What is beat length and why is it often specified for PM fiber, instead of polarization extinction ratio?
It is difficult for manufacturers to specify a polarization extinction ratio (PER) for light output by polarization-maintaining (PM) fibers, since this parameter depends on the length of the fiber, how it is routed, and the polarization and alignment of the input light. Beat length is independent of these factors, which makes it a convenient parameter for quantifying the fiber's potential to preserve polarization. A smaller beat length is better, and it is a useful parameter to reference when choosing a PM fiber and its operating temperature. While beat length provides information about a PM fiber's potential to perform well, its actual performance and the PER of the light output by the fiber ultimately depend on the details of the fiber's deployment.
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Figure 1: The red and blue curves represent waves that are polarized parallel to the PM fiber's slow and fast axes, respectively. Since the slow axis has the higher refractive index, the wave polarized parallel to this axis (red wave) oscillates at a higher rate. This can be seen by looking at the waves' amplitudes (indicated by spheres) at differernt propagation distances.
Beat Length of a PM Fiber
The concept of a PM fiber's beat length can be visualized by considering waves propagating along the fiber's two orthogonal axes, fast and slow. One way these waves can be excited in the fiber is by coupling in monochromatic, linearly polarized light, with the input light's polarization angle oriented midway between the fiber's fast and slow axes.
If this is done, the orthogonally polarized waves will have the same amplitude and be in phase when they enter the fiber. But, the waves do not stay in phase as they propagate, since the refractive index of the slow axis is larger than that of the fast axis (nslow > nfast ). The wave polarized parallel to the slow axis has a shorter period, and therefore propagates over a shorter distance for a given number of oscillations, than the light polarized parallel to the fast axis.
The phase difference between the two waves increases linearly with propagation distance (Figure 2). Locations at which there is a factor of 2 difference between the phases of the two waves are indicated by the green markers. In the figure, the phases and phase difference angles were folded so that they could be plotted on a scale of 0 to 2. The beat length is the periodic distance over which the phase difference increases to an amount equal to 2. Note that while the accumulated phase difference over a beat length is 2, this phase difference is typically not acquired over a whole number of oscillations by either wave.
The beat length (Lp ),
is proportional to wavelength () and inversely proportional to the fiber's birefringence (B = nslow - nfast ).
Typical Beat Lengths
The larger the refractive index difference between the two fiber axes, the larger the birefringence, the shorter the beat length, and the better the polarization-preserving performance of the fiber. The beat length remains constant along the length of the fiber, as long as the fiber's birefringence does not change. Manufacturers often specify beat length for selected wavelengths and limited temperature ranges.
To date, PM fibers with beat lengths <1 mm have had elliptical cores and mode field diameters (MFDs) significantly smaller than those of standard single mode optical fibers. Many applications require fibers with circular cores and MFDs close to those of standard single mode fibers. Typical PM fibers that meet these criteria and perform well have beat lengths between 1 mm and a few millimeters. It is interesting to note that standard single mode fibers also have measurable beat lengths, although they are meters long. This is due to their cores not having a perfectly circular cross section. Since the ellipticity of their cores is slight and changes randomly along the length of the fiber, standard single mode fibers are not useful as PM fibers.
The Amplitude Does not Beat
In the case of PM fibers, beat length refers to a repeating phase relationship between waves polarized parallel to the orthogonal slow and fast axes of a PM fiber. The sum of these waves at any point along the fiber determines the polarization state of the light beam at that point. For example, when the waves are in phase, the light is linearly polarized, and the waves are out of phase by /2 (90°), the light is circularly polarized.
An amplitude beat pattern does not occur, since these waves are polarized orthogonal to one another. Two waves only produce an amplitude beat pattern when they have components polarized parallel to one another. For the same reason, a signal with an interference term equal to zero will result when a photodetector is used to measure the combined intensity of two orthogonally polarized waves with different periods.
 Chris Emslie, in Specialty Optical Fibers Handbook, edited by Alexis Mendez and T. F. Morse (Elsevier, Inc., New York, 2007) pp. 243-277.
 Malcolm P. Varnham et al., "Analytic Solution for the Birefringence Produced by Thermal Stress in Polarization-Maintaining Optical Fibers," J. Lightwave Technol., LT-1(2), 332-339 (1983).
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Date of Last Edit: July 30, 2021
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