## Abstract

We present an efficient, low-cost modulation transfer function (MTF) measurement approach, optimized for characterization of tunable micro-lenses; the MTF may easily be measured at a variety of different focal lengths. The approach uses a conventional optical microscope with an optimized approach for lens illumination and the measurement results have been correlated with a commercial MTF measurement system. Measurements on fixed-focus and tunable micro-lenses were performed; for the latter, resolution for lenses with back focal length of 11 mm was 55 lines/mm, decreasing to 40 lines/mm for a back focal length of 4 mm. In general, it was seen that performance was better for lenses with longer focal lengths.

©2010 Optical Society of America

## 1. Introduction

Tunable micro-lenses are of interest for a wide variety of applications, particularly beam shaping, optical interconnection, and biomedical systems [1]. For the latter application, tunable micro-optical systems for use in extremely small fiber endoscopes have been demonstrated [2,3] and it has been shown that tunable lenses allow an improvement of the lateral resolution as well as the signal-to-noise ratio in imaging systems. As a result, the optical properties of the variable micro-lenses are of particular interest and an efficient means to determine the modulation transfer function (MTF) is essential in system design.

Two easily determined lens parameters are the back focal length (bfl) [4–6] and the resolution [7]; these have been measured for fluidic adaptive lenses. However, resolution specifies only the limiting spatial frequency, and can be a misleading performance criterion due to the lack of performance information at other spatial frequencies. As a result, MTF test systems are becoming more popular for electro-optical systems and optical imaging systems, since this parameter provides an indication of the performance over a wide range of spatial frequencies [8–10]. Such test systems tend to be expensive and complex, and are difficult to apply to measurements of non-standard, tunable micro-lens samples which need to be evaluated at a range of focal lengths.

We present here a low cost, highly-efficient measurement setup that overcomes a number of these limitations. A complete optical MTF measurement is implemented using a conventional optical microscope (Zeiss Axioplan 2) and the approach can be used to measure MTF as well as point spread function (PSF), line spread function (LSF) and back focal length. The setup is suitable for measuring micro-lenses with small focal lengths, below 20 mm, and for different diameters without the need for standardized mountings as is often the case for commercial equipment.

## 2. Tunable micro-lens

Figure 1 shows a liquid-filled plano-convex tunable micro-lens, with a clear aperture diameter of 2 mm, of the type which has been characterized in this paper [11]. This hybrid optical component consists of a transparent substrate (BK7 glass), a structured silicon chip and a PDMS membrane. The glass substrate forms the planar surface of the microlens, and the PDMS membrane forms an aspherical convex surface. The silicon cavity is filled with water or another suitable optical liquid with an appropriate refractive index. Applying different pneumatic pressures to the tunable microlens through the pressure inlet, the curvature of the PDMS membrane will change, resulting in different focal lengths.

Lenses of this type have been developed for use in endoscopic optical coherence tomography (OCT) systems, and the attainable focal lengths range from millimeters to infinity without any moving optical components. Using tunable lenses in an OCT system has been shown to result in a high depth-independent lateral resolution, the characterization of which, however, requires knowledge of the MTF of the optical system.

Figure 2 shows images of a test object at different applied pressures under broad-band illumination, and hence different focal lengths of the tunable lens. These images had been taken with one additional lens of fixed focal length in the optical path for a better adjustment of the aperture. It is only a qualitative description of the lens with no significant differences between the images. The development of the tunable lens requires a quantitative analysis too, and this is described in chapter 5. For all four pressure values, the distances of the optical setup were adjusted in the way that the total magnification is roughly the same.

## 3. Measurement setup

The measurement of MTF as proposed here relies on an initial measurement of the point spread function (PSF) at a given focal length, using a two-dimensional imager and an approximately point-like source. Based on the knowledge of the transfer function of the optical system, a one- or two-dimensional Fourier transform of the PSF data can be used to calculate the associated MTF.

Figure 3 shows the measurement setup employed to realize this measurement. A multimode fiber (Ø = 62.5 µm) is used to approximate a point source; the emitted light is collimated and the resultant approximately planar wavefront (redirected by a fixed mirror) illuminates the lens under test, placed on the z-stage of the microscope, which can be adjusted with a precision of roughly 200 nm, but the actual position can be read with a resolution of 25 nm. The objective lens of the microscope directs the resultant image onto a planar CCD imager.

According to different illuminations, there are two kinds of transfer functions for optical imaging systems, amplitude transfer function (ATF) and optical transfer function (OTF). The first one is a result of coherent illumination, because the complex amplitudes of all object points vary in unison, and the various impulse responses in the image plane also vary in unison. Therefore, imaging with coherent illumination is linear in the complex amplitude. In case of incoherent illumination, there is no fixed phase or coherence relation between light from adjacent points of the object, and the various impulse responses in the image plane vary in an uncorrelated way. The irradiance of the image sums up statistically, so the imaging is linear in intensity or power [12]. In this case, the optical transfer function (OTF), which is the Fourier transform of the impulse-response function (PSF or LSF), results. MTF is the absolute value or modulus of OTF. For the majority of lens applications, it is sufficient to assume that the illumination is incoherent, usually realized by a broad-wavelength band pass source of large extent. Therefore, in the present measurement approach, MTF is measured as the transfer function of an incoherent imaging system. An extended white light source is used and the performance of tunable lenses under broad-band illumination is determined.

The collimated emission from the optical fiber (an infinite conjugate of the system) illuminates the sample and, as different pressures are applied to the lens, the focal length varies and the vertical stage is used to place the focal point on the CCD in the image plane of the microscope. The resultant intensity distribution (the PSF) is then recorded in a given sampling area of the CCD and the two-dimensional intensity distribution corresponds to the raw data from which the MTF is calculated, using the procedure described below.

Accurate determination of the focal position is essential for a reliable measurement, since any defocus will strongly affect the resulting calculated MTF. Alternatively, the optimal determined MTF may be used to predict the position of the focal point [13].

## 4. Theoretical basics

The Fourier transform is typically used in linear systems to calculate the transfer function in the frequency domain, and in the present case this method is applied for the frequency analysis of the optical imaging system. We first show how the OTF can easily be determined, followed by considerations of calibration and corrections required for an accurate measurement.

#### 4.1 System response

The image intensity distribution (irradiance) *I _{i}(x_{i}, y_{i})* is given by the convolution of the ideal image and the system impulse response as

**$$\begin{array}{l}{I}_{i}({x}_{i},{y}_{i})={\displaystyle \int {\displaystyle {\int}_{-\infty}^{+\infty}{I}_{g}}}({x}_{0,}{y}_{0}){h}_{I}({x}_{i}-{x}_{0},{y}_{i}-{y}_{0})d{x}_{i}d{y}_{i}\\ ={I}_{g}({x}_{i},{y}_{i})\ast {h}_{I}({x}_{i},{y}_{i})\end{array}$$**

**where**

*I*is the real intensity distribution in the image plane,_{i}(x_{i}, y_{i})*h*is the system impulse response (PSF or LSF), and represents the convolution operation [14]._{I}(x_{i}, y_{i})*I*is the intensity distribution of the ideal image predicted by geometrical optics (considering the magnification of the optical system). If there are no diffraction and aberration effects in the optical system, the system impulse response_{g}(x_{i}, y_{i})*h*is a delta function, and this perfect optical system is capable of generating a point image of a point object._{I}(x_{i}, y_{i})Taking the Fourier transforms of the real image and ideal image intensity spectra, we can thus write

**$${F}_{i}({f}_{x},{f}_{y})={F}_{g}({f}_{x},{f}_{y})\times {H}_{I}({f}_{x},{f}_{y})$$ [REMOVED MACROBUTTON FIELD]**

**where**

*F*corresponds to the frequency spectrum of the real image,_{i}(f_{x}, f_{y})*F*is the frequency spectrum of the ideal image, and_{g}(f_{x}, f_{y})*H*is the OTF; the latter can thus easily be calculated by a point-by-point division of_{I}(f_{x}, f_{y})*F*and_{i}(f_{x}, f_{y})*F*. The MTF, our ultimate goal, is then simply found as the absolute value of_{g}(f_{x}, f_{y})*H*The intensity distribution is recorded by the integrated CCD camera, readout by the microscope software and analyzed by Matlab routines._{I}(f_{x}, f_{y}).#### 4.2 Relationship between PSF and LSF

For the determination of the line spread function using commercial instruments, a slit is often used to obtain the LSF in a particular direction; therefore one needs to change the direction of the slit to measure this parameter in various orientations. In the measurement approach outlined here, a multimode fiber is used to approximate a two-dimensional (2D) point source, which allows us to obtain the PSF directly. The entire 2D MTF information for the lens may thus be obtained by a two-dimensional Fourier transform, requiring only one measurement. The LSF can then be determined by integrating the PSF along the direction of interest. For example, the LSF in the horizontal (x) direction is a convolution of a delta-function in vertical (y) direction and the PSF, namely [14 ]

**$\delta (x)1(y)$represents a line source object, a delta function in**

*x*and constant in*y*.The advantage of using the PSF rather than the LSF as a basis for the measurement and thus determination of the modulation transfer function is that the MTF can then be determined for all directions. Hence, it is easy to provide information over the full microlens aperture and the approach may be used, for example, for optimizing real time alignment during the assembly of the microlens.

#### 4.3 Spatial frequency calibration

The MTF represents the optical characteristic in the frequency domain. Every artifact, imperfection or deviation from ideality has to be taken into account, making a calibration of the frequency components of the system necessary. This calibration has to be done for the real image as well as the ideal image frequency domain.

### 4.3.1 Calibration of the image-receiver MTF (real image frequency correction)

The calibration or scaling of the detector is based on the pixel size of the CCD and the magnification of the objective lens. These two factors determine the sampling interval (the actual length of each pixel), but they are not easy to determine accurately; a good approximation is necessary to determine the exact sampling interval.

Using a grating with a known period in the focal plane of the objective lens, the sampling interval may be adequately calibrated. For the system used here, the sampling interval is 1.06 µm for a 10x objective lens, 0.53 µm for 20x, 0.214 µm for 50x, and 0.107µm for 100x magnification. The corresponding Nyquist frequencies using the different objective lenses are then 472 lines per mm, 943 l/mm, 2336 1/mm, and 4673 1/mm, respectively. The measured spatial frequencies should thus be below half the Nyquist frequency, otherwise aliasing effects can be introduced and errors will occur [14]. Therefore, before measuring the MTF of the lens, the sampling interval of the test system and the cutoff frequency (diffraction limit) of the lens [15] have to be determined to decide which objective lens is to be used in the microscope.

### 4.3.2 Calibration of the finite illumination source (ideal image frequency correction)

In the ideal case, an infinitely small point source should be used to make sure the ideal image is a delta function. The spectrum of the ideal image is then constant in the frequency domain. The transfer function *H _{I}(f_{x}, f_{y})* is directly the image spectrum

*F*However, in reality one has a source of finite size in order to obtain sufficient flux, hence a non-delta-function ideal image limits the bandwidth of the input source spectrum.

_{i}(f_{x}, f_{y}).In the present system, the diameter *D’* of the ideal image is related to the diameter *D* of the multimode fiber which illuminates the lens under test, the focal length of the collimator lens *f _{c}* and the focal length of the lens under test

*f*as

_{t}**A multimode fiber with a diameter of 62.5 µm and a collimator lens with a focal length of 100 mm are used; the focal length of the tunable lens will change according to the applied pressure. As a reference value, we may assume, that the ideal image is represented by a circular step function (a cylindrical “flattop”) with a diameter equal to D'. The resultant frequency spectrum of the ideal image is then given by [12]**

**$${F}_{g}({f}_{x},{f}_{y})=2({J}_{1}(\pi D{\text{'}}_{({f}_{x},{f}_{y})})/\pi D{\text{'}}_{({f}_{x},{f}_{y})})$$ [REMOVED MACROBUTTON FIELD]**

**where***J*represents a Bessel function which is plotted in Fig. 4 . The figure illustrates the normalized ideal image frequency spectrum calculated for different circular apertures using this ideal flattop image. In Fig. 4(a), the solid line shows the ideal image frequency spectrum of a commercial glass lens (3 mm aperture,_{1}*f*= 6 mm) with an ideal image of 3.75 µm diameter, and the dashed line shows the ideal image frequency spectrum of a tunable micro-lens (2 mm aperture,*f*= 11 mm, at 1 kPa) with a 7 µm ideal image according to Eq. (4). This shows that a shorter focal length produces a smaller ideal image (3.75 μm) and thus a wider frequency band. Figure 4(b) illustrates the MTF value for different diameters*D’*at a resolution of 100 lines/mm. When the diameter of the ideal image increases to 12.3 µm, this value drops down to zero.#### 4.4 Correction of non-idealities

Several non-ideal aspects of the measurement system lead to artifacts in the measured frequency spectrum which need to be corrected. These are in particular the finite size of the pinhole, the sampling area of the image on the CCD and the MTF of the objective lens.

### 4.4.1 Finite pinhole size

Although it is desirable to use an illumination from as close to a point source as possible, a multimode fiber rather than a single mode fiber is used as a “point” source in the employed experimental setup. The relatively large aperture of the fiber limits the range of MTF which can be measured. As illustrated in Fig. 4(b), it is impossible to measure MTF above 100 lines/mm, for example, when the diameter of the ideal image is larger than 12 µm. However, a smaller pinhole (as would result if a mono-mode fiber were used) results in lower illumination intensity and thus in a reduced signal-to-noise ratio at the detector. The signal quality is then severely affected by electronic noise in the test setup.

As a result, a practical tradeoff in choosing an appropriate pinhole size optimizes this situation. The source should be small enough to generate sufficiently high spatial frequencies but large enough to provide an adequate flux and a high signal-to-noise ratio for the system.

As shown in Fig. 4(a), the ideal image frequency spectrum of a lens illuminated by a pinhole of finite size is not constant in the frequency range of interest; therefore the pinhole size needs to be included in the calibration. The diameter *D'* and its frequency spectrum can be obtained from Equations Eq. (4) and Eq. (5), respectively. To remove the influence of the finite pinhole size, the overall optical system MTF may be calculated by a point-by-point division of the initially measured image frequency spectrum and the calculated ideal image frequency spectrum, namely

**$$MTF({f}_{x},{f}_{y})=\left|{H}_{I}({f}_{x},{f}_{y})\right|={F}_{i}({f}_{x},{f}_{y})/{F}_{g}({f}_{x},{f}_{y})$$ [REMOVED MACROBUTTON FIELD]**

**4.4.2 Influence of spatial sampling area and noise**

**A final measurement issue refers to system noise, particularly at the imaging CCD. Taking measurement data over different square areas, as shown in Fig. 5
, results in different amplitudes of the fundamental frequency since different amounts of noise in the nominally dark regions are included in the sampled data. Since an MTF measurement is strongly affected by normalization, an error in the fundamental frequency will influence the MTF significantly. As shown in the figure, the sum of all measured intensities increases linearly with the number of sampled pixels when the sampling area exceeds a certain minimum value. This linear intensity increase results from the sum of background noise.**

**Therefore, the sampling area should be taken to be sufficiently large to be in the region of linear increase; this case implies that all of the relevant signal has been measured and any further outlying regions only contribute to noise. The linear range is found by calculating the correlation coefficient starting from the right and moving step-by-step to the left, i.e. from the linear to the nonlinear area. As a stop criterion, a value of 0.9999 for the correlation coefficient of a linear regression is taken. Left from this point the sampling area is defined, whereas the right part is used for the determination of the noise which will be subtracted from the measured data.**

**4.4.3 Influence of the objective lens and summary of the calculation process**

**Taking the ratio of F_{i} to F_{g} compensates for the effect of both the finite size of the light source as well as the influence of the focal lengths of the collector and the lens under test. Since the system is linear, the MTF of the lens under test can be inversely calculated by a point-by-point division of the overall optical system MTF and the MTF of the objective lens [14]. For the used high performance microscope objective lenses, the diffraction limited MTF of the microscope objectives are used as a suitable approximation, because the lens errors are assumed to be relatively small.**

**The entire calculation process for the MTF is summarized in Fig. 6
. The PSF is measured and either subject to a 2D Fourier transform to yield F _{i} or integrated to yield the LSF, from which F_{i} can be calculated using a (faster) 1D Fourier transform. The resultant value of F_{i} is then corrected by a division by F_{g}, yielding H_{I}
_{;} taking the absolute value, finally yields the MTF.**

**5. Experimental results**

**5.1 Reference measurements**

**To validate the performance of this measurement approach, the MTF of two commercial achromatic glass lenses (Linos G322250000 (A), G322201000 (B)) was determined using the setup shown in Fig. 3. Lens A has a focal length of 6 mm and an aperture of 3 mm whereas lens B has a focal length of 20 mm and an aperture of 10 mm. For external corroboration, the same measurement was carried out using an high-accuracy commercial MTF setup (Trioptics ImageMaster® HR).**

**The measurement procedure and data analysis was carried out as outlined above: The generated PSF was measured, the noise level subtracted and the LSF calculated by integration. F_{i} was then determined by taking the Fourier transform and the correction due to the finite pinhole size was applied.**

**As shown in Fig. 7(a)
, the MTF result of lens A after the two-step correction (4.4.1 and 4.4.2) agrees quite well with that generated by the commercial setup. For lens B, Fig. 7(b) shows that the measurement results do not agree as well as in case of lens A. This discrepancy is due to the limited beam diameter of the illumination field, whose diameter is about 12 mm, not much larger than the 10 mm aperture of lens B. Therefore, for a large aperture lens, a much wider incoming parallel light beam is required to avoid system errors. The parameters of lens A and the tunable micro-lenses which are our primary interest are comparable, indicating that the measurement setup is suitable for MTF measurements of micro-lenses.**

**5.2 Measurements of the tunable micro-lens**

**Before measuring the MTF of the micro-lens, one needs to measure its focal length at different pressures to allow the correction of the finite pinhole size using Eq. (4). Herein, f_{t} represents the effective focal length (efl), the distance between the focal point and the principle plane of the lens. Since the tunable lens (as seen in Fig. 1) is a thick plano-convex lens, the efl cannot be measured directly, only the back focal length (bfl) is directly available. The difference between efl and bfl depends on the focal value, thus on the applied pressure or curvature. bfl is smaller than efl, and for an operating pressure of 10 kPa, for example, the difference is roughly 0.7 mm. At lower pressures, the difference becomes smaller. Generally, the influence of this small difference is not significant and thus bfl is used in the correction.**

**Figure 8(a)
shows the lens profiles measured under different pressures by a profilometer. The profile data are imported into ZEMAX to determine the expected back focal length. These bfl values are compared with directly measured values at different pressures, which are determined by the microscope setup using the movable z-stage. The simulation and measured values are compared in Fig. 8(b) and show good agreement.**

**The images of the focal points at different pressures, representing the PSF, are shown in Fig. 9(a)
. The color-coded z-axis is displayed using a logarithmic scale to make even small amplitudes near the central peak visible.**

**Integrating the PSF along the vertical direction leads to the LSF for the horizontal direction; the LSF at different pressure values is plotted in Fig. 9(b), indicating that for lower pressure the intensity is concentrated into a smaller area. The narrower the LSF, the less blurring occurs in the image forming process, and the resultant compact impulse response indicates a better image quality. The degradation of performance at higher pressure is due to the (undesirable) aspheric deformation of the lens near the edge of the aperture, an effect which leads to strong aberrations.**

**A two-dimensional MTF may be generated by a 2D Fourier transform of the PSF and gives an indication of the resolution in different radial directions; this characteristic may be of value in determining the performance of micro-lenses with non-radially symmetric profiles, imperfections, or assembly errors. Figure 10(a)
shows a 2D MTF result of the tunable micro-lens operated at 1 kPa corresponding to f = 11 mm. In Fig. 10(b), the 2D MTF result becomes more clear at different angles in polar coordinates. One can easily see deviations from the axial symmetry, indicating lens errors which are not rotationally symmetric, as shown in Fig. 11
. From the shape of the PSF or MTF curves, it is possible to examine limitations in the performance of the micro-lens due to fabrication and design errors.**

**The measured MTF of the tunable micro-lens at 1 kPa is compared with the result using the commercial setup and ZEMAX simulations. These three results agree reasonably well, as seen in Fig. 12(a)
, although the commercial equipment seems to overestimate the resolution at high spatial frequencies (above 30 lines/mm) when comparing with our measurements and simulations. Finally, Fig. 12(b) shows the MTF of the micro-lens operated at different pressures, showing that the performance degrades at higher pressure. This behavior agrees with that shown in Fig. 9(b), where aberrations increase due to non-ideal lens deformation at high pressure. The differences between the MTF curves for different focal lengths of the tunable lens are not very large. This indicates that the imaging quality should be similar. Figure 2 confirms this behavior for the qualitative conclusion derived from the images at different focal length.**

**6. Conclusion**

**A low cost, high efficiency optical test setup for measuring MTF based on conventional optical microscopy has been proposed and used for the determination of the MTF of tunable micro-lenses. The approach allows easy determination of BFL, PSF and LSF. Reference measurements using commercial lenses and comparison with a commercial MTF setup, as well as simulation results, show the validity of the method. The experimental results show that the performance of the tunable micro-lenses is best at low actuation pressures, corresponding to large focal lengths, and that the resolution degrades for shorter focal lengths. The resulting aberrations can partly be corrected by using additional optical components, such as fixed-focus lenses or additional tunable micro-lenses [16]. Using the approach outlined here, it is possible to realize an automated measurement system, automatically scanning the focal length while concomitantly determining the best MTF at different pressures.**

**In summary, the system is highly flexible and is able to qualify variable micro-lenses over a wide focal range by modification of various modules, such as the diameter of the pinhole source, the aperture of the collimator lens, or the magnification of the objective lens. Therefore, MTF measurements can be performed for a wide variety of lenses by simply modifying modular components of the measurement system.**

**Acknowledgment**

**The crosscheck experiment was supported by the German company SICK AG who provided the commercial MTF setup (Trioptics ImageMaster HR). The authors would like to thank Peter Liebetraut, Philipp Müller and Dr. Wolfgang Mönch for technical discussions.**

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