## Abstract

Nanophotonic chip coupling using an optical thin-film stack forming a micro graded-refractive-index (GRIN) lens with a super-high numerical aperture (NA) that is highly compact (tens of micron long) and can be directly integrated is presented. We explore the lens’ integration on the surface of Silicon-On-Insulator (SOI) platform with an asymmetric GRIN profile. We show that to achieve high efficiency for optical coupling between an optical fiber and a nanophotonic waveguide with a sub-wavelength (λ/n) beam size, conventional asymmetric parabolic GRIN profile is no longer adequate due to the super-high NA needed (>3.1), which results in severe spatial beam aberration at the focal plane. We present an efficient algorithm to computationally generate the ideal GRIN profile that is completely aberration free even at super-high NA, which improves the coupling efficiency from ~66% (parabolic case) to ~95%. A design example involving an optical thin-film stack using an improved dual-material approach is given. The performance of the thin-film stack is analyzed. This thin-film stack based GRIN lens is shown to be high in coupling efficiency, wavelength insensitive and compatible with standard thin-film process.

©2010 Optical Society of America

## 1. Introduction

Optical coupling of light beam between fiber and nanophotonic chip with waveguide having a sub-wavelength <λ/n (λ is the wavelength in free-space and n is the refractive index of waveguide core) thickness has been a challenge. To achieve a high coupling efficiency between the nanophotonic waveguide and optical fiber, various coupling schemes have been proposed and demonstrated numerically or experimentally in the published literature recently, which include using an inverse taper, a grating structure, and a bilevel mode converter [1–7]. These proposed structures improved the coupling efficiency somewhat but bear with some specific requirements. For example, the coupling scheme using an inverse tapered nanophotonic waveguide requires a thick buried oxide layer and the approach using the grating structure needs a vertical / tilted fiber-alignment.

In this paper nanophotonic chip coupling using an optical thin-film stack based micro graded-refractive-index lens with a super-high numerical aperture and aberration-free focusing that is highly compact (tens of micron long) and can be directly integrated is presented. This GRIN lens with a super-high NA or super-focusing capability will be referred to as super-GRIN lens. A geometry of interest in this paper involves the coupling of light beam from an optical fiber into a silicon-on-insulator platform with nanophotonic waveguides having a silicon waveguide core thickness of about 300 nm, which is a singlemode waveguide for 1550 nm wavelength applications with a waveguide width of ~450 nm. To achieve the beam coupling, a super-GRIN lens is fabricated on top of the nanophotonic waveguide at the end as shown schematically in Fig. 1(a) . The nanophotonic waveguide is tapered first so that the light beam is expanded to match the eigenmode of optical fiber in the horizontal direction. The super-GRIN lens has the same width as the tapered waveguide but expands the light beam from sub-wavelength spotsize to match the optical fiber in the vertical direction within a very short distance. In the present paper, expanding the light beam by the super-GRIN lens is investigated with a two-dimensional modal in the vertical direction.

The super-GRIN lens is realized by depositing a stack of dielectric thin films as shown in Fig. 1. The thickness for each layer is typically at below one tenth of the optical wavelength in the material in order to reduce scattering losses [8]. For λ = 1550 nm, the film thickness of single layer will be nanometer in dimensions and typically below 100 nm (assuming film refractive index of n>1.5). As discussed in our previous publication [8], such an optical thin-film stack made up of only two alternating materials with different refractive indices (n_{H} and n_{L}) can be used to realize a GRIN profile with an effective arbitrary refractive index variation from n_{0} to n_{R}, which is referred to as the dual-material approach [see Fig. 1(b)]. With a proper design, the super-GRIN lens realized with the dual material approach can have a high numerical aperture [8].

For ease of integrating the super-GRIN lens on the SOI platform, the lens is placed on the surface of the silicon nanophotonic waveguide of interest. This vertically asymmetric geometry requires the GRIN profile to be asymmetric as shown in Fig. 1(b). Using an asymmetric GRIN lens to couple light from a waveguide to optical fiber was demonstrated in Refs. [9,10]. However, in Ref. [9], the light beam was expanded from 1.45 μm to 6.8 μm using an asymmetric parabolic profile with a small refractive index contrast of ~0.2. Our interest here is to couple light from an optical fiber with a mode-field diameter (MFD) of ~10.4 μm to a nanophotonic waveguide with a thickness of ~300 nm (at λ = 1550 nm, the MFD is ~380 nm), which requires the light beam to be expanded by over 25 times. To achieve ultra-compact size for the lens, it would be desirable to achieve this very high 25x transformation in the beam size within as short a propagation distance as possible, which needs the lens to have a very short focusing length. The compact lens size is important to enable ease of fabrication and saving of valuable real-estate space on the photonic chip. To achieve the small focused spot size at the sub-wavelength nanophotonic waveguide, the super-GRIN lens shall have as high a numerical aperture as possible. For coupling light beam into silicon waveguide with refractive index of 3.5, our example below shows that NA >3.1 is required for the GRIN lens.

In Section 2, we show that for optical coupling to a nanophotonic waveguide, an asymmetric GRIN lens with a low refractive index contrast is no longer adequate due to the fact that the low refractive index contrast will give a low NA for the lens and will not be able to bend around the light rays that are propagating at large angles. In addition, we show that at high NA, the usual parabolic refractive index profile is no longer adequate due to severe spatial aberration for the beam at the focal plane caused by the large-angle light rays of the focusing beam. These two issues make it impossible for the conventional low-index-contrast GRIN lens or even high-index-contrast GRIN lens but with the conventional parabolic profile to achieve good coupling efficiency from an optical fiber to a nanophotonic waveguide.

In Section 3, we show that an ideal asymmetric GRIN lens profile can be obtained using an efficient algorithm to computationally generate a GRIN profile that is completely aberration free for the focusing beam even at high NA. For the aberration-free GRIN lens, the rays at different propagating angles emitted from the nanophotonic waveguide end shall have the same focusing length, which means they shall have horizontal propagating direction when reaching the optical fiber facet. This ideal graded refractive index profile differs from commonly used parabolic profile and can focus the light beam from optical fiber to sub-wavelength spotsize at the nanophotonic waveguide without phase aberration for the focusing beam. With this ideal graded refractive index profile, the coupling efficiency from nanophotonic waveguide to optical fiber can be optimized to be ~95%, but is only ~66% for the GRIN lens with a parabolic profile.

The super-GRIN lens designed for optical coupling between fiber and nanophotonic waveguide requires a super-high numerical aperture (NA=3.167). However, the current approaches using ion-exchange technology or varying material composition can only produce a low refractive index contrast. Hence, in Section 4, we present the design example of a dual material method [8] that will allow the high-refractive-index contrast to be effectively achieved with conventional thin-film deposition technique. An improved design approach is presented in this paper as compared to that in Ref [8]. The performance of thin-film stack based GRIN lens is analyzed. Our numerical result indicates that with the optimized thin-film stack, the coupling efficiency between nanophotonic chip and optical fiber can be ~92%. It is also shown that this thin-film stack based GRIN lens is wavelength insensitive within a wide wavelength range, which will enable it to be applied to photonic subsystem on chip employing wavelength-division-multiplexing (WDM) with a broad optical bandwidth.

## 2. Numerical aperture requirement, aberration issue and coupling between nanophotonic waveguide and optical fiber

In this section, we will first show that the GRIN lens should have a sufficiently high NA to collimate the light from the nanophotonic waveguide based on “refractive lensing”. We will then show that the usual parabolic GRIN lens profile has a severe aberration issue when the NA is high and consequently is no longer adequate to achieve high coupling efficiency between nanophotonic waveguide and optical fiber.

For the discussion below, the nanophotonic waveguide considered has a refractive index contrast of 3.5 and 1.5 between the waveguide core and cladding. In the ray picture, the light beam from the waveguide is represented as rays with different propagation angles. The angle spanned by the rays is basically given by the beam’s divergence angle due to diffraction. Table 1
shows the beam’s divergence angle ${\theta}_{d}$, defined here as the half-angle that encompasses 95% of the beam energy, versus the beam’s mode field diameter (MFD, which is defined as the beam diameter at 1/e^{2} of the beam intensity) at the wavelength of 1550 nm under different thickness of the nanophotonic waveguide (from 800 nm to 300 nm with a decrement of 100 nm). From Table 1, we see that when the thickness of the waveguide is 300nm (it is a singlemode waveguide used in nanophotonic integration with a waveguide width ~450 nm), the mode-field diameter is 380 nm and the divergence angle is as large as 64° (to encompass the 95% energy point).

In order to collimate the light beam with a sub-wavelength spotsize in the GRIN lens based on “refractive lensing”, we need the ray that propagates at the largest angle to bend around and become perpendicular to the facet of GRIN lens (i.e. to become collimated) before coupling into the optical fiber. We can take the largest angle ray to be propagating at angle ${\theta}_{d}$. In order to meet this requirement, a simple geometrical picture shown in Fig. 2
indicates that based on Snell’s law of refraction, the refractive index n_{R} at the top of the GRIN lens shall be given by ${n}_{R}={n}_{0}\mathrm{cos}{\theta}_{d}$ in order to bend this largest-angle beam to be collimated (by observing that the total bending angle experienced by the ray is independent of thickness *D* of the GRIN lens provided that the refractive index contrast between n_{0} and n_{R} is given). The numerical aperture of a GRIN lens is given by $NA=\sqrt{{n}_{0}^{2}-{n}_{R}^{2}}$. The required refractive index n_{R} and the corresponding numerical aperture of the GRIN lens for different mode-field diameter of the waveguide mode is presented in Table 1, from which it can be seen that when the MFD is decreased to 380 nm, which corresponds to the sub-wavelength case ($\raisebox{1ex}{$\lambda $}\!\left/ \!\raisebox{-1ex}{$n$}\right.=443$ nm in this example), the required refractive index n_{R} is 1.53 and the corresponding NA of the GRIN lens is ~3.15.

For this super-high NA, we will show that light propagation in an asymmetric parabolic GRIN profile suffers from severe aberration. The GRIN lens has the following design parameters: n_{0}=3.5, n_{R}=1.5 (NA=3.16) and thickness D=10 μm. Light propagation in the parabolic asymmetric GRIN lens are investigated using ray tracing as well as full electromagnetic simulation using two-dimensional finite-difference-time-domain (FDTD) method respectively [11]. Figure 3(a)
shows the ray-tracing in the asymmetric GRIN lens (from −60° to 60°, with a 5° step), from which we can see the rays are bent in the GRIN lens due to the super-high NA but there is substantial spatial aberration of the focused spot. The large-angle rays have shorter focusing length and they do not focus at the same point as the small-angle rays. In particular, the different focusing lengths make the paths of different rays to cross with each other after they are bent down from the peak. In the wave picture for an ideal GRIN lens, the beam shall form a “beam waist” with plane wavefront at both the nano-waveguide and the lens facet. However, the full electromagnetic simulation using FDTD method in the GRIN lens [see Fig. 3(b)] shows the severe interference while light beam propagating in the GRIN lens due to the different rays crossing with each other in the ray picture. In particular, the wavefront when the beam expended to near its maximum size never becomes a perfect vertical plane wavefront. The aberration in the wavefront results in poor coupling of the beam energy into the optical fiber.

To use the asymmetric parabolic GRIN lens for nanophotonic waveguide coupling, we need to optimize the height *D* and length *L* of the GRIN lens and also the fiber position (center of the fiber core) *H* from the bottom of the GRIN lens. The location of the beam’s maximum size is approximately where the lens facet shall be or where one shall couple into an optical fiber. Using the coupling between a standard singlemode fiber (SMF28) and nanophotonic waveguide with a thickness of 300 nm as an example. The parameters for SMF28 fiber used are n_{co}=1.4554, n_{cl}=1.4447 and R=4.15 μm and with the parameter-space scanning approach, it is found that the maximal coupling efficiency can be achieved when D=15 μm, H=5.3 μm, and L=24.18 μm. With the optimized parameters, the corresponding normalized transverse light intensity profiles at the GRIN lens facet and the eigenmode of the optical fiber are shown in Fig. 4
. The severe ripples of the intensity profile at the facet of the GRIN lens are caused by the interference of light while propagating, as discussed above, due to the non-ideal parabolic graded refractive index profile. The maximal coupling efficiency from the nanophotonic waveguide to the optical fiber using this parabolic GRIN lens is estimated to be only 66%.

## 3. Aberration-free asymmetric super-GRIN lens: design algorithm and fiber to nanophotonic waveguide coupling

Below we describe how an aberration-free asymmetric GRIN lens is designed with a computational algorithm. A light beam from the nano-waveguide with a sub-wavelength spotsize is taken as the input light source *P*. As shown in Fig. 5
, the design is to generate an ideal graded refractive index profile so that each ray emitting from the point source will be bent to be parallel exactly at the lens facet. This will occur at the half distance of the so-called “beat length” for a GRIN lens. This half-distance of the beat length will be referred to as the focusing length L_{f} below.

For one version of the design algorithms, the asymmetric GRIN lens is first divided into *N* layers with a constant thickness of *h* (*h=D/N*) for each layer [see Fig. 5(a)]. Each layer is sufficiently thin so that the refractive index for each layer is regarded as a constant and denoted by n_{i} (i=1,2,3…) for the i-th layer. The entire GRIN lens is designed if the refractive index of each layer n_{i} is designed and then we let the layer thickness *h* to become arbitrarily small to result in the design of a continuous index profile for the GRIN lens. For the asymmetric GRIN lens, the GRIN profile shall have gradually decreasing refractive index from the nano-waveguide to the lens’ top surface so that n_{i}>n_{i + 1}.

The refractive index *n _{1}* of the first layer is first chosen to be equal to the refractive index of the nano-waveguide core, which is the refractive index of the material medium on the left of the GRIN lens containing the light source

*P*. This is done so that the optical reflection between the GRIN lens and nano-waveguide medium on its left is minimized when light propagates from the left side into the GRIN lens. After the first layer’s

*n*is chosen, the refractive index

_{1}*n*for the second layer can be determined using the condition that the ray shall become horizontal at distance

_{2}*L*from the point source [see Fig. 5(b)], thus its incident angle from layer

_{f}*n*to layer

_{1}*n*shall be at the critical angle ${\theta}_{c1}$ so that the propagating angle θ

_{2}_{1}for the ray in layer

*n*shall be given by ${\theta}_{\text{1}}=\frac{\pi}{2}-{\theta}_{c1}=\mathrm{arc}\mathrm{cos}\left(\frac{{n}_{2}}{{n}_{1}}\right)$so that the ray will become horizontally oriented at the interface between layer

_{1}*n*and layer n

_{1}_{2}(Note that this is equivalent to

*n*, which is obtained from Snell’s law at the critical angle). From Fig. 5(b), we can also see that ${L}_{f}=\frac{h}{tg\left({\theta}_{1}\right)}$. Hence,

_{1}cos (θ_{1})=n_{2}cos(0=n_{2}*n*can be directly calculated by using this equation to solve for $\mathrm{cos}\left({\theta}_{1}\right)$ and substitute it into the preceding Snell’s law equation at the critical angle, which results in ${n}_{2}=\frac{{L}_{f}}{\sqrt{{L}_{f}^{2}+{h}^{2}}}{n}_{1}$.

_{2}Following this strategy, ray tracing for calculating the refractive index *n _{i}* for the

*i-th*layer is presented in Fig. 5(c) and the incidence angle from the layer

*n*to layer

_{i-1}*n*is at the critical angle. We also have the relationship ${n}_{1}\mathrm{cos}{\theta}_{1}={n}_{2}\mathrm{cos}{\theta}_{2}=\mathrm{...}={n}_{i}$ based on the Snell’s law and hence ${\theta}_{k}=\mathrm{arc}\mathrm{cos}\left(\frac{{n}_{i}}{{n}_{1}}\right)$ (k=1,2,…,i-1). The focus length ${L}_{f}=l{f}_{1}+l{f}_{2}+\mathrm{...}+l{f}_{i-2}+l{f}_{i-1}$ and correspondingly $l{f}_{1}=\frac{h}{tg({\theta}_{1})}=\frac{h}{\sqrt{{\left(\frac{{n}_{1}}{{n}_{i}}\right)}^{2}-1}}$, $l{f}_{2}=\frac{h}{tg({\theta}_{2})}=\frac{h}{\sqrt{{\left(\frac{{n}_{2}}{{n}_{i}}\right)}^{2}-1}}$, $l{f}_{i-2}=\frac{h}{tg({\theta}_{i-2})}=\frac{h}{\sqrt{{\left(\frac{{n}_{i-2}}{{n}_{i}}\right)}^{2}-1}}$ and $l{f}_{i-1}=\frac{h}{tg({\theta}_{i-1})}=\frac{h}{\sqrt{{\left(\frac{{n}_{i-1}}{{n}_{i}}\right)}^{2}-1}}$. Therefore, the refractive index

_{i}*n*for the

_{i}*i*-th layer can be calculated with the equation below:

Using this simple algorithm, the refractive index profile of the GRIN lens can be generated layer by layer provided that *D*, ${L}_{f}$ and ${n}_{1}$ are given.

As a numerical example of the above algorithm for generating the ideal refractive index profile, we let the parameter *D* giving the GRIN lens’ height to be 10 μm and the desired focal length of the GRIN lens to be *L _{f}*=10 μm. The layer numbers of the GRIN lens is chosen to be N=1000 (which is verified to be sufficiently large to converge to a continuous curve). After solving Eq. (1) for each layer, the refractive index profile obtained for the asymmetric GRIN lens is presented in Fig. 6
, for which the refractive index variation is from 3.5 to 1.458. As a comparison, the refractive index distribution based on a parabolic profile with the same refractive index variation range is also plotted in Fig. 6, from which it can be seen that the refractive index profile generated differs distinctly from the conventional parabolic profile.

Light propagation within this aberration-free GRIN lens designed is simulated using ray tracing and 2D FDTD respectively. The ray-tracings in the aberration-free GRIN lens are shown in Fig. 7(a) . Compared to the ray trajectories shown in Fig. 3(a) for the parabolic GRIN lens, it can be seen that all the rays are well focused in this new graded refractive index profile. The electromagnetic simulation using 2D FDTD is presented in Fig. 7(b), which shows that the input beam is expanded and focused as desired and there is no interference as seen in the case of parabolic GRIN lens as shown in Fig. 3(b).

With this aberration-free super-GRIN lens, the coupling between a nanophotonic waveguide and optical fiber can be optimized by choosing appropriate structural parameters (height *D*, length L and fiber position *H*). In the design, the coupling efficiency to the SMF28 optical fiber is estimated using the overlap integral between the eigenmode of SMF28 and the expanded field through a 2D wide-angle beam propagation method (WA-BPM) [12], which is fast but has a good agreement with the 2D FDTD simulation. This is an estimation as it is a 2D calculation by assuming that the main loss in efficiency is in the vertical mode matching with the optical fiber (i.e. the horizontal mode mismatch, while not zero, is assumed to be smaller than the vertical mode mismatched). As WA-BPM is fast, it enables us to quickly optimize the mode coupling by vertically translating the fiber position and varying the designed parameters D and L_{f} until the computed coupling efficiency is the highest. The WA-BPM method, while good for use in parameter space optimization, however, cannot account for reflection losses, which can be accounted for by subsequently re-calculated with the FDTD method. For this numerical example, it is found that the maximal coupling efficiency is achieved when using the refractive index profile generated by D=15.6 μm and L_{f}=16 μm. It has a non-parabolic refractive index profile ranging from 3.5 to 1.49 with a corresponding numerical aperture of NA=3.167. The length of GRIN lens is L=16.85 μm, and the vertical position of the center of the fiber core is 4.6 μm from the bottom of the GRIN lens. With this optimized parameters the corresponding maximal coupling efficiency is 96.4% according to the WA-BPM simulation.

To verify the optimization result, numerical simulation of light propagation in the GRIN lens is carried out using the 2D FDTD method and the maximal coupling efficiency is found to be 94.80% when the length of GRIN lens L=16.84 μm, which shows that the two simulation methods have a good agreement. The corresponding transverse intensity profile at the facet of the GRIN lens simulated by FDTD is plotted in Fig. 8 , from which we can see that the light beam from nanophotonic waveguide can be effectively expanded to match that of the optical fiber. No severe ripples are observed as in the case of the parabolic GRIN lens. The slight intensity distortion is due to the asymmetric nature of the GRIN lens’ refractive index profile.

## 4. Practical implementation of the super-GRIN lens using a thin-film stack, effect of layer thickness, and wavelength sensitivity

Following Ref [8]. we use a thin-film stack to effectively realize the graded refractive index profile. Two materials are deposited alternatively to form a thin film stack acting as a GRIN lens, referred to as the dual-material method. One material has a high refractive index *n _{H}* such as silicon and the other material has a low refractive index

*n*such as silicon dioxide. The thickness of each layer is ranged from a few nanometers to tens of nanometers depending on the desired equivalent refractive index we want to achieve.

_{L}The actual thin-film layer design under the dual material method will be illustrated below. For the illustration, we will use the above super-focusing GRIN lens designed in Section 3 as the numerical example. The continuous graded refractive index profile *n(y)* generated in Section 3 for the aberration-free super-GRIN lens is shown in Fig. 9(a)
as a function of the vertical coordinate “*y*”, where y=0 is at the bottom of the super-GRIN lens. The super-GRIN lens has a super-high numerical aperture with a refractive index variation *n(y)* ranging from 3.5 to 1.49. The method to transform this continuous GRIN profile *n(y)* into a binary refractive index profile under the dual-material method is consisted of two steps. Step one: divide the GRIN lens into *M* discretized layer and the thickness of the *j ^{th}* discretized-layer is denoted as

*f*. Within each discretized layer the refractive index is regarded as constant n

_{j}_{j}(j = 1,2,3…

*M*) as shown in Fig. 9(a). Step two [see Fig. 9(b)]: under the dual-material method, each discretized layer is replaced with an effective material with total thickness

*f*and an effective refractive index

_{j}*n*

_{j}_{.}This effective material is made up of alternation of two thin-film layers with respective refractive indices ${n}_{H}$ and ${n}_{L}$ (${n}_{H}>{n}_{j}>{n}_{L}$) and thicknesses ${f}_{Hj}$ and ${f}_{Lj}$, respectively. The total thickness of each dual material pair is

*f*(${f}_{Hj}+{f}_{Lj}={f}_{Tj}$). The thicknesses ${f}_{Hj}$ and ${f}_{Lj}$ are determined based on averaging of dielectric permittivity

_{Tj}In general *f _{j}* can be different for different discretized layer

*j*and

*f*is chosen so that

_{Tj}*(f*, where

_{j}/f_{Tj})=C_{j}*C*is an integer describing the number of dual material pair within the j

_{j}^{th}discretized layer. There are thus many ways to implement the dual-material design for a given GRIN profile

*n(y).*To further simplify the design algorithm, in an algorithm we adopt, we choose

*C*=1 so each discretized layer is approximated only by one dual material pair and we further choose

_{j}*f*to be the same value for each of the discretized layer. Thus if we have

_{j}=f*M*discretized layer then

*f=D/M*. The thin-film stack designed has a total layer number of

*N*as each discretized layer is replaced by two thin-film layers.

_{f}=2MThe present design determines the thickness of two materials within each layer through averaging the dielectric permittivity. As compared to the previous approach in Ref [8], in which it is determined through averaging the refractive index. Our numerical example below indicates that with the present approach, the thin-film stack designed has a better performance in terms of coupling efficiency between optical fiber and nanophotonic waveguide.

In this paper, we choose silicon as the high refractive index material and silicon dioxide as the low refractive index material, which is due to the high refractive index contrast required by the GRIN lens designed above and also the consideration of compatibility with the CMOS fabrication in the electronic integration. For the above aberration-free GRIN lens example, the layout for the silicon and silicon dioxide thin-film stack is calculated using Eq. (2) under different layer thickness *f* (different thin-film layer number N_{f}). Light propagation within the thin-film stack based GRIN lens is simulated using 2D FDTD. Corresponding light propagation patterns are shown in Fig. 10
from (a) to (d) When *f=300 nm* (*N _{f}=104*),

*f=250 nm (N*). These numerical results indicate that the two-material thin-film stack act as a GRIN lens and expands the light beam with a sub-wavelength spotsize from nanophotonic waveguide as expected.

_{f}=124), 200 nm (N_{f}=156), 150 nm (N_{f}=208For these four cases, the corresponding coupling efficiency calculated from the propagation simulation is presented in Fig. 11
with blue circle marks. The thin-film stack designed by averaging of refractive index proposed in Ref. [8]. is also evaluated and the corresponding coupling efficiency is given in Fig. 11 with red squares for comparison. These numerical results indicate that the thin-film stack based GRIN lens designed by the present approach results in a higher coupling efficiency. The coupling efficiency between the fiber and nano-waveguide using thin-film stack based GRIN lens shown in Fig. 11 indicates that when *f* is around 150 nm, the coupling efficiency reaches the maximal value of ~92%. When increasing the *f* value to be over 250 nm, i.e., using a thin-film stack with a thicker layer but fewer layer number N_{f}, the coupling efficiency decreases because of the scattering loss. According to these simulation results, the thin-film stack with a coupling efficiency >90% requires 100~200 layer, which is compatible with current optical thin-film (such as DWDM device) coating technology. To further increase the layer number of the thin-film stack will result in less scattering loss, but will increase the fabrication challenge. Hence, there is an optimal point due to practical consideration. Based on this layout design and performance evaluation, we can choose the optimal layout for this example i.e., f=150 nm and the corresponding binary refractive index profile is presented in Fig. 12
, with which the coupling between the nanophotonic waveguide and optical fiber is ~92%.

Coupling efficiency between optical fiber and nanophotonic waveguide through this thin-film stack based GRIN lens is calculated under different wavelength using 2D FDTD. The corresponding wavelength dependence of the coupling efficiency for the thin-film stack is presented in Fig. 13 , which shows that it is almost wavelength insensitive (<0.5%) over this wide wavelength range from 1500 nm to 1600 nm. This wavelength-independence enables it to be applied to photonic subsystem on chip employing wavelength-division-multiplexing with a broad optical bandwidth.

## 5 Conclusion

We have proposed and designed the nanophotonic chip coupling using an optical thin-film stack forming a micro graded-refractive-index lens with a super-high numerical aperture that is highly compact and can be directly integrated. The thin-film stack based GRIN lens integrated on the surface of nanophotonic chip has an asymmetric GRIN profile.

We have shown that to achieve high efficiency for optical coupling from an optical fiber into a nanophotonic waveguide with a sub-wavelength beam size, a super-high numerical aperture is needed and conventional asymmetric parabolic GRIN profile is no longer adequate due to the severe spatial beam aberration. An efficient algorithm has been developed to computationally generate the ideal GRIN profile that is completely aberration free even at super-high NA. With the aberration-free GRIN lens the coupling efficiency between optical fiber and nanophotonic waveguide can be improved from ~66% (parabolic case) to ~95%.

A design example involving an optical thin-film stack has been presented using an improved dual-material approach. The optical coupling between fiber and nano-waveguide using thin-film stack based GRIN lens has been evaluated and the numerical example has shown a coupling efficiency of ~92%. The thin film stack based GRIN lens is also shown to be wavelength insensitive in a broad wavelength range, which enables it to be applied to WDM system on chip.

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