Fast light in unbalanced low-loss Mach-Zehnder interferometers

An analytical approach is reported that describes previously observed fast-light regimes in linear and passive Mach-Zehnder interferometers (MZI) where the optical path difference is due to a different length of the branches. Approximate expressions are developed for the transmission coefficient and group delay spectral functions valid for frequencies close to the transmission minima ${\omega }_{\min }$, where these regimes occur. It is found that the group delay at ${\omega }_{\min }$ verifies a simple scaling law. We demonstrate that slow light cannot arise in this system, and that tunneling and superluminal regimes appear only for low-loss devices, where the attenuation drives the change in the propagation regimes. The propagation of a sinusoidally modulated pulse train through the MZI is described, and relevant figures of merit, which are intrinsic to the system and universal for any operative spectral range, are determined. The theoretical approach is illustrated by simulations of a silicon-based interferometer designed for advancing pulses at 1.55 μm. Also, previously reported experimental results in the radiofrequency range are interpreted in the framework of the model.

Figure 1

(a) Schematic of an unbalanced MZI and (b) its typical transmission spectrum with (dashed line) and without (solid line) attenuation.

Figure 2

(a) Transmission coefficient magnitude and (b) group delay in units of phase delay through an Si-based MZI with length difference between arms $\Delta =L/10$ and refractive index $n=3.48$ for two values of the attenuation coefficient $\alpha$.

Figure 3

Pulse propagation regimes at the transmission minima of an Si-Based MZI with $\alpha \Delta =0.015$ as a function of the total attenuation $\alpha L$. Group delay plotted from Eq. (14a) (solid line) and phase delay through a vacuum (dashed line). The delays are given in units of the phase delay over the system's effective length.

Figure 4

(a) Fractional advancement (solid line) and pulse compression (dashed line) versus normalized modulation frequency, and (b) pulse compression (dashed line) and secondary to main peak amplitude ratio (short-dashed line) as a function of the fractional advancement, for a 100% sinusoidally modulated pulse train with carrier frequency tuned at one of the transmission minima.

Figure 5

Numerical simulation of an Si-based MZI with parameters Δ = 200 μm, α = 6.5 dB/cm, and $n$ = 3.48 for three values of the effective length: $L$ = 2 mm (solid line), $L$ = 1.5 cm (dashed line), and $L$ = 2 cm (dotted line). (a) Magnitude of the transmission coefficient and group delay. (b) Normalized traces of pulses with carrier frequency tuned at $f_{\min }$ and transmitted through each interferometer. The incident pulse has its peak at $t$ = 0.

Figure 6

Normalized traces of pulses transmitted through the interferometer of $L$ = 2 mm for two carrier frequencies: tuned at $f_{\min }$ (solid line) and tuned at $f_{\max }$ (dashed line). The incident pulse has its peak at $t$ = 0.

Figure 7

Experimental and model results of an RF MZI with the indicated parameters around a transmission minimum. (a) Magnitude and phase of the transmission coefficient and (b) group delay.

Figure 8

Experimental pulse power traces and their fitted envelopes for two different carrier frequencies: (a) 58.3 MHz (tunneling) and (b) 50 MHz (normal regime). Each trace is normalized to its maximum amplitude. The fitted envelopes have modulation index (a) $M$ = 0.95 and (b) $M$ = 0.65.