MEMS compatible micro-GRIN lenses for fiber to chip coupling of light

Graded-Index (GRIN) lenses with a diameter of 125 μm are presented. This diameter enables the assembly of the GRIN lenses onto an optical micro-system using the same passive alignment grooves as used for the light carrying optical fibers. In contrast to refractive lenses, GRIN lenses have flat endfaces and the focal distance of a GRIN lens is defined by its length. Therefore, GRIN lenses can be diced from a selected multimode optical fiber with a regular wafer dicing machine. The effects of the resulting surface roughness are reduced by immersing the optical parts into index matching oil, which can not be applied for refractive lenses. This has a further advantage since an anti-reflective coating becomes dispensable. The coupling efficiency of a collimator set-up using the GRIN lenses is studied using paraxial ray calculations. The calculated minimal coupling losses of less than 0.3 dB are in excellent agreement with the measured results. Losses smaller than 2 dB over a coupling length of 2 mm have been measured. © 2006 Optical Society of America OCIS codes: 060.0060 Fiber optics and optical communication, 230.0230 Optical devices References and links 1. W. Noell, P-A. Clerc, L. Dellmann, B. Guldimann, H.P. Herzig, O. Manzardo, C. Marxer, K. Weible, R. Dändliker, N. de Rooij, “Applications of SOI-based optical MEMS,” IEEE J. Sel. Top. Quantum Electron. 8, 2002, 148-154. 2. H. Toshiyoshi, H. Fujita, “Electrostatic Micro Torsion Mirrors for an Optical Switch Matrix,” IEEE Journal of Microelectromechanical Systems 5, 1996, 231-7. 3. Corning SMF-28 product information. 4. C. Marxer et, C. Thio, M-A. Gretillat, N. de Rooij, R. Bättig, O. Anthamatten, B. Valk, P. Vogel, “Vertical Mirrors Fabricated by Deep Reactive Ion Etching for Fiber-Optic Switching Apllications,” JMEMS 6, No. 3, 1997, 277-285. 5. Ph. Nussbaum, R. Völkel, H.P. Herzig, M. Eisner, S. Haselback, „Design, Fabrication and Testing of Microlens Arrays for Sensors and Microsystem,“ Pure Appl. Opt. 6, 1997, 617-636. 6. W. R. Cox, C. Guan, D. J. Hayes, “Microjet Printing of Micro-optical Interconnects and Sensors,” SPIE Phot. West Proc., V.3852, 2000, 400. 7. J. Bähr, U. Krackhardt, K.-H. Brenner, “Fabrication and Testing of planar Micro Lens Arrays by Ion Exchange Technique in Glass,” SPIE-Int. Soc. Opt. Eng. Proceedings of Spie 4455, 2001, 281-92. 8. F. Zhu, Y. Sato, Y. Sasaki, K. Hamanaka, “Fiber Collimator Arrays for Optical Devices/Systems,” Proc. SPIE 4564, 2001, 175-182. 9. Corning Product Information: http://www.corning.com 10. Patent: US4701011, Emkey William L, Jack Curtis A, “Multimode fiber-lens optical coupler” 20.10.1987. 11. Webpage: http://www.oki.com/en/press/2002/z02062e.html 12. S. Yuan, A. Riza, “General formula for coupling-loss characterization of single-mode fiber collimators by use of gradient-index rod lenses,” Appl. Opt. 38, No. 15, 1999. 13. Saleh B, Teich M, Foundametals of photonics, “New York, Wiley, 1991”. 14. International Patent Application: W. Noell, M. Zickar, C. Marxer, N. de Rooij, “Optical Coupling Element and Method of Manufacturing the Optical Coupling Element,“ PCT/IB2004/001611. 15. M. Zickar, W. Noell, A. Oudalov, C. Marxer, N. de Rooij, “4x4 & 8x8 Optical Matrix Switch with 250 μm Mirror Pitch,” OMEMS ‘04, Takamatsu, Japan, 162-163. #68089 $15.00 USD Received 21 February 2006; revised 26 April 2006; accepted 26 April 2006 (C) 2006 OSA 15 May 2006 / Vol. 14, No. 10 / OPTICS EXPRESS 4237


Introduction
The small nature of light beams, which are guided in Telecom single mode fibers, make relevant several optical effects that are negligible in the macro-world.Due to high divergence angles of small optical beams the optical losses increase drastically if the light beam has to travel over a long distance.Microlenses enable the beam waist to be broadened and increase the coupling length between two waveguides.
The compatibility of an optical element to a Micro Electro-Mechanical System (MEMS) is defined by its dimensions and the accuracy with which it can be aligned to the microchip.Optical fibers are widely used to guide light to a very well defined place on an Optical MEMS (OMEMS).They can be aligned to the optical element using U-or V-grooves etched into the MEMS device.In an ideal case the grooves are etched during the same step as the optical element itself, which enables a passive alignment of the optical components with very high accuracy [1,2].The light beam emerging from a typical single mode fiber (SMF) has a mode field diameter of about 10 µm [3].The high diffraction angle of this beam allows free space propagation over only several tens of micrometers where the light can be processed before being coupled into another waveguide.This is only suitable for very thin optical components such as vertical optical mirrors, optical attenuators or thin film filters [4].For longer free space propagation the beam waist has to be broadened using collimating lenses.
Different fiber-lens systems have been proposed.The shape of a classical refractive lens is ideally a radial hyperbolic function resulting in different optical path lengths along its radius due to the varying thickness of the material.These lenses can be fabricated by reflowing patterned photo-resist shapes on a transparent substrate.The lens shape of the photo-resist can be transferred into the substrate using dry etching techniques, usually into silicon or glass [5].Printing technologies have been reported using microjets to control the amount of dispensed liquid thus controlling the shape of the lens [6].
Another technique to create an optical path difference along a lens is changing its refractive index in radial direction.Planar lenses can be fabricated by ion exchange technique in glass where a radial refractive index variation is achieved by molecular diffusion [7,8].
An inconvenience for all planar lenses is the difficult alignment to optical fibers or microchips.Due to their planar nature it is difficult to integrate passive alignment structures.Particularly angular misalignments are very likely.
Ball lenses are transparent balls assembled to a chip.Ball lensed fibers are fabricated by heating the tip of an optical fiber.The surface tension of melted glass forms a ball lens directly attached to the fiber [9].
Cylindrical micro-lenses consist of a rod of transparent material with a graded refractive index (GRIN) profile in radial direction.The length of these rods defines the focal distance of the lens.Optical multi-mode fibers (MMF) with GRIN core are commercial products.If such a GRIN fiber is fusion spliced to a single mode fiber (SMF), cut and polished to a desired length the result is a lensed fiber [10].This method leaves a little bead at the splicing area.
In conclusion, all these lenses suffer from their non-uniform lateral dimension, which makes it difficult to align them to optical fibers.Additionally, their fabrication is complicated, needs a clean-room environment or they can not be fabricated in parallel causing a high production cost per piece.Therefore, it is the subject of this paper to describe the design, fabrication and characterisation of cylindrical lenses having the same diameter as single mode fibers (125 µm).These lenses can be easily assembled into U-or V-grooves etched on a MEMS chip.This enables passive alignment to optical fibers and other micro-optical devices.

Optical fibers
Optical fibers can be divided into two classes: Step index fibers consist of a fiber core and a surrounding cladding.The refractive index of the core is slightly higher.Paraxial light rays in the core are totally reflected at the boundary keeping the light confined in the core (Fig. 1(a)).
The second class is graded index fibers, which have a slowly varying refractive index along its radial axis (Fig. 1(b)).If the refractive index follows a parabolic distribution it can be expressed by where n 0 is the refractive index in the centre of the fiber, r is the distance from the centre and α is the parabolic constant.The trajectory of a paraxial ray entering a GRIN core is a sinusoidal oscillation with a pitch of 2π/α .If the GRIN material is cut at a certain length d it can act as a lens with desired focal distance.For an incoming plane wave the GRIN lens has a focal length of

Micro-GRIN lenses as optical collimators
The furthermost usage of micro-lenses on a chip is light collimation.In order to have larger free space coupling lengths the incoming beam waist has to be broadened.A broader beam waist has a smaller divergence angle, θ 0 (θ 0 = λ/πω 0 ) therefore, a double diameter yields a four times higher coupling length having the same losses [12].A schematic of the investigated arrangement is shown in Fig. 2. A light beam emerging from the SMF1 is collimated by GRIN lens 1, travels through the free space region and is refocused by GRIN lens 2 into the outgoing SMF 2.

ABCD law
A convenient way for calculating paraxial rays in an optical system is the ABCD matrix method [13].The basic principle of this approach is to split the system in a number of regions with a known ABCD matrix.For each region the coordinates x (radial position of the ray) and u (normalized slope) of the output can be related to the coordinates of the input by means of a matrix characterizing the system: Such a transfer matrix is called the ABCD matrix.For the given problem a number of different matrices are used.For a homogeneous medium the matrix is defined as where d is the geometrical length along the z-axis.For a discontinuity between two plane dielectrics where 1 n is the refractive index of the input medium and 2 n is the refractive index of the output medium.Finally, a GRIN lens can be represented by 1 cos( ) sin( ) sin( ) cos( ) where Z is the length of the GRIN lens.The parabolic constant α can be expressed by where NA is the numerical aperture of the fiber and r core the radius of the GRIN fiber core.By multiplying the transfer matrices we can obtain the description of the whole system.The complex curvature parameter q is altered by the optical system as where j is the index for the corresponding medium.Because the q parameter identifies the width ω and curvature R of a Gaussian beam, this simple law allows us to calculate the effect of an arbitrary paraxial system on the Gaussian beam.

Coupling efficiency
The coupling efficiency η is calculated by the so-called "overlap integral" as

Beam waist GRIN lens 1 GRIN lens 2 SMF 2
Fig. 2. Schematic of the collimating setup.A light beam emerges from the SMF and is collimated by GRIN lens 1. GRIN lens 2 refocuses the beam into the other SMF.
where η is a fraction of energy from the incident beam that couples into the output fiber.E i (x,y) is the complex amplitude of the fiber mode to be coupled and E f (x,y) is the complex amplitude of the receiver fiber mode.The mathematical description of the beam clipping becomes visible in the non-infinite boundaries of the integral in the nominator of equation 9.

Optical simulations
Many applications in OMEMS require optical collimators with a coupling length in the order of a millimetre.The collimating lens system is needed in order to broaden the beam waist coming out of the single mode fiber.The schematic of such a collimation system is shown in Fig. 2. Cylindrical GRIN lenses have a big advantage in that they can be easily aligned to the cylindrical shape of optical fibers.As an example, a prototype of an optical switch is shown in Fig. 3.The GRIN lenses were integrated and aligned to single mode fibers and optical mirrors using U-grooves and microfabricated springs that were defined by one lithographic mask.
A new and inexpensive approach to fabricate micro-GRIN lenses with a diameter of 125 µm is based on the dicing of selected multimode fibers with a GRIN core distribution [14].An ideal fiber should have a smooth, parabolic GRIN distribution and a small NA, which results in a larger length tolerance of the GRIN lens.The focal distance of the GRIN lens is adjusted by its length.Therefore, the coupling losses in function of the GRIN lens length and coupling length were studied and are presented hereafter.
The coupling efficiency of the collimating system outlined in Fig. 2 is calculated using ABCD matrixes.The simulations show the influence on the coupling efficiency of the following parameters: • Length of the GRIN rods • Tolerance of the GRIN rod length • Distance between SMF and GRIN lens • Achievable coupling length for a given maximal coupling loss For all simulations a symmetrical setup as depicted in Fig. 2 was studied.The free space region was assumed to be a medium matching approximately the refractive index of the GRIN fiber.
Fabrication tolerances of the fiber manufacturer can change slightly.To avoid having to repeat the simulations for different fibers, we decided to purchase a larger quantity of the selected GRIN fiber, thus ensuring a constant quality of the diced GRIN lenses.The measurement of the refractive index is shown in Fig. 4.

GRIN lenses
The GRIN fiber core had a diameter of 64 µm and a NA of 0.269, which results in a parabolic index α = 0.006 mm -1 (n 2 = n 0 2 [1-α 2 r 2 ]).All important parameters for the fiber are summarized in Table 1.

Coupling efficiency using the paraxial approximation
The ABCD law is a straightforward method to calculate the coupling efficiency for Gaussian beams.However, it does not take into account the losses originating from diffraction and its paraxial nature makes it unsuitable for calculating angular misalignment losses.
Figure 5 shows the calculated propagation of the beam waist in the collimator setup.The light beam exiting SMF1 diverges and is collimated by GRIN lens 1.The distance between SMF1 and GRIN1 determines the size of the beam waist in the lenses as well as in the free space region where other optical elements can be placed.Care has to be taken that the beam waist does not exceed the size of the lenses.With a gap distance of 400 µm the beam waist already extends to a beam diameter of 60 µm in GRIN1.Figure 6 shows four calculations for different gap lengths between SMF and GRIN lens.Each graph represents the coupling losses of the whole system in function of GRIN-and coupling length.To reduce Fresnel reflection and diffraction losses the setup is assumed to be immersed into index matching fluid with a refractive index of 1.45 closely matching the refractive index of the lens.The centre of the GRIN lens has a refractive index n 0 = 1.48.The utilized wavelength λ is 1.55 µm.
Several conclusions can be drawn from these graphs.First, the length tolerances for a collimating lens are rather tight.A length variation of 10 µm increases the losses by several dB.It is apparent that the wider the gap between the GRIN lens and the SMF the larger the region of acceptable collimation.This is explained by an overall broadening of the propagating beam.However, the mathematical model does not take into account the losses that occur when the beam exceeds the core of the GRIN lens.Therefore, it is only valid for relatively small beam waists that are confined within the core of the lens.
The comparison of the performance of collimator systems having different gap sizes between the SMF and the GRIN lens is shown in Fig. 7.For a fixed coupling distance the optical losses can be minimized by choosing the right lens length.However, in cases a small loss over a large coupling length is required [15].A gap size of 350 µm in conjunction with a 610 µm GRIN lens delivers the best results over 2.3 mm keeping the losses relatively constant and lower than 3 dB.
A collimation system based on GRIN fibers with a core of 125 µm yields about four times the coupling length as the used GRIN fiber with a core diameter of 64 µm.
According to the simulations we should be able to fabricate lenses with a coupling length of 10 mm having overall losses less than 3dB.After consultation with the fiber manufacturer we decided to study GRIN lenses with a core diameter of 110 µm.A custom-made fiber would allow us to obtain a fiber with an optimal refractive index distribution.Since the parabolic index α is coupled to the NA, a series of simulations with variable NA was performed and is shown in Fig. 8.The NA has to be larger than the NA of the SMF (being approximately NA ≈ θ 0 = λ/πω 0 ≈ 0.1).The whole setup was again immersed in index matching fluid.During the first simulations the gap between SMF and GRIN lens was kept constant at 450 µm (Fig. 8(a), Fig. 7. Simulated effect of the gap width between SMF and GRIN lenses on the coupling loss.A large gap leads to a smaller overall loss which is due to the overall broadening of the beam waist.However, a small gap can result in a smaller loss for a fixed, short coupling length.

8(b), 8(c))
. The maximum beam diameter in GRIN 1 is 79 µm.The GRIN lens length tolerance, which is represented by the width of the blue region, decreases with the NA.The achievable coupling length at a given coupling loss also decreases if the gap is kept constant.However, at a larger NA the beam in GRIN 1 is focalized stronger which results in a smaller maximum beam waist.To be able to compare the performance of the GRIN lenses with different NA the gap between SMF and GRIN lens should be adjusted in order to have the same maximum beam diameter in GRIN 1. Fig. 8(d) shows such a simulation for a NA of 0.3 and a beam diameter of 80 µm, which is achieved by setting the gap length to about 550 µm.
If it is compared to Fig. 8(a) it is visible that the coupling length is practically equal but the lens length tolerance of the lens with larger NA is smaller.

Fabrication of the GRIN lenses
The fabrication of the GRIN lenses requires only a minimal set of equipment when compared to the conventional methods used for the fabrication of GRIN lenses.The fabrication sequence is plotted in Fig. 9.The first step is the dicing of V-grooves in the silicon substrate.This is done by means of a V-shaped dicing blade and a wafer dicing machine.Then, the fibers are cut to appropriate lengths and inserted into the V-grooves.The fibers are fixed by means of thermal glue.
The glue has to match a number of requirements, the foremost being the ease with which it can be removed.For the current application, a thermal glue was used.This glue has the property of turning into a liquid when heated to a certain temperature and solidifying again when cooled.Best results were obtained by heating the silicon wafer on a hot plate to 130°C.The glue is then spread over a second wafer which is then pressed on top of the first one carrying the GRIN fiber pieces in the grooves.This ensures a good penetration of the glue into the grooves as well as an extra mechanical fixation for the fibers.
After the glue has cooled down the lenses can be fabricated.The fabrication utilizes the same dicing machine that was used for the creation of the V-grooves.GRIN rod pieces of defined length are cut by dicing in the perpendicular direction to the fiber pieces (Fig. 10).The x-step distance of the dicing machine is composed by the desired lengths of the GRIN lenses plus the width of the dicing blade.
In order to have a small surface roughness on the diced lenses a fine diamond blade with grit size of 3 to 6 µm was taken.Based on the microscope image of the lens one can conclude that a rather high degree of surface roughness is present.This is confirmed by an SEM image of the lens (Fig. 11(a)) and an AFM measurement indicating an RMS roughness of over 1 µm.Such a degree of surface roughness does not necessarily mean that the lens will not operate in index matched fluid.Since the optical rays do not experience a refractive index variation at the lens-fluid interface losses due to scattering can be avoided.However, operation in air will be accompanied with high losses.
The glue is removed by immersing the diced lenses into a solvent.The individual lenses can be filtered from the solution by a conventional filter paper and can now be assembled into microfabricated U-grooves with a semi-automated assembly stage.With appropriate tooling the assembly of the lenses is simple and fast.U-grooves equipped with springs allow for passive alignment to other microfabricated elements on the chip.
In order to reduce the surface roughness a number of approaches have been investigated.One is based on etching the GRIN lenses in a hydrofluoric acid solution.The reasoning behind this is that although the acid will etch the fiber isotropically, the rougher outer edges will most likely be smoothed out by the process.The fibers were fixed by photoresist and not by glue that dissolves faster in the acid.After being etched for 10 minutes in BHF an AFM measurement of the fiber indicated a slightly lower surface roughness.Based on these AFM measurements, one would suspect that there is an improvement of the flatness.However, inspection under an optical microscope leads to a different conclusion.
On the micro-scale the edges have indeed been smoothed out, whereas on the macro scale a very clear deterioration of the surface is visible.A moon crater landscape with significant height variations was created, probably due to micro-cracks in the glass (Fig. 11(b)).
A different solution to the roughness problem is mechanical polishing of the endfaces.This technique has yielded very good results with regard to the surface roughness (Fig. 12).However, the fixation of the lenses during the polishing or the precise control of the length of  the lens is very difficult.Moreover, this process, although suitable for small batches, does not meet the requirements of mass production.
The surface roughness of the non polished GRIN lenses is too high to be considered as a good optical element.An optically flat surface should have a roughness smaller than the wavelength of the processed light divided by 20.It is evident that the coupling losses of an optical element with such a surface will be high due to the scattering of the light.The question now is what is the influence of the surface roughness if the GRIN lenses are surrounded by a material having a similar refractive index?Are the losses only noticeable in the presence of surface roughness and refractive index mismatch?

Measurements
The basis of the measurement setup was formed by a set of straight rectangular U-grooves etched into a silicon wafer.The grooves had the width of an optical fiber.A set of microfabricated springs on the walls of the grooves ensured precise alignment.The GRIN lenses were positioned into the grooves by means of a micro-assembly stage and a special vacuum tool.The stage was equipped with a microscope.The lenses were placed manually into the grooves.Finally, when both the fibers and the GRIN lenses were in place, a glass lid was placed on top and gently loaded with a needle.This ensured that the components were all tightly in place.
Initially, the GRIN lenses were placed as far apart as possible.The length of the free space region between the lenses could then be tuned by pushing the lenses with the SM fibers.However, the lenses can only be pushed in one direction.By retracting the fibers we could precisely determine the length of the SMF-GRIN lens gap.The precision of this movement was estimated to be in the order of 10 µm.The actual distance between the GRIN lenses was determined by means of a ruler present in the microscope above the setup with a precision of 50 µm.
The experiment was performed in index matching fluid covering the whole assembly.The fluid decreases Fresnel reflection losses that are due to a mismatch of refractive index between two surfaces.Additionally, the surface roughness of both fibers and GRIN lenses were smoothed out by the oil.
The output signal was measured by a photo detector that was previously calibrated in order to provide the value of the insertion loss.

Results
The simulations indicate that the length of an ideal GRIN lens is dependent on the gap size used between the lens and the SMF.The paraxial simulation for a fixed gap size of 250 µm between the GRIN lens and the SMF is shown in Fig. 13(a).A typical example of a measured GRIN lens collimator is shown in Fig. 13(b).The measured values follow the same trend as the simulations.The length of a GRIN lens that yields the lowest cumulative losses of the required working distance is 645 µm according to the measurements and 650 µm according to the simulations.A typical "window" where the collimator scheme operates well is clearly visible.The measurements carried out for different gap sizes and GRIN lens lengths exhibit similar behaviour and good agreement with the simulations.
For the gap size of 350 µm the optimal GRIN lens length has been found to be 610 µm.The measured losses for different coupling lengths are shown in Fig. 14.In the same plot, the performance of a 645 µm long GRIN lens with a gap size of 250 µm is shown.In agreement with the simulations, the longer GRIN lens features lower insertion losses for a narrow operating region.However, the overall cumulative losses of the shorter lens are lower.It seems that although the optical beam in the GRIN lens extended the size of the core, no substantial loss is observed.To further confirm this claim even shorter GRIN lenses have been taken whereas the gap size has been increased to above 350 µm.A GRIN lens with a length of 595 µm and a gap size of 400 µm has been characterized.The losses of the setup remained remarkably constant in the range of 1.3 -2.0 dB for coupling lengths of 0.2 to 2 mm.It is likely that light is reflected at the core-cladding interface causing smaller losses as predicted.Therefore, taking a fiber with a substantially bigger core, i.e. smaller cladding, would enable efficient coupling over even longer distances, which will be verified in the near future.Finally, we would like to note the relationship between surface roughness and the observed losses deduced from the experiments.As stated above, the collimator setup was immersed into index matching fluid that virtually eliminated all negative consequences of surface roughness.The latter can be concluded from the fact that the best insertion losses measured with the collimating setup were only 0.3 dB.Butt-to-butt coupling of two single mode fibers in the same grooves yielded insertion losses of 0.15 dB.The excess loss contributed by the lenses is therefore only 0.15 dB.Spreading this loss over the 4 interfaces of the lens means that each one contributes a negligible amount of loss.However, once a similar experiment was carried out without the presence of oil, the losses rose dramatically.With oil a measured loss was 3.6 dB for a certain setup.Without oil the loss increased to 8 dB.

Conclusion
This paper describes the simulation, fabrication and characterization of micro-optical GRINlenses that can be passively aligned to optical fibers and other micro-optical elements on a microchip.The cylindrical lenses have the same diameter as commercially available optical fibers (125 µm) and can be placed into a U-groove, which is etched during the same step as the reflective mirrors.This ensures a passive alignment of the micro-optical components.The GRIN lenses are diced using a conventional wafer dicing machine and without the need of an expensive clean room environment.The surface roughness on the GRIN lenses is in the order of 1 µm.Since GRIN lenses have flat end faces, index-matching fluid can be used to smooth out the surface roughness.This has an additional advantage since Fresnel reflection can be avoided and the need of an anti-reflective coating becomes dispensable.Optical measurements in index matching oil confirmed that a negative influence of the surface roughness is not measurable.A collimator setup based on GRIN lenses showed minimal losses of only 0.3 dB and coupling lengths of 2 mm where a loss lower than 2 dB could be achieved.Calculations showed that a custom-made fiber with a GRIN core of 110 µm would achieve coupling lengths of 8 mm at the same loss.The excellent measurement results make these GRIN lenses a promising candidate for many applications in OMEMS.

Fig. 1 .
Fig. 1.Trajectory of a ray and refractive index distribution of a step-index fiber (a) and a GRIN fiber with parabolic index distribution (b).

Fig. 3 .
Fig. 3. GRIN lenses with a diameter of 125 µ m passively aligned to an optical fiber ribbon and a MEMS mirror matrix: The microfabricated U-grooves and holding springs assure a perfect passive alignment between the micro-optical parts.

Fig. 5 .
Fig. 5. Calculation of the beam waist in a collimator setup.The light beam emerging from the SMF1 diverges and is collimated by GRIN lens 1.

Fig. 4 .
Fig. 4. Measurement of the refractive index distribution of the selected MM fiber.

Fig. 6 .
Fig. 6.Coupling loss as function of GRIN lens length and coupling distance: a) 200 µm, b) 250 µ m, c) 300 µm, d) 350 µ m.A larger gap between SM fiber and GRIN lens increases the coupling length and shorter GRIN lenses need to be taken.

Fig. 8 .
Fig. 8. Coupling loss as function of GRIN lens length and coupling distance for GRIN fiber with core diameter of 110 µ m: (a) NA = 0.2, (b) NA = 0.25, (c) NA = 0.3 µ m, (d) NA = 0.3 with corrected gap SMF-GRIN.Coupling length of 8 mm with losses < 3 dB are achievable.A smaller NA yields a larger length tolerance which facilitates the lens fabrication.

Fig. 9 .
Fig. 9. Fabrication sequence for GRIN lens rods: (a) dicing of the V-grooves, (b) gluing of the GRIN fiber pieces, (c) dicing of the GRIN lenses.

Fig. 10 .
Fig. 10.Surface roughness of the GRIN lens after dicing.The roughness was measured to be about 1 µm RMS.

Fig. 11 .
Fig. 11.(a) AFM image of a diced GRIN lens after dicing.(b) GRIN lens after etching for 60 minutes in BHF.The crater-like surface is probably caused by micro-cracks in the glass after dicing.

Fig. 13 .
Fig. 13.Comparison between simulations and measurements of the GRIN lenses in index matching oil: (a) Simulation of the losses with a gap size of 250 µm , (b) measured losses of the collimation setup with gap size of 250 µm.

Fig. 14 .
Fig. 14.Comparison of the losses for two different GRIN lens and gap lengths.Even though the beam diameter exceeds the GRIN lens core the setup yields low losses over a large coupling length.

Table 1 .
Summarized parameters of the selected MM fiber.