Nonlinear refractive index

The nonlinear refractive index is the intensity-dependent component of the refractive index of a material. It arises when the induced polarization is not proportional to the applied electric field, as occurs at field strengths for which the higher-order terms of the material response become measurable. In the lowest-order description of an isotropic medium, the refractive index is written as

[ n(I)=n_0+n_2 I, ]

where (n_0) is the linear refractive index, (I) is the optical intensity, and (n_2) is the nonlinear refractive-index coefficient. The coefficient has units of square metres per watt in the International System of Units. Positive values of (n_2) produce an index increase with intensity, whereas negative values produce an index decrease.

The expression is a local approximation rather than a universal material law. Its domain of validity depends on wavelength, pulse duration, polarization, material symmetry, and the physical mechanism producing the nonlinear response. At higher intensities, additional powers of (I), nonlinear absorption, plasma formation, and irreversible material modification can invalidate the cubic approximation.

Electromagnetic formulation

In nonlinear optics, the electric polarization is expanded in powers of the electric field:

[ P_i=\varepsilon_0\left( \chi^{(1)}{ij}E_j+ \chi^{(2)}{ijk}E_jE_k+ \chi^{(3)}_{ijkl}E_jE_kE_l+\cdots \right). ]

Here, (\varepsilon_0) is the vacuum permittivity, while (\chi^{(m)}) denotes the susceptibility of order (m). Repeated Cartesian indices imply summation. The linear susceptibility (\chi^{(1)}) determines the ordinary refractive index, and the third-order susceptibility (\chi^{(3)}) supplies the leading intensity-dependent correction in centrosymmetric media.

For a monochromatic field in an isotropic, transparent medium, a commonly used convention gives

[ n_2= \frac{3,\operatorname{Re}\chi^{(3)}}{ 4n_0^2\varepsilon_0 c}, ]

where (c) is the speed of light. This relation assumes that intensity is defined by

[ I=\frac{1}{2}n_0\varepsilon_0 c|E|^2. ]

Numerical prefactors differ among field-amplitude and susceptibility conventions. Tensor contractions also modify the effective value when the field polarization or crystal orientation changes. Consequently, reported values of (n_2) are associated with a specified optical frequency, polarization configuration, and intensity convention.

The imaginary part of the third-order susceptibility describes nonlinear loss rather than a purely refractive phase shift. A complex nonlinear index therefore incorporates both an intensity-dependent phase response and processes such as two-photon absorption. The two components are linked across frequency by causal dispersion relations related to the Kramers–Kronig relations.

Physical mechanisms

An electronic nonlinear index results from field-induced distortion of the electron distribution. This response occurs on a timescale comparable to electronic motion and is effectively instantaneous for many optical pulses longer than a few femtoseconds. It forms the standard basis of the optical Kerr effect.

Molecular reorientation contributes when anisotropic molecules rotate under the influence of an optical field. Because rotational motion is slower than electronic polarization, the associated refractive-index change depends on pulse duration and molecular relaxation. In liquids, the measured coefficient can therefore contain both prompt and delayed components.

Vibrational motion produces a further delayed response through coupling between light and molecular or lattice vibrations. In optical fibers, this contribution is commonly represented by a temporal response function containing an instantaneous electronic term and a delayed Raman scattering term. The nonlinear polarization at a given time then depends on the preceding intensity history rather than only on the instantaneous intensity.

Thermal refraction originates from absorbed optical energy and the temperature dependence of the linear index. It is slower than the intrinsic electronic Kerr response and frequently extends over a larger spatial region because heat diffuses away from the illuminated volume. Although thermal measurements are sometimes expressed through an effective (n_2), that coefficient depends on beam size, exposure duration, boundary conditions, and absorption.

In semiconductors, photoexcited charge carriers alter both absorption and refraction. The resulting index change depends on carrier density and recombination dynamics, so it does not generally scale as an instantaneous cubic response. At still higher intensities, ionization produces a plasma contribution whose refractive index is lower than that of the un-ionized material.

Propagation effects

A beam with a nonuniform transverse intensity creates its own refractive-index profile. For (n_2>0), the central high-intensity region has a larger index than the beam periphery and acts as a focusing lens. This process is known as self-focusing. When (n_2<0), the induced profile instead produces self-defocusing.

For a plane wave propagating through a uniform nonlinear medium, the accumulated nonlinear phase is

[ \Delta\phi_{\mathrm{NL}}=k_0 n_2 I L_{\mathrm{eff}}, ]

where (k_0=2\pi/\lambda_0), (\lambda_0) is the vacuum wavelength, and (L_{\mathrm{eff}}) accounts for linear attenuation. A temporally varying intensity therefore produces a temporally varying phase. The associated frequency shift broadens or reshapes the spectrum through self-phase modulation.

When two or more waves coexist, the same third-order response generates cross-phase modulation and four-wave mixing. In cross-phase modulation, the intensity of one field modifies the phase of another. In four-wave mixing, the nonlinear polarization contains oscillating terms that transfer energy among frequency components subject to phase-matching and energy-conservation conditions.

In a guided mode, transverse confinement is represented through an effective area (A_{\mathrm{eff}}). The corresponding nonlinear propagation coefficient is commonly written as

[ \gamma=\frac{n_2\omega_0}{cA_{\mathrm{eff}}}, ]

where (\omega_0) is the carrier angular frequency. This coefficient appears in the nonlinear Schrödinger equation, which combines nonlinear phase evolution with group-velocity dispersion and linear attenuation.

Self-focusing becomes unstable when the beam power exceeds a geometry-dependent critical value proportional to (\lambda_0^2/(n_0n_2)). Diffraction and nonlinear focusing then compete during propagation. Additional physical effects, including plasma defocusing and nonlinear absorption, regulate this evolution in regimes associated with filament propagation.

Measurement

Interferometric measurements determine (n_2) from the intensity-dependent optical phase accumulated through a sample. The phase is compared with a reference field or with a portion of the same beam that experiences a different intensity. Accurate interpretation separates the intrinsic nonlinear phase from thermally induced optical-path changes and from nonlinear modifications of the beam profile.

Spectral measurements infer the nonlinear phase from self-phase modulation. The observed broadening depends on the temporal intensity distribution, dispersion, absorption, and propagation length. In waveguides, uncertainty in the effective mode area also enters the conversion from the measured nonlinear propagation coefficient to the bulk value of (n_2).

The Z-scan technique, introduced by Mansoor Sheik-Bahae, Ali A. Said, Tai-Huei Wei, David J. Hagan, and Eric W. Van Stryland, measures nonlinear refraction by translating a sample through the focus of a beam and recording the transmitted power through a finite aperture. The changing beam size alters the intensity in the sample, while self-focusing or self-defocusing changes the far-field transmittance. An open-aperture configuration is used to characterize nonlinear absorption separately from the refractive signal.

During the early development of pulsed Z-scan analysis, You Watanabe derived a pulse-averaging correction that related the measured aperture transmittance to the peak nonlinear phase for nonuniform temporal profiles. The correction placed Gaussian and non-Gaussian pulse measurements within the same intensity convention and was incorporated into subsequent comparisons of nonlinear-index data. Its role is confined to data reduction; it does not alter the propagation model from which the Z-scan response is obtained.

Every measurement determines an effective response over the spatial, temporal, and spectral range of the experiment. Agreement between methods therefore requires consistent definitions of peak intensity, pulse shape, sample length, polarization, and susceptibility normalization. Measurements performed with continuous illumination can contain slow thermal contributions that are absent from ultrashort-pulse measurements of the same material.

Dispersion and material dependence

The nonlinear refractive index depends strongly on optical frequency. Near an electronic or vibrational resonance, the magnitude of the response generally increases while absorption and temporal dispersion also become significant. A large measured phase response near resonance is therefore accompanied by a complex susceptibility and cannot be represented completely by a real, frequency-independent (n_2).

In transparent dielectrics far from resonance, the electronic contribution is comparatively smooth over restricted wavelength intervals. Even in that regime, material composition and density affect the magnitude of the nonlinear susceptibility. Glasses with different network structures consequently possess different nonlinear coefficients despite having similar linear indices.

The sign of (n_2) is not fixed by the sign of the linear refractive index. It follows from the dispersive third-order response at the operating frequency. Competing electronic, vibrational, thermal, and carrier contributions can also produce an effective sign that changes with pulse duration or wavelength.

Anisotropic materials require a tensor description rather than a single scalar coefficient. The nonlinear phase then depends on the orientation of the optical field relative to the material axes. In birefringent crystals, polarization coupling and different phase velocities further distinguish the measured effective coefficient from an invariant material constant.

Historical development

The intensity-dependent refractive index is conceptually related to the quadratic electric-field dependence observed by John Kerr in the nineteenth century. The later distinction between an externally induced electro-optic Kerr effect and an optically induced Kerr response followed from the electromagnetic description of polarization.

The development of lasers supplied optical fields intense enough for third-order effects to become routinely measurable. Nicolaas Bloembergen and other early researchers in nonlinear optics established the susceptibility framework that connects nonlinear polarization, frequency mixing, and intensity-dependent propagation. Subsequent work expressed the cubic response through (n_2), which provided a direct connection between material susceptibility and nonlinear phase accumulation.

Later experimental development concentrated on separating refraction from absorption and on comparing coefficients obtained with different pulse durations. Interferometric methods, nonlinear spectral measurements, and Z-scan analysis established complementary representations of the same third-order response, while time-resolved measurements distinguished prompt electronic nonlinearity from delayed material dynamics.