The Thin Grating Regime
When an acoustic wave creates a refractive index modulation in a thin crystal slab, the optical path length varies sinusoidally across the beam aperture. Unlike the Bragg regime where only one diffracted order satisfies phase matching, the thin grating imposes no such selectivity — all diffraction orders are generated simultaneously. Raman and Nath first described this phenomenon in 1935, providing the theoretical framework that explains the multi-order pattern as a natural consequence of sinusoidal phase modulation.
Bessel Functions and Intensity Distribution
The mathematical elegance of Raman-Nath diffraction lies in its connection to Bessel functions. A pure sinusoidal phase grating with modulation depth Δφ produces diffraction orders whose amplitudes are exactly the Bessel functions J_m(Δφ). The intensities are therefore J_m²(Δφ). As the modulation depth increases, energy redistributes from the zeroth order into higher orders in a characteristic oscillatory pattern — the zeroth order first goes to zero at Δφ ≈ 2.40 (the first zero of J₀).
Frequency Shifting
Each diffraction order experiences a frequency shift equal to m times the acoustic frequency: the m-th order light has frequency ν_opt + m × f_acoustic. This occurs because diffraction from a moving grating imparts a Doppler shift. The positive orders are up-shifted and the negative orders are down-shifted in frequency. This property makes Raman-Nath devices useful as multi-frequency optical sources and for heterodyne detection schemes.
Limitations and the Transition Regime
The Bessel function model is strictly valid only when the Q parameter is much less than 1 — meaning the grating is optically thin enough that light does not significantly diffract while still inside the crystal. As Q increases toward 1, the higher orders begin to weaken relative to Bessel function predictions, and asymmetries appear between positive and negative orders. By Q ≈ 4π, only the first order remains significant, marking the transition to Bragg behavior.