## Abstract

We report on the microfabrication of continuous aspherical optical surfaces with a single-mask process, using anisotropic etching of silicon in a KOH water solution. Precise arbitrary aspherical surfaces with lateral scales on the order of several millimeters and a profile depth on the order of several micrometers were fabricated using this process. We discuss the factors defining the precision of the formed component and the resulting surface quality. We demonstrate 1 mm and 5 mm replicated aspherical phase plates, reproducing defocus, tilt, astigmatism and high-order aberrations. The technology has a potential for serial production of reflective and refractive arbitrary aspherical micro-optical components.

©2003 Optical Society of America

## 1. Introduction

Aspherical micro-optical components are used in optical telecommunications to couple light to fibers, in optical memory pickups [1], in wavefront sensors [2], in beam forming optics for semiconductor lasers and in display applications.

Microfabrication of aspherical surfaces represents a non-trivial task [3]. Traditional methods such as grinding and polishing are not applicable due to the small size of the components. Moreover, micro-optical components are usually produced in very large quantities, therefore preference should be given to replication methods rather than to direct fabrication. A number of replication technologies was developed, such as molding, coining, casting and embossing, whereas the mastering methods are still quite complicated [3].

We present an inexpensive single-mask method to fabricate aspherical master surfaces, based on KOH etching of silicon. The method can be used for batch production of reflective (master) and refractive (replicated) aspherical surfaces.

The method relies on the experimental evidence that maskless anisotropic KOH etching, started from a pyramidal pit in the silicon bulk eventually results in a rounded, almost spherical, depression [4]. The depth and the radius of curvature of the depression depend on the initial pit size and on the etch parameters.

We used a specific geometry of an array of initial pits to obtain an arbitrary aspherical surface by etching in KOH.

## 2. Anisotropic etching of spherical depressions

Anisotropic etching of<100> silicon through a circular mask produces a pyramidal pit formed by four (111) planes. The pit depth is *d*
_{0}/√2, where *d*
_{0} is the diameter of the opening (Fig. 1). Kendall [4] has shown that stripping of the mask and further maskless anisotropic etching of bare silicon eventually transforms the pit into a nearly spherical concave surface with low roughness and high surface quality.

Figure 2 illustrates the process. The oxide mask is stripped away and subsequent immersion of the sample in the etchant solution favors the etching of (411) planes, whose etch rate is faster than that of the (111) planes. The (411) sidewalls eventually reach a depth in which another inverted pyramid is formed, whose depth - with respect to the <100>Si top surface -determines the sagitta *s* of the rounded profile formed later on.

An expression for the sagitta is given by (1):

where *θ* is the angle between the facets of the new pyramid and the top surface (*θ*=19.47 deg between the planes (411) and (100)), and *m* is the etch ratio between these same two planes, i.e., *m*=*R*
_{<114>}/*R*
_{<100>}, where *R*
_{(pqr)} indicates the etch rate of the planes <*pqr*> for a given KOH concentration.

Further etching results in a rounded surface at the bottom of the dip, possibly as a result of the faster etching of the {*n*11} planes, which overtake each other consecutively. In the process, the remaining (411) facets keep moving laterally until they eventually disappear, while the diameter of the spherical section increases. When the top silicon surface has been etched to a depth *h*, the diameter *D* of the spherical depression is given by an empirical formula (2), obtained in [4]:

This formula is only conceptually valid for *h*>2.5 *d*
_{0}, at which depth there are no remains of the (411) planes. The top contour evolves from squarish to circular as the etch depth increases; for *h*>7.0 *d*
_{0} the diameter uniformity lies within 5 %. The radius of curvature of the resulting spherical section is given by Eq. (3), as follows:

We experimentally found out that placing pits close to each other in a regular structure results in an array of overlapping spherical depressions, with very sharp interfaces. Using this property we fabricated orthogonal and hexagonal micromirror and microlens arrays - see Fig. 3 - for Hartmann-Shack wavefront sensors [5]. This hexagonal array has 127 microlenses with a 300 *µ*m pitch. The focal length is 17 mm and the spot diameter ~85 *µ*m. The Si template was etched anisotropically in KOH to a 175 *µ*m depth.

The *rms* surface roughness measured for different micromirrors in the etched array was in the range between 15 and 25 nm. Wavefront deformation caused by this roughness in the replicated optical component with refractive index *n* is 1/(*n*-1) times smaller - in the range between 8 and 13 nm (for *n*=1.5), which is acceptable for visible light.

The normalized focal length of a microstructure is given by:

where *α* is the expression inside brackets in (1) and *β*=0.5 for a micromirror and *β*=1/(*n*-1) for a microlens with refraction index *n*. The minimum focal length for a structure is limited by the ratio *h*/*d*
_{0}=2.5, for which we obtain spherical bottoms, and is about 30 *d*
_{0}. The maximum focal length is limited by the wafer thickness and by the minimum hole size, which is defined by the lithographic mask. A micromirror etched with a 33-wt% KOH:H_{2}O solution (85°*C*) can have focal lengths ranging from 0.03 mm to 13 mm, for a maximum etch depth of 400 *µ*m. A microlens with n=1.4 imprinted on this mirror template can have focal lengths ranging from 0.15 mm to 65 mm.

The *F*
_{#}=*f* /*D*, where *f* is the focal length and *D* is the diameter of a single element, is given by:

We observe that the smallest possible *F*
_{#} is restricted by the *h*/*d*
_{0} ratio, which must not be smaller than 2.5 and should preferably be larger than 7. Therefore, the smallest *F*
_{#} for this technology results from the shortest possible etching and is about *F*
_{#}=2.5, when the structure top edge is still squarish, introducing pincushion distortion. For *h*/*d*
_{0}=7, where the top edge is fairly circular, we obtain *F*
_{#}=4.3. In the case of an array of micromirrors with 100% fill factor, the micromirrors should overlap laterally, resulting in *F*
_{#} >4.3.

## 3. Approximation of aspherical surfaces by bulk micromachining

When the etching process was started with several pits positioned very close to each other, the profile was formed by laterally overlapping spherical surfaces as illustrated in Fig. 4. By choosing the right starting pit positions and pit sizes, one can attempt to approximate an arbitrary aspherical surface by overlapping spherical depressions produced by the anisotropic etching. The process can be modelled as if imprinting the surface profile using a set of balls with different diameters. It is easy to show that the minimum approximation error of a spherical surface with radius of curvature *R _{a}* by a set of spherical depressions of radius

*R*, positioned in a uniform orthogonal grid with pitch

_{d}*p*, has an

*rms*deviation error σ

*:*

_{s}where *R _{d}* is given by Eq. (3) with positive radius of curvature corresponding to a concave surface. Analysis of Eq. (6) shows that:

- a concave spherical surface with
*R*=_{a}*R*can be approximated with zero error;_{d} - a flat surface is better approximated with smaller pitch values and larger etch depths, resulting in larger values of
*R*, as the approximation quality is less dependent on the initial pit size*d*_{0}, where*d*_{0}<*p*; - convex surfaces are approximated with maximum error.

The *rms* error of approximation is given in Table 1 for some typical values of *h*, *d*
_{0} and *p*. For a good approximation of profiles with high spatial frequencies, the radius *R _{d}* must be relatively small, while to obtain a smooth optical surface

*R*has to be large to minimize the error (6). The optimal approximation is obtained from a compromise between these conditions. The grid pitch

_{d}*p*, the etch depth

*h*and the pit size

*d*

_{0}should satisfy the compromise chosen.

The metamorphosis of the bottom profile during the etching of the sample is illustrated in Fig. 5, which shows the interferometric pattern of the desired surface and that of the etched silicon chip (21×21 pits) at various etch depths. We observe how the initially isolated cavities evolve into a more seamless pattern as we etch deeper. According to calculations, the best fit was found at a depth around 150*µm*, although there we can still distinguish a discrete pit pattern. Further etching smoothens the interface profile at the expense of a higher global deviation from the desired shape. A smoother surface at the optimal etch depth can be obtained by using a higher pit density. A numerical model demonstrates that the accuracy of the approximation of a particular surface can also improve with a higher KOH concentration.

To produce a mask pattern that approximates an arbitrary surface after etching, we need to solve an inverse problem, i.e., given the surface function *S*(*x*,*y*), we need to find the optimal distribution and sizes of the initial pits, which will be simultaneously etched for a certain time *t _{etch}*=

*h*/

*R*

_{<411>}.

We developed a simple computer code based on the above “ball-stamping” principle where the depth of imprint and the radii are defined by the relations (1–3). The user supplies the two-dimensional profile and etching parameters. To generate the approximation, we used a condition in which each spherical segment has only one common point with the approximated surface. The error of approximation produced by this approach is somewhat higher than described by the expression (6) and can reach √10*p*
^{2}/15*R*.

The code - illustrated by Fig. 6 - first looks for the maximum of the arbitrary two-dimensional function *S*(*x*,*y*) that has to be approximated. This maximum corresponds to the maximum sagitta *s _{max}*. Then, with the use of Eqs. (1) and (2), the maximum initial pit size

*d*

_{0}

*and the etch depth*

_{max}*h*are calculated to ensure that only the bottom of the etched depression touches the aimed profile

*S*(

*x*,

*y*). Next, a grid of initial pits is defined and the initial pit sizes are calculated from Eqs. (1) and (2) for the whole grid. The aimed profile is approximated by the lateral superposition of etched depressions.

## 4. Experimental demonstration

The fabrication of an aspherical surface consists of the following steps:

- design of a lithographic mask and transfer of the pattern to an oxide layer deposited on the silicon wafer;
- KOH etching to form the pyramidal pits;
- removal of the oxide mask and further anisotropic etching to form the aspherical surface.

We used<100>525 *µ*m silicon wafers with the 1 *µ*m SiO_{2} mask on the front surface and a protective nitride layer on the back surface. The etchant for the whole process is a 33-wt% KOH:H_{2}O solution at 85°*C*.

We designed a number of 1×1 mm and 5×5 mm structures to reproduce surfaces described by Zernike polynomials [6]. The 1 mm structures were approximated with arrays of 41×41 pits and the 5mmstructures with 101×101 pits. The structures were processed according to the chart shown in Fig. 2 and the resulting silicon chips were used as a mold to replicate the aspherical components in a layer of polymer with refractive index *n*~1.5 deposited on a glass surface.

The “intrinsic” micro-roughness of the etched silicon surface σ* _{i}* increases with the etch depth [7]. After removing the oxide, directly etching of the silicon surface to a depth of h=150

*µ*m resulted in a σ

*~15 to 25 nm*

_{i}*rms*roughness, which is slightly higher than the experimentally measured roughness reported in [7]. In practice, the micro-roughness of the etched aspherical surface is the composition of two uncorrelated contributions:

The resulting *rms* roughness of the etched surface can be found as
${\sigma}_{r}=\sqrt{\left({\sigma}_{i}^{2}+{\sigma}_{s}^{2}\right)}$
.

The fabricated test phase plates were tested interferometrically in a single-pass Mach-Zender interferometer [6]. Figure 7 shows 5×5 mm aspherical surfaces obtained with a 101×101 pit approximation. The relatively large pitch (50*µ*m) of the initial grid and shallow etching depth resulted in a somewhat higher structural roughness σ* _{s}*, clearly visible in the interferometric patterns.

Figure 8 shows the interferometric pattern corresponding to a sample design including different low-order 1×1mm phase correctors fabricated using a 41×41 mask and etched to a depth of 50 *µ*m. It is seen that the optical quality of the replicated surfaces is high. All interferometric patterns clearly correspond to the desired aspherical surfaces.

## 5. Conclusion

We proposed, theoretically described and implemented a single-mask bulk micromachining technology for micro-fabrication of reflective and refractive aspherical optical components.

The technology is based on the micromachining of the desired surface profile as a combination of spherical patches produced by anisotropic etching of silicon in KOH water solution.

The theoretical analysis shows that the method allows micromachining of high-quality smooth aspherical surfaces with lateral sizes on the order of hundreds of micrometers to several millimeters with *rms* errors on the order of several nm.

Several aspherical reflective surfaces and transparent phase plates with sizes of 1×1 mm and 5×5 mm were fabricated. They approximate tilt, defocus, astigmatism and a couple of higherorder aberrations with good optical quality.

The technology has a potential for serial production of reflective and refractive arbitrary aspherical micro-optical components.

## Acknowledgements

The authors are thankful to O. Soloviev, J. Groeneweg and C. Visser for their assistance.

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