Adjoint optimization of polarization -splitting grating couplers PENG SUN1 THOMAS VAN VAERENBERGH 2 SEAN HOOTEN 1 AND

2025-04-24 0 0 1.73MB 15 页 10玖币

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Adjoint optimization of polarization-splitting

grating couplers

PENG SUN,1, †,* THOMAS VAN VAERENBERGH,2 SEAN HOOTEN,1 AND

RAYMOND BEAUSOLEIL1

1Hewlett Packard Labs, 820 N McCarthy Blvd, Milpitas, CA 95035, USA

2Hewlett Packard Labs, HPE Belgium, B-1831 Diegem, Belgium

†Currently at NVIDIA Corporation, 2788 San Tomas Expy, Santa Clara, CA 95051

*psun@outlook.com

Abstract: We have designed a polarization-splitting grating coupler (PSGC) in silicon-on-

insulator (SOI) that has 1.2 dB peak loss in numerical simulations, which is the best simulated

performance of PSGCs without a bottom reflector to the best of our knowledge. Adjoint

method-based shape optimization enables us to explore complex geometries that are intractable

with conventional design approaches. Physics-based process-independent knowledge of

PSGCs is extracted from the adjoint optimization and can be transferred to other platforms with

a minimum of effort.

1. Introduction

Polarization-splitting grating couplers (PSGCs) interface silicon photonic integrated circuits

with single-mode optical fibers that have random polarization states [1-4]. A PSGC consists

of a two-dimensional (2D) array of scattering elements that scatter light from an optical fiber

into two orthogonal waveguides, or vice versa. The scattering elements, which typically have

the quantity of ~800 to match the mode size of an SMF-28 fiber, are arranged on collinear

lattices or elliptical lattices to generate planar phase front in the scattered mode to match that

of the optical fiber. For collinear lattices, the optical fiber mode is converted to planar phase

front in the silicon slab, and then a pair of long waveguide tapers convert the slab mode (~10

um in width) to a pair of orthogonal single-mode silicon rib waveguides (~400 nm in width).

For elliptical lattices, the optical fiber mode is converted to cylindrical phase front in the silicon

slab that are focused at the two focal points of the elliptical lattices, where a pair of single-mode

silicon rib waveguides are placed.

Optimizations of PSGCs should include two objectives: to maximize coupling efficiency

and to minimize polarization dependent loss. First, the coupling efficiency can be improved by

tailoring the grating’s scattering strength to match the fiber mode profile, which is commonly

referred to as apodization. The ideal scattering strength of one-dimensional (1D) gratings that

generate gaussian mode has closed-form solution as shown in eqn. (4) in [5], which requires

weak scattering at the beginning of the grating, and strong scattering at the end. The layout of

a PSGC possesses mirror symmetry to split the incident light from the optical fiber into two

identical, orthogonal silicon rib waveguides, and therefore the apodization of the grating must

be described by a mirror-symmetric 2D profile. For PSGCs, intuitively the scattering strength

should have similar characteristics that the scattering is weaker at the edge of the grating area

and stronger at the center. The scattering strength can be modulated by the scatterers’ aspect

ratio [6], that is, to stretch a polygon along one silicon waveguide and compress it along the

orthogonal waveguide while preserving the area of the polygon such that the phase matching

condition is not perturbed. Higher aspect ratio generates weaker scattering for light propagation

along the stretching axis, and stronger scattering along the compressing axis. Second, the

optical fibers are often tilted/polished at a small angle to suppress reflection of the gratings and

reflection at the fiber-chip interface. As a consequence, breaking of the circular symmetry in

the fiber-grating mode overlap introduces disparity in the coupling efficiencies between the two

orthogonal polarization states. It has been shown that the polarization-dependent loss (PDL)

and polarization-dependent wavelength (PDW) can be effectively modulated by stretching the

scattering elements along the S- and P-polarization axes [7-9].

A thorough optimization of PSGCs will require designers to choose the aspect ratio and

the shape parameters for each and every scatterer, which is infeasible given the sheer number

of scatterers. The adjoint method enables exploration of high-dimensional parameter spaces

by calculating the gradient of a Figure-of-Merit (FOM) function of a linear system with fixed

and low computational cost, regardless of the number of parameters [10-13]. The highly

complicated PSGC problem is uniquely suited for adjoint method. In this work, we designed

PSGCs with adjoint method, and demonstrated peak coupling loss of 1.2 dB without a bottom

reflector. Section 2 details the parameterization, optimization, and validation of the PSGCs.

We also present our efforts to interpret and understand the physical insights behind the adjoint-

optimized PSGCs. Section 3 studies the dimensionality reduction and manufacturability of the

adjoint-optimized designs. Section 4 concludes the paper with discussions on future works.

2. Adjoint optimization of PSGCs

We study PSGCs based on a collinear lattice, and the technique developed in this work can

be generalized to PSGCs on an elliptical lattice. An example PSGC on a collinear lattice is

shown in Fig. 1(a). The PSGC is excited from a waveguide source on one of the two symmetric

arms. Scattered fields are captured by a monitor on top of the grating area and matched to a

tilted gaussian mode to compute the coupling efficiency. The coupling efficiency of a fiber-

excited PSGC is identical to that of a waveguide-excited PSGC of the same design, as dictated

by the reciprocity of the device. The tilted gaussian mode, which emulates the mode of an

angle-polished optical fiber, is centered at the intersection of the two orthogonal waveguide

arms. Projection of the fiber wavevector on the grating, as well as the S- and P-polarization

vectors, are schematically shown in Fig. 1(a). Figure 1(b) shows the example of a circular

scatterer under the transformation of aspect ratio AR=1.2, where all the points are scaled by a

factor of AR along X axis and 1/AR along Y axis.

Fig.1 (a) Top view of a PSGC on a collinear lattice. (b) Illustration of transforming a circle by aspect ratio.

We expect scatterers at different locations to have different aspect ratio. To optimize

apodization of the PSGC, we synthesize the 2D map of aspect ratio across the grating area using

generalized Fourier series. We choose 2D Chebyshev polynomials as the basis functions for

two reasons. First, as an intuitive guess inspired by the ideal 1D scattering strength, the

scattering strength of 2D gratings should increase monotonically from the starting edge to the

center, and therefore can be approximated by polynomials. Second, Chebyshev polynomials

yield the best polynomial approximation of all the polynomials with identical principal

coefficients [14]. Figure 2 shows the contour maps of 2D Chebyshev polynomials on a unit

square [-1,1]×[-1,1] up to the 2nd order, with mirror symmetry enforced on the diagonal axis.

The first two terms with p=0, q=0, and p=1, q=0 determine a linearly ramped AR profile along

the two waveguides, and intuitively they should play a more important role than other

coefficients.

Fig.2 2D Chebyshev polynomials up to the 2nd order, with mirror symmetry enforced on the diagonal axis.

For shape optimization of the scatterers, we synthesize each individual polygon with sine

and cosine basis functions in polar coordinates as shown in eqn. (1), where cn and sn are the

trigonometric coefficients and r0 is the DC component:

  



 (1)

Area of the polygon is given by eqn. (2):

  





 



 







 (2)

Fill Factor of the grating, which is defined as the ratio of the polygon area over the unit cell

area, is forced to be constant regardless of the polygon shape to generate uniform phase front.

The DC component r0 needs to be calculated for each polygon shape, as shown in eqn. (3):





 







 (3)

The first 6 orders of sine and cosine functions in polar coordinates, with coefficient of 40

nm on top of a 200-nm-radius circle for all the basis functions, are illustrated in Figure 3. Each

of the basis functions serves a specific purpose in modulating the light scattering. For example,

the 2nd order cosine function stretches the scattering elements along one of the two orthogonal

waveguides, and primarily increase or decrease the scattering intensity [6]. The 2nd order sine

and 4th order cosine stretch the polygon towards the four corners of the unit cell along the S-

and P-polarization in the optical fiber, which can modulate the grating’s PDW/PDL [7]. We

expect the 2nd order sine and 4th order cosine to play an important role in modulating the

PDL/PDW. The 4th order sine function stretches the polygon towards the four corners of the

unit cell but off by an angle of 22.5°, and therefore we do not expect it to be as important as the

4th order cosine function in modulating the PDL/PDW. We stop at the highest order of 6, as

higher order trigonometric basis functions correspond to fine features and sharp angles in the

scattering elements, which will have worse manufacturability and diminishing impact on the

optical performance as the order increases.

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Adjointoptimizationofpolarization-splittinggratingcouplersPENGSUN,1,†,*THOMASVANVAERENBERGH,2SEANHOOTEN,1ANDRAYMONDBEAUSOLEIL11HewlettPackardLabs,820NMcCarthyBlvd,Milpitas,CA95035,USA2HewlettPackardLabs,HPEBelgium,B-1831Diegem,Belgium†CurrentlyatNVIDIACorporation,2788SanTomasExpy,SantaClara,CA95051*...

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