# Microbending Loss

This is a continuation from the previous tutorial - ** polarization mode dispersion**.

Fibers often exhibit excess loss when they are spooled or cabled as the result of small deflections of the fiber axis that are of random amplitude and are randomly distributed along the fiber. The loss induced in optical fiber by these small random bends and stress in the fiber axis is called microbending loss.

Figure 2.16 cartoons the impact of a single microbend, at which, analogous to a splice, power can be coupled from the fundamental mode into higher order leaky modes.

Because external forces are transmitted to the glass fiber through the polymer coating material, the coating material properties and dimensions, as well as external factors, such as temperature and humidity, affect the microbending sensitivity of a fiber.

Further, microbending sensitivity is also affected by coating irregularities such as variations in coating dimensions, the presence of particles such as those in the pigments of color coatings, and inhomogeneities in the properties of the coating materials that vary along the fiber axis.

Coating surface slickness can also affect the mechanical state into which a fiber relaxes after spooling or within a cable structure, thereby affecting microbending loss.

The fiber axis perturbations that cause microbending loss are random in magnitude and are randomly distributed along the fiber. The perturbations are, therefore, modeled by a stochastic process characterized by broadband and randomly phased spatial frequency components.

The statistics of fiber deformation may be unknown, in contrast to the limiting case of a deterministic long-period fiber grating. The statistics of microbends can only be measured indirectly. Profilometry can be used to measure the roughness of the inner surface of a cable core tube.

Fourier analysis of the roughness profile of the extruded polymer surface might yield spatial frequency content in a range around 500 microns. However, the properties of the fiber as a stiff beam serve to impose a low pass filter on the spatial frequency content, with fairly sharp cutoff characteristics varying as the fourth power.

Mode coupling results when microbends occur, transferring power from guided modes to radiation modes. Many approximate analyses of microbending loss have been proposed.

One of the simplest metrics used for parameterizing the sensitivity of fibers to microbending loss is the MAC factor defined in the earlier discussion of macrobending.

Theory and experiment predict decreasing microbending sensitivity with decreasing MFD and increasing cutoff, and thus, decreasing MAC factor. When using MAC factor for the microbending sensitivity analysis, care has to be taken to compare fibers with similar refractive index profiles.

Analyzing microbend loss more rigorously requires time-intensive calculations of the modal coupling coefficients between the guided and unguided modes. Unguided modes can be represented as cladding, leaky, or radiation modes.

Marcuse predicted the microbend loss of single mode fibers as a function of wavelength based on mode coupling theory between guided modes and cladding mode using analytic expressions for the LP modes.

A stochastic model was developed for the random bends assuming a Gaussian-shaped auto-correlation function with rms perturbation amplitude, \(\sigma\), and correlation length, \(L_\text{c}\).

The magnitude and wavelength dependence of the predicted microbending loss was found to be strongly dependent on the value of the correlation length of the bends.

In general, we follow Marcuse’s approach except that we use finite element techniques to find the solutions for the fiber modes. In Marcuse’s derivation, the fundamental mode field shape was approximated as Gaussian, while a number of simplifying assumptions were used in approximating the cladding mode solutions.

Bjarklev improved the accuracy of the microbending calculations by using the more accurate approach of solving for the cladding modes of a coated fiber surrounded by air. Here, we also use the more accurate approach of coupling to the set of leaky modes, rather than cladding modes.

Microbend loss following coupled mode theory can be represented as

\[\tag{2.34}2\alpha_m=\sum_{p=1}^{\infty}C_{1p}^2\Phi(\beta_{01}-\beta_{1p})\]

where \(C_{1p}\) is the coupling coefficient between the fundamental guided mode and \(p\)-th cladding mode and \(\Phi\) is the power spectrum of the axis deformation function. \(\beta_{01}\) is the propagation constant of the guided mode, and \(\beta_{1p}\) is the propagation constant of \(p\)-th leaky mode.

The coupling coefficient is calculated as

\[\tag{2.35}C_{1p}^2=\frac{k^2}{2}\frac{\left(\displaystyle\int_0^{\infty}\frac{\text{d}n}{\text{d}r}E_{01}E_{1p}r\text{d}r\right)^2}{\displaystyle\int_0^{\infty}E_{01}^2r\text{d}r\int_0^{\infty}E_{1p}^2r\text{d}r}\]

where \(E_{01}\) and \(E_{1p}\) are electric fields of guided mode and leaky mode, respectively.

The power spectrum of the Gaussian deformation is

\[\tag{2.36}\Phi(\beta_{01}-\beta_{1p})=\sqrt{\pi}\sigma^2L_\text{c}\cdot\exp\left\{-\left[\frac{1}{2}(\beta_{01}-\beta_{1p})L_\text{c}\right]^2\right\}\]

where \(\sigma\) is the rms deviation of the distortion function, and \(L_\text{c}\) is the correlation length.

The physical significance of the correlation length \(L_\text{c}\) can be understood through a discussion of the spectral analysis of the fiber axis deformation \(f(z)\).

The spectral analysis of aperiodic signals is often accomplished by taking the Fourier transform of the auto-correlation function of the signal. The autocorrelation function of a periodic signal is also periodic with the same frequency spectrum as the original signal.

In contrast, the auto-correlation function of a aperiodic signal, with randomly varying amplitude and phase, will decay monotonically from the maximum of one to zero for large shift.

The environmental roughness impressed on an optical fiber, resulting in deformation of the fiber axis, will be a function of this type. The power spectrum of such an autocorrelation will reflect the length scales present in the environmental roughness and decay to zero at higher frequencies.

For simplicity in an analytic formulation, Marcuse assumed that the auto-correlation of the unspecified function \(f(z)\) is Gaussian of the form \(R(u)=\sigma^2\exp\left[-\left(\frac{u}{L_\text{c}}\right)^2\right]\), where \(\sigma\) is the rms deviation of the distortion function \(f(z)\) and \(L_\text{c}\) is its correlation length.

Therefore, the Fourier transform must be a Gaussian function of spatial frequency, leading to the simple expression of Eq. (2.36). The argument to the power spectrum of the perturbations is the difference in propagation constants \((\beta_{01}-\beta_{1p})\), of the modes that are exchanging power.

The strength of the coupling is proportional to power spectrum evaluated at \(\beta_{01}-\beta_{1p}\) and the overlap integral of the electric fields of the coupled modes, as shown in Eq. (2.35).

The correlation length is, thus, a measure of the shortest length scale represented in the random perturbation function of the fiber axis.

As an example of the dependence of microbending loss on fiber profile when the statistics of the fiber axis perturbations are known, we describe microbend loss for typical matched clad (MC) and depressed clad (DC) fiber designs. Figure 2.17 shows calculated microbend loss for the MC and DC fibers as a function of wavelength for the cases where correlation length, \(L_\text{c}\), is 50, 450, and 800 μm.

The microbending loss is largest for small \(L_\text{c}\), = 50 μm, and is slowly decreasing as wavelength increases. The large magnitude microbending loss occurs for short correlation length because the broad width of the perturbation power spectrum results in coupling to several leaky modes.

The variation with wavelength is low because the wavelength-dependent coupling of individual modes is averaged across several modes. However, in practice, perturbations on this length scale are only weakly transmitted to the fiber core because of the strong spatial filtering from the coating and the stiffness of the fiber.

For a more physically relevant correlation length of 450 μm, the microbend loss is seen to increase as the wavelength increases, as is typical of microbend added loss spectra, as observed in real fibers in cables or modules.

In this regimen, the depressed cladding design shows lower microbend sensitivity than standard matched clad fiber. This is also observed in practice. The calculated increase with wavelength in this range is also reasonable.

For a longer correlation length of 800 μm, the trend with wavelength is similar to the 450-μm case, but with an overall reduction in magnitude. The nearest leaky mode, to which microbending may power out of the fundamental LP_{01} mode, is usually LP_{11}.

As \(L_\text{c}\rightarrow\infty\), \(2\pi/\Lambda\) becomes too small to couple even these two most closely spaced modes, and microbending loss becomes negligible.

The dual coating is used to protect the fiber and to reduce the stress at the fiber when external force exists to the fiber due to cabling or spooling.

Yang describes the impact of coating properties on microbending loss: Microbending losses decrease with increasing thickness, Young’s modulus, and Poisson’s ratio of the primary coating.

Similarly, changes in refractive index in the glass fiber decrease with the increasing Young’s modulus and Poisson’s ratio of the secondary coating.

Figure 2.18 shows a calculation of the stress at the fiber axis as a function of primary modulus and secondary modulus following Yang’s derivation.

As the modulus of the secondary coating increases, the stress decreases. However, as the modulus of the primary coating increases, the stress increases. The combination of soft primary and hard secondary are desirable for best microbending performance.

Since the microbend loss is proportional to the stress, we can design coating geometry and materials to reduce the microbend loss. However, other coating performance metrics must also be kept in balance, so there are limits to the improvement available in microbending performance by tailoring coating properties.

The next tutorial introduces ** laser pumping and population inversion**.