Unusual Impulse-Momentum in Non-Reciprocal Optical Solitons

By Dr. Noopur JainReviewed by Louis CastelFeb 27 2026

A recent study published in Light: Science & Applications reports an unusual impulse-momentum relationship in a non-reciprocal optical solitary wave system. The research reveals that when an external impulse is applied to one component of a two-beam optical solitary wave, the resulting total momentum of the composite system can exceed, or even reverse, the expected response.

These findings challenge conventional intuition based on reciprocal interactions and offer new insight into momentum conservation in complex optical systems.

Image Credit: synthetick/Shutterstock.com

Understanding Non-Reciprocity in Physical Systems

Non-reciprocity is fundamental to systems where interactions between agents are asymmetric, a concept well recognized in predator-prey dynamics, neuroscience, and collective motion. Artificial materials exploit non-reciprocity to realize exotic phenomena such as the non-Hermitian skin effect and non-reciprocal quantum phase transitions.

In optics, non-reciprocal propagation stems from mechanisms like magneto-optical effects, time-modulated materials, nonlinearities, or topological features, but the exploration of non-reciprocal nonlinear wave interactions remains scarce. Recent work established an optical platform with stroboscopic nonlinearity causing asymmetric, attraction-repulsion type interactions between two optical beams, analogous to predator-prey chase-and-run dynamics. This system opens the door to studying how non-reciprocal interactions affect fundamental relationships such as impulse and momentum.

Modeling a Two-Beam Non-Reciprocal Optical System

The optical system studied consists of two coupled paraxial wave equations (nonlinear Schrödinger-type under the paraxial approximation) describing two optical beams (labeled A and B) propagating along the longitudinal axis (z) with transverse coordinate (x).

The beams experience competing nonlinearities: beam A undergoes self-focusing nonlinearity, while beam B experiences self-defocusing. The nonlinear refractive index change is modeled as:

 Δn = γ |ψ_A|²/(1 + |ψ_A|²) - γ |ψ_B|²/(1 + |ψ_B|²),

where γ is the nonlinear coefficient. This form results from the stroboscopic nonlinear response of the medium, causing beam A to attract beam B via a waveguide effect, while beam B repels beam A acting as an anti-waveguide, thus producing strong non-reciprocal internal interactions.

Stationary solitary wave solutions are found by assuming the beams share a common nonlinear waveguide mode, yielding shape-invariant profiles with propagation constants β_A and β_B. External impulses are implemented as small angular tilts on either beam, representing transverse momentum kicks.

Both numerical simulations based on the coupled equations and experiments using a strontium barium niobate (SBN) crystal with an AC electric field bias are conducted. The two beams are temporally separated to experience self-focusing and defocusing nonlinearities in alternating intervals, interacting through the crystal’s memory effect without spatial overlap, forming optical solitary waves. Beam tilts are used to apply impulses selectively to either component, and the resultant shifts and momentum changes of the solitary wave are measured and analyzed.

When Momentum Defies Intuition

The theoretical analysis predicts and numerical simulations confirm that when an impulse is applied to beam A (self-focusing), the solitary wave acquires a momentum larger than the impulse itself, deviating from the classical expectation that momentum gain equals the applied impulse. Mathematically, the momentum change Δk_x relates to the impulse J by a coefficient exceeding one.

Conversely, when the impulse acts on beam B (self-defocusing), the solitary wave moves counter to the direction of the impulse, implying a negative coefficient in the impulse-momentum relationship. Both cases show a linear dependency between the solitary wave momentum change and the applied impulse as long as the wave remains intact. Analytical expressions derived from the model quantify this behavior in terms of the spatial widths of the beams, tied to the non-reciprocal interactions.

Experimentally, the phenomenon is realized in the SBN crystal by synchronizing two stripe-shaped beams with a blinking positive and negative voltage bias, yielding self-focusing and defocusing nonlinearities, respectively. The solitary wave formed exhibits nearly invariant propagation profiles over several millimeters. Applying a small tilt to beam A results in a leftward shift of the combined solitary wave that exceeds the classical impulse expectation by a factor of approximately 1.61. Applying the same tilt to beam B causes the solitary wave to move rightward, opposite to the applied impulse, with a proportionality coefficient near −0.59. These experimental results closely align with simulations, validating the model’s predictions and confirming that the unusual impulse-momentum relationship stems from the asymmetric nonlinear coupling. Slight deviations between experiment and theory are attributed to diffusion effects in the real crystal.

Conclusion

This work uncovers unconventional impulse-momentum relations for optical solitary waves composed of two nonlinear, non-reciprocally interacting components. The momentum change induced by an external impulse can significantly exceed or even invert relative to the impulse depending on which component is affected. The effect arises from the asymmetric internal waveguide and anti-waveguide forces between the self-focusing and self-defocusing beams mediated by stroboscopic nonlinearities. Experimental implementation in an SBN crystal confirms the theoretical predictions, prompting deeper fundamental consideration of momentum exchange in non-reciprocal optical systems. These results open up avenues for fundamental research into non-reciprocal light interactions and may inspire novel non-Hermitian photonic device concepts exploiting non-Hermitian physics for enhanced control over wave momentum and propagation.