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# Polarization Beat Length

Author: the photonics expert

Definition: the period of interference effects in a birefringent medium

When two waves with different linear polarization states propagate in a birefringent medium, their phases will evolve differently. We assume that the polarization of each wave is along one of the principle directions of the medium, so that this polarization will be preserved during propagation. For monochromatic waves with identical optical frequencies, the difference in phase delay will be proportional to the propagation length <$L$> and to the difference in wavenumbers <$\Delta k$> (for a bulk medium) or in the imaginary parts of the propagation constants <$\Delta \beta$> (for a waveguide):

$$\Delta \varphi = \Delta \beta \;L$$

This means that the phase relation between both waves is restored after integer multiples of the so-called polarization beat length (or polarization beat distance):

$${L_{\rm{b}}} = \frac{{2\pi }}{{\Delta \beta }} = \frac{\lambda }{{\Delta {n_{{\rm{eff}}}}}}$$

where <$\lambda$> is the vacuum wavelength (not the wavelength in the medium). More generally, the polarization evolution is periodic, with the polarization beat length being the period of evolution.

## Calculator for Polarization Beat Length

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If the two waves were in phase originally (<$L = 0$>), the total resulting polarization state was linear. After a quarter of the polarization beat length, the phase difference is <$\pi / 2$>, and we have circularly polarized light. After half the beat length, we have linearly polarized light again, but with the direction rotated by 90°. After one full beat length, the original polarization direction is obtained.

## Typical Beat Length Values

Polarization-maintaining fibers can have polarization beat lengths of a few centimeters or even only a few millimeters. This is a much stronger birefringence than achievable by strongly bending a fiber with radially symmetric design. Even sub-millimeter beat lengths are possible with some photonic crystal fibers. Tentatively, a short beat length, as results from strong birefringence, reduces the sensitivity of the fiber to mode coupling effects, which may result from imperfections in manufacturing or from mechanical stress.

Fibers with a nominally symmetric design would theoretically have an infinitely large polarization beat length. In practice, they can have some birefringence due to imperfections and mechanical stress. Do to the randomness of such influences, they do not have a well-defined polarization beat length, and not even a well-defined polarization axis. One may nevertheless define a statistical value of the polarization beat length; one typically finds values of a few meters or some tens of meters. One may also specify a birefringence correlation length beyond which the birefringence changes substantially.

## Measurement of Polarization Beat Length of Fibers

There are different methods for measuring the polarization beat length of an optical fiber. As it is difficult to access the light within the fiber or to vary the fiber length, the preferred methods work with a fixed length <$L$> of fiber. One method is to inject linearly polarized broadband light (for example, from a superluminescent source) and to record the optical spectrum after transmission through the fiber and another polarizer. That spectrum will exhibit oscillations with a period <$\Delta \lambda$>, from which the polarization beat length can be calculated:

$${L_{\rm{b}}} = \frac{{\Delta \lambda }}{\lambda }L$$

This equation is based on the usually well justified assumption that the difference of effective indices depends only weakly on the wavelength.

There is a number of more sophisticated measurement techniques, often exploiting fiber nonlinearities such as Brillouin scattering. Some of them can spatially resolve the birefringence in the fiber.

## Relation to Polarization Mode Dispersion

Polarization mode dispersion is the polarization dependence of the group delay of light. The differential group delay is essentially the derivative of the difference in propagation constant with respect to optical frequency. One can thus not directly calculate PMD from the polarization beat length, at least not if one has that only for one particular wavelength.

## More to Learn

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### Bibliography

 [1] V. N. Filippov et al., “Measurement of polarisation beat length in single-mode optical fibres with a polarisation modulator”, Electron. Lett. 26 (10), 658 (1990); https://doi.org/10.1049/el:19900431

## Questions and Comments from Users

2020-12-03

Can one observe beats with non-polarized light?