2.1.

Substrate Refractive Index

The refractive index can span a large range, from about 1.44 for glass to 3.5 for silicon, and its variation is usually measured by interferometry.^18,19 The quantitative evaluation of this variation is defined by standards,^20,21 and materials are sorted into different classes. The three best classes available from wafer manufacturers have a refractive index variation $\mathrm{\Delta }n<5·{10}^{-6}$, giving an optical path difference peak-to-valley (PV) smaller than 5 nm for wafers with a thickness of 1 mm. Such variation can thus reasonably be neglected for micro-optics with substrate thicknesses in the range of 0.5 to 5 mm. For this reason, one can occult this control step in the microlens metrology process when high-quality substrates are used.

2.2.

Substrate Thickness

A variation in the substrate thickness mainly changes the effective back focal length, as can be seen in Eq. (1). Controlling the substrate thickness is thus particularly important in systems where the microlens position cannot be adjusted to compensate for defocusing. For example, a tilt between the front and back surfaces of the substrate is problematic for an MLA as it creates a linear variation of the microlenses focal length.

To do so, thanks to the long history of the semiconductor industry, many different tools are commercially available to measure the substrate thickness and ensure its uniformity. Likewise, there exist many wafer suppliers. Usually, the thickness wafer tolerance is typically in the range $\pm 10$ to $25\mu \mathrm{m}$ and the total thickness variation in the range 1 to $10\mu \mathrm{m}$ depending on the material and the wafer size. Measuring and monitoring these parameters does not present any particular difficulty and is thus not the heart of the microlens metrology although it has to be done.

2.3.

Microlens Surface

The surface of a plano-convex microlens is the functional component of a microlens and its fabrication is the main know-how of manufacturers. As the refractive index variation is negligible and the thickness variation produces only a focal shift, the performance of a microlens is mostly dictated by its surface. For this reason, the characterization of microlenses is very closely linked to surface metrology and the assessment of their performance by surface measurement is possible. More specifically, micro-optics deals with (a) spherical micro-surfaces that are very smooth.

The surface of microlenses can be divided into two qualitatively different components:²² first, the surface form or figure,²³ which contains the low spatial frequencies of the surface and shapes the wavefront; and second, the roughness, which contains the high spatial frequencies and is mainly responsible for light scattering.^24–26 This specific division is not absolute and there is no general admitted rule to determine the cutoff frequency between these two regimes. One approach is to use the Fresnel number to estimate the limit.²⁷

As the measurement of high frequencies usually requires a different measuring system than measuring low frequencies, we discuss the measurement of the two components separately in the following sections.

2.3.2.

Surface roughness

Typical values for microlens roughness are in the range $S_q=1$ to 10 nm. The vertical resolution of the measuring tool must thus be sufficient to measure this. Also, the highest spatial frequency that the instrument should be able to measure to cover all the roughness range²⁶ must be in the order of the light working wavelength, thus about $0.5\mu \mathrm{m}$.

For optical surface profilers, measuring the roughness is possible when they are equipped with high magnification microscope objectives ($100\times$ or higher). Indeed, this allows accessing high frequencies components. Moreover, it is only with these magnifications that confocal microscopes can achieve a vertical nanometric resolution. For interferometric techniques, this is possible for all magnifications.

One also has to note that the values for the vertical resolution are given for flat measurements. For slopes, the ratio signal-to-noise decreases meaning a decrease of the vertical resolution and more difficulty to measure small height differences. Measuring the roughness of steep surfaces is thus also more difficult for optical profilers.

Contact probes can also be used to measure surface roughness. However, the tip acts as a low-pass filter and, consequently, the tip radius must be small enough to access high spatial frequencies. Typically, mechanical styluses have a tip whose radius is $>2\mu \mathrm{m}$, thus preventing a complete collection of the surface roughness information. As an alternative, atomic force microscopes (AFMs) that have smaller tips, can be used. Different works report on the microlens surface measurement by AFM and a comparison of the different techniques can be found in Refs. 40 and 41 (Macro and Micro-Tribology and Mechanics of Magnetic Storage Systems).

It has to be noted that microlenses manufactured by photoresist reflow with subsequent RIE⁴² have usually a roughness with a random shape and with an amplitude $S_q\approx 1\mathrm{nm}$, see Fig. 2(a), which is optically good enough. On the other hand, microlenses that are manufactured based on micro-machining mastering^43–45 have typically a roughness with a specific shape caused by the tool, see Fig. 2(b). In this case, more care should be taken to specify the roughness tolerance as the amplitude might significantly degrade the optical performance.

Fig. 2

Examples of roughness measurement by a CSI (Nexview from Zygo) equipped with a $100\times$ microscope objective. Spatial sampling is $0.09\mu \mathrm{m}/\text{pixel}$ and the spatial resolution is $0.34\mu \mathrm{m}$. A high-pass spline filter whose cut-off period is $20\mu \mathrm{m}$ is applied. (a) For a reflow-RIE microlens, the roughness is seen to be random. $S_q=0.5\mathrm{nm}$. (b) In the case of a micro-machined microlens, concentric circles created by the manufacturing tool are visible. $S_q=8.2\mathrm{nm}$.

3.1.

Wavefront

The most common method for phase measurement is interferometry. Two types of interferometers have mainly been applied to microlens testing: the Mach–Zehnder type^46–49 and the Twyman–Green type.^37,47–49 Both interferometers are used in different configurations: usually, the probing wave is either plane or spherical, and the on-axis wavefront aberration is recorded.

Another technique that can be used to record the wavefront is using Shack–Hartman sensors. Those instruments typically consist of a 2D MLA placed in front of a camera and serve to sample the incoming wavefront. Each microlens produces a focal spot whose position in the image plane is a function of the incoming wavefront tilt at this particular microlens location. Thus, recording the focus position shift of all microlenses allows the reconstruction of the entire incoming wavefront.

With this technique, there is no need for multiple images recording contrary to interferometry, but it has the drawback that wavefront sampling is usually rather low. This technique is commercially available (e.g., Ref. 50).

3.2.

Point Spread Function

The PSF of a microlens provides information about its performance.^39,51 For microlenses used in imaging systems, the knowledge of the PSF leads to important information such as the resolution, the Strehl ratio, or the depth of field. This measurement is particularly important for high-quality imaging optics used in astronomical instruments. An example of the measurement of a microlens PSF is shown in Fig. 3.

Fig. 3

Measured 3D PSF of a spherical microlens illuminated by an incident plane-wave. The effect of spherical aberration can be observed as the presence of many smaller intensity peaks before the main focal spot.

This type of measurement is relatively straightforward to setup compared to interferometry as it consists mainly of a series of images taken with the help of a microscope objective at different locations along the optical axis and around the focal spot of the tested microlens.⁵² This operation is typically realized by mounting the microscope objective on a piezo actuator.

PSF measurements give a good insight into the optical performance of high-quality microlenses. Another advantage is the concentration of the focal spot on a very small area ($<10\times 10\mu \mathrm{m}$) allowing the use of high magnification and high NA microscope objectives. This condition is important since the NA of the microscope objective must be larger than the NA of the tested lens to resolve correctly the PSF and not omit its fine structure.

3.3.

Discussion

The advantage of functional testing is a direct assessment of the optical performance provided the test is performed under valid conditions. The drawbacks of this approach are the cost to setup these valid conditions and the difficulty to make a link with the surface.

Valid conditions mean that the testing conditions must be equivalent to the working conditions. In particular, the test illumination must be similar to the working one, in terms of wavelength, wavefront, and irradiance. Unless it has a very low NA, the surface of a microlens is generally aspheric to reduce the aberrations. As the asphericity depends on a specific illumination, using a different illumination will result in artificial aberrations. This means that the standard plane-wave and spherical wave illuminations are not sufficient to test aspheres. This poses also difficulties when the material is non-transparent to visible light, e.g., silicon. All this limits thus the possible implementation of such a method in an industrial environment where dozens of different microlenses are processed and tested in parallel.

Another drawback of these measurements is the accuracy assessment. As these types of characterization take into account not only the microlens geometrical parameters but also the physical ones, one could think that the performance evaluation would be more accurate in comparison with surface measurements. However, one also has to take into consideration the error caused by the optical setup. Indeed, relay optics between the camera and the microlens add errors. In particular, a microscope objective is required to expand the field-of-view (FOV) and additional aberrations are induced. This aspect is discussed more deeply in Sec. 5.2. In summary, we can say that functional testing is less practical to setup than surface measurement.

5.1.

Geometrical Limitations

Here, we discuss the influence of the microlens geometry, mainly its diameter, height, and maximum surface slope, on the measurement process. Indeed, these quantities are fundamental factors that limit the usability of the different measurement techniques. We have to stress that these geometrical limitations concern only surface profilers, both mechanical and optical, as functionality testing is performed under working conditions and thus does not face this problem.

For stylus tools, the diameter of a refractive microlens, typically $<1\mathrm{mm}$, and its height, usually below $300\mu \mathrm{m}$, are not limiting factors. Indeed, profiles over several centimeters and with a vertical range over 1 mm can be recorded with standard commercial instruments. For surface slopes, the limiting factor lies in the geometry of the stylus tip, which is a cone ended by a portion of a ball. Indeed, the maximum surface angle of the rounded part of the tip must be higher than the slope of the measured surface to have proper contact. A common value for the maximum slope of the tip is 60 deg, but tips exist with higher values. This value is sufficient for almost all microlenses used in real systems nowadays. As a consequence, the microlens geometry is not a concern for stylus instruments.

For optical surface profilers working in reflection, the story is completely different. Indeed, the geometrical limitations depend almost entirely on the imaging system, and especially on its central compound, the microscope objective. Indeed, its FOV gives the largest measurable microlens diameter, its working distance (WD) gives the maximum measurable height, and its NA gives the limit of the maximum surface slope that can be measured using specular reflection, see Fig. 6(a). As shown previously, the microlens surfaces are very smooth, and going above this limit by using scattered light seems to us almost unfeasible even if some work is currently done.^57,58 We have thus to look at the microscope objectives properties that are used in those optical surface profilers to estimate the limitations.

Fig. 6

(a) Illustration (not to scale) of the maximum surface slope of a reflow-RIE based microlens. In this example, the NA of the microscope objective is not sufficient to collect the light that is reflected at the steepest point of the surface. (b) Comparison between different optical surface profilers in terms of surface geometry limitation. Confocal microscopes present an advantage over CSIs for FOV below $800\mu \mathrm{m}$ as the maximum measurable slope is higher.

The number of available microscope objectives is relatively limited. Here, we briefly compare the microscope objectives that equip selected commercial optical profilers: the confocal microscope $\mu \mathrm{surf}$ from Nanofocus, the CSI Nexview from Zygo, and the combined instrument (confocal + CSI) S-neox from Sensofar.^59–61 For the $10\times$ magnification, the NA is similar for all objectives. For the $20\times$, interferometric objectives have a NA of 0.4. For confocal, objectives with an NA of up to 0.6 can be obtained. This suggests an advantage of confocal microscopes over CSIs. This is also confirmed for $50\times$ and $100\times$ objectives. For instance, the $50\times$ Mirau objective has a NA of 0.55, whereas confocal microscopes can be equipped with objectives whose NA is 0.8.

These considerations are summarized in Fig. 6(b), which presents the estimated maximum measurable slope as a function of the FOV. Due to non-interferometric microscope objectives, confocal microscopes have a better capability to measure surface slopes. It has to be stressed that the given values represent only an estimation. Other factors, such as the sampling in the case of interferometric measurements, or the signal-to-noise ratio are other important limiting factors.

To put these considerations in a more practical context, we use three instruments based on different techniques to measure the surface of a 2-mm diameter reference ball. First, we use the CSI Nexview from Zygo equipped with a $100\times$ NA 0.85 WD 0.5-mm microscope objective. Because the FOV is reduced for high magnifications, we also use stitching with this instrument. Second, we use the stylus Talysurf from Taylor Hobson, which can record 3D surface information and not only surface profiles. Finally, we use the confocal microscope $\mu \mathrm{surf}$ custom from Nanofocus equipped with a $20\times$ NA 0.6 WD 0.9-mm microscope objective. Stitching is not used in this case.

Figure 7 shows the reference ball surface information that can be collected by these three instruments. As expected, the stylus probes the largest surface area for a comparable time frame but with the drawback of low sampling, which prevents the acquisition of the high spatial frequency information of the surface.

Fig. 7

Surface measurements of a 1-mm ROC reference ball by different measuring instruments. (a) Profiles. (b) Top view: FOV. The stylus tool allows gathering more information about the surface.

To evaluate the quality of these measurements, the best-sphere fit residuals (irregularities) are shown in Fig. 8. All three measurements present artifacts when used in the manufacturer configuration. For the confocal measurement, it has been shown that this error can be corrected.³⁶

Fig. 8

Form of the best-sphere fit residual for the three instruments calibrated following the manufacturer guidelines. (a) Nexview $100\times$, $\mathrm{RMS}=34\mathrm{nm}$, sampling $x$: $0.17\mu \mathrm{m}$, $y$: $0.17\mu \mathrm{m}$. (b) Talysurf, $\mathrm{RMS}=55\mathrm{nm}$, sampling $x$: $10\mu \mathrm{m}$, $y$: $40\mu \mathrm{m}$. (c) $\mu \mathrm{surf}$ $20\times$, $\mathrm{RMS}=78\mathrm{nm}$, sampling $x$: $1.6\mu \mathrm{m}$, $y$: $1.6\mu \mathrm{m}$. As the ball surface has a deviation from a spherical surface below 10 nm RMS, these residuals are considered measurement artifacts. In particular, the vertical lines visible in panel (a) are the consequence of the stitching operation (it is worth noting that the $x$ and $y$ scales are different).

This comparison is summarized in Table 1. Information about measurement time is also included. Like this, any user can select the measuring system based on his needs in terms of surface information, quality of the measurement, and measurement time.

Table 1

Comparison between three different measurement methods using three different instruments. The amount of information that is collected (surface area) and its quality (RMS value) are different from one method to the other, but the measurement time as well. A trade-off between these different parameters has to be found.

In addition to their relative difficulty to measure steep slopes, one disadvantage of interferometric systems is that another type of errors occurs during the phase reconstruction, which causes a localized artificial high roughness. This is shown in Fig. 9(a), which presents the measured irregularity (best asphere fit residual) of an aspheric surface. Even though the exact origin of this issue might be difficult to determine, this problem clearly limits the capability of the instrument. However, this error can be well corrected when the surface is smooth⁵⁵ as it is shown in Fig. 9(b), which presents the same exact measurement after being processed.

Fig. 9

(a) The measured irregularity of an aspheric microlens ($R=460\mu \mathrm{m}$, $\kappa =-4$) puts in evidence artifacts in CSI measurements due to the bad reconstruction of the surface from phase information. (b) The same measurement after correction.

This discussion would not be complete without a comment on the range of geometries spanned by real microlenses. First, we can say from our experience that a large majority of the microlenses fabricated by resist reflow with a subsequent RIE process are measurable by optical surface profilers without stitching. However, the more recent fabrication technique based on polymer replication with direct writing molds, which allows creating steeper and higher microlenses, represents a real challenge for surface metrology. Currently, the only available option for this technology is to use mechanical styluses that perform multiple scans to obtain the full 3D surface information. As mentioned previously, the main drawbacks of this approach are the measurement time and the difficulty to measure sticky surfaces (e.g., polydimethylsiloxane).

5.2.

Accuracy and Precision

Measurement accuracy and precision⁶² are fundamental limits and determine the highest microlens quality that can be achieved as these limits restrain the feedback information needed to optimize the fabrication process. On the one hand, it is easy to assess precision by running multiple measurements under repeatable or reproducible conditions. Then, a comparison with the desired tolerance allows us to decide whether the instrument is capable to resolve what we are looking for. Figure 10 shows the ROC precision under repeatable conditions for two different instruments for the same 2-mm diameter reference ball. The ROC precision mainly depends on the quality of the actuators in the systems as well as on the vibrations present during the measurement.

Fig. 10

Measured ROC distribution under repeatable conditions with two different tools. (a) Nexview: 100 occurences, $\sigma =0.2\mu \mathrm{m}$. (b) Talysurf: 15 occurences, $\sigma =0.05\mu \mathrm{m}$. The ROC nominal value is $1000\mu \mathrm{m}$.

On the other hand, accuracy assessment for any measured parameter is trickier as most of the measuring systems are complex and a model for the measurement process is not available. A simple workaround is to estimate the measurement accuracy using a reference object or calibration artifact. This discussion about accuracy does not concern functional testing in real working conditions.

Several calibration artifacts for surface measurements exist. First, flat reference mirrors are used to correct the residual flatness of optical surface profilers. However, these references are of limited use for microlenses, whose surface largely deviates from a plane. More interesting are the reference balls of very high quality that are commercially available (e.g., Ref. 63). A random ball test⁶⁴ performed with those balls provides a reference surface with arbitrary accuracy.⁵⁵ Thereby, accuracy can be assessed and measurement error corrected. The limitation of this approach is the limited number of available ball ROCs. Also, this approach is of limited usefulness for aspheric microlenses.

For the functional tests performed in transmission, i.e., wavefront of PSF measurements, which require relay optics between the tested microlens and the camera, the situation is even more tricky. Indeed, no artifact exists for transmission, meaning that one cannot evaluate easily the accuracy. In macro-optics, one option is to use computer-generated holograms. This option is not applicable to micro-optics for scale reasons, with the exception of cylindrical microlenses.⁶⁵

This issue leaves us with the assumption that the error caused by the relay optics is negligible compared to the aberrations provoked by the tested microlens. This is however a strong assumption that is likely not valid in practice. An alternative would be to evaluate the error based on ray-tracing and perfect knowledge of the relay optics, which is not easy to do. This difficulty to evaluate the measurement uncertainty leaves us in the dark and is a real drawback to increase the microlens quality. We believe this is one of the main drawbacks of testing in transmission.

5.3.

Industrial Perspective

In an industrial manufacturing environment, microlens surface measurements are preferred over functional testing (wavefront, PSF) for multiple reasons: first, because the surface must be known to develop, optimize and control the fabrication process. Second, microlenses with different materials, working at different wavelengths, having different functionalities thus different surface forms are manufactured. This would require many different optical setups to test the optical function of all these microlenses, whereas a single optical surface profiler can handle most of them.

Second, as the surface profilers have initially been developed for the semiconductor industry, many suppliers exist that propose instruments designed for wafer-level inspection. Most of the techniques can be commercially found.^66–72 On the other hand and to our knowledge, there is no commercially available instruments that measure the microlens functionality in transmission and that are equipped for wafer-level optics. The fact that building in-house an instrument that can handle wafer-level micro-optics is extremely complicated also pushes microlens manufacturers to use already existing surface profilers.

5.4.

Guidelines

Based on the previous discussions, we propose some guidelines about the practice of microlens metrology. First, we discuss what are the steps that must be undertaken to obtain information for fabrication process optimization. In a second time, we discuss the evaluation of the microlens performance.