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We present a formalism to calculate the nonlinear terahertz response of monolayer graphene using a two-band tight-binding model of graphene. We develop the dynamic equations for the electron density matrix in the length gauge to calculate the interband and intraband carrier dynamics at terahertz frequencies. Using the calculated nonlinear interband and intraband current densities, we obtain the nonlinear transmitted and reflected terahertz fields. We find that when the conditions are such that the interband and intraband current densities are comparable, strong generation of odd harmonics is observed. We examine the response of undoped graphene as a function of a wide variety of system parameters, including operating temperature, Fermi velocity, pulse duration, and terahertz central frequency. We find that the response is generally most sensitive to the operating temperature and the terahertz central frequency. In particular, changing the ambient temperature from 10 to 100 K reduces the third harmonic field by a factor of 20, while increasing the terahertz central frequency from 0.5 to 5 THz results in an increase in the ratio of the generated third harmonic field to the fundamental of the reflected field from 19% to 49%. The very strong dependence on the temperature and central frequency is found to be due to the strong dependence of the nonlinear generation on the strength of the interplay between the intraband and interband dynamics. Our work demonstrates that, under the right conditions of low temperature and moderate THz field central frequency, third harmonic generation should be experimentally observable in undoped monolayer graphene at moderate THz field amplitudes of only 1 kV cm−1.

In this paper, a left-handed metamaterial coplanar-waveguide bandpass filter with a compact electrical size is proposed. The filter features a strongly improved out-of-band rejection ratio compared with conventional cascaded split ring resonator (SRR) structures while all dimensions remain well below a third of the guided design wavelength. Especially, the problem of low upper stopband attenuation is overcome. This is achieved by combining an SRR with a complementary SRR cell. We also demonstrate that this approach enables flexible filter designs for a broad range of bandwidth ratios with small insertion losses at the centre frequency of the device.