastomer meniscus. The method uses analytical methods, FEA simulations and ray-tracing analysis for the elimination of spherical aberration in the vicinity of an optimum operating focal length. Using this method, we also presented the design of a tunable lens of 3 mm diameter and demonstrated that compared to a conventional tunable lenses with uniform elastomer membranes, the spherical aberration over the focal length range from 6 mm to 12 mm is significantly reduced. Calculated over 80% of the full aperture, the RMS surface error remains below λ/4 limit from 7.7 mm to 8.5 mm focal length range, within which the lens is diffraction-limited. The effects of potential fabrication errors on the optical performance were also analyzed and tolerance limits for those types of errors were defined for the designed tunable lens. It was demonstrated that flatness of the flat side of the meniscus surface is the most critical feature to ensure diffraction-limited lens performance.

References and links

1. S. Manzanera, C. Canovas, P. M. Prieto, and P. Artal, “A wavelength tunable wavefront sensor for the human eye,” Opt. Express 16, 7748–7755 (2008). [CrossRef]   [PubMed]  

2. H. B. Yu, G. Y. Zhou, F. S. Chau, F. W. Lee, and S. H. Wang, “A tunable Shack–Hartmann wavefront sensor based on a liquid-filled microlens array,” J. Micromech. Microeng 18, 105017 (2008). [CrossRef]  

3. N. Chronis, G. L. Liu, K. H. Jeong, and L. P. Lee, “Tunable liquid-filled microlens array integrated with mi-crofluidic network,” Opt. Express 11, 2370–2378 (2003). [CrossRef]   [PubMed]  

4. K. Aljasem, A. Werber, A. Seifert, and H. Zappe, “Fiber optic tunable probe for endoscopic optical coherence tomography,” J. Opt. A: Pure Appl. Opt. 10, 044012 (2008). [CrossRef]  

5. N. Weber, H. Zappe, and A. Seifert, “Optical micro-system with highly flexible tunability for endoscopic micro-probes,” in “International Conference on Optical MEMS and Nanophotonics,” (2011), pp. 51–52.

6. N. Weber, “Highly flexible micro-bench system for endoscopic micro-probes,” Ph.D. thesis, University of Freiburg (2013).

7. k. Mishra, C. Murade, B. Carreel, I. Roghair, J. M. Oh, G. Manukyan, D. van den Ende, and F. Mugele, “Optofluidic lens with tunable focal length and asphericity,” Sci. Rep. 4, 6378 (2014). [CrossRef]   [PubMed]  

8. H. B. Yu, G. Y. Zhou, H. M. Leung, and F. S. Chau, “Tunable liquid-filled lens integrated with aspherical surface for spherical aberration compensation,” Opt. Express 18, 9945–9954 (2010). [CrossRef]   [PubMed]  

9. Y. K. Fuh, M. X. Lin, and S. Lee, “Characterizing aberration of a pressure-actuated tunable biconvex microlens with a simple spherically-corrected design,” Optics and Lasers in Engineering 50, 1677–1682 (2012). [CrossRef]  

10. A. Santiago-Alvarado, J. González-García, F. Itubide-Jiménez, M. Campos-Garcíab, V.M. Cruz-Martineza, and P. Rafferty, “Simulating the functioning of variable focus length liquid-filled lenses using the finite element method (FEM),” Optik - International Journal for Light and Electron Optics 124, 1003–1010 (2013). [CrossRef]  

11. N. Weber, H. Zappe, and A. Seifert, “High-precision optical & fluidic micro-bench for endoscopic imaging,” in “International Conference on Optical MEMS and Nanophotonics,” (2010), pp. 85–86.

12. P.-H. Cu-Nguyen, A. Grewe, M. Hillenbrand, S. Sinzinger, A. Seifert, and H. Zappe, “Tunable hyperchromatic lens system for confocal hyperspectral sensing,” Opt. Express 21, 27611–27621 (2013). [CrossRef]  

13. P. Waibel, D. Mader, P. Liebetraut, H. Zappe, and Andreas Seifert, “Chromatic aberration control for tunable all-silicone membrane microlenses,” Opt. Express 19, 18584–18592 (2011). [CrossRef]   [PubMed]  

14. S. Thiele, A. Seifert, and A. M. Herkommer, “Wave-optical design of a combined refractive-diffractive varifocal lens,” Opt. Express 22, 13343–13350 (2014). [CrossRef]   [PubMed]  

15. H. B. Yu, G. Y. Zhou, F. S. Chau, F. W. Lee, S. H. Wang, and H. M. Leung, “A liquid-filled tunable double-focus microlens,” Opt. Express 17, 4782–4790 (2009). [CrossRef]   [PubMed]  

16. D. Y. Zhang, N. Justis, V. Lien, Y. Berdichevsky, and Y. H. Lo, “High-performance fluidic adaptive lenses,” Appl. Opt. 43, 783–787 (2004). [CrossRef]   [PubMed]  

17. Q. D. Yang, P. Kobrin, C. Seabury, S. Narayanaswamy, and W. Christian, “Mechanical modeling of fluid-driven polymer lenses,” Appl. Opt. 47, 3658–3668 (2008). [CrossRef]   [PubMed]  

18. A. Werber and H. Zappe, “Tunable microfluidic microlenses,” Appl. Opt. 44, 3238–3245 (2005). [CrossRef]   [PubMed]  

19. P. M. Moran, S. Dharmatilleke, A. H. Khaw, K. W. Tan, M. L. Chan, and I. Rodriguez, “Fluidic lenses with variable focal length,” Appl. Phys. Lett. 88, 041120 (2006). [CrossRef]  

20. G. R. Lemaitre, Astronomical Optics and Elasticity Theory: Active Optics Methods (Springer Science & Business Media, 2008).

21. S. P. Timosenko and S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, 1996).

22. G. R. Lemaitre, “Optical design and active optics methods in astronomy,” Opt. Rev. 20, 103–117 (2013). [CrossRef]  

23. M. Ferrari, G. R. Lemaitre, S. P. Mazzanti, O. von der Luehe, B. di Biagio, P. Montiel, D. Revest, P. Joulie, and J. F. Carré, “Highly variable curvature mirrors for the Very Large Telescope Interferometer,” Proc. SPIE2201, 811–820 (1994). [CrossRef]  

24. W. Greger, T. Hsel, T. Fellner, A. Schoth, C. Mueller, J. Wilde, and H. Reinecke, “Low-cost deformable mirror for laser focussing,” Proc. SPIE 6374, 63740F (2006). [CrossRef]  

25. S. T. Choi, B. S. Son, G. W. Seo, S.-Y. Park, and K.-S. Lee, “Opto-mechanical analysis of nonlinear elastomer membrane deformation under hydraulic pressure for variable-focus liquid-filled microlenses,” Opt. Express 22, 6133–6146 (2014). [CrossRef]   [PubMed]  

26. P. P. Zhao, Ç. Ataman, and H. Zappe, “An endoscopic microscope with liquid-tunable aspheric lenses for continuous zoom capability,” Proc. SPIE9130, 913004–11 (2014). [CrossRef]  

References

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  • |

  1. S. Manzanera, C. Canovas, P. M. Prieto, and P. Artal, “A wavelength tunable wavefront sensor for the human eye,” Opt. Express 16, 7748–7755 (2008).
    [Crossref] [PubMed]
  2. H. B. Yu, G. Y. Zhou, F. S. Chau, F. W. Lee, and S. H. Wang, “A tunable Shack–Hartmann wavefront sensor based on a liquid-filled microlens array,” J. Micromech. Microeng 18, 105017 (2008).
    [Crossref]
  3. N. Chronis, G. L. Liu, K. H. Jeong, and L. P. Lee, “Tunable liquid-filled microlens array integrated with mi-crofluidic network,” Opt. Express 11, 2370–2378 (2003).
    [Crossref] [PubMed]
  4. K. Aljasem, A. Werber, A. Seifert, and H. Zappe, “Fiber optic tunable probe for endoscopic optical coherence tomography,” J. Opt. A: Pure Appl. Opt. 10, 044012 (2008).
    [Crossref]
  5. N. Weber, H. Zappe, and A. Seifert, “Optical micro-system with highly flexible tunability for endoscopic micro-probes,” in “International Conference on Optical MEMS and Nanophotonics,” (2011), pp. 51–52.
  6. N. Weber, “Highly flexible micro-bench system for endoscopic micro-probes,” Ph.D. thesis, University of Freiburg (2013).
  7. k. Mishra, C. Murade, B. Carreel, I. Roghair, J. M. Oh, G. Manukyan, D. van den Ende, and F. Mugele, “Optofluidic lens with tunable focal length and asphericity,” Sci. Rep. 4, 6378 (2014).
    [Crossref] [PubMed]
  8. H. B. Yu, G. Y. Zhou, H. M. Leung, and F. S. Chau, “Tunable liquid-filled lens integrated with aspherical surface for spherical aberration compensation,” Opt. Express 18, 9945–9954 (2010).
    [Crossref] [PubMed]
  9. Y. K. Fuh, M. X. Lin, and S. Lee, “Characterizing aberration of a pressure-actuated tunable biconvex microlens with a simple spherically-corrected design,” Optics and Lasers in Engineering 50, 1677–1682 (2012).
    [Crossref]
  10. A. Santiago-Alvarado, J. González-García, F. Itubide-Jiménez, M. Campos-Garcíab, V.M. Cruz-Martineza, and P. Rafferty, “Simulating the functioning of variable focus length liquid-filled lenses using the finite element method (FEM),” Optik - International Journal for Light and Electron Optics 124, 1003–1010 (2013).
    [Crossref]
  11. N. Weber, H. Zappe, and A. Seifert, “High-precision optical & fluidic micro-bench for endoscopic imaging,” in “International Conference on Optical MEMS and Nanophotonics,” (2010), pp. 85–86.
  12. P.-H. Cu-Nguyen, A. Grewe, M. Hillenbrand, S. Sinzinger, A. Seifert, and H. Zappe, “Tunable hyperchromatic lens system for confocal hyperspectral sensing,” Opt. Express 21, 27611–27621 (2013).
    [Crossref]
  13. P. Waibel, D. Mader, P. Liebetraut, H. Zappe, and Andreas Seifert, “Chromatic aberration control for tunable all-silicone membrane microlenses,” Opt. Express 19, 18584–18592 (2011).
    [Crossref] [PubMed]
  14. S. Thiele, A. Seifert, and A. M. Herkommer, “Wave-optical design of a combined refractive-diffractive varifocal lens,” Opt. Express 22, 13343–13350 (2014).
    [Crossref] [PubMed]
  15. H. B. Yu, G. Y. Zhou, F. S. Chau, F. W. Lee, S. H. Wang, and H. M. Leung, “A liquid-filled tunable double-focus microlens,” Opt. Express 17, 4782–4790 (2009).
    [Crossref] [PubMed]
  16. D. Y. Zhang, N. Justis, V. Lien, Y. Berdichevsky, and Y. H. Lo, “High-performance fluidic adaptive lenses,” Appl. Opt. 43, 783–787 (2004).
    [Crossref] [PubMed]
  17. Q. D. Yang, P. Kobrin, C. Seabury, S. Narayanaswamy, and W. Christian, “Mechanical modeling of fluid-driven polymer lenses,” Appl. Opt. 47, 3658–3668 (2008).
    [Crossref] [PubMed]
  18. A. Werber and H. Zappe, “Tunable microfluidic microlenses,” Appl. Opt. 44, 3238–3245 (2005).
    [Crossref] [PubMed]
  19. P. M. Moran, S. Dharmatilleke, A. H. Khaw, K. W. Tan, M. L. Chan, and I. Rodriguez, “Fluidic lenses with variable focal length,” Appl. Phys. Lett. 88, 041120 (2006).
    [Crossref]
  20. G. R. Lemaitre, Astronomical Optics and Elasticity Theory: Active Optics Methods (Springer Science & Business Media, 2008).
  21. S. P. Timosenko and S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, 1996).
  22. G. R. Lemaitre, “Optical design and active optics methods in astronomy,” Opt. Rev. 20, 103–117 (2013).
    [Crossref]
  23. M. Ferrari, G. R. Lemaitre, S. P. Mazzanti, O. von der Luehe, B. di Biagio, P. Montiel, D. Revest, P. Joulie, and J. F. Carré, “Highly variable curvature mirrors for the Very Large Telescope Interferometer,” Proc. SPIE2201, 811–820 (1994).
    [Crossref]
  24. W. Greger, T. Hsel, T. Fellner, A. Schoth, C. Mueller, J. Wilde, and H. Reinecke, “Low-cost deformable mirror for laser focussing,” Proc. SPIE 6374, 63740F (2006).
    [Crossref]
  25. S. T. Choi, B. S. Son, G. W. Seo, S.-Y. Park, and K.-S. Lee, “Opto-mechanical analysis of nonlinear elastomer membrane deformation under hydraulic pressure for variable-focus liquid-filled microlenses,” Opt. Express 22, 6133–6146 (2014).
    [Crossref] [PubMed]
  26. P. P. Zhao, Ç. Ataman, and H. Zappe, “An endoscopic microscope with liquid-tunable aspheric lenses for continuous zoom capability,” Proc. SPIE9130, 913004–11 (2014).
    [Crossref]

2014 (3)

2013 (3)

G. R. Lemaitre, “Optical design and active optics methods in astronomy,” Opt. Rev. 20, 103–117 (2013).
[Crossref]

A. Santiago-Alvarado, J. González-García, F. Itubide-Jiménez, M. Campos-Garcíab, V.M. Cruz-Martineza, and P. Rafferty, “Simulating the functioning of variable focus length liquid-filled lenses using the finite element method (FEM),” Optik - International Journal for Light and Electron Optics 124, 1003–1010 (2013).
[Crossref]

P.-H. Cu-Nguyen, A. Grewe, M. Hillenbrand, S. Sinzinger, A. Seifert, and H. Zappe, “Tunable hyperchromatic lens system for confocal hyperspectral sensing,” Opt. Express 21, 27611–27621 (2013).
[Crossref]

2012 (1)

Y. K. Fuh, M. X. Lin, and S. Lee, “Characterizing aberration of a pressure-actuated tunable biconvex microlens with a simple spherically-corrected design,” Optics and Lasers in Engineering 50, 1677–1682 (2012).
[Crossref]

2011 (1)

2010 (1)

2009 (1)

2008 (4)

Q. D. Yang, P. Kobrin, C. Seabury, S. Narayanaswamy, and W. Christian, “Mechanical modeling of fluid-driven polymer lenses,” Appl. Opt. 47, 3658–3668 (2008).
[Crossref] [PubMed]

S. Manzanera, C. Canovas, P. M. Prieto, and P. Artal, “A wavelength tunable wavefront sensor for the human eye,” Opt. Express 16, 7748–7755 (2008).
[Crossref] [PubMed]

H. B. Yu, G. Y. Zhou, F. S. Chau, F. W. Lee, and S. H. Wang, “A tunable Shack–Hartmann wavefront sensor based on a liquid-filled microlens array,” J. Micromech. Microeng 18, 105017 (2008).
[Crossref]

K. Aljasem, A. Werber, A. Seifert, and H. Zappe, “Fiber optic tunable probe for endoscopic optical coherence tomography,” J. Opt. A: Pure Appl. Opt. 10, 044012 (2008).
[Crossref]

2006 (2)

W. Greger, T. Hsel, T. Fellner, A. Schoth, C. Mueller, J. Wilde, and H. Reinecke, “Low-cost deformable mirror for laser focussing,” Proc. SPIE 6374, 63740F (2006).
[Crossref]

P. M. Moran, S. Dharmatilleke, A. H. Khaw, K. W. Tan, M. L. Chan, and I. Rodriguez, “Fluidic lenses with variable focal length,” Appl. Phys. Lett. 88, 041120 (2006).
[Crossref]

2005 (1)

2004 (1)

2003 (1)

Aljasem, K.

K. Aljasem, A. Werber, A. Seifert, and H. Zappe, “Fiber optic tunable probe for endoscopic optical coherence tomography,” J. Opt. A: Pure Appl. Opt. 10, 044012 (2008).
[Crossref]

Artal, P.

Ataman, Ç.

P. P. Zhao, Ç. Ataman, and H. Zappe, “An endoscopic microscope with liquid-tunable aspheric lenses for continuous zoom capability,” Proc. SPIE9130, 913004–11 (2014).
[Crossref]

Berdichevsky, Y.

Campos-Garcíab, M.

A. Santiago-Alvarado, J. González-García, F. Itubide-Jiménez, M. Campos-Garcíab, V.M. Cruz-Martineza, and P. Rafferty, “Simulating the functioning of variable focus length liquid-filled lenses using the finite element method (FEM),” Optik - International Journal for Light and Electron Optics 124, 1003–1010 (2013).
[Crossref]

Canovas, C.

Carré, J. F.

M. Ferrari, G. R. Lemaitre, S. P. Mazzanti, O. von der Luehe, B. di Biagio, P. Montiel, D. Revest, P. Joulie, and J. F. Carré, “Highly variable curvature mirrors for the Very Large Telescope Interferometer,” Proc. SPIE2201, 811–820 (1994).
[Crossref]

Carreel, B.

k. Mishra, C. Murade, B. Carreel, I. Roghair, J. M. Oh, G. Manukyan, D. van den Ende, and F. Mugele, “Optofluidic lens with tunable focal length and asphericity,” Sci. Rep. 4, 6378 (2014).
[Crossref] [PubMed]

Chan, M. L.

P. M. Moran, S. Dharmatilleke, A. H. Khaw, K. W. Tan, M. L. Chan, and I. Rodriguez, “Fluidic lenses with variable focal length,” Appl. Phys. Lett. 88, 041120 (2006).
[Crossref]

Chau, F. S.

Choi, S. T.

Christian, W.

Chronis, N.

Cruz-Martineza, V.M.

A. Santiago-Alvarado, J. González-García, F. Itubide-Jiménez, M. Campos-Garcíab, V.M. Cruz-Martineza, and P. Rafferty, “Simulating the functioning of variable focus length liquid-filled lenses using the finite element method (FEM),” Optik - International Journal for Light and Electron Optics 124, 1003–1010 (2013).
[Crossref]

Cu-Nguyen, P.-H.

Dharmatilleke, S.

P. M. Moran, S. Dharmatilleke, A. H. Khaw, K. W. Tan, M. L. Chan, and I. Rodriguez, “Fluidic lenses with variable focal length,” Appl. Phys. Lett. 88, 041120 (2006).
[Crossref]

di Biagio, B.

M. Ferrari, G. R. Lemaitre, S. P. Mazzanti, O. von der Luehe, B. di Biagio, P. Montiel, D. Revest, P. Joulie, and J. F. Carré, “Highly variable curvature mirrors for the Very Large Telescope Interferometer,” Proc. SPIE2201, 811–820 (1994).
[Crossref]

Fellner, T.

W. Greger, T. Hsel, T. Fellner, A. Schoth, C. Mueller, J. Wilde, and H. Reinecke, “Low-cost deformable mirror for laser focussing,” Proc. SPIE 6374, 63740F (2006).
[Crossref]

Ferrari, M.

M. Ferrari, G. R. Lemaitre, S. P. Mazzanti, O. von der Luehe, B. di Biagio, P. Montiel, D. Revest, P. Joulie, and J. F. Carré, “Highly variable curvature mirrors for the Very Large Telescope Interferometer,” Proc. SPIE2201, 811–820 (1994).
[Crossref]

Fuh, Y. K.

Y. K. Fuh, M. X. Lin, and S. Lee, “Characterizing aberration of a pressure-actuated tunable biconvex microlens with a simple spherically-corrected design,” Optics and Lasers in Engineering 50, 1677–1682 (2012).
[Crossref]

González-García, J.

A. Santiago-Alvarado, J. González-García, F. Itubide-Jiménez, M. Campos-Garcíab, V.M. Cruz-Martineza, and P. Rafferty, “Simulating the functioning of variable focus length liquid-filled lenses using the finite element method (FEM),” Optik - International Journal for Light and Electron Optics 124, 1003–1010 (2013).
[Crossref]

Greger, W.

W. Greger, T. Hsel, T. Fellner, A. Schoth, C. Mueller, J. Wilde, and H. Reinecke, “Low-cost deformable mirror for laser focussing,” Proc. SPIE 6374, 63740F (2006).
[Crossref]

Grewe, A.

Herkommer, A. M.

Hillenbrand, M.

Hsel, T.

W. Greger, T. Hsel, T. Fellner, A. Schoth, C. Mueller, J. Wilde, and H. Reinecke, “Low-cost deformable mirror for laser focussing,” Proc. SPIE 6374, 63740F (2006).
[Crossref]

Itubide-Jiménez, F.

A. Santiago-Alvarado, J. González-García, F. Itubide-Jiménez, M. Campos-Garcíab, V.M. Cruz-Martineza, and P. Rafferty, “Simulating the functioning of variable focus length liquid-filled lenses using the finite element method (FEM),” Optik - International Journal for Light and Electron Optics 124, 1003–1010 (2013).
[Crossref]

Jeong, K. H.

Joulie, P.

M. Ferrari, G. R. Lemaitre, S. P. Mazzanti, O. von der Luehe, B. di Biagio, P. Montiel, D. Revest, P. Joulie, and J. F. Carré, “Highly variable curvature mirrors for the Very Large Telescope Interferometer,” Proc. SPIE2201, 811–820 (1994).
[Crossref]

Justis, N.

Khaw, A. H.

P. M. Moran, S. Dharmatilleke, A. H. Khaw, K. W. Tan, M. L. Chan, and I. Rodriguez, “Fluidic lenses with variable focal length,” Appl. Phys. Lett. 88, 041120 (2006).
[Crossref]

Kobrin, P.

Lee, F. W.

H. B. Yu, G. Y. Zhou, F. S. Chau, F. W. Lee, S. H. Wang, and H. M. Leung, “A liquid-filled tunable double-focus microlens,” Opt. Express 17, 4782–4790 (2009).
[Crossref] [PubMed]

H. B. Yu, G. Y. Zhou, F. S. Chau, F. W. Lee, and S. H. Wang, “A tunable Shack–Hartmann wavefront sensor based on a liquid-filled microlens array,” J. Micromech. Microeng 18, 105017 (2008).
[Crossref]

Lee, K.-S.

Lee, L. P.

Lee, S.

Y. K. Fuh, M. X. Lin, and S. Lee, “Characterizing aberration of a pressure-actuated tunable biconvex microlens with a simple spherically-corrected design,” Optics and Lasers in Engineering 50, 1677–1682 (2012).
[Crossref]

Lemaitre, G. R.

G. R. Lemaitre, “Optical design and active optics methods in astronomy,” Opt. Rev. 20, 103–117 (2013).
[Crossref]

G. R. Lemaitre, Astronomical Optics and Elasticity Theory: Active Optics Methods (Springer Science & Business Media, 2008).

M. Ferrari, G. R. Lemaitre, S. P. Mazzanti, O. von der Luehe, B. di Biagio, P. Montiel, D. Revest, P. Joulie, and J. F. Carré, “Highly variable curvature mirrors for the Very Large Telescope Interferometer,” Proc. SPIE2201, 811–820 (1994).
[Crossref]

Leung, H. M.

Liebetraut, P.

Lien, V.

Lin, M. X.

Y. K. Fuh, M. X. Lin, and S. Lee, “Characterizing aberration of a pressure-actuated tunable biconvex microlens with a simple spherically-corrected design,” Optics and Lasers in Engineering 50, 1677–1682 (2012).
[Crossref]

Liu, G. L.

Lo, Y. H.

Mader, D.

Manukyan, G.

k. Mishra, C. Murade, B. Carreel, I. Roghair, J. M. Oh, G. Manukyan, D. van den Ende, and F. Mugele, “Optofluidic lens with tunable focal length and asphericity,” Sci. Rep. 4, 6378 (2014).
[Crossref] [PubMed]

Manzanera, S.

Mazzanti, S. P.

M. Ferrari, G. R. Lemaitre, S. P. Mazzanti, O. von der Luehe, B. di Biagio, P. Montiel, D. Revest, P. Joulie, and J. F. Carré, “Highly variable curvature mirrors for the Very Large Telescope Interferometer,” Proc. SPIE2201, 811–820 (1994).
[Crossref]

Mishra, k.

k. Mishra, C. Murade, B. Carreel, I. Roghair, J. M. Oh, G. Manukyan, D. van den Ende, and F. Mugele, “Optofluidic lens with tunable focal length and asphericity,” Sci. Rep. 4, 6378 (2014).
[Crossref] [PubMed]

Montiel, P.

M. Ferrari, G. R. Lemaitre, S. P. Mazzanti, O. von der Luehe, B. di Biagio, P. Montiel, D. Revest, P. Joulie, and J. F. Carré, “Highly variable curvature mirrors for the Very Large Telescope Interferometer,” Proc. SPIE2201, 811–820 (1994).
[Crossref]

Moran, P. M.

P. M. Moran, S. Dharmatilleke, A. H. Khaw, K. W. Tan, M. L. Chan, and I. Rodriguez, “Fluidic lenses with variable focal length,” Appl. Phys. Lett. 88, 041120 (2006).
[Crossref]

Mueller, C.

W. Greger, T. Hsel, T. Fellner, A. Schoth, C. Mueller, J. Wilde, and H. Reinecke, “Low-cost deformable mirror for laser focussing,” Proc. SPIE 6374, 63740F (2006).
[Crossref]

Mugele, F.

k. Mishra, C. Murade, B. Carreel, I. Roghair, J. M. Oh, G. Manukyan, D. van den Ende, and F. Mugele, “Optofluidic lens with tunable focal length and asphericity,” Sci. Rep. 4, 6378 (2014).
[Crossref] [PubMed]

Murade, C.

k. Mishra, C. Murade, B. Carreel, I. Roghair, J. M. Oh, G. Manukyan, D. van den Ende, and F. Mugele, “Optofluidic lens with tunable focal length and asphericity,” Sci. Rep. 4, 6378 (2014).
[Crossref] [PubMed]

Narayanaswamy, S.

Oh, J. M.

k. Mishra, C. Murade, B. Carreel, I. Roghair, J. M. Oh, G. Manukyan, D. van den Ende, and F. Mugele, “Optofluidic lens with tunable focal length and asphericity,” Sci. Rep. 4, 6378 (2014).
[Crossref] [PubMed]

Park, S.-Y.

Prieto, P. M.

Rafferty, P.

A. Santiago-Alvarado, J. González-García, F. Itubide-Jiménez, M. Campos-Garcíab, V.M. Cruz-Martineza, and P. Rafferty, “Simulating the functioning of variable focus length liquid-filled lenses using the finite element method (FEM),” Optik - International Journal for Light and Electron Optics 124, 1003–1010 (2013).
[Crossref]

Reinecke, H.

W. Greger, T. Hsel, T. Fellner, A. Schoth, C. Mueller, J. Wilde, and H. Reinecke, “Low-cost deformable mirror for laser focussing,” Proc. SPIE 6374, 63740F (2006).
[Crossref]

Revest, D.

M. Ferrari, G. R. Lemaitre, S. P. Mazzanti, O. von der Luehe, B. di Biagio, P. Montiel, D. Revest, P. Joulie, and J. F. Carré, “Highly variable curvature mirrors for the Very Large Telescope Interferometer,” Proc. SPIE2201, 811–820 (1994).
[Crossref]

Rodriguez, I.

P. M. Moran, S. Dharmatilleke, A. H. Khaw, K. W. Tan, M. L. Chan, and I. Rodriguez, “Fluidic lenses with variable focal length,” Appl. Phys. Lett. 88, 041120 (2006).
[Crossref]

Roghair, I.

k. Mishra, C. Murade, B. Carreel, I. Roghair, J. M. Oh, G. Manukyan, D. van den Ende, and F. Mugele, “Optofluidic lens with tunable focal length and asphericity,” Sci. Rep. 4, 6378 (2014).
[Crossref] [PubMed]

Santiago-Alvarado, A.

A. Santiago-Alvarado, J. González-García, F. Itubide-Jiménez, M. Campos-Garcíab, V.M. Cruz-Martineza, and P. Rafferty, “Simulating the functioning of variable focus length liquid-filled lenses using the finite element method (FEM),” Optik - International Journal for Light and Electron Optics 124, 1003–1010 (2013).
[Crossref]

Schoth, A.

W. Greger, T. Hsel, T. Fellner, A. Schoth, C. Mueller, J. Wilde, and H. Reinecke, “Low-cost deformable mirror for laser focussing,” Proc. SPIE 6374, 63740F (2006).
[Crossref]

Seabury, C.

Seifert, A.

S. Thiele, A. Seifert, and A. M. Herkommer, “Wave-optical design of a combined refractive-diffractive varifocal lens,” Opt. Express 22, 13343–13350 (2014).
[Crossref] [PubMed]

P.-H. Cu-Nguyen, A. Grewe, M. Hillenbrand, S. Sinzinger, A. Seifert, and H. Zappe, “Tunable hyperchromatic lens system for confocal hyperspectral sensing,” Opt. Express 21, 27611–27621 (2013).
[Crossref]

K. Aljasem, A. Werber, A. Seifert, and H. Zappe, “Fiber optic tunable probe for endoscopic optical coherence tomography,” J. Opt. A: Pure Appl. Opt. 10, 044012 (2008).
[Crossref]

N. Weber, H. Zappe, and A. Seifert, “Optical micro-system with highly flexible tunability for endoscopic micro-probes,” in “International Conference on Optical MEMS and Nanophotonics,” (2011), pp. 51–52.

N. Weber, H. Zappe, and A. Seifert, “High-precision optical & fluidic micro-bench for endoscopic imaging,” in “International Conference on Optical MEMS and Nanophotonics,” (2010), pp. 85–86.

Seifert, Andreas

Seo, G. W.

Sinzinger, S.

Son, B. S.

Tan, K. W.

P. M. Moran, S. Dharmatilleke, A. H. Khaw, K. W. Tan, M. L. Chan, and I. Rodriguez, “Fluidic lenses with variable focal length,” Appl. Phys. Lett. 88, 041120 (2006).
[Crossref]

Thiele, S.

Timosenko, S. P.

S. P. Timosenko and S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, 1996).

van den Ende, D.

k. Mishra, C. Murade, B. Carreel, I. Roghair, J. M. Oh, G. Manukyan, D. van den Ende, and F. Mugele, “Optofluidic lens with tunable focal length and asphericity,” Sci. Rep. 4, 6378 (2014).
[Crossref] [PubMed]

von der Luehe, O.

M. Ferrari, G. R. Lemaitre, S. P. Mazzanti, O. von der Luehe, B. di Biagio, P. Montiel, D. Revest, P. Joulie, and J. F. Carré, “Highly variable curvature mirrors for the Very Large Telescope Interferometer,” Proc. SPIE2201, 811–820 (1994).
[Crossref]

Waibel, P.

Wang, S. H.

H. B. Yu, G. Y. Zhou, F. S. Chau, F. W. Lee, S. H. Wang, and H. M. Leung, “A liquid-filled tunable double-focus microlens,” Opt. Express 17, 4782–4790 (2009).
[Crossref] [PubMed]

H. B. Yu, G. Y. Zhou, F. S. Chau, F. W. Lee, and S. H. Wang, “A tunable Shack–Hartmann wavefront sensor based on a liquid-filled microlens array,” J. Micromech. Microeng 18, 105017 (2008).
[Crossref]

Weber, N.

N. Weber, H. Zappe, and A. Seifert, “Optical micro-system with highly flexible tunability for endoscopic micro-probes,” in “International Conference on Optical MEMS and Nanophotonics,” (2011), pp. 51–52.

N. Weber, “Highly flexible micro-bench system for endoscopic micro-probes,” Ph.D. thesis, University of Freiburg (2013).

N. Weber, H. Zappe, and A. Seifert, “High-precision optical & fluidic micro-bench for endoscopic imaging,” in “International Conference on Optical MEMS and Nanophotonics,” (2010), pp. 85–86.

Werber, A.

K. Aljasem, A. Werber, A. Seifert, and H. Zappe, “Fiber optic tunable probe for endoscopic optical coherence tomography,” J. Opt. A: Pure Appl. Opt. 10, 044012 (2008).
[Crossref]

A. Werber and H. Zappe, “Tunable microfluidic microlenses,” Appl. Opt. 44, 3238–3245 (2005).
[Crossref] [PubMed]

Wilde, J.

W. Greger, T. Hsel, T. Fellner, A. Schoth, C. Mueller, J. Wilde, and H. Reinecke, “Low-cost deformable mirror for laser focussing,” Proc. SPIE 6374, 63740F (2006).
[Crossref]

Woinowsky-Krieger, S.

S. P. Timosenko and S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, 1996).

Yang, Q. D.

Yu, H. B.

Zappe, H.

P.-H. Cu-Nguyen, A. Grewe, M. Hillenbrand, S. Sinzinger, A. Seifert, and H. Zappe, “Tunable hyperchromatic lens system for confocal hyperspectral sensing,” Opt. Express 21, 27611–27621 (2013).
[Crossref]

P. Waibel, D. Mader, P. Liebetraut, H. Zappe, and Andreas Seifert, “Chromatic aberration control for tunable all-silicone membrane microlenses,” Opt. Express 19, 18584–18592 (2011).
[Crossref] [PubMed]

K. Aljasem, A. Werber, A. Seifert, and H. Zappe, “Fiber optic tunable probe for endoscopic optical coherence tomography,” J. Opt. A: Pure Appl. Opt. 10, 044012 (2008).
[Crossref]

A. Werber and H. Zappe, “Tunable microfluidic microlenses,” Appl. Opt. 44, 3238–3245 (2005).
[Crossref] [PubMed]

P. P. Zhao, Ç. Ataman, and H. Zappe, “An endoscopic microscope with liquid-tunable aspheric lenses for continuous zoom capability,” Proc. SPIE9130, 913004–11 (2014).
[Crossref]

N. Weber, H. Zappe, and A. Seifert, “Optical micro-system with highly flexible tunability for endoscopic micro-probes,” in “International Conference on Optical MEMS and Nanophotonics,” (2011), pp. 51–52.

N. Weber, H. Zappe, and A. Seifert, “High-precision optical & fluidic micro-bench for endoscopic imaging,” in “International Conference on Optical MEMS and Nanophotonics,” (2010), pp. 85–86.

Zhang, D. Y.

Zhao, P. P.

P. P. Zhao, Ç. Ataman, and H. Zappe, “An endoscopic microscope with liquid-tunable aspheric lenses for continuous zoom capability,” Proc. SPIE9130, 913004–11 (2014).
[Crossref]

Zhou, G. Y.

Appl. Opt. (3)

Appl. Phys. Lett. (1)

P. M. Moran, S. Dharmatilleke, A. H. Khaw, K. W. Tan, M. L. Chan, and I. Rodriguez, “Fluidic lenses with variable focal length,” Appl. Phys. Lett. 88, 041120 (2006).
[Crossref]

J. Micromech. Microeng (1)

H. B. Yu, G. Y. Zhou, F. S. Chau, F. W. Lee, and S. H. Wang, “A tunable Shack–Hartmann wavefront sensor based on a liquid-filled microlens array,” J. Micromech. Microeng 18, 105017 (2008).
[Crossref]

J. Opt. A: Pure Appl. Opt. (1)

K. Aljasem, A. Werber, A. Seifert, and H. Zappe, “Fiber optic tunable probe for endoscopic optical coherence tomography,” J. Opt. A: Pure Appl. Opt. 10, 044012 (2008).
[Crossref]

Opt. Express (8)

N. Chronis, G. L. Liu, K. H. Jeong, and L. P. Lee, “Tunable liquid-filled microlens array integrated with mi-crofluidic network,” Opt. Express 11, 2370–2378 (2003).
[Crossref] [PubMed]

H. B. Yu, G. Y. Zhou, H. M. Leung, and F. S. Chau, “Tunable liquid-filled lens integrated with aspherical surface for spherical aberration compensation,” Opt. Express 18, 9945–9954 (2010).
[Crossref] [PubMed]

P.-H. Cu-Nguyen, A. Grewe, M. Hillenbrand, S. Sinzinger, A. Seifert, and H. Zappe, “Tunable hyperchromatic lens system for confocal hyperspectral sensing,” Opt. Express 21, 27611–27621 (2013).
[Crossref]

P. Waibel, D. Mader, P. Liebetraut, H. Zappe, and Andreas Seifert, “Chromatic aberration control for tunable all-silicone membrane microlenses,” Opt. Express 19, 18584–18592 (2011).
[Crossref] [PubMed]

S. Thiele, A. Seifert, and A. M. Herkommer, “Wave-optical design of a combined refractive-diffractive varifocal lens,” Opt. Express 22, 13343–13350 (2014).
[Crossref] [PubMed]

H. B. Yu, G. Y. Zhou, F. S. Chau, F. W. Lee, S. H. Wang, and H. M. Leung, “A liquid-filled tunable double-focus microlens,” Opt. Express 17, 4782–4790 (2009).
[Crossref] [PubMed]

S. Manzanera, C. Canovas, P. M. Prieto, and P. Artal, “A wavelength tunable wavefront sensor for the human eye,” Opt. Express 16, 7748–7755 (2008).
[Crossref] [PubMed]

S. T. Choi, B. S. Son, G. W. Seo, S.-Y. Park, and K.-S. Lee, “Opto-mechanical analysis of nonlinear elastomer membrane deformation under hydraulic pressure for variable-focus liquid-filled microlenses,” Opt. Express 22, 6133–6146 (2014).
[Crossref] [PubMed]

Opt. Rev. (1)

G. R. Lemaitre, “Optical design and active optics methods in astronomy,” Opt. Rev. 20, 103–117 (2013).
[Crossref]

Optics and Lasers in Engineering (1)

Y. K. Fuh, M. X. Lin, and S. Lee, “Characterizing aberration of a pressure-actuated tunable biconvex microlens with a simple spherically-corrected design,” Optics and Lasers in Engineering 50, 1677–1682 (2012).
[Crossref]

Optik - International Journal for Light and Electron Optics (1)

A. Santiago-Alvarado, J. González-García, F. Itubide-Jiménez, M. Campos-Garcíab, V.M. Cruz-Martineza, and P. Rafferty, “Simulating the functioning of variable focus length liquid-filled lenses using the finite element method (FEM),” Optik - International Journal for Light and Electron Optics 124, 1003–1010 (2013).
[Crossref]

Proc. SPIE (1)

W. Greger, T. Hsel, T. Fellner, A. Schoth, C. Mueller, J. Wilde, and H. Reinecke, “Low-cost deformable mirror for laser focussing,” Proc. SPIE 6374, 63740F (2006).
[Crossref]

Sci. Rep. (1)

k. Mishra, C. Murade, B. Carreel, I. Roghair, J. M. Oh, G. Manukyan, D. van den Ende, and F. Mugele, “Optofluidic lens with tunable focal length and asphericity,” Sci. Rep. 4, 6378 (2014).
[Crossref] [PubMed]

Other (7)

M. Ferrari, G. R. Lemaitre, S. P. Mazzanti, O. von der Luehe, B. di Biagio, P. Montiel, D. Revest, P. Joulie, and J. F. Carré, “Highly variable curvature mirrors for the Very Large Telescope Interferometer,” Proc. SPIE2201, 811–820 (1994).
[Crossref]

P. P. Zhao, Ç. Ataman, and H. Zappe, “An endoscopic microscope with liquid-tunable aspheric lenses for continuous zoom capability,” Proc. SPIE9130, 913004–11 (2014).
[Crossref]

N. Weber, H. Zappe, and A. Seifert, “High-precision optical & fluidic micro-bench for endoscopic imaging,” in “International Conference on Optical MEMS and Nanophotonics,” (2010), pp. 85–86.

N. Weber, H. Zappe, and A. Seifert, “Optical micro-system with highly flexible tunability for endoscopic micro-probes,” in “International Conference on Optical MEMS and Nanophotonics,” (2011), pp. 51–52.

N. Weber, “Highly flexible micro-bench system for endoscopic micro-probes,” Ph.D. thesis, University of Freiburg (2013).

G. R. Lemaitre, Astronomical Optics and Elasticity Theory: Active Optics Methods (Springer Science & Business Media, 2008).

S. P. Timosenko and S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, 1996).

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Figures (10)

Fig. 1
Fig. 1 (a) Schematic depiction of a liquid-filled tunable lens consisting of a rigid frame as the lens chamber sealed by an elastomer membrane from the top and rigid substrate from the bottom. The chamber is filled with a liquid. By applying external pressure through pressure inlet, the membrane can be inflated (deflated) to have a convex (concave) profile. (b) Schematic diagram of the tunable lens developed in this work, where a meniscus of variable thickness replaces the uniform membrane. The meniscus has a plano-convex shape with a base thickness tb and sag thickness of ts.
Fig. 2
Fig. 2 Flow chart of the design procedure to obtain the meniscus thickness distribution for a given profile.
Fig. 3
Fig. 3 (a) Normalized meniscus thickness profiles calculated at different pressures and identical t0 of 38 µm. All three configurations produce the same lens profile z(r). As the pressure decreases, the meniscus decreases more gradually towards the edge, lowering the effective rigidity. (b) Normalized meniscus thickness profiles calculated for different t0 and an identical pressure of 1 KPa. In this case, the meniscus thickness at the center exhibits a more gradual decrease towards the edge. In all cases, the meniscus material is PDMS (E = 1.37 MPa, ν = 0.499).
Fig. 4
Fig. 4 Analytically calculated and optimized initial meniscus thickness distribution for the tunable aspherical lens. With the diameter of 3 mm, the normalized initial meniscus profile has a base thickness of 20 µm, and a maximum sag height of 38 µm.
Fig. 5
Fig. 5 (a) FEA results depicting the meniscus deflection as a function of pressure ranging from 0.2 KPa to 3 KPa within steps of 0.2 KPa. The inset plots the axial deflection at the center as a function of pressure, which is highly nonlinear due to the large deformation effects. (b) The deviation from the desired aspherical profile at different focal lengths. The deviation is minimum at the design focal length of 8 mm and grows significantly as the lens is tuned. Due to the base thickness, the pressure necessary for the optimum focal length is 1.2 KPa instead of 1 KPa (the value used to obtain the analytical profile).
Fig. 6
Fig. 6 MTF curves of the tunable lens with variable thickness meniscus as our designed for different focal lengths, by comparison, the blue curves are the MTF of the tunable lens with 30 µm uniform thickness meniscus. (a) Focal length of 6 mm. (b) Focal length of 8 mm. (c) Focal length of 10 mm. (d) Focal length of 12 mm.
Fig. 7
Fig. 7 The RMS error in the MTF as a function of effective lens focal length. The λ/4 limit is denoted by the red line. Diffraction limited performance can be obtained between 7.7 mm and 8.5 mm focal lengths, which corresponds to an effective tuning range 10%.
Fig. 8
Fig. 8 MTF curves for tunable lenses with base thickness errors of ±0.5 µm, ±1 µm, ±2 µm. (a) Focal length of 6 mm. (b) Focal length of 8 mm. (c) Focal length of 10 mm (d) Focal length of 12 mm.
Fig. 9
Fig. 9 MTF curves for tunable lenses with meniscus tilt errors of 0.01°, 0.02° and 0.04°. (a) Focal length of 6 mm. (b) Focal length of 8 mm. (c) Focal length of 10 mm. (d) Focal length of 12 mm.
Fig. 10
Fig. 10 The RMS error in the MTF as a function of effective lens focal length. The λ/4 limit is denoted by the black line. (a) shows the RMS due to the meniscus base thickness error. (b) shows the RMS due to the meniscus tilt.

Equations (9)

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Z ( r ) = C r 2 1 + 1 ( 1 + K ) C 2 r 2 + k = 2 α k r 2 k ,
D ( r ) = E t 3 ( r ) 12 ( 1 ν 2 ) ,
d 2 u d r 2 + ( 1 t d t d r + 1 r ) d u d r + ( ν t d t d r 1 r ) u r + 1 2 ( 1 t d t d r + 1 ν r ) ( d u d r ) 2 + d 2 u d r 2 d u d r = 0 ,
d ( 2 z ) d r + ( d 2 z d r 2 + ν r d z d r ) 1 D d D d r 1 t 2 ( d u d r + ν u r + 1 2 ( d z d r ) 2 ) d z d r Pr 2 D = 0.
t ( ρ ) ρ = 0 = 1 , t ( ρ ) ρ = 1 = 0 , u ( ρ ) ρ = 0 = 0 , ( d u d r ) r = 0 = ε .
t o ( r ) = m 1 exp ( r n 1 ) + m 2 exp ( r n 2 ) + t b ,
Z P ( r ) = A n r 8 + B n r 7 + C n r 6 + D n r 5 + E n r 4 + F n r 3 + G n r 2 + H n r + I n ,
A n ( P ) = i = 0 8 a i P i ,
Z P ( r ) = i = 0 8 a i P i r 8 + i = 0 8 b i P i r 7 + i = 0 8 c i P i r 6 + i = 0 8 d i P i r 5 + i = 0 8 e i P i r 4 + i = 0 8 f i P i r 3 + i = 0 8 g i P i r 2 + i = 0 8 h i P i r + i = 0 8 i i P i .

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