## Single-mask microfabrication of aspherical optics using KOH anisotropic etching of Si

Optics Express, Vol. 11, Issue 18, pp. 2244-2252 (2003)

http://dx.doi.org/10.1364/OE.11.002244

Acrobat PDF (2073 KB)

### 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

2. C. Paterson and J. C. Dainty, “A hybrid curvature and gradient wavefront sensor,” Opt. Lett. **25** (23), 1687–1689 (2000). [CrossRef]

4. D. L. Kendall, W. P. Eaton, R. Manginell, and T. G. Digges Jr., “Micromirror arrays using KOH:H_{2}O micromachining of silicon for lens templates, geodesic lenses, and other applications,” Opt. Eng. **33** (11), 3578–3588 (1994). [CrossRef]

## 2. Anisotropic etching of spherical depressions

*d*

_{0}/√2, where

*d*

_{0}is the diameter of the opening (Fig. 1). Kendall [4

4. D. L. Kendall, W. P. Eaton, R. Manginell, and T. G. Digges Jr., “Micromirror arrays using KOH:H_{2}O micromachining of silicon for lens templates, geodesic lenses, and other applications,” Opt. Eng. **33** (11), 3578–3588 (1994). [CrossRef]

*s*of the rounded profile formed later on.

*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.

*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

4. D. L. Kendall, W. P. Eaton, R. Manginell, and T. G. Digges Jr., “Micromirror arrays using KOH:H_{2}O micromachining of silicon for lens templates, geodesic lenses, and other applications,” Opt. Eng. **33** (11), 3578–3588 (1994). [CrossRef]

*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:

5. G. Vdovin, O. Akhzar-Mehr, P. M. Sarro, D. W. de Lima Monteiro, and M. Y. Loktev, “Arrays of spherical micromirrors and molded microlenses fabricated with bulk Si micromachining,” in *MEMS/MOEMS: Advances in Photonic Communications, Sensing, Metrology, Packaging and Assembly*, U. Behringer, B. Courtois, A. M. Khounsary, and D. G. Uttamchandani, Proc. SPIE4945, 107–111 (2003). [CrossRef]

*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.

*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.

*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.

*F*

_{#}=

*f*/

*D*, where

*f*is the focal length and

*D*is the diameter of a single element, is given by:

*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

*R*by a set of spherical depressions of radius

_{a}*R*, positioned in a uniform orthogonal grid with pitch

_{d}*p*, has an

*rms*deviation error σ

*:*

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

_{d}- 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.

*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*must be relatively small, while to obtain a smooth optical surface

_{d}*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.

*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.

*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>}.

*p*

^{2}/15

*R*.

*S*(

*x*,

*y*) that has to be approximated. This maximum corresponds to the maximum sagitta

*s*. Then, with the use of Eqs. (1) and (2), the maximum initial pit size

_{max}*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

- 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.

*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*.

*n*~1.5 deposited on a glass surface.

*rms*roughness of the etched surface can be found as

*µ*m) of the initial grid and shallow etching depth resulted in a somewhat higher structural roughness σ

*, clearly visible in the interferometric patterns.*

_{s}*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

*rms*errors on the order of several nm.

## Acknowledgements

## References and links

1. | I. D. Nikolov, K. Goto, S. Mitsugi, Y. J. Kim, and V. I. Kavardjikov, “Nanofocusing recording probe for an optical disk memory,” Nanotechnology |

2. | C. Paterson and J. C. Dainty, “A hybrid curvature and gradient wavefront sensor,” Opt. Lett. |

3. | H.P. Herzig (ed), |

4. | D. L. Kendall, W. P. Eaton, R. Manginell, and T. G. Digges Jr., “Micromirror arrays using KOH:H |

5. | G. Vdovin, O. Akhzar-Mehr, P. M. Sarro, D. W. de Lima Monteiro, and M. Y. Loktev, “Arrays of spherical micromirrors and molded microlenses fabricated with bulk Si micromachining,” in |

6. | D. Malacara and S. L. DeVore, “Optical interferogram evaluation and wavefront fitting,” in |

7. | G. Findler, J. Muchow, M. Koch, and H. Munzel, “Temporal evolution of silicon surface roughness during anisotropic etching processes,” in |

**OCIS Codes**

(220.1000) Optical design and fabrication : Aberration compensation

(220.1250) Optical design and fabrication : Aspherics

(220.3620) Optical design and fabrication : Lens system design

(220.4000) Optical design and fabrication : Microstructure fabrication

(220.4610) Optical design and fabrication : Optical fabrication

(350.3850) Other areas of optics : Materials processing

**ToC Category:**

Research Papers

**History**

Original Manuscript: June 26, 2003

Revised Manuscript: September 1, 2003

Published: September 8, 2003

**Citation**

D. W. de Lima Monteiro, O. Akhzar-Mehr, P. M. Sarro, and G. Vdovin, "Single-mask microfabrication of aspherical optics using KOH anisotropic etching of Si," Opt. Express **11**, 2244-2252 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-18-2244

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### References

- I. D. Nikolov, K. Goto, S. Mitsugi, Y. J. Kim, V. I. Kavardjikov, �??Nanofocusing recording probe for an optical disk memory,�?? Nanotechnology 13, 471477 (2000).
- C. Paterson and J. C. Dainty, �??A hybrid curvature and gradient wavefront sensor,�?? Opt. Lett. 25 (23), 1687-1689 (2000). [CrossRef]
- H.P. Herzig (ed), Micro-Optics: elements, systems and applications, (London, Taylor & Francis, 1998).
- D. L. Kendall, W. P. Eaton, R. Manginell and T. G. Digges Jr., �??Micromirror arrays using KOH:H2O micromachining of silicon for lens templates, geodesic lenses, and other applications,�?? Opt. Eng. 33 (11), 3578-3588 (1994). [CrossRef]
- G. Vdovin, O. Akhzar-Mehr, P. M. Sarro, D. W. de Lima Monteiro and M. Y. Loktev, �??Arrays of spherical micromirrors and molded microlenses fabricated with bulk Si micromachining,�?? in MEMS/MOEMS: Advances in Photonic Communications, Sensing, Metrology, Packaging and Assembly, U. Behringer, B. Courtois, A. M. Khounsary and D. G. Uttamchandani, Proc. SPIE 4945, 107-111 (2003). [CrossRef]
- D. Malacara and S. L. DeVore, �??Optical interferogram evaluation and wavefront fitting,�?? in Optical Shop Testing, D. Malacara, 2nd ed. (Wiley Interscience, New York, 1992).
- G. Findler, J. Muchow, M. Koch and H. Munzel, �??Temporal evolution of silicon surface roughness during anisotropic etching processes,�?? in Micro Electro Mechanical Systems, pp.62-66 (New York, Institute of Electrical and Electronics Engineers, 1992).

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