The Bragg Frequency Convertor: A Meeting Between Spatial and Temporal Periodicities For Selective Parametric Frequency Translation

Abstract

This study introduces the Bragg Frequency Convertor, a spatial-temporal-periodic grating that extends the concept of conventional Bragg gratings into the dynamic domain to achieve pure parametric frequency conversion. By time-modulating either the high-index or low-index layers of a quarter-wave Bragg grating, we demonstrate selective and directional frequency conversion: Modulation of the high-index layers selectively yields down-conversion, whereas modulation of the low-index layers leads to up-conversion. Such a pure frequency conversion emerges from the synergistic interplay of spatial and temporal periodicities. The output is thus dominated by the converted frequency, with the carrier and undesired time harmonics suppressed. We derive a coupled-mode theory explaining such a layer-dependent phase matching and validate it with full-wave simulations, showing tunable conversion efficiency via modulation phase. The advance presented here lies in recognizing the conventional Bragg grating not as a passive partner but as an enabling scaffold: its intrinsic spatial periodicity is exploited to impose the precise spectral constraints required for efficient temporal modulation and the generation of pure parametrically converted signals. This work establishes temporal Bragg gratings as a versatile platform for reconfigurable and spurious-free frequency conversion, with applications in optical systems, signal processing, spectral engineering, and quantum photonics.

Figures (7)

• Figure 1: Schematic of the Bragg Frequency Convertor. (a) Up-conversion configuration: low-index layers are time-modulated. The device consists of $N$ periods of alternating high-index and low-index layers, each satisfying the quarter-wave condition at the design frequency. The low-index layers are modulated targeting up-conversion to $f_{+1} = f_0 + f_m$, while the high-index layers remain static. An incident wave at $f_0$ enters from the left. The static Bragg grating formed by the periodic index contrast creates a stopband that strongly reflects the carrier. Simultaneously, the modulated low-index layers generate sidebands through parametric interaction. Due to the Bloch mode structure of the quarter-wave stack, modulating the low-index layers preferentially couples to the upper band edge, enhancing up-conversion to $f_{+1}$. This favored sideband lies outside the stopband and propagates freely through the structure, emerging in both transmission and reflection. (b) Down-conversion configuration: high-index layers are time-modulated while the low-index layers remain static. This configuration preferentially couples to the lower band edge, enhancing down-conversion to $f_{-1}$. The down-converted sideband lies outside the stopband and propagates freely, while the carrier $f_0$ and other time harmonics suppressed by the Bragg stopband.
• Figure 2: Space-time diagram of a temporal Bragg grating (up-conversion mode). The diagram tracks wave propagation through multiple spatial periods (horizontal axis $z$) and multiple modulation periods (vertical axis $ct$). Low-index layers are time-modulated (striped regions) and high-index layers are static (solid). Layer thicknesses $d_\text{L}$ and $d_\text{H}$ with corresponding transit times $\tau_\text{L} = n_\text{L} d_\text{L}/c$ and $\tau_\text{H} = n_\text{H} d_\text{H}/c$. Spatial period $\Lambda = d_\text{L} + d_\text{H}$. Modulation period $T_\text{m} = 2\pi/\omega_\text{m}$ indicated by vertical spacing $cT_\text{m}$. Temporal scattering events recur with period $T_\text{m}$ within each modulated layer.
• Figure 3: Multiple-scattering diagram of a temporal Bragg grating operating in the up-conversion regime, where the low-index layers are time-modulated. An incident carrier at frequency ($f_0$) undergoes distributed frequency conversion within the modulated low-index regions, generating the up-converted sideband ($f_{+1}$). Constructive interference of the converted components across successive periods leads to the dominance of the ($f_{+1}$) harmonic at both output ports, while the original carrier is suppressed. The diagram illustrates the principal reflection and transmission pathways responsible for the distributed up-conversion process.
• Figure 4: Multiple-scattering diagram of a temporal Bragg grating operating in the down-conversion regime, where the high-index layers are time-modulated. An incident wave at the carrier frequency ($f_0$) undergoes distributed frequency conversion within the modulated high-index regions, generating the down-converted sideband ($f_{-1}$). Constructive interference of the converted components across successive periods leads to the dominance of the ($f_{-1}$) harmonic at both output ports, while the original carrier is suppressed. The figure illustrates the principal reflection and transmission pathways responsible for the distributed frequency conversion process. The dominant sideband in a quarter-wave temporal Bragg grating is determined by the relative spatial phase between the temporal modulation profile and the carrier standing-wave distribution. Modulating alternating layers shifts the effective mixing profile by half a period, reversing the sign of the distributed phase-matched coupling coefficient and thereby switching the dominant conversion process between up- and down-conversion.
• Figure 5: Transmittance as a function of frequency for the passive (unmodulated) Bragg grating consisting of $N = 8$ periods of alternating high-index ($n_H = 2.5$) and low-index ($n_L = 1.5$) layers, each satisfying the quarter-wave condition at the design frequency $f_0 = 150$ THz.