# Iterative approach to arbitrary nonlinear optical response functions of graphene

## TL;DR

This paper introduces a simple iterative method to derive arbitrary nonlinear optical response functions in graphene, enabling analytical calculations up to ninth order and accommodating various band gaps and gating potentials.

## Contribution

It presents a novel iterative approach for calculating nonlinear optical responses in graphene, extending to high orders and different electronic configurations.

## Key findings

- Analytical expressions for harmonic generation up to ninth order.
- Demonstrated gate-tunable third- and fifth-harmonic generation.
- Explored intensity-dependent refractive index and high-harmonic generation.

## Abstract

Two-dimensional materials constitute an exciting platform for nonlinear optics with large nonlinearities that are tunable by gating. Hence, gate-tunable harmonic generation and intensity-dependent refraction have been observed in e.g. graphene and transition-metal dichalcogenides, whose electronic structures are accurately modelled by the (massive) Dirac equation. We exploit on the simplicity of this model and demonstrate here that arbitrary nonlinear response functions follow from a simple iterative approach. The power of this approach is illustrated by analytical expressions for harmonic generation and intensity-dependent refraction, both computed up to ninth order in the pump field. Moreover, the results allow for arbitrary band gaps and gating potentials. As illustrative applications, we consider (i) gate-dependence of third- and fifth-harmonic generation in gapped and gapless graphene, (ii) intensity-dependent refractive index of graphene up to ninth order, and (iii) intensity-dependence of high-harmonic generation.