In this section, we have a closer look at the recirculating modulator operation and what considerations need to be taken into account to achieve an increased modulation efficiency. When a sinusoidal signal V(t)=A⁡sinωst is applied to a single pass phase modulator, the output optical field from the device can be expressed asEout=EoeiωcteiR⁡sinωst=Eoeiωct∑n=−∞n=+∞JnReinωst,(1)where Eo and ωc are the amplitude and angular frequency of the optical carrier, ωs is the angular frequency of the radio frequency (RF) modulating signal, R is the modulation index, which is equal to πA/Vπ, where A is the amplitude of microwave signal and Vπ is the voltage required to shift the phase of the carrier by π, and Jn is the Bessel function of the first kind. In the case of a recirculating modulator, as introduced in Fig. 1(a), the light is modulated multiple times, which for N recirculations can be expressed asEout=EoeiωcteiR0⁡sinωsteiR1⁡sinωst−ΔT1×eiR2⁡sinωst−ΔT2…eiRN⁡sinωst−ΔTN,(2)where Rj (j = 0, 1, 2, …, N) is the modulation index of the jth recirculation and ΔTj (j = 1, 2, …, N) is the loop transit time for light to re-enter the modulator, which is effectively the time delay between two different modes be modulated by the RF electrode. The loop transit time is given by ΔTj = ΔLj/vg, where ΔLj is the length of loop j and vg is the group velocity of the fundamental mode TE0 in the loop waveguides. If one assumes that the modulation index is the same for all modes, Eq. (2) can be written asEout=EoeiωcteiR*κ*sinωst−φ,(3)where φ=arc⁡tansinωsΔT1+sinωsΔT2+⋯+sinωsΔTn1+cosωsΔT1+cosωsΔT2+⋯+cosωsΔTn and the κ is the enhancement factor of the recirculating modulator. In the case of Nth recirculation, the enhancement factor can be expressed asκ=1+cosωsΔT1+cosωsΔT2+⋯+cosωsΔTN2+sinωsΔT1+sinωsΔT2+⋯+sinωsΔTN2.(4)Then, expanding Eq. (3) by Bessel functions of the first kind, the output signal can be described asEout=Eoeiωct*∑n=−∞n=+∞JnR*κ*ein(ωst−φ).(5)As one can see from Eqs. (4) and (5), the output signal is strongly dependent on the enhancement factor, which is determined by modulation frequency ωs and the delay time ∆Tj for each recirculating loop. Maximum enhancement factor can be achieved when ΔTj = 2mπ/ωs, where m is an integer. For N recirculations, the maximum enhancement factor is equal to N + 1.