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Inverse-designed photonic fibers and metasurfaces

for nonlinear frequency conversion [Invited]

CHAWIN SITAWARIN,

WEILIANG JIN,

ZIN LIN,

AND ALEJANDRO W. RODRIGUEZ

Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA

John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA

*Corresponding author: arod@princeton.edu

Received 22 November 2017; revised 19 March 2018; accepted 20 March 2018; posted 21 March 2018 (Doc. ID 313902);

published 20 April 2018

Typically, photonic waveguides designed for nonlinear frequency conversion rely on intuitive and established

principles, including index guiding and bandgap engineering, and are based on simple shapes with high degrees

of symmetry. We show that recently developed inverse-design techniques can be applied to discover new kinds of

microstructured fibers and metasurfaces designed to achieve large nonlinear frequency-conversion efficiencies. As

a proof of principle, we demonstrate complex, wavel ength-scale chalcogenide glass fibers and gallium phosphide

three-dimensional metasurfaces exhibiting some of the largest nonlinear conversion efficiencies predicted thus far,

e.g., lowering the power requirement for third-harmonic generation by 10

and enhancing second-harmonic

generation conversion efficiency by 10

. Such enhancements arise because, in addition to enabling a great degree

of tunability in the choice of design wavelengths, these optimization tools ensure both frequency- and

phase-matching in addition to large nonlinear overlap factors.

OCIS codes: (050.1755) Computational electromagnetic methods; (060.4370) Nonlinear optics, fibers; (190.2620) Harmonic

generation and mixing; (190.4360) Nonlinear optics, devices; (350.4238) Nanophotonics and photonic crystals.

https://doi.org/10.1364/PRJ.6.000B82

1. INTRODUCTION

Nonlinear frequency conversion plays a crucial role in many

photonic applications, including ultra-short pulse shaping

[1,2], spectroscopy [3], generation of novel optical states

[4–6], and quantum information processing [7–9]. Although

frequency conversion has been studied exhaustively in bulky

optical systems, including large ring resonators [10] and etalon

cavities [11], it remains largely unstudied in micro- and nano-

scale structures where light can be confined to lengthscales of

the order of or even smaller than its wavelength. By confining

light over long a time and to small volumes, such highly com-

pact devices greatly enhance light – matter interactions, enabling

similar as well as new [12] functionalities compared to those

available in bulky systems but at much lower power levels.

Several proposals have been put forward based on the premise

of observing enhanced nonlinear effects in structures capable of

supporting multiple resonances at far-away frequencies

[13–21], among which are micro-ring resonators [22,23]

and photonic crystal (PhC) cavities [24,25]. However, to date,

these conventional designs fall short of simultaneously meeting

the many design challenges associated with resonant frequency

conversion, chief among them being the need to support multi-

ple modes with highly concentrated fields, exactly matched

resonant frequencies, and strong mode overlaps [26]. Recently,

we proposed to leverage powerful, large-scale optimization

techniques (commonly known as inverse design) to allow

computer-aided photonic designs that can address all of these

challenges.

Our recently demonstrated optimization framework allows

automatic discovery of novel cavities that support tightly local-

ized modes at several desired wavelengths and exhibit large non-

linear mode overlaps. As a proof-of-concept, we proposed

doubly resonant structures, including multi-layered, aperiodic

micro-post cavities and multi-track ring resonators, capable of

realizing second-harmonic generation efficiencies exceeding

−1

[27,28]. In this paper, we extend and apply this op-

timization approach to design extended structures, including

micro-structured optical fibers and PhC three-dimensional

metasurfaces, as shown in Fig. 1, for achieving high-efficiency

(second- and third-harmonic) frequency conversion. Harmonic

generation, which underlies numerous applications in science,

including coherent light sources [29], optical imaging and

microscopy [30,31], and entangled-photon generation [32],

is now feasible at lower power requirements thanks to the avail-

ability of highly nonlinear χ

2

and χ

3

materials such as III–V

semiconductor compounds [33,34] and novel types of

chalcogenide glasses [35]. In combination with advances in

materials synthesis, emerging fabrication technologies have also

B82

Vol. 6, No. 5 / May 2018 / Photonics Research

Research Article

enabled demonstrations of sophisticated micro-structured

fibers [36] and metasurfaces [37–44], paving the way for

experimental realization of inverse-designed structures of

increased geometric and fabrication complexity, which offer

orders-of-magnitude enhancements in conversion efficiencies

and the potential for augmented functionalities.

Given a material system of intrinsic χ

2

or χ

3

nonlinear

coefficient, the efficiency of any given frequency-conversion

process in a resonant geometry will be determined by a few

modal parameters. The possibility of confining light within

small mode volumes over a long time or distance leads to sig-

nificant gains in efficiency (i.e., lower power requirements),

stemming from the higher intensity and cascadability of non-

linear interactions (compensating for the otherwise small bulk

nonlinearities). In particular, the efficiency of such resonant

processes depends on the product of mode lifetimes and a

nonlinear coefficient β, given by Eqs. (6) and (8) below, which

generalizes the familiar concept of quasi-phase-matching to sit-

uations that include wavelength-scale resonators [26]. For

propagating modes, leaky or guided, the existence of a propa-

gation phase further complicates this figure of merit, with

optimal designs requiring: (i) phase-matching and frequency-

matching conditions, (ii) large nonlinear mode overlaps β, and

(iii) large dimensionless lifetimes Q (low material absorption

and/or radiative losses in the case of leaky modes). The main

design challenge is the difficult task of forming a doubly res-

onant cavity with far-apart modes that simultaneously exhibit

long lifetimes and large β, along with phase and frequency

matching. To date, the majority of prior works on frequency

conversion in fibers [45–47 ] and metasurfaces [38–40,42,48–51]

have focused on only one of these aspects (usually phase match-

ing) while ignoring the others. The geometries discovered by our

optimization framework, in contrast, address the above criteria,

revealing complex fibers and metasurfaces supporting TE or

TM modes with guaranteed phase and frequency matching, long

lifetimes Q, and enhanced o verlap factors β at any desired

propagation wavevector, and resulting in orders-of-magnitude

enhancements in conversion efficiencies.

2. OVERVIEW OF OPTIMIZATION

The possibility of fine-tuning spatial features of photonic de-

vices to realize functionalities not currently achievable by con-

ventional optical design methodologies based on index guiding

and bandgap confinement (which work exceedingly well but

are otherwise limited for narrowband applications) has been

a major drive behind the past several decades of interest in

the topic of photonic opt imization [52,53]. Among these tech-

niques are probabilistic Monte Carlo algorithms, e.g., particle

swarms, simulated annealing, and genetic algorithms [54–56].

Though sufficient for the majority of narrowband (single-

mode) applications, many of these gradient-free methods are

limited to typically small sets of design parameters [57] that

often prove inadequate for handling wideband (multi-mode)

problems. On the other hand, gradient-based inverse-design

techniques are capable of efficiently exploring a much larger

design space by making use of analytical derivative information

of the specified objective and constraint functions [58], dem-

onstrated to be feasible for as many as 10

design variables [59].

Recently, the development of versatile mathematical program-

ming methods and the rapid growth in computational power

have enabled concurrent progress in photonic inverse

design, allowing theoretical (and more recently, experimental)

demonstrations of complex topologies and unintuitive geom-

etries with unprecedented functionalities that would be

arguably difficult to realize through conventional intuition

alone. However, to date, most applications of inverse design

in photonics are confined to linear devices such as mode con-

verters, waveguide bends, and beam splitters [57,58,60–65].

We believe that this paper along with our recent works

[27,28] provides a glimpse of the potential of photonic

optimization in nonlinear optics.

A typical optim ization problem seeks to maximize or

minimize an objective function f , subject to certain constraints

g, over a set of free variables or degrees of freedom (DOFs) [66].

Generally, one can classify photonic inverse design into two

different classes of optimization strategies, based primarily

on the nature or choice of DOF [67]. Given a computational

domain or grid, the choice of a finite-dimensional parameter

space not only determines the degree of complexity but also

the convergence and feasibility of the solutions. One possibility

is to exploit each DOF in the computational domain as an op-

timization parameter, known as topology optimization (TO),

in which case one typically (though not always) chooses the

dielectric permittivity of each pixel ϵr as a DOF (known

as a continuous relaxation parameter [68]). Another possib ility,

known as shape optimization, is to expand the optimization

parameter space in a finite set of shapes (independent of the

computational discretization), which may be freeform contours

represented by so-called level sets [69] (the level-set method) or

basic geometric entities with simpler parametrizations (e.g., pol-

ytopes) [70]. In the level-set method, the zeros of a level-set

“function” Φr define the boundaries of “binary shapes”;

the optimization then proceeds via a level-set partial differential

equation characterized by a velocity field, which is, in turn,

constructed from derivative information [69]. A much simpler

variant (which we follow) is to choose a fixed but sufficient

number of basic binary shapes whose parameters can be made

to evolve by an optimization algorithm. Essentially, for such a

parametrization, the mathematical representations of the shapes

must yield continuous (analytic) derivatives, which is not fea-

sible a priori due to the finite computational discretization and

Fig. 1. Schematic illustration of third-harmonic generation and

second-harmonic generation processes in inverse-designed microstruc-

tured fibers and metasurfaces, respectively.

Research Article

Vol. 6, No. 5 / May 2018 / Photonics Research B83

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