Beat length

The beat length is the distance along a waveguide over which two modes with slightly different effective indices accumulate a phase difference of $2\pi$. Most often the two modes are the orthogonal polarizations of a single-mode fiber, and the beat length measures its modal birefringence $B = n_x - n_y$:

$$L_B = \frac{\lambda}{B} = \frac{2\pi}{\beta_x - \beta_y}.$$

Light launched at 45° to the axes changes from linear to circular to the orthogonal linear state and back again over one beat length. Polarization-maintaining fiber with $B$ of 3 to 5 × 10⁻⁴ has $L_B$ of 3.1 to 5.2 mm at 1550 nm. Standard single-mode fiber has only residual birefringence, of order $10^{-7}$, and beat lengths of order 10 m (15.5 m for $B = 10^{-7}$), varying along the fiber and with how it is laid.

Polarization holding and retardance

A short beat length means strong birefringence. Bends, twists, stress and temperature gradients move light between the two polarization axes only if they vary along the fiber with a spatial period close to $L_B$. In PM fiber, with a few millimeters of beat length, the slow disturbances of cabling and handling are far off that period, so light launched on one axis stays there; the residual cross-coupling is then specified through the h-parameter and the polarization extinction ratio. In standard fiber, with meters of beat length, ordinary bends and twists couple the polarizations freely and the output state wanders.

The phase difference after a length $z$ is the retardance $2\pi z/L_B$. A 1 cm piece of PM fiber with $B = 5 \times 10^{-4}$ at 1550 nm therefore has a retardance of about 1160°, more than three full waves, which is why a fiber section only acts as a stable waveplate when its length and temperature are both controlled.

Relation to group delay and PMD

The two polarizations also travel at different group velocities. For a fiber whose group birefringence $B_g$ is close to $B$, as in stress-rod fibers, the differential group delay per unit length is

$$\frac{\Delta\tau}{L} \approx \frac{B_g}{c},$$

1.67 ps per meter for $B_g = 5 \times 10^{-4}$. In standard fiber the same formula gives 0.33 ps/km for $B = 10^{-7}$ over a short, uniformly birefringent section, but over long lengths the axes rotate randomly, the delays partly cancel, and the delay grows as the square root of length. That statistical delay is polarization mode dispersion, and the beat length together with the length over which the axes stay correlated sets the PMD coefficient.

Measurement

Two methods are standard.

• Side scattering. Visible light launched at 45° to the axes scatters by Rayleigh scattering, and because a dipole does not radiate along its own axis, the scattered intensity seen from one side oscillates along the fiber with period $L_B$. At 633 nm a fiber with $B = 5 \times 10^{-4}$ shows fringes 1.27 mm apart, which can be photographed and measured directly.
• Wavelength scanning. Light launched at 45° through a fiber of length $L$ and an output polarizer gives a transmission that oscillates with wavelength, with period $\Delta\lambda = \lambda^2/(B_g L)$: 4.8 nm for 1 m of fiber with $B_g = 5 \times 10^{-4}$ at 1550 nm. This measures the group birefringence, which can differ noticeably from the phase birefringence in elliptical-core and microstructured fibers.

Polarization-resolved optical frequency-domain reflectometry maps the birefringence along a fiber with millimeter-scale resolution.

Beat length between other modes

The same idea applies to any two modes. In a directional coupler the even and odd supermodes beat, and power returns to the input guide after $\lambda/(n_e - n_o)$; with a supermode index difference of 0.05 at 1550 nm that is 31 µm. Integrated-optics texts often call the half of that, $L_\pi = \lambda/[2(n_e - n_o)]$, the beat length: it is the full cross-over length (15.5 µm here) and equals $\pi/(2\kappa)$ with the coupling coefficient $\kappa$. The MMI coupler beat length $L_\pi = \pi/(\beta_0 - \beta_1)$ follows the same $\pi$ convention. Taper and mode-converter design uses the $2\pi$ version between a guided mode and the cladding, as in the inverse taper adiabaticity criterion. A quoted beat length therefore needs its convention, since the two differ by a factor of two.

Common questions

What is a typical beat length of PM fiber?

About 3 to 5 mm at 1550 nm for PANDA and bow-tie designs, from $B$ = 3 to 5 × 10⁻⁴. Because $B$ changes only slowly with wavelength in stress-induced fibers, the beat length scales roughly in proportion to wavelength and is about 1.3 mm at 633 nm for $B = 5 \times 10^{-4}$.

Is a shorter beat length better?

For holding polarization, yes: a shorter beat length means stronger birefringence and weaker coupling between the axes from external disturbances. It also means a larger differential group delay between the axes and a fiber retardance that is more sensitive to temperature.

What is the beat length of standard single-mode fiber?

Of order meters to tens of meters, from residual birefringence near $10^{-7}$. The value is not a fixed property of the fiber type: it depends on core ellipticity, internal stress, bends and twist, and changes along the length.

References: J. Noda, K. Okamoto and Y. Sasaki, "Polarization-maintaining fibers and their applications," Journal of Lightwave Technology 4, 1071 (1986); K. Okamoto, Fundamentals of Optical Waveguides, 2nd ed. (Academic Press, 2006); G. P. Agrawal, Nonlinear Fiber Optics, 5th ed. (Academic Press, 2013); L. B. Soldano and E. C. M. Pennings, "Optical multi-mode interference devices based on self-imaging: principles and applications," Journal of Lightwave Technology 13, 615 (1995).