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Abstract
Tolerancing is an important step toward the fabrication of high-quality and cost-effective lens surfaces. It is critical for wafer-level optics, when up to tens of thousands microlenses are fabricated in parallel and whose surfaces cannot be formed individually. However, approaches developed for macro-optics cannot be directly transposed for microlenses because of differences in fabrication and testing techniques. In particular, microlens surfaces are usually limited to conical surfaces. Here, we study the connection between the microlens optical performance and the form of its surface, suggesting surface form representations suited for tolerancing purposes. Then, we compare them with common representations for tolerancing real optical systems. Measured surface forms of microlenses are also provided to make the tolerancing procedure realistic. In addition, we propose term definitions for micro-optics, complements to typical terms for macro-optics, to ease the communication between optical designers and manufacturers. Based on the results presented in this paper, guidelines are proposed for tolerancing microlenses. We suggest applying them as a first step toward a more effective and comprehensive tolerancing procedure.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. INTRODUCTION
Refractive wafer-level optics (WLO) [1] are a well-established technology that enables important applications such as wavefront sensing [2], laser beam shaping [3], or fiber coupling [4]. The fundamental component of refractive microlenses is aspheric surfaces. An efficient tolerancing of such surfaces is thus an essential process toward successful micro-optical systems.
The subject of tolerancing aspheric surfaces has been explored by different authors [5,6] and is summarized by a standard from the international organization for standardization (ISO) [7]. However, these approaches are mostly applicable for conventional macrolenses, and their application to micro-optics requires some adaptations, which, to our knowledge, have not been discussed yet. Indeed, fabrication, testing, and the type of surface that can be produced are different between the macro and micro regimes.
WLO fabrication techniques allow the manufacturing in parallel of up to tens of thousands microlenses on a single wafer. An avoidable drawback of WLO is that the surface of the individual microlenses cannot be formed independently. It may happen that over one substrate there are elements that meet the optical specification and others which do not. For this reason, it is particularly important to have an effective tolerancing for WLO aspheric microlenses. Indeed, tighter tolerances than needed results in an unnecessary discarding of microlenses and in an increase of the fabrication cost. Another drawback of WLO is its difficulty to produce arbitrary surface forms. This is why microlenses are usually restricted to conical surfaces, which are considered in this work.
The metrology of microlenses is also different from its macroscopic counterpart. Microlens surface form characterization usually relies on optical surface profilers, whereas for macrolenses, this operation is mainly based on wavefront deformation measured by interferometry [8]. This leads to differences in the definitions of surface form tolerances.
The process of tolerancing starts with the choice of a surface form representation, i.e., a set of parameters that represents the surface form. To motivate this choice, we investigate the link between the optical performance and the surface form for simple systems. Then, the performance degradation of the optical system under investigation is evaluated as a function of perturbations of the surface form. To perform this step as accurately and realistically as possible, we include, as a manufacturer of WLO, real distributions of surface form deviations. This step allows the determination of the tolerances for each parameter of the surface representation. This is an optimization problem, and we solve it by using global optimization algorithms. Finally, we compare the different surface form representations for simple optical systems as well as for real systems. Based on these results, we propose guidelines to perform an efficient tolerancing of aspheric microlenses. Moreover, all parameters and terms that are used are defined and motivated in order to avoid any confusion and to stress the need for standardization in this domain.
This paper starts with a discussion about the representation and the fabrication of aspheric surfaces in Section 2. Typical surface form deviations are also provided in this section. In Section 3, we try to derive ideal surface form tolerances by linking optical merit functions and surface forms for simple systems. The comparison between selected surface form representations is done for simple systems in Section 4 and in Section 5 for real systems. Discussion of the results as well as general guidelines are presented in Section 6. This paper also contains a series of annexes containing the mathematical derivations that support the discussion.
2. ASPHERIC SURFACE REPRESENTATION AND FABRICATION
A. Surface Form Representation and Tolerances
In this work, by aspheric surface, we mean a rotationally symmetric conical surface defined by its sag [9],
The measurement of such surfaces is commonly performed by optical surface profilers such as confocal microscopes or coherence scanning interferometers or by mechanical stylus profilers, which directly provide the surface sag. This is different from macro-optics where the surface is usually tested by interferometry [8,10]. In the case of interferometry, the surface form deviation is measured as wavefront deformation and defined perpendicularly to the surface [7]. For this reason, all the surface form tolerances established for macro-optics, except the sagitta deviation, are defined to be perpendicular to the nominal surface. In the context of micro-optics, we propose to reuse most of the terms defined in ISO 10110-5 [7] but to define them to consider deviations along the $z$ axis as well as to adapt them to conical surfaces. Sagitta deviation and surface form deviation are thus equivalent for microlenses.
Another difference with ISO 10110-5 comes from the fact that the microlens is attached to the substrate, which is taken to be the reference flat. Tilt is thus considered in the total surface deviation.
Another parameter that we want to clarify is irregularity because it is no more defined for conical surfaces. We propose to keep the term irregularity for the surface deviation from the best aspheric fit with $ R $ as a fit parameter but $ \kappa $ being fixed at its nominal value $ {\kappa _n} $. Indeed, only a change of ROC can be compensated by refocusing, whereas a change of conic constant provokes spherical aberration that cannot be compensated. We propose to use the term aspheric irregularity for the similar case where $ \kappa $ is also considered as a variable of the fit. All these quantities are summarized in Table 1. This list is not exhaustive and should be completed when needed.
Table 1. Explanation of the Terms Used in This Papera
B. Fabrication Methods and Typical Surface Form Deviations
Many different techniques exist for the fabrication of microlenses [11]. However, for volume production, wafer techniques are the most cost-effective methods thanks to their capacity to produce up to tens of thousands microlenses in parallel. Among them is the well-known resist reflow with the subsequent etching fabrication process [12], which is widely applied for the fabrication of fused silica and silicon microlenses. Another important technology is the replication by UV imprint lithography [13], which allows the fabrication of polymer-based microlenses. Imprint templates can be manufactured by different techniques: resist reflow with or without subsequent etching, micromachining [14,15], or direct laser writing [16,17]. Typical microlenses have a diameter in the range of 100–1000 µm and a sag in the range of 1–300 µm. The substrate thickness is usually in the range of 0.3–3 mm.
It is always an advantage to have knowledge about the typical surface form deviation caused by the manufacturing process. This enables an accurate modeling of surface form deviations that are essential to evaluate the degradation of the optical system performance during the tolerancing procedure. Figure 1(a) presents typical surface form deviations produced by the reflow with the subsequent reactive ion etching (RIE) fabrication method. One advantage of this approach is that the produced surfaces have high spatial frequencies with very low amplitudes, and, thus, the 30 first Zernike polynomials are usually sufficient for an accurate modeling. Indeed, the root mean square (RMS) value of the remaining part of the Zernike representation (roughness) is typically a few nanometers and is thus negligible. This distribution of surface form deviations is obtained by measuring the surface of 300 silicon microlenses with a confocal microscope. We observe that defocus is clearly the main component. This is explained by the nonuniformity of the etching chamber and can be partially corrected as discussed in Ref. [18]. Also, the measured second and third orders have a quasi-null value, which confirms that the microlenses are not tilted. These measured surface form deviations are used to create a random distribution of surfaces during the tolerancing process.
Fig. 1. (a) Zernike representation of the surface form deviation for a typical measured distribution over one wafer of silicon microlenses manufactured by reactive ion etching. Value is given as the distribution average. Error bar half-length is one standard deviation. Indexing follows Noll’s convention [20]. It is seen that process nonuniformity is mainly translated into a spread of focal lengths. (b) Distributions of coefficients used to simulate the surface form deviation. Only the relative value is given since the coefficients are scaled in order to obtain the desired total surface deviation RMS.
C. Random Distribution of Microlenses
Zernike representation of the surface allows for an easy simulation of surface perturbation in order to perform tolerancing: the coefficient of each Zernike polynomial can be assumed to be a random variable with a certain distribution. For simplicity, we assume that all random variables are independent. We also assume that they all have the same uniform distribution with a mean of zero. The uniform distribution is chosen because it is considered more conservative than the normal distribution [19]. Defocus is assumed to have a standard deviation 8 times higher [see Fig. 1(b)], in order to mimic the distribution in Fig. 1(a).
In order to obtain perturbations with increasing amplitude, for each trial, only the relative value of the coefficients is used. Each coefficient is normalized to produce a surface form deviation with the desired RMS value.
It has to be stressed that the choice of the distribution matters only when the correlation between the surface form tolerances and the optical figure of merit (FOM) is weak. In this case, the optical designer is encouraged to get insight about the surface form deviation from the manufacturer.
3. SURFACE FORM DEVIATION AND OPTICAL PERFORMANCE
We start our discussion by restricting ourselves to the case of a plane wave focused on-axis. The on-axis restriction is motivated by several reasons: First, important applications of microlenses are on-axis systems such as fiber coupling or beam shaping. Second, it is a simple case that allows analytical derivations and consequently offers a good platform to discuss concepts. Finally, it can be seen as the starting point for off-axis focusing description, when the latest is considered to be a small perturbation of the on-axis case, which is reasonable for small angles. The off-axis case is nevertheless considered in Section 5.B.
In order to make a link between the lens surface and the lens optical performance, we have first to establish what is usually optical performance. Among the most widespread optical FOM, we can mention [21] the RMS spot size, the modulation transfer function (MTF), the wavefront aberration, and the Strehl ratio.
The optical system considered in this section and in Section 4 is depicted in Fig. 2(b). Typical geometrical parameters are chosen for the microlens. As defined in ISO 14880-1 [22], they are diameter $ 2a = 300\;{\unicode{x00B5}{\rm m}} $, ROC $ R = 500\;{\unicode{x00B5}{\rm m}} $, conic constant $ \kappa = - 0.54 $, refractive index $ n = 1.5 $, and thickness $ T = 500\;{\unicode{x00B5}{\rm m}} $.
Fig. 2. Schematic of the optical systems considered in Sections 3 and 4. A plane wave is focused on-axis by a microlens toward its (a) front side or its (b) back side. The microlens parameters are diameter $ 2a = 300\;{\unicode{x00B5}{\rm m}} $, ROC $ R = 500\;{\unicode{x00B5}{\rm m}} $, refractive index $ n = 1.5 $, and thickness $ T = 500\;{\unicode{x00B5}{\rm m}} $. Conic constant optimal value is $ \kappa = - 0.54 $ for back side focusing and $ \kappa = - 2.25 $ for front side focusing.
A. Ideal Aspheres
First, we focus on “ideal” aspheres, which means aspheric surfaces whose ROCs and conic constants deviate from their nominal values but without any irregularity. This makes the situation easy to work with because the surface is defined by only two parameters and allows for a graphical representation.
In this example, the FOM is chosen to be the on-axis RMS spot size. It is calculated as a function of $ R $ and $ \kappa $ for the system considered in Fig. 2(b). Figure 3(a) presents the results without focus compensation. The RMS spot size is almost constant along a diagonal in the $ R/\kappa $ space. By looking at Fig. 3(e), we observe that the RMS spot size is mainly a consequence of defocus, which seems to be a linear function of $ \Delta R $ and $ \Delta \kappa $. Appendix B confirms this linear relation and shows how to derive it. A change of conic constant provokes thus a defocus on top of spherical aberration. For this reason, the RMS spot size provoked by a change of ROC can be partially compensated by a change of conic constant. Finally, Fig. 3(c) shows that RMSt is well correlated to RMS spot size in this particular case.
Fig. 3. RMS spot size with and without focus compensation as well as different surface form tolerances for ideal aspheres in the range $ R = 500 \pm 20\;{\unicode{x00B5}{\rm m}} $ and $ \kappa = - 0.54 \pm 1 $. It is seen that RMS spot size is well correlated with RMSt and Zernike defocus $ {c_4} $. Likewise, if focus compensation is used, RMS spot size is well correlated to RMSi and Zernike primary spherical $ {c_{11}} $. Relation between these parameters can be mathematically explained (see Appendix B).
With focus compensation [see Fig. 3(b)], the RMS spot size becomes almost uniquely a function of the conic constant. This is explained by the fact that the level of spherical aberration is almost uniquely defined by the conic constant [see Fig. 3(f)]. Finally, Fig. 3(d) shows that RMSi is well correlated to RMS spot size in this particular case.
These results lead to ways of tolerancing ideal aspheres: for systems without focus compensation, because the defocus is a linear function of $ \Delta \kappa $ and $ \Delta R $, tolerances of the form $ R = {R_n} \pm \Delta R $ and $ \kappa = {\kappa _n} \pm \Delta \kappa $ are inefficient. Graphically, this can be seen by the fact that the tolerance bands can only represent a rectangle in the $ R/\kappa $ space, which cannot fit the diagonal observed in Figs. 3(a), 3(c), and 3(e). To increase the efficiency of the tolerancing, different ways are possible: tolerance RMSt or tolerance $ {c_4} $ and $ {c_{11}} $ at the same time, meaning constructing a trapezoid in the $ R/\kappa $ space. An example of rectangle and trapezoid is found in Fig. 3(a). Another option is to create a function of $ R $ and $ \kappa $ that can be used to perform the tolerancing. On the other hand, for systems with focus compensation, tolerances of the form $ R = {R_n} \pm \Delta R $ and $ \kappa = {\kappa _n} \pm \Delta \kappa $ are efficient. Indeed, the optical performance is well correlated to the conic constant.
B. Real Aspheric Surface Forms
The surface of a real microlens needs an infinite number of parameters to be defined. Or, practically, a high-enough number of coefficients for a given accuracy. An intuitive approach to tolerance the form of real surfaces would be to transfer the methods developed for ideal aspheres to tolerance of the aspheric irregularity, which is the deviation from the best asphere fit, $ R $ and $ \kappa $ being the fit parameters (see Table 1).
Another option is to express the optical FOM as a function of the surface and investigate the link between surface form and the optical FOM. Unfortunately, this is impossible in practice for most cases because the merit function cannot be analytically expressed as a function of the surface. An exception is the case of a plane wave focused toward the front focal spot of a plano–convex microlens [see Fig. 2(a)], when the FOM is represented by the geometrical lateral aberration or by the wavefront aberration.
In one dimension, the lateral aberration $ \delta u $ is approximated by (see Appendix A1)
This suggests that slope deviation should be toleranced as already mentioned by ISO 10110-5. This is confirmed in Fig. 4(a), which shows the excellent correlation between the on-axis RMS spot size and the slope deviation RMS for the simulated distribution of microlenses explained in Section 2.C.
Fig. 4. On-axis RMS spot size as a function of slope deviation for the simulated distribution of surfaces as described in Section 2.C. Without focus compensation, the correlation between RMS spot size and total slope deviation RMS is almost perfect, thus suggesting that total slope deviation RMS might be a powerful surface form tolerance. With compensation, the correlation between irregular slope deviation and RMS spot size is less ideal but still a good candidate for tolerancing. (a) No compensation; (b) focus compensation.
Likewise, Fig. 4(b) presents the RMS spot size as a function of the irregular slope deviation RMS for the same distribution of microlenses but when focus is compensated. In this case, the correlation is not as good as before. This is due to the fact that the fit parameter $ R $ and focus compensation do not exactly represent the same degree of freedom.
In the case of this optical system, the wavefront can also be computed analytically (see Appendix A2). The phase aberration as a first-order perturbation is approximated by
4. COMPARISON BETWEEN DIFFERENT SURFACE FORM REPRESENTATIONS
An ideal tolerancing process is a method that accepts all the elements that respect the specification on the optical FOM and which rejects all the elements that are out-of-spec. To quantitatively evaluate the different surface form representations, which we define to be the set of parameters and surface form tolerances to which tolerances are assigned, we introduce the notion of tolerancing efficiency. We define the tolerancing efficiency to be the ratio of the number of elements that are within the tolerances to the number of the elements that are optically in-spec. This definition makes sense only if there is at the same time a control of the out-of-spec elements that lie within the tolerance bands (false positive). Here we adopt the convention that consists of rejecting all out-of-spec elements.
In this paper, we consider the process of tolerancing as follows: we simulate a distribution of microlenses (see Section 2.C) that is supposed to represent well the potentially manufactured microlenses. Then, for a given surface form representation, the tolerances are found for each parameter. We restrict ourselves to find tolerances for each parameter independently and in a symmetrical form for nonzero target values (e.g., $ R \pm \Delta R $, $ RM{S_i} \lt $ Threshold,…).
Mathematically, the problem is to determine the tolerance bands that maximize the number of accepted elements under the constraint that all elements within the tolerance bands must be optically in-spec. When the surface representation consists of a single parameter, this can easily be done by hand. However, when using multiple parameters, finding the tolerances becomes an optimization problem in a multidimensional space that may have a multitude of local optimums. Global optimization methods are thus needed. Here, we use both a genetic algorithm [25] and simulated annealing [26] to find the optimal tolerances. The optimization run is done several times with both techniques to increase the chance of finding the global optimum. Any additional constraint that the manufacturer may set, for instance, a minimum tolerance of 1% on the ROC value, can directly be integrated in the global optimization process.
The comparison of the different surface representations is done for the following case: the merit function is the on-axis RMS spot size of the system presented in Fig. 2(b). Practically, a distribution of 2500 simulated microlenses is considered, and their total surface deviation RMS ranges from 0 to 1000 nm. First, no focus compensation is assumed. Arbitrarily, elements with a RMS spot size smaller than 15 µm are considered to meet the optical specification.
Tolerances for selected representations are given in Table 2 as well as their tolerancing efficiency. The simplest approach is to tolerance RMSt, but its efficiency is only 54.9%, which is the lowest in this test. Another representation with a single parameter is the total slope deviation. As shown in Section 3.B, this parameter is well correlated to the on-axis RMS spot size that we consider here as our FOM. This is confirmed since the efficiency is 100%, which makes it the best approach for this specific case. The representation based on $ R $ and RMSi gives an efficiency of 80.9%. The approach based on the generalization of ideal asphere with additional RMSai has an efficiency of 74.0%. Replacing $ R $ and $ \kappa $ by defocus and using the RMS of the surface form deviation without the defocus component, $ {{\rm RMS}^*} $, allows for an improvement of ${\sim}9\% $, giving an efficiency of 82.6%.
Some lessons can be learned from these results: first, RMSt is not an efficient parameter. Second, the total slope deviation RMS is ideal for tolerancing this system. Third, the conic constant should not be toleranced together with the ROC. Finally, increasing the number of parameters, for instance, from $R/{\rm RMSi}$ to $ R/\kappa /{\rm RMSai} $, does not necessarily increase the efficiency.
The same example is again considered, but this time, focus compensation is assumed. We arbitrarily fix that elements with a RMS spot size smaller than 10 µm meet the optical specification. Results are provided in Table 3. Most of the conclusions drawn in the case without focus compensation are also valid in this case: RMSt has a low efficiency of 45.5% and should not be used. This was expected because it is the only representation that does not take into account the degree of freedom provided by focus compensation. Representation based on slope deviation is again the most efficient representation even though it has only 86.9% of efficiency this time. All remaining representations are equivalent in this case.
A few more points need to be discussed: one might think that increasing the number of parameters increases the tolerancing efficiency, but this is not correct. Also, for a high number of parameters, the tolerancing procedure becomes more complex because of the optimization process. In any case, it is better to reduce as much as possible the number of parameters.
On a single wafer, microlenses do not have RMSt from 0 to 1000 nm, but are more condensed around a certain value. It may happen that the entire wafer is largely in-spec and the representation has no real importance. But, it may also happen that the microlens distribution center is not far from the spec limit, and in this case, the representation is even more critical than in the present example.
5. APPLICATIONS TO REAL SYSTEMS
Until now, the discussion focused on a very simple optical system. The complexity of real systems comes from different parts: first, the illumination. For instance, the apodization might be nonuniform, and the incoming light may have a certain angular spectrum. On-axis considerations need thus to be generalized. It also comes from the physical model needed to propagate the light: physical optics may be needed. Finally, more complex FOM such as the coupling efficiency (CE) into a waveguide might also be considered.
To show how to incorporate such advanced considerations, we present two examples inspired from real systems. The first one, presented in Section 5.A, presents a microlens that couples an on-axis Gaussian beam into a single-mode optical fiber (SMF). The second example, found in Section 5.B, presents the case of a microlens array (MLA) that acts as a projector. It is an off-axis example.
Fig. 5. Schematic of a collimated Gaussian beam coupled into a single-mode optical fiber. In order to achieve a good coupling efficiency, the microlens has to be actively aligned.
Fig. 6. Reduction of the coupling efficiency as a function of RMSi and weighted RMSi. With a Gaussian weight, the correlation between coupling efficiency and RMSi is increased, allowing a more efficient tolerancing (see Table 4). (a) Uniform weight; (b) Gaussian weight.
A. Fiber Coupling
One major application of refractive microlenses is to couple the light into waveguides and into optical fibers. Aspheric microlenses can do it perfectly in theory [27]. Here, we consider a collimated Gaussian beam that is coupled into a standard SMF (Cornell SMF-28) (see Fig. 5). The microlens is in fused silica, and its geometrical properties are diameter $ 2a = 500\;{\unicode{x00B5}{\rm m}} $, ROC $ R = 600\;{\unicode{x00B5}{\rm m}} $, conic constant $ \kappa = - 0.49 $, and thickness $ T = 500\;{\unicode{x00B5}{\rm m}} $. The light wavelength $ \lambda $ is 1.55 µm. In this configuration, the optimized CE is 95%.
Fiber coupling imposes tight tolerances, and focus compensation is usually performed. For this reason, we only compare representations that take into account this degree of freedom. Based on the results of Table 3, we consider irregularity RMS and irregular slope deviation RMS for the tolerancing. To go one step further, the apodization should be considered as well. For this reason, we also weight the RMS value of the surface form tolerances according to the beam apodization.
The CE is calculated for a simulated microlens distribution, as described in Section 2.C. The calculation is done with the software FRED from Photon Engineering [28]. We decided to set the optical specification to be a reduction of the CE of less than 20%. Tolerances are found according to the procedure described in Section 4.
Results of this tolerancing are presented in Table 4. RMSi provides the best tolerancing efficiency in this case. For this reason, we also use this representation with weighted RMSi. As expected, the tolerancing efficiency is increased with the weight. Figure 6 shows how the correlation between CE and RMSi increases when weighted according to the beam apodization.
In conclusion, for fiber coupling application, a simple and efficient way to tolerance the microlens is to use the ROC and the irregularity RMS with a weight that corresponds to the apodization of the beam.
B. Microprojector
One recent application of MLAs is pattern projection [29,30]. The working principle is presented in Fig. 7. A chromium layer buried in the substrate is illuminated and imaged at infinity by a projection microlens. This operation is performed in parallel by every microlens through the MLA. The final projection is thus the sum of the single projections.
Fig. 7. Representation of a single channel of a micro-optical array projector. A chromium pattern buried in the substrate is imaged at infinity and can thus be projected on a screen. The final image is the superposition of the projection of all channels. Microlens properties are diameter $ 2a = 300\;{\unicode{x00B5}{\rm m}} $, ROC $ R = 500\;{\unicode{x00B5}{\rm m}} $, conic constant $ \kappa = - 0.55 $, refractive index $ n = 1.5 $, and thickness $ T = 500\;{\unicode{x00B5}{\rm m}} $.
From a design and tolerancing point of view, such system can be described using backpropagation: the microlens is illuminated by plane waves with different angles and imaged on the chromium layer, which lies in the focal plane of the microlens. Because of the angular spectrum, the imaging is performed within a certain field of view and not only on-axis. If the angular spectrum is large, images in the field cannot be diffraction limited, and aberrations are mainly geometrical. In this case, it is advised to take the RMS spot size as the performance metric. Practically, the merit function should be the average RMS spot size over the field of view.
For each angular component, the RMS spot size is computed with respect to the RMS spot centroid produced by the nominal surface. The system distortion is thus not taken into account. This is not a limitation for such a projection system since this distortion can be compensated in the chromium pattern design. It has also to be said that such system is not compensated for defocus due to fabrication and utilization constraints. Microlens properties are diameter $ 2a = 300\;{\unicode{x00B5}{\rm m}} $, ROC $ R = 500\;{\unicode{x00B5}{\rm m}} $, conic constant $ \kappa = - 0.55 $, refractive index $ n = 1.5 $, and thickness $ T = 500\;{\unicode{x00B5}{\rm m}} $.
The average RMS spot size is computed for the simulated microlens distribution, as explained in Section 2.C. Here, we are interested in generalizing the results obtained for the on-axis system. For this, the average RMS spot size is compared to the total surface deviation RMS and the total slope deviation RMS. Results are presented in Fig. 8. As for the on-axis system, the correlation between RMSt and average RMS spot size is poor, whereas the total slope deviation RMS is extremely well correlated to the average RMS spot size.
Fig. 8. RMS spot size averaged over the full field of view as a function of the total surface deviation RMS and of the total slope deviation RMS for the simulated distribution of surface described in Section 2.C. Like for the on-axis case, the correlation between the RMS spot size and the total slope deviation RMS is excellent.
In conclusion, these results show that the off-axis case can be handled similarly to the on-axis case. In particular, total slope deviation RMS is also a very efficient parameter to tolerance microlens surfaces used in off-axis optical systems whose performance is well described by the RMS spot size.
6. DISCUSSION
A. Application Domain of the Results
In this paper, we focus on rotationally symmetric surfaces. However, the results of this work can be extended without problem to circular and noncircular cylindrical microlenses. Nevertheless, the useful tool of the Zernike polynomials cannot be used anymore. Historically, because of fabrication process limitations, microlenses have been limited to spherical or conical surfaces. Nowadays, thanks to micromachining and imprint technology as well as direct writing techniques, it is also possible to manufacture freeforms. Going from spherical to aspheric surfaces requires an additional parameter: the conic constant. For freeforms, a high number of parameters might be required, or the surface cannot even be defined by an explicit equation. In these situations, certain surface form tolerances such as irregularity are no more defined nor relevant. For all these reasons, tolerancing freeforms is out of the scope of this work even if some of the presented conclusions, notably that the slope deviation is an effective tolerancing parameter, could probably be applied to such complex surfaces.
B. Guidelines
Some lessons can be drawn from the results presented in this paper. The first one is that the parameters used for the surface design, $ R $ and $ \kappa $ in the present case, are not necessarily the best ones for tolerancing (see Fig. 3 and Tables 2 and 3). Indeed, the slope deviation RMS is demonstrated to be a more effective surface form tolerance in many situations.
A second lesson, as it is seen in Table 2, is that tolerancing the conic constant might be inefficient. Moreover, throughout the demonstrated examples, no case is found where it is judicious to do it. Also, the Zernike representation of the surface form deviation presented in Fig. 1(a) does not show that Zernike primary spherical, which represents the asphericity set by the conic constant, has a larger amplitude than any other specific coefficient. This suggests there is no reason why $ {c_{11}} $ or $ \kappa $ should be toleranced.
We have to mention that these recommendations are not specific to micro-optics. Indeed, similar conclusions have been derived for macrolenses [6,19,31]. This is not surprising since in both cases the light beam is mostly shaped by means of refraction.
ISO standards about surface form tolerances have not been developed or adapted for microlenses. In this paper, we show some of the differences between micro- and macro-optics and why it is difficult to use the exact ISO definitions for aspheric microlenses. The lack of standards in micro-optics might be an obstacle for the communication between manufacturer and designer as most surface form tolerances have to be redefined and clarified. One goal of this paper is to stress this problem and to advocate an extension of the ISO standards to micro-optics. As the tolerances we propose here as an alternative are, however, not fundamentally different from the ISO standards, they could be a first step toward such an extension.
C. Tolerancing and Machine Learning
Besides the challenges of surface form representation determination, and of tolerances determination, there is one more assumption that limits the effectiveness of the tolerancing procedure: the independent tolerancing of the parameters.
This is highlighted for ideal aspheres without focus compensation: the problem is not the choice of the parameters $ R $ and $ \kappa $, because by definition the surface is entirely defined by them, but the independent tolerancing of these parameters. As suggested, tolerancing the correct function of $ R $ and $ \kappa $ could solve this problem. However, finding such function would be difficult in a higher dimensional space.
Classification problems can be approached using machine learning methods [32], which could take care of the aforementioned challenges: determination of an optimal surface representation and determination of the optimal decision boundaries (tolerances). By using deep learning methods, input data could be in basic shape, for instance, Zernike coefficients or even meshed surfaces, like images that feed convolutional neural networks (CNNs). In these cases, however, the set of training data has to be pretty large.
Nonetheless, approaching the tolerancing process with machine learning methods has few drawbacks: this is nonstandard, and nothing can be put on a technical drawing. Moreover, the algorithm should be shared between the optical designer and the manufacturer, which poses issues in terms of software compatibility.
7. CONCLUSION
In this work, we review, introduce, and comment on approaches to tolerance of the surface form of aspheric microlenses. In particular, we compare them for typical micro-optical systems leading to important guidelines. For instance, we show that slope deviation should be toleranced as it has a strong correlation with the optical performance, when represented by the RMS spot size. We also emphasize that parameters used for the design, the ROC and the conic constant, are not ideal for tolerancing.
This work is not an exhaustive treatment of the topic, but proposes guidelines based on fundamental considerations about the optics of microlenses and the importance of the surface representation. It is more an extension and an adjustment of ISO 10110-5 developed for macro-optics rather than a complete new approach. In the future, this work could be extended to more complex systems, for instance, to systems that use several microlenses or to recently developed freeform microlenses.
Performing an effective tolerancing is important to guarantee the quality of the microlenses but also to allow the fabrication at the best possible cost. For these reasons, using the advice developed in this work should help any optical designer in the task of tolerancing their optical system, in particular if they are not familiar with micro-optics.
APPENDIX A: LINK SURFACE FORM DEVIATION AND OPTICAL MERIT FUNCTION
Here we consider the optical system depicted in Fig. 2(a), and we want to link the microlens surface form deviation to the optical performance. We use the on-axis RMS spot size and the on-axis RMS wavefront as the optical merit functions.
The incoming plane wave can be represented by rays. Any of these rays travel parallel to the optical axis and are thus only defined by its $ y $ position $ \tilde y $. Then, this ray hits the surface $ z = s(y) $ at a position $ ( {\tilde y,s(\tilde y)} ) $. After refraction, this ray is a linear function $ y = m(z - s) + \tilde y $ with $ m $ given by
A1. RMS Spot Size
When the surface is optimized, all rays have zero lateral aberration $ u $ at the focal spot $ z = {f_{{\rm E},f}} $ and thus satisfy the following condition:
Now, a real surface has a certain surface form deviation, which is usually small compare to the surface itself. One can take a first-order perturbative approach and rewrite the previous equation,
By identification, when the second orders are neglected, the lateral aberration at the focal spot is given by
By having a look at the different orders of magnitude for typical microlenses, we observe that the dominant term is
and thus that lateral aberration at focus $ \delta u $ is mainly caused by slope deviation.A2. Wavefront Aberration
Following the same idea, it is possible to perform a perturbative approach to approximate the wavefront aberration for the same optical system. We start by considering the phase at the vertex plane for a ray $ \tilde y $. Using optical path length, it is written as
In the case of the paraxial approximation, all the trigonometric functions are linearized, and $ m \approx (n - 1)s^\prime $. Also, if the square root is replaced by its Taylor expansion, the phase becomes
Now, the wavefront aberration can be approximated by the total differential, $ \delta \phi \approx \frac{{\partial \phi }}{{\partial s}}\delta s + \frac{{\partial \phi }}{{\partial s^\prime }}\delta s^\prime $. Finally, the wavefront aberration is explicitly given by
APPENDIX B. RELATION BETWEEN $ R $, $ \kappa $ AND ZERNIKE DEFOCUS COEFFICIENT $ {c_4} $
Here, we investigate the link between three parameters: $ R $, $ \kappa $, and Zernike defocus coefficient $ {c_4} $. To allow analytical derivations, we consider the Taylor expansion of Eq. (1) to the fourth order,
For a microlens with $ R = 500\;{\rm um} $, $ 2a = 300\;{\unicode{x00B5}{\rm m}} $, and $ \kappa = - 0.54 $, the RMS value of the difference between the Taylor expansion and the complete expression is 1.2 nm. This is acceptable for tolerancing, where only small perturbations of the surface around its nominal form are considered.
The defocus $ {c_4} $ is given by the scalar product of the defocus polynomial expression and the aspheric formula. The scalar product is the integral over a unit disk, and for this reason, the variable change $ h = a\rho $ is made, $ a $ being the semiaperture and $ \rho $ the radial position. This is expressed by
And after integration,
Again, it has to be stressed that we are interested in small perturbations, and thus in the differential form of this equation, which reads
The subscript “$n $” is here to stress that we consider a deviation from the nominal surface. Defocus is thus a linear function of $ \delta R $ and $ \delta \kappa $, which is confirmed in Fig. 3(e). This means that a change of ROC can be compensated by a change of conic constant. This also means that if a lens suffers from spherical aberration, it is possible to reduce it by slightly moving the lens out of its paraxial focus.
What is also interesting to look at is the amount of information carried by $ R $ and $ \kappa $ that is actually supported by $ {c_4} $. To do so, we calculate the projection of the fourth order in Eq. (B1) on $ {c_4} $. Because this is independent of the vector length or polynomial coefficient, it is given by
This shows that defocus contains most of the information represented by $ R $ and $ \kappa $. This explains why, in Fig. 3, RMSt and $ {c_4} $ have an excellent correlation.
Acknowledgment
The authors want to thank Sophiane Tournois, Chen Yan, and Raoul Kirner for useful discussions.
Disclosures
The authors declare no conflicts of interest.
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