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The DC Kerr Effect in Nonlinear Optics

Nikolas Eptaminitakis, Plamen Stefanov

TL;DR

The paper develops a weakly nonlinear geometric-optics framework for Maxwell’s equations with a cubic Kerr nonlinearity to model the DC Kerr effect and polarization rotation. It proves the existence of exact solutions near carefully constructed asymptotic expansions that combine a strong stationary field with a propagating beam, and it analyzes how the Kerr nonlinearity induces a phase retardation that causes birefringence. A key contribution is an inverse-result showing that the polarization change along light rays yields the X-ray transform of the nonlinear susceptibility $\chi^{(3)}$, enabling recovery of $\chi^{(3)}$ from measurements. The approach relies on a second-order hyperbolic reduction, a structured multi-scale expansion, and Guès’ theorem to justify the approximations, with implications for Kerr-cell devices and nonlinear-optics inverse problems.

Abstract

We use weakly nonlinear geometric optics to study a model for the DC Kerr effect (the Kerr electro-optic effect), in which a light beam propagating through a material with strong nonlinear optical properties can have its polarization rotated by applying a strong external electric field. This effect is used to build fast switches (Kerr cells). We prove existence of an exact solution of the nonlinear Maxwell system with a cubic Kerr nonlinearity, with the wavelength $h$ being a small parameter. We justify the effect within this model, and also solve the inverse problem of recovery of the nonlinear susceptibility $χ^{(3)}$ from the change of the polarization.

The DC Kerr Effect in Nonlinear Optics

TL;DR

The paper develops a weakly nonlinear geometric-optics framework for Maxwell’s equations with a cubic Kerr nonlinearity to model the DC Kerr effect and polarization rotation. It proves the existence of exact solutions near carefully constructed asymptotic expansions that combine a strong stationary field with a propagating beam, and it analyzes how the Kerr nonlinearity induces a phase retardation that causes birefringence. A key contribution is an inverse-result showing that the polarization change along light rays yields the X-ray transform of the nonlinear susceptibility , enabling recovery of from measurements. The approach relies on a second-order hyperbolic reduction, a structured multi-scale expansion, and Guès’ theorem to justify the approximations, with implications for Kerr-cell devices and nonlinear-optics inverse problems.

Abstract

We use weakly nonlinear geometric optics to study a model for the DC Kerr effect (the Kerr electro-optic effect), in which a light beam propagating through a material with strong nonlinear optical properties can have its polarization rotated by applying a strong external electric field. This effect is used to build fast switches (Kerr cells). We prove existence of an exact solution of the nonlinear Maxwell system with a cubic Kerr nonlinearity, with the wavelength being a small parameter. We justify the effect within this model, and also solve the inverse problem of recovery of the nonlinear susceptibility from the change of the polarization.
Paper Structure (14 sections, 12 theorems, 130 equations, 3 figures)

This paper contains 14 sections, 12 theorems, 130 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Reduction to a 2nd order hyperbolic system
  4. The weakly nonlinear geometric optics
  5. The general setup
  6. Initialization of the problem, the linear solution
  7. The asymptotic solution: the leading term U zero
  8. Interpretation and Kerr cells
  9. The asymptotic solution: full expansion
  10. The strong electric field E
  11. The Profile Equations -- Proofs of Lemmas \ref{['lm:eikonal_transport']}-\ref{['lm:C0']}
  12. Exact Solutions
  13. Proof of Theorem \ref{['thm_inverse']}
  14. Concluding remarks

Key Result

Theorem 1

Under the assumptions above, we have the following. In parts item:b and item:c, the solution is unique among functions in a sufficiently small neighborhood of an asymptotic solution with finitely many terms, see Proposition prop_Gues for a precise statement.

Figures (3)

  • Figure 1: The setup. The backward (downward) propagating free solution is not shown.
  • Figure 2: The polarization changes from linear to ellipsoidal along the way. The two axes are preserved. Left: The polarization starts as a linear one but the nonlinear interaction converts it into elliptical. Right: Frontal view.
  • Figure 3: Polarization change with a constant $\chi^{(3)}$. The parameters are chosen so that the total rotation is by $\pi/2$. The round polarizing filters have orientations along the dotted lines, and the one on the right transmits all the light.

Theorems & Definitions (23)

  • Theorem 1
  • Theorem 2
  • Proposition 1
  • Lemma 1
  • Remark 1
  • Lemma 2
  • Lemma 3
  • Remark 2
  • Lemma 4
  • Proposition 2
  • ...and 13 more