## Abstract

A monolithically integrated asymmetric graded index (GRIN) or step-index (GRIN) mode converters for microphotonic waveguides are proposed and described. The design parameters and tolerances are calculated for amorphous silicon (a-Si) couplers integrated with silicon-on-insulator waveguides. The GRIN and step-index couplers operate over a wide wavelength range with low polarization dependence, and the lithographic resolution needed is only ±1 μm. Finally, experimental results are presented for a single layer 3 μm thick step-index a-Si coupler integrated on a 0.8 μm thick SOI waveguide. The measured variation of coupling efficiency with coupler length is in agreement with theory, with an optimal coupling length of 15 μm for this device.

©2006 Optical Society of America

## 1. Introduction

The poor coupling of light into small high index contrast waveguides presents a barrier to the acceptance and deployment of microphotonic waveguide technology in communications, information processing and optical sensing. In silicon-on-insulator (SOI) and SiO_{x}N_{y} waveguides, the two commonly used platforms for compact high index contrast waveguide circuits and photonic crystal devices, the waveguide mode diameter is often of the order of one micron or less. In the absence of an effective mode converter, the mode mismatch losses between the waveguide and typical fiber mode or free space beam are high. Many coupler designs have been proposed and demonstrated, based for example on two dimensional structures such as inverse tapers [1–3] or photonic crystals and gratings [4], which employ e-beam or deep-UV lithography to pattern features of the order of 100 nm in size. Gray scale lithography has also been used to create vertically tapered 3-dimensional waveguide couplers [5]. These approaches, though effective, rely on specialized processes and fabrication tools. The inverse taper coupler scheme also requires that the lower waveguide cladding is thick enough to avoid coupling the expanded mode into the underlying substrate, or that the taper is itself embedded in larger low-index contrast waveguide [2,3].

An alternative approach using planar graded index (GRIN) lens structures has been suggested. For example, a digital index profile structure has been proposed [6], and a freestanding planar GRIN lens structure with a quadratic index profile has been fabricated and demonstrated using an amorphous Si_{x}O_{y} structure [7]. Such GRIN structures are the planar analogue of the cylindrical GRIN lens used in fiber optic and optoelectronics packaging. Conventional and planar GRIN lenses ideally will have a quadratic index profile peaked at the center on the lens [8]. Coupling is achieved by aligning the center plane of the symmetric GRIN structure with the waveguide core of the device, either in a hybrid integration scheme [7], or by the overgrowth of the lens structure in a previously formed trench with the correct depth to achieve the required alignment [6]. Although feasible, either approach presents considerable difficulties in material growth, fabrication, or in packaging.

It has not been widely recognized that an asymmetric GRIN structure that terminates in the high index layer, as shown in Fig. 1, can be equally effective in coupling light to an underlying waveguide. Such asymmetric-GRIN lens designs result in an enormous simplification for the monolithic integration of a coupler with a waveguide, since the GRIN lens can now be grown directly on top of an underlying waveguide using standard deposition techniques such as plasma enhanced chemical-vapor deposition (PECVD). The need for precise alignment of the waveguide and GRIN lens, or etching of a recessed trench and overgrowth in the case of a monolithic GRIN coupler, is thereby eliminated. Although accurate calibration of the layer refractive indices and thicknesses is essential, this is already routinely achieved by the optical coatings industry. In preliminary theoretical work, GRIN lens structures consisting of continuous graded index structure, or of only a few discrete layers with uniform index, were both found to be effective [10].

In this paper we present the theoretical description of light propagation in an asymmetric GRIN and step-index structures in terms of waveguide mode analysis. Coupling efficiencies and tolerances are calculated for GRIN structures designed to couple light into SOI waveguides. As we show below, the lateral spatial resolution required to pattern these monolithic couplers is approximately ±1 μm, well within the range of standard photolithographic tools. Layers with the necessary refractive index values to create the desired index profile for silicon waveguides (3.3 < *n* < 3.47 at λ = 1550 nm) or for Si_{3}N_{4} waveguides (1.5 < *n* < 2.0 at λ = 1550 nm) are easily obtained using well-calibrated PECVD deposition of amorphous silicon (a-Si) or SiO_{x}N_{y}. Finally we present an experimental demonstration of a single layer amorphous silicon (a-Si) step-index waveguide coupler monolithically integrated with an SOI waveguide.

## 2. Theory

The operation of a conventional GRIN lens is usually described in terms of a ray tracing calculation [8]. However, in the context of waveguides optics it is more natural to consider the GRIN lens as an example of an imaging multimode interference (MMI) system. A GRIN lens is in fact a waveguide with a quadratic index profile

where the constants n_{0} and x_{0} determine the magnitude and spatial variation of the refractive index. A waveguide with this index profile will support orthogonal waveguide modes with field profiles given by Hermite-Gaussian functions

where H_{v}(x) is the Hermite polynomial of order *v* and *k*
_{0} is the free space wavevector [8]. The corresponding mode wavevectors are given by

Finally, the total guided field propagating in a quadratic index waveguide can be expressed as a superposition of these modes with the form

Here *c _{v}* are the expansion coefficients determined by initial conditions (i.e. the input field profile launched into the structure), while the term exp(-

*iβ*) gives the phase evolution of the field along the z-axis or propagation direction. The focusing properties of a quadratic index waveguide or GRIN lens result from the evolution of the relative phase of these modes as they propagate.

_{v}zFor a full GRIN lens excited by a symmetric input field, the symmetric modes in Eq. (4) are excited and *c _{v}* = 0 for odd values of

*v*. If the combined expansion coefficients and phase term,

*c*exp(-

_{v}*iβ*), in Eq. (4) are all real and positive, then the mode superposition generates the tightly focused field profile shown in Fig. 2(a). Note that in this pedagogic example the expansion coefficients were arbitrarily chosen to satisfy

_{v}z*c*= (1/

_{v}*v*!). On the other hand when the phase of

*c*(-

_{v}exp*iβ*) for each successive symmetric mode differs by π (i.e. successive terms alternate in sign) the broad field distribution in Fig. 2(a) is generated for the same

_{v}z*c*values. For the lower order modes where

_{v}*v*<<

*k*, the wavevectors of successive even modes obtained from Eq. (3) are spaced by approximately equal increments of

_{0}n_{0}x_{0}*Δβ*~ 2/

*x*. Hence, if either one of the two field distributions shown in Fig. 2(a) are launched into the GRIN waveguide, as the light propagates the total field profile will periodically oscillate between the these two distributions with a period

_{0}*ΔL*= π

*x*, giving an effective lens focal length of

_{0}*f*= π

*x*/2. This is the same focal length obtained by the usual ray tracing calculation for a GRIN lens [10]. Note that for the silicon GRIN couplers discussed later in this paper,

_{0}*k*~ 100, and the assumption of equally spaced modes is valid for at least the first five modes.

_{0}n_{0}x_{0}If the quadratic index waveguide is truncated at the center plane by a highly reflecting layer such that the electric field is zero at *x* = 0 as in Fig. 2(b), only the odd modes in Eq. (4) are allowed waveguide modes. When the weighting coefficients of these odd modes are all real and in phase, the narrow field distribution near the bottom edge of the waveguide in Fig. 2(b) is produced. When the successive modes are π out of phase, the broad distribution is generated. As in the case for the full GRIN structure, as the modes propagate in the asymmetric-GRIN structure the field distribution will oscillate between the focused and wide distribution shown in Fig. 2(b) with the same focal length *f* = π*x _{0}*/2 as the symmetric GRIN lens, and the starting field profile is reproduced at the self-imaging length 2π

*x*.

_{0}The preceding qualitative discussion illustrates that the asymmetric GRIN and full GRIN lens are similar in working principle to MMI couplers [11] and related self-imaging structures [12], in which the equal spacing of propagation wavevectors leads to periodic focusing or imaging. More importantly, the results in Fig. 2(b) suggest that the truncated quadratic index waveguide has the same focusing properties as a full GRIN lens, and therefore may be an effective waveguide coupling element when fabricated on and properly matched with an underlying waveguide.

We now consider an integrated asymmetric-GRIN waveguide and the corresponding coupler shown in Fig. 1(a) and (b). The waveguide coupler consists of a GRIN layer on top of the waveguide, adjacent to the input facet. The GRIN layer is removed beyond the focal length of the coupler such that any light coupled into the final waveguide remains confined. The field inside the coupler is given by the same general modal expansion of Eq. (4), where the *M _{v}(x)* are now the normalized orthogonal modes of the combined coupler-waveguide structure. The amplitude of each excited mode

*c*is determined from a field overlap integral of the orthogonal waveguide modes with the source field

_{v}*S(x)*that is launched into the waveguide.

Since we can assume that the coefficients *c _{v}* are all real provided the launched field profile

*S(x)*is real at

*z*= 0, the input coupling efficiency

*η*is just the sum of the light power coupled to each waveguide mode, and can be expressed as a sum of the squared mode expansion coefficients:

_{in}Letting L be the length of the GRIN coupler section, the total coupling efficiency *η _{c}* into the output waveguide is given by the overlap integral of the field

*E(x,L)*with the final output waveguide mode profile

*G(x)*at the end of the coupler.

$$\phantom{\rule{.2em}{0ex}}={\mid \sum _{v}{c}_{v}{g}_{v}\mathrm{exp}\left(-{\mathrm{i\beta}}_{v}L\right)\mid}^{2}$$

Here *g _{v}* are the expansion coefficients of the output mode profile G(x) in terms of the GRIN waveguide modes M

_{v}(x). As discussed above, in an ideal quadratic index waveguide the mode coefficients

*c*required to match the wide input field profile

_{v}*S(x)*will tend to have alternating signs with incrementing mode number

*v*, while the coefficients

*g*needed to match the narrow output waveguide mode

_{v}*G(x)*will have the same sign. As a result the product

*c*in Eq. (7) will also alternate in sign as the mode index

_{v}g_{v}*v*increments. Hence from Eq. (7) the maximum coupling efficiency is obtained if the wave vectors

*β*are equally spaced by

_{v}*Δβ*=

*π/L*. Here the coupler length

*L*=

*π/Δβ*is just the effective focal length

*f*of the GRIN lens. On the other hand, in a realizable waveguide the lower waveguide boundary is not a perfect reflector, and the electric field will have an evanescent tail that extends into the lower cladding. Finally, the target waveguide under the GRIN layers will usually have a uniform refractive index which does not match the lower edge of the GRIN lens structure.

In the following section we demonstrate that an asymmetric-GRIN waveguide coupler can effectively couple light into waveguides despite these unavoidable deviations from the ideal quadratic index profile. The four GRIN structures shown in Figs. 3(a) to 3(d) are considered. A wavelength of λ = 1550 nm and (transverse electric) TE polarized (in the layer plane) light are assumed. The structures in Fig. 3(c) and 3(d) are optimized to couple light into a 0.5 μm thick Si waveguide positioned between *x* = 0 and *x* = 0.5 μm. The lowest order modes of each of these structures are also shown. The first coupler in Fig. 3(a) is an ideal quadratic index profile waveguide with *n _{0}* = 3.405 and

*x*= 8.23 μm. Only the lowest five odd, or symmetric, mode profiles are shown. The second waveguide is 4 μm thick truncated version of the first structure, with a SiO

_{0}_{2}lower cladding (

*n*=1.46) and an air upper cladding above x = 4 μm. The coupler-waveguide structure in Fig. 3(c) consists of a conventional SOI based 0.5 μm thick silicon waveguide (

*n*= 3.47) on a SiO

_{2}cladding, over which an asymmetric-GRIN coupler structure has been fabricated to form an integrated coupler. The GRIN layer is 3.5 μm thick with a quadratic index profile (

*x*= 8.21 μm,

_{0}*n*= 3.372) chosen to maximize coupling to the underlying Si waveguide, using a simple optimization algorithm. A BPM simulation of the propagation of light through this coupler is shown in Fig. 1(a). The final structure in Fig. 3(d) consists of a similar 0.5 μm thick Si waveguide, with an overlying coupler layer of uniform index of

_{0}*n*= 3.365. The waveguide modes and propagation wavevectors for these structures were calculated using a multilayer algorithm [9], and subdividing the graded index region into a sufficient number of discrete steps to achieve the required accuracy.

The lowest five modes shown in Fig. 3 for the symmetric quadratic, truncated quadratic, and quadratic asymmetric-GRIN coupler waveguides are almost identical, as are their modal wave vectors *β _{v}* plotted in Fig. 4. Examination of the mode shapes clearly shows that a linear in-phase superposition of modes must result in a tightly focussed field distribution, while an anti-phase superposition will yield a broad profile. The similarity in the modes in Fig. 3(a), (b) and (c), and those for an ideal quadratic index lens occurs because the quadratic index gradient confines the lower order modes to a region well away from the upper cladding interface, while the small evanescent tail at the bottom Si – SiO

_{2}interface represents a negligible perturbation to the mode shape and wave vector. The wave vector differences between successive modes

*β*in Fig. 4 are nearly constant for the first several modes because

_{v}*n*~ 100, with obvious deviations from linearity only occurring for

_{0}x_{0}k_{0}*v*> 6. This satisfies the criteria for focusing and imaging. This comparison demonstrates that the deviations from a quadratic index structure necessary to create a monolithic asymmetric-GRIN waveguide coupler of Fig. 3(c) have a negligible effect on the mode profiles and effective indices of the lowest order modes in a GRIN structure. In summary, when the lowest order modes dominate the launched field profile, the asymmetric-GRIN coupler waveguides in Fig. 3 retain the useful imaging and focusing properties of the ideal symmetric and asymmetric quadratic index structures.

In the next section we calculate and compare the coupling efficiencies of asymmetric GRIN couplers in Fig. 3(b), (c), and (d) using the mode expansion method outlined in Eqs. (5), (6), and (7). Each GRIN coupler structure is excited by a TE polarized incident field distribution *S(x)* having a Gaussian profile with a FWHM of 3.5 μm, centered on the coupler/waveguide structure. Only the power in the fundamental mode of the 0.5 μm thick output waveguide is considered. A coupling efficiency of 19% is obtained by direct coupling of this input mode to the 0.5 μm waveguide. Losses due to reflection at the waveguide facet are not included in these calculations.

Table 1 presents the mode coefficients *c _{v}* and

*g*for the first 10 modes in the truncated quadratic index structure and the GRIN waveguide coupler in Figs. 3((b) and (c) respectively. Also given are the calculated (using Eq. (7)) coupling efficiencies

_{v}*η*from the input mode to the coupler structure,

_{in}*η*from the coupler structure to the 0.5 μm silicon output waveguide (

_{out}*η*=∑

_{out}*g*

_{v}^{2}), and the overall coupling

*η*from input mode to the output waveguide. Both structures described in Table 1 behave much the same way as the idealized quadratic index lenses discussed previously. Since the signs of successive coefficients

_{c}*c*alternate, while the coefficients

_{v}*g*all have the same sign, Eq. (7) requires that successive values of the phase

_{v}*β*must increment by π to achieve efficient coupling. This condition is approximately satisfied for only the first six modes at the optimum coupler lengths of L = 11 μm for the truncated quadratic index structure and L = 12 μm for the GRIN coupler, but the power in the higher modes is small and contributes a negligible amount to the overall coupling efficiency. A coupling efficiency of

_{v}L*η*= 88% is calculated for the ideal truncated quadratic index structure of Fig. 3(b), while for the optimized asymmetric-GRIN coupler of Fig. 3(c), a coupling efficiency into the Si waveguide of

_{c}*η*= 78% is predicted.

_{c}Previous simulations [10] predict that step-index waveguide couplers, in which the quadratic index profile is approximated by a small number of layers with uniform index, can be almost as effective as a true quadratic index profile coupler. Here we only consider the simplest example of the single layer step-index coupler structure shown in Fig. 3(d). This coupler is much simpler both in terms of structure and fabrication requirements. Table 2 presents the calculated *c _{v}* and

*g*for the 5 first modes for the same input mode

_{v}*S(x)*used in the preceding section, as the index step Δn between the coupler layer and the silicon waveguide is varied. From the table, two observations are evident: the incident field couples primarily with the two first modes shown in Fig. 3(d), while higher modes (

*v*≥2) contribute little to the power coupled to the output waveguide. Equation (7) simplifies to

*η*~ |

_{c}*c*| + |

_{0}g_{0}*c*| and the coupling length L depends on the modes spacing

_{1}g_{1}*L*=

*π/k*where

_{0}Δβ*Δβ*can here be taken as the mode spacing between the first two modes. Despite the simplicity of this structure, a coupling efficiency of η

_{c}= 45% is achieved at

*Δn*= 0.10. As the number of layers forming the GRIN region is increased to more fully approximate a true quadratic index profile, the coupling efficiency will improve. Our previous simulations [10] have shown that an optimized three layer GRIN coupler has a coupling efficiency almost indistinguishable from a coupler with a true quadratic index profile.

In the preceding analysis, we have treated coupling in the vertical direction, whereas coupling to a ridge or photonic wire waveguide must also involve lateral mode conversion of the input light. However, vertical and lateral coupling can be effectively accomplished in two distinct stages. Once the light is coupled from the GRIN coupler to the thin Si slab waveguide, lateral mode conversion with little additional loss is usually easily accomplished using adiabatic tapers that are defined in the same lithography and fabrication steps as the ridge or wire waveguide. For photonic wire waveguides such lateral tapers may be as short as 10 μm [13]. Using the BPM method, we have evaluated the coupling efficiency from a 4 μm wide GRIN coupler to a 0.5 μm wide Si ridge waveguide [10] using the layer specifications shown in Fig. 3(c). Once the light has been coupled to the 0.5 μm Si layer, only 0.01 dB excess loss is incurred as the light propagates through a 100 μm long waveguide that tapers from 4 μm to the final waveguide width of 0.5 μm.

## 3. Alignment and fabrication tolerances

The usefulness of a waveguide coupler depends on both the ease of coupler fabrication, and on the tolerance of the coupler to changes in the launched input field properties. In this section we evaluate and compare the variation of coupling efficiency with coupler length, as well as the input beam position, angular alignment, wavelength and polarization for the optimized asymmetric-GRIN coupler waveguide of Fig. 3(c) and the single layer coupler of Fig. 3(d). The input mode was again assumed to be TE polarized with a 3.5 μm FWHM Gaussian mode profile. Although smaller than the mode size of a standard telecommunications fibre, this mode is comparable to typical spot profiles that would occur at the focal plane of a tapered fiber tip. The coupling efficiencies to the fundamental mode of the 0.5 μm thick Si waveguide were evaluated using the semi-vectorial beam-propagation method (BPM). Ease of fabrication depends in large part on the tolerance of the coupler to small changes in coupler length, shown in Fig. 5. The coupling length for an optimized quadratic index coupler is 12 μm, with variations within ±1 μm resulting in less than -1 dB change in coupling loss. The 50 μm coupling length for the single layer coupler is longer but less sensitive to changes in coupling length. The length tolerances on these couplers are well within the range of standard photolithographic tools.

Fig. 6 shows the variation of coupling efficiency with polarization, wavelength, input beam position and input beam angle. Both the quadratic GRIN structure and the single layer GRIN coupler are remarkably insensitive to wavelength, a distinct advantage compared with grating and prism based waveguide coupling methods. The coupling dependence on input mode position and angular alignment are similar to those expected when coupling to a waveguide of comparable dimensions to the total 4 μm coupler thickness. Finally, the polarization dependent loss (PDL) of the coupler is shown in Fig. 6(c). Near λ = 1500 nm the coupler PDL is less than 1 dB for both the single layer coupler and the optimized GRIN design. However, the PDL varies much more rapidly with wavelength for the single index coupler. The PDL will be larger for couplers composed of one or more discrete uniform index layers, than for smoothly varying index profiles, because the different electromagnetic boundary conditions for fields normal and parallel to the discontinuous interfaces will play a significant role in shaping the propagating field distribution. Nevertheless, the PDL reported here is comparable to or significantly better than previously reported coupling geometries for small Si waveguides.

## 4. Experiment

The single layer a-Si/SOI step-index coupler shown in Fig. 7 was fabricated and tested to demonstrate the GRIN (or step-index) coupler concept. Although the coupling efficiency of continuous quadratic index profile coupler in Fig. 3(c) is almost twice that of the single layer structure, a single layer coupler is significantly easier to fabricate using a-Si PECVD, since calibration and control of the a-Si growth rate and index is only required at one operating point. Fig. 7(a) shows a schematic cross-section of the fabricated coupler, and Fig. 7(b) shows a scanning electron microscope image of the facet area of several fabricated coupler structures.

The SOI waveguides were formed in a 0.8 μm silicon layer on a 0.4 μm buried SiO_{2} layer. Ridges were etched using reactive ion etching, to an etch depth of approximately 0.5 μm. The ridge waveguide width was 2.0 μm, but 400 μm long adiabatic tapers expanded to a 10 μm wide waveguide at the input facets shown in Fig. 7(b). Three-dimensional BPM simulations indicate that the coupling of such adiabatically tapered GRIN lenses is almost identical to that predicted by calculations for the case of a slab waveguide with an overlying GRIN layer.

To form the coupler, a 0.5 μm thick layer of SiO_{2} was first deposited on top of the SOI waveguide layer by PECVD. The waveguide coupler sections were initially defined by etching a window in this SiO_{2} layer over the waveguide adjacent to the eventual input facet position. Finally a 3 μm layer of a-Si was deposited by PECVD over the entire wafer, and subsequently removed everywhere except over the coupler sections. The remaining a-Si regions slightly overlap the boundaries of the oxide window so that the exact coupler length is defined by the oxide window dimensions, as indicated in Fig. 7(a). Since the dimensional uncertainty in patterning windows in the thin SiO_{2} layer is much less than in the removal of the 3 μm thick a-Si layer, it is much easier to fabricate couplers of a specified length using this window masking process. Couplers of lengths ranging from 5 μm to 200 μm were fabricated, so that the dependence of coupling efficiency on coupler length could be measured. Although the monolithic GRIN coupling scheme does not require specialized lithography, it is clear from Fig. 6 that the final coupler length should be within a few microns of the optimum value. To achieve this objective, the vertical input facets were fabricated using a lithographically defined two step vertical etch process. A 10 μm deep inductively coupled plasma (ICP) etch was used to create the vertical waveguide facets, and a second deep etch was used to facilitate dicing. The details of facet fabrication will be described elsewhere. Once the facets were formed, the Si wafers were diced into 4 mm wide bars to expose the etched facet and allow coupling of light into the waveguide from an optical fiber.

Measurements of the relative coupling efficiency of step-index couplers were carried out using a broad-band erbium-doped fiber source (λ ~ 1525–1560 nm) coupled to the waveguide input facet from the cleaved end of a polarization maintaining fiber. An in-line polarizer was used to select the polarization of light launched into the waveguide. After propagating through a 4 mm long waveguide, light was collected from the opposite facet and collimated using a microscope objective lens, and projected through a polarizing filter onto a photodetector. Fig. 8 shows the measured output power using TE polarized light, for a series of waveguides with increasing coupler length from 0 μm to 70 μm. The coupling efficiency increases by approximately four times at the optimal coupling length near 15 μm. Although it is not possible to separate the effects of polarization dependent waveguide loss and coupler loss in this experiment, the combined polarization dependent loss of the optimal coupler and contiguous waveguide is better than -0.4 dB. We attribute the observed scatter in measured intensities to defects present at the Si/a-Si interface near the etched facet, and at the surface of the a-Si film. These defects and roughness can cause loss and mode conversion in the coupler sections.

The expected field evolution and coupling efficiency in the fabricated structure were calculated for TE polarized light, using a semi-vectorial beam-propagation method (BPM) simulation. The refractive index of the deposited PECVD a-Si was measured using ellipsometry at a wavelength of λ =1550 nm on unpatterned witness wafers. The refractive index over the wafer area showed some spatial variation, ranging from *n* = 3.36 to 3.40. Using an assumed a-Si refractive index of n = 3.365, the calculated variation of coupling efficiencies with coupler length shown in Fig. 8 is in qualitative agreement with experiment with respect to both the coupling periodicity and variation of coupling magnitude. Future work will be directed at optimizing the a-Si deposition conditions and facet fabrication process in order to reduce the number of the defects, improved calibration of the a-Si refractive index of a-Si deposition, and finally the fabrication of multilayer GRIN lens for optimized fiber to waveguide coupling.

## 5. Summary

We have proposed and described the design of a monolithically integrated waveguide coupler structure based on an asymmetric GRIN or step-index lens waveguide. The coupler design is based on the ability of asymmetric half-GRIN lens structures to focus and couple light in a similar way that conventional symmetric GRIN lenses do. The asymmetric GRIN design can be used to make monolithically integrated waveguide couplers for small high index contrast silicon-on-insulator and Si_{x}N_{y} waveguides, and does not depend on high-resolution lithography or 3-dimensional fabrication techniques. The index range required to fabricate the asymmetric GRIN and step-index couplers is available using PECVD deposition of a-Si and a-Si_{x}O_{y} [7], as well as SiO_{x}N_{y} for which a similar step-index structure has recently been reported [14]. While maximum coupling efficiencies are predicted for an optimized quadratic index coupler, much simpler structures such as a step-index waveguide coupler employing a single uniform index layer also achieve significant improvement in coupling to the 0.5 μm Si waveguides examined in this work. Asymmetric GRIN couplers have very wide tolerances with respect to input beam alignment, wavelength and polarization. In particular, the calculated tolerance on coupler length of approximately ±1 μm (for -1 dB variation in coupling efficiency) is well within the capability of standard photolithography. However, well calibrated film deposition tools capable of producing the necessary film thicknesses and refractive indices, such as those employed in the optical coatings industry, are essential for routinely manufacturing fully optimized couplers. Although the theoretical examples given in this paper are based on coupling to a 0.5 μm thick Si waveguide, the asymmetric GRIN and step index coupler can be optimized for coupling to thicker waveguides, or waveguides as thin as 0.2 μm. In general, coupling to thinner waveguides requires steeper index gradients, but otherwise the operating principles and attainable coupling efficiencies are similar. Finally, we have fabricated and characterized a monolithically integrated asymmetric step-index coupler formed using a single uniform a-Si layer deposited on a 0.8 μm thick SOI waveguide. The measured variation of coupling efficiency with GRIN coupler length is in good qualitative agreement with theoretical calculations. The monolithically integrated asymmetric-GRIN lens coupler is a simple, manufacturable solution for improving the optical coupling of light into high index contrast microphotonic waveguides.

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