Micro gradient-index conical lenses: simulationand fabrication methods

Deqiang Song, Michael Sanchez, Matthias Gross, and Sadik Esener

In this paper, a micro gradient-index conical lens, which has a larger acceptance angle than a conven-tional microlens, is presented. Methods on how to simulate these lenses in commercial optical designsoftware CodeV are introduced, and the effects of several index profiles and cone shapes are compared insimulation. Results show that a micro gradient-index conical lens has a four times larger acceptanceangle compared with a microlens. Additionally, conical lenses with a Gaussian-index profile show a largeracceptance angle than those with a solid refractive index. Fabricated conical lenses show an acceptanceangle of more than 27° for a detection threshold of 50%, which agrees with the simulation result. © 2005Optical Society of America

OCIS codes: 220.1770, 110.2760, 350.3950.

1. Introduction

In optical fiber communication and parallel opticalinterconnections, the data rate has been increasingdramatically over the past years.1–4 Consequently,photodetectors must be scaled down to reduce para-sitic capacitance. The standard method to collectlight for small photodetectors is to use a microlensbefore the photodetector to focus the light beam ontothe detector. In this approach, the acceptance angle(only the light within this angle can be focused ontodetector) of the microlens is roughly two or threedegrees, under the assumption that the microlensdiameter is 250 �m, detector diameter is 50 �m, andF-number of the microlens is 2. However, in someapplications, such as board-to-board optical intercon-nects and wireless optical communications, a largeracceptance angle is desired to increase the link dis-tance to achieve looser misalignment tolerances.5

To get a larger acceptance angle, a micro gradient-index conical lens (MGCL) is a good alternative to themicrolens, in terms of collecting light. A MGCL is alens in a tapered shape made of gradient refractive-index material. As shown in Fig. 1, the shape profileis generated by rotating a curve with equation y

� R�z� around axis z, and the refractive index at acertain location follows n�r, z� � f�r, R�z��, where R�z�is the radius of the cross section at position z.

When R�z� is a straight line, this description de-grades to a normal conical lens with a gradient index(GRIN). Several studies have addressed the ray prop-agations in normal conical GRIN lenses, especiallywith Gaussian- or parabolic-index profiles.5,6 For anormal conical GRIN lens with a parabolic-index pro-file, the ray trajectories can be obtained analyticallyby solving the Eikonal equation. The solution showsthat the behavior of a ray in a normal conical lens issinusoidal in radius, decreasing as the ray movestoward the conical tip.5 This shows that the GRINlens is capable of converging light.

In contrast, solid-index conical lenses have a con-stant refractive index with a variable shape profile.Also known as an “optical concentrator,” solid-indexconical lenses have been studied for many years.7–11

An optimal shape profile for solid-index conical lenseswas solved as a compound parabolic concentrator(CPC). For any predetermined acceptance angle, aCPC can be designed, so that behind this CPC, thedetector will collect all the rays entering the lenswithin the acceptance angle. The maximum accep-tance angle for an ideal CPC is nearly 90°.7

Although the CPC is an optimal solution for lightfunneling, it has a very stringent requirement oncurvature profile fabrication. In micro optical sys-tems, such as optical interconnects or fiber commu-nication, the lens dimensions are of the order of tensto hundreds of micrometers, which makes the cur-vature requirement impractical to satisfy. In this

The authors are with the Department of Electrical and Com-puter Engineering, University of California, San Diego, 9500 Gil-man Drive, La Jolla, California 92093. D. Song can be reached bye-mail at desong@soliton.ucsd.edu.

Received 18 August 2004; accepted 15 September 2004.0003-6935/05/183747-05$15.00/0© 2005 Optical Society of America

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paper, we combine the properties of both GRIN lensand optical concentrator and form a new structure.The resulting lens can be fabricated at small dimen-sions,12 although some sacrifice of acceptance angleis required. We investigate the effects of differentshapes and several refractive-index profiles, andimplement the simulation for conical lenses usingCode V.

2. Simulation Methods for Micro Gradient-IndexConical Lens

The best simulator is determined by the dimensionand shape of the device. MGCL has a 250��m en-trance and a 50��m exit, and this is 294 and 59 timeslarger than the feature size of the optical wave�0.8 �m�. In this dimension, geometrical optics is sup-posed to give the correct result. Although vector-waveoptics is a more rigid simulator (for example, Full-WAVE from RSoft, Inc.), it is not practical to simulatea 2.5�mm-long device. For some scalar-wave simula-tors (BeamPROP from RSoft, Inc.), which is moreefficient than a vector-wave simulator, the side wallslope might break the slowly varying envelope ap-proximation of the simulator.

Based on current commercial optical simulationsoftware for ray tracing, GRIN lenses with a conicalshape cannot be simulated directly, although severalmodels can provide an approximate simulation. Thethree models are the sequential surfaces model [SSmodel, as in Fig. 2(a)], the nested cones model [NCmodel, as in Fig. 2(b)], and the segmented nestedcones model [SNC model, as in Fig. 2(c)].

The SS model simulates the conical lens by a seriesof sequential cylindrical layers with different diame-ters, where the index profile of each surface is definedaccording to the location of the surface. In the NCmodel, the conical lens is decomposed into a numberof nonsequential surfaces, layered from axis to sideboundary. The refractive indices keep constant ineach single surface, but change from surface to sur-face. The SNC model is a combination of the SS modeland the NC model. First the conical lens is cut into anumber of segments longitudinally, similar with theSS model. Then each segment is modeled by the NCmodel.

The SS model works very well for paraxial-beampropagations, while it has difficulty in dealing withreflections on the side boundary. The limitation ofthis model is that the side boundary of each thincylindrical surface is always parallel to the cone axis,

which is not true in reality. The NC model is suitablefor side-boundary reflection, and it can simulate ahigh-order curve cone surface. The NC model worksmore accurately for beams with a large incident angleto the conical lens. For paraxial-beam simulation us-ing the NC model, reflection could occur on the inter-face between adjacent cone layers because of thegrazing incidence. The reflection is generally not in-cluded in the simulation; thus the simulation accu-racy is affected. But with a larger incident angle, thereflection will reduce. The SNC model is an improvedNC model. Since each segment has the property of anNC model, it has no difficulty in dealing with reflec-tions at the boundary; since the whole conical lensconsists of several segments, it has more flexibilityand can represent almost any shape profile.

In order to build a MGCL in Code V as accuratelyas possible, a large number of surfaces are used. Gen-erally the surface number is several hundred. Be-cause these surfaces follow some common rules, weused a helper program (written in C��) to generatethese surfaces in Code V. After this model is built inCode V, the acceptance angle can be measured.

3. Simulation Results

The purpose of our simulations is to investigatethe acceptance angles of MGCLs with differentrefractive-index and shape profiles for various di-ameters of the incoming light beam. The parame-ters for the MGCL are as follow. The basicrefractive index is n0 � 1.50. For Gaussian-indexprofiles, the refractive-index change from the center

Fig. 1. Gradient-index conical lens.

Fig. 2. Three available models for gradient-index conical lens: (a)sequential surface model (SS model), (b) nested cones model (NCmodel), and (c) segmented nested cones model (SNC model).

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to the boundary is �n � 0.05, i.e., the index at thecenter is 1.55 ��n0 � �n�, and the index at the bound-ary is 1.518 ��n0 � �n�1�e��. The cone length is2.5 mm, and the diameter of the entry and exit sur-faces are 250 �m and 50 �m, respectively. The entryhas the same diameter as a microlens, and the exithas the same dimension as a photodetector.

A. Comparison of Solid-Index and Gaussian-IndexProfiles

The transmission efficiency, defined as the percent-age of the optical power received at the detector sur-face to the total incident optical power, is used tocompare the different structures. The efficiency withdifferent normalized diameters and at different inci-dent angles is plotted in Fig. 3, where the normalizedbeam diameter is defined as the ratio of the diameterof the incident beam to that of the lens entrance.

It is seen that for both index profiles, for a givenbeam diameter, the efficiency drops with increasingincident angles. From the results, we can see thatthe efficiency of a Gaussian-index-profile lens fallsoff more slowly than that of a solid-index-profilelens. Specifically, when the normalized beam diam-eter is 0.5 �125 �m�, for a threshold of 50%, theacceptance angle for the Gaussian-index lens ismore than 17°, and more than 13° for the solid-index lens. In either case, the acceptance angle isover four times larger than that with a simple mi-crolens, which is �3°. So we can conclude here thatthe Gaussian-index profile realizes an improvementin acceptance angle, as compared to a solid-indexprofile.

B. Comparison of Different Shape Profiles

Since the Gaussian-index profile has a larger accep-tance angle than a solid-index profile, we further in-vestigated the shape effects for conical lenses with aGaussian-index profile. Three typical shapes are ex-plored: normal, concave, and convex cones.

The normal cone has straight side walls, while theconvex and concave cones are given an outward orinward curvature at the side walls. As shown in Fig.4, these three different shape profiles have similarperformance. In a close comparison at high efficiency

level ��70%�, the concave cone shape has a slightlysmaller acceptance angle than the normal cone, whilethe convex has a slightly bigger acceptance angle.Considering the fabrication difficulty, the normalcone shape is identified as the optimum because it ismuch easier to fabricate than cone lenses with curvedshapes.

4. Fabrication and Measurement

Among all the index profiles and shape profiles, solid-index normal conical lenses can be fabricated themost accurately, so we choose this lens to experimen-tally verify the simulation result of acceptance angle.For a primary experiment, fiber etching is an efficientand flexible way to attain a solid-index conical lens.We used 48% HF acid to etch a multimode fiber corewith a 200��m diameter (FT-200-EMT, Thorlabs,Inc.). The fiber is dipped vertically into the etchingsolution and attached to a linear motion actuator,which is used to pull the fiber out from the etchantslowly in order to form a conical shape. The slope ofthe cone can be controlled by the pulling speed (e.g.,a fast pulling generates a small slope on the fiber). Bythis means, we obtain a tapered fiber with a desiredslope, as shown in Fig. 5. After cutting the etchedfiber at proper positions and polishing, it can work asa conical lens. The conical lens we fabricated has anentrance diameter of 200 �m, an exit diameter of60 �m, and a length of 2 mm. For mass production ofconical lens arrays, injection molding might be a bet-ter approach, considering the high precision and po-tentially low cost.

In order to measure the acceptance angle of thisconical lens, a rotating stage was used to provide

Fig. 3. Transmission efficiency of conical lenses for various nor-malized incident beam diameters.

Fig. 4. Efficiency of gradient-index cone lenses for various nor-malized incident beam diameters for three cone shapes.

Fig. 5. Etched multimode fiber.

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angular movement, as shown in Fig. 6. The colli-mated incident laser beam has a beam diameter of120 �m, measured with a laser beam profiler (DigitalBeamView Analyzer Lasercam, Coherent, Inc.). Theconical lens is located in the rotating center of thestage, followed by a pinhole and detector. The use ofthe pinhole is to block the light leaked from the sidesurface of the conical lens.

The normalized optical power, defined by the ra-tio of received power to the maximum receivedpower, is plotted against the incident angle, asshown in Fig. 7. When we choose 50% as the detec-tion threshold, this conical lens provides an accep-tance angle from �12° to �15°. This demonstratesthat the acceptance angle of the conical lens is sig-nificantly larger than that of a simple microlens(shown as a dashed curve in Fig. 7). In this exper-iment, the full angle of 27° is over 4 times largerthan that of an equivalent microlens. For compari-son, the simulation result for this cone lens is alsoshown in Fig. 7 (round symbols connected by dottedcurves). It can be seen that the experimental accep-tance angle is very close to the simulation result,except that the experimentally detected energy fallsoff more gradually with incident angle than doesthe simulation result. A main reason for this differ-ence is the reflection loss from the fabrication de-fects. The number of reflections increases with theincident angle; thus the reflection loss takes more

effect. This will make a gradual drop in efficiency.Another possible reason is that we used geometricray tracing in the simulation, which assumes a uni-form energy distribution on the whole beam crosssection and that all the rays are ideally collimated.But an actual beam has a Gaussian energy distri-bution. Nonideal collimation and diffraction mayalso contribute to the discrepancy. So in practice,the detected energy is more likely to drop graduallywhen increasing incident angles, instead of fallingoff suddenly after a critical incident angle, as insimulation.

5. Conclusion

By using practical models (SS, NC, and SNC) tosimulate micro gradient-index conical lenses(MGCL) in Code V, we verified that a MGCL has alarger acceptance angle than a simple microlensdoes. The shapes and refractive-index profiles bothhave effects on the acceptance angles of MGCL.Conical lenses with a Gaussian-index profile have alarger acceptance angle than solid-index-profilelenses; convex, normal, and concave cones havesimilar acceptance angles, although convex conesslightly outperform at a high efficiency level. Con-sidering the fabrication, a normal cone with aGaussian-index profile appears to be the best solu-tion for increasing the acceptance angle in a prac-tical micro optical system. A Gaussian-indexnormal MGCL can provide an acceptance angle ofapproximately �17° for a 50% detection threshold.

A fabricated conical lens shows a �12° to �15°acceptance angle, similar to the simulation result of�14° for this device. Under the same conditions, anideal simple lens can only provide an acceptanceangle of 3°. Thus the MGCL is an attractive alter-native to the simple microlens for increasing theacceptance angle.

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