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Definition: a parameter for quantifying the Kerr nonlinearity of a medium

German: nichtlinearer Brechungsindex

Category: nonlinear optics

How to cite the article; suggest additional literature

When light with high intensity propagates through a medium, this causes nonlinear effects. The simplest of these is the Kerr effect, which can be described as a change (usually an increase) in the refractive index in proportion to the optical intensity I

index change via SPM

with the nonlinear refractive index n2. The units of that quantity are m2/W (or cm2/W) in the SI system, but in older literature one finds n2 values in esu units. For the conversion of such units, the equation

n2 in m2/W from esu units

can be used, where n is the refractive index.

Normally, nonlinear indices are measured for linearly polarized light. For circularly polarized light in a not birefringent medium, the Kerr effect is weaker by one third.

Conversion of Nonlinear Index Values

Refractive index:
n2 in SI units: calc
n2 in esu units: calc

After you have modified some values, click a "calc" button to recalculate the field left of it.

At very high optical intensities, the equation above for the nonlinear index change may need a higher-order correction. For example, one may have a term proportional to the index squared, with a negative coefficient, so that the Kerr effect saturates.

In addition to the Kerr effect (a purely electronic nonlinearity), electrostriction can significantly contribute to the value of the nonlinear index [4, 5]. Here, the electric field of light causes density variations (acoustic waves) which themselves influence the refractive index via the photoelastic effect. That mechanism, however, involves a significant time delay and is thus relevant only for relatively slow power modulations, but not for ultrashort pulses. In optical fibers, the contribution of electrostriction at low (megahertz) frequencies is typically of the order of 10–20% of that of the Kerr effect, but can strongly depend on the material.

For transparent crystals and glasses, n2 is typically of the order of 10−16 cm2/W to 10−14 cm2/W. Silica, as used e.g. in silica fibers, has a relatively low nonlinear index of 2.7 · 10−16 cm2/W for wavelengths around 1.5 μm, whereas some chalcogenide glasses exhibit several hundred times higher values. Semiconductor materials also have very high nonlinear index values. It has been shown that the nonlinear index scales in proportion to the inverse fourth power of the bandgap energy, but also depends on the proximity of the test wavelength to the bandgap [2]. The nonlinearity can also be negative (self-defocusing nonlinearity), particularly for photon energies above ≈ 70% of the bandgap energy.

Materials with a high nonlinear index often have a small bandgap energy, and therefore also often exhibit strong two-photon absorption (TPA). For some applications such as channel conversion in telecom systems, this is detrimental, and a figure of merit such as n2 / β (where β is the TPA coefficient) can be used to compare different materials.

The measurement of the nonlinear index of bulk samples is often done with the z-scan technique, which is based on self-focusing via the Kerr lens.

The nonlinearity of optical fibers can be quantified by measuring spectral broadening resulting from self-phase modulation. Note, however, that the polarization state may be scrambled in a non-polarization-maintaining fiber, and this can affect the result. Also, the result is a kind of average over the material properties of the fiber core and the cladding.

Bibliography

[1]M. Sheik-Bahae et al., “Sensitive measurement of optical nonlinearities using a single beam”, IEEE J. Quantum Electron. 26 (4), 760 (1990)
[2]M. Sheik-Bahae et al., “Dispersion and band-gap scaling of the electronic Kerr effect in solids associated with two-photon absorption”, Phys. Rev. Lett. 65 (1), 96 (1990)
[3]K. S. Kim et al., “Measurement of the nonlinear index of silica-core and dispersion-shifted fibers”, Opt. Lett. 19 (4), 257 (1994)
[4]E. L. Buckland and R. W. Boyd, “Electrostrictive contribution to the intensity-dependent refractive index of optical fibers”, Opt. Lett. 21 (15), 1117 (1996)
[5]E. L. Buckland and R. W. Boyd, “Measurement of the frequency response of the electrostrictive nonlinearity in optical fibers”, Opt. Lett. 22 (10), 676 (1997)
[6]I. P. Nikolakakos et al., “Broadband characterization of the nonlinear optical properties of common reference materials”, IEEE Sel. Top. Quantum Electron. 10 (5), 1164 (2004)
[7]A. Major et al., “Dispersion of the nonlinear refractive index in sapphire”, Opt. Lett. 29 (6), 602 (2004)
[8]I. Dancus et al., “Single shot interferometric method for measuring the nonlinear refractive index”, Opt. Express 21 (25), 31303 (2013)

(Suggest additional literature!)

See also: Kerr effect, self-phase modulation, Kerr lens, refractive index, B integral, nonlinearities, z-scan measurements

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RP Fiber Power – the versatile Fiber Optics Software

An Amazing Tool

RP Fiber Power software

This amazing tool is extremely helpful for the development of passive and active fiber devices.

ASE

Watch our quick video tour!

Single-mode and Multi­mode Fibers

fibers

Calculate mode properties such as

  • amplitude distributions (near field and far field)
  • effective mode area
  • effective index
  • group delay and chromatic dispersion

Also calculate fiber coupling efficiencies; simulate effects of bending, nonlinear self-focusing or gain guiding on beam propagation, higher-order soliton propagation, etc.

Arbitrary Index Profiles

A fiber's index profile may be more complicated than just a circle:

special fibers

Here, we "printed" some letters, translated this into an index profile and initial optical field, propagated the light over some distance and plotted the output field – all automated with a little script code.

Fiber Couplers, Double-clad Fibers, Multicore Fibers, …

fiber devices

Simulate pump absorption in double-clad fibers, study beam propagation in fiber couplers, light propagation in tapered fibers, analyze the impact of bending, cross-saturation effects in amplifiers, leaky modes, etc.

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fiber amplifier

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eye diagram

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fiber laser

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fiber laser

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Apply any sequence of elements to your pulses!

… and even Bulk Devices

regenerative amplifier

For example, study

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RP Fiber Power is an extremely versatile tool!

Mode Solver

fiber modes

For example, calculate

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All this is calculated with high efficiency!

Beam Propagation

beam propagation

Propagate optical field with arbitrary wavefronts through fibers. These may be asymmetric, bent, tapered, exhibit random disturbances, etc.

See our demo video for numerical beam propagation.

Laser-active Ions

level scheme

Work with the standard gain model, or define your own level scheme!

Can include different ions, energy transfers, upconversion and quenching effects, complicated pumping schemes, etc.

Multiple Pump and Signal Waves, ASE

optical channels

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signal parameters

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