Nonlinearity can significantly influence topologically protected photonic states in various ways. These states, typically realized in photonic crystals, waveguide arrays, or metamaterials, are protected against backscattering and disorder due to their underlying topological invariants (like Chern numbers or Zak phase). When nonlinear optical effects (e.g., Kerr nonlinearity, saturable nonlinearity, χ² processes) are introduced, the situation becomes richer and more complex.

Here’s a structured explanation:

1. Background: Topologically Protected Light States

• These states exist at edges or interfaces of topological photonic systems.
• They rely on linear band topology for their robustness, meaning their behavior is predictable under linear wave dynamics.
• Examples: Edge states in photonic topological insulators, valley-Hall edge modes, Floquet topological phases.

2. Role of Nonlinearity

When nonlinear effects enter the system, the superposition principle no longer holds, and the governing equations (e.g., the nonlinear Schrödinger equation for photonic lattices) change the nature of the modes.

a) Self-Induced Topology

• Nonlinearity can modify the effective refractive index locally, leading to a self-induced change in topology.
• Example: In a Kerr medium, higher intensity shifts local bands → this can open/close topological gaps dynamically.

b) Nonlinear Edge Solitons

• Topologically protected edge modes can transform into soliton-like states due to the balance between diffraction and nonlinearity.
• These nonlinear edge solitons are robust but no longer purely topological; their stability comes from nonlinearity plus topology.

c) Nonlinear Frequency Shifts

• Nonlinear interaction (like χ³ Kerr effect) can cause intensity-dependent frequency shift of edge states.
• This can move the state in or out of the bandgap, altering protection.
• At high power, states may delocalize or couple to bulk modes, reducing topological robustness.

d) New Topological Phases

• Nonlinearity can generate new topological invariants dynamically.
• Example: In nonlinear Floquet systems, periodic modulation combined with Kerr nonlinearity can create exotic phases not accessible in linear systems.

e) Instabilities and Bistability

• Nonlinearity often introduces bistability (two stable intensity states for the same input) or modulational instability, which can destabilize topological protection.
• However, nonlinear symmetry breaking can also create new edge-localized modes.

3. Physical Consequences

• Light Routing: Nonlinearity allows intensity-dependent control of edge transport (optical diodes, isolators).
• All-Optical Switching: Use topological edge solitons for switching between channels without backscattering.
• Topological Lasers: Nonlinear gain saturation plays a key role in stability.

4. Open Questions in Research

• Can strong nonlinearity destroy topological protection?
• How to define topological invariants in nonlinear systems?
• What happens in quantum nonlinear regimes (photon-photon interaction)?