# Working Definitions of Cutoff Wavelength

This is a continuation from the previous tutorial - ** introduction to lenses for image formation and manipulation**.

### 1. Introduction

The cutoff wavelength of a single-mode optical fiber is the wavelength above which only a single bound mode, the fundamental LP_{01} mode, propagates.

For numerous reasons concerning transmission performance (bandwidth, multipath interference, modal noise, etc.), it is desirable to operate fibers in the regime where only the fundamental mode propagates. (This discussion does not address the intentional use of multimoded fibers for short-reach applications, where as many as 10- to 18-mode groups may be allowed to propagate at the operating wavelength.)

In this section, we discuss the theoretical and effective cutoff wavelengths of step-index single-mode fibers.

### 2. Theoretical Cutoff Wavelength

It has already been noted that the well-known weakly guiding analysis by Gloge shows that a matched cladding optical fiber supports the propagation of only the fundamental LP_{01} mode when the \(V\) number of the waveguide is less than \(2.405\). Therefore, the theoretical cutoff wavelength for a step-index fiber, \(\lambda_\text{c}^\text{th}\), is defined as

\[\tag{2.16}\lambda_\text{c}^\text{th}=\frac{2\pi{n_1}a}{2.405}\sqrt{2\cdot\Delta}\]

where \(n_1\) is the refractive index of the core, \(a\) is the core radius, and \(\Delta\) is the relative index difference between the core and cladding.

At wavelengths greater than \(\lambda_\text{c}^\text{th}\), the transverse propagation constant \(\beta_{t2}\) of the first higher order LP_{11} mode in the cladding region becomes a real number.

This changes the solution for the electric field in the cladding from a decaying, evanescent field to an oscillatory, propagating field, thus resulting in radial energy flow (i.e., one that carries energy away from the fiber axis). The bound mode becomes a leaky mode.

### 3. Effective Cutoff Wavelength

Consider the behavior of the LP_{11} at wavelengths shorter than \(\lambda_\text{c}^\text{th}\). Far below \(\lambda_\text{c}^\text{th}\) the LP_{11} mode is tightly confined within the core region and losses will generally be comparable to those of the fundamental mode. As the wavelength increases, the LP_{11} mode becomes less tightly confined to the core.

The decreasing mode confinement gives rise to excess LP_{11} mode loss when axial imperfections, such as microbends or macrobends, are present.

Microbends are defined as small-scale random deflections of the fiber axis, small relative to the core size, such as would be present when the fiber is pressed against a rough surface.

Macrobends are large-scale deflections in the fiber axis, such as loops or the bends associated with placing fiber on a spool.

Generally as one approaches, wavelengths 100 nm or so below \(\lambda_\text{c}^\text{th}\), the LP_{11} mode becomes very loosely confined to the core and very lossy if the fiber axis is not maintained perfectly straight. Even at wavelengths well below LP_{11}, the losses on the order of 10 dB/m or more readily occur, so that the LP_{11} does not effectively transmit energy over distances of more than a few meters.

This has led to the concept of the effective cutoff wavelength, \(\lambda_\text{c}^\text{eff}\), of a single mode fiber, which is defined phenomenologically as described later in this tutorial.

Figure 2.4 shows the power as a function of wavelength that is observed at the output of a short (2 meter) length of fiber at wavelengths that span the fiber’s LP_{11} mode cutoff.

At the input of the fiber, the launch conditions are such that power P_{01} and P_{11} launched into the LP_{01} and LP_{11} modes, respectively, for all wavelengths. If we assume that the loss of the LP_{01} mode experiences in the 2-m length of fiber is negligible, then the power at the output of the fiber as a function of wavelength is

\[\tag{2.17}P_\text{out}(\lambda)=P_{01}+P_{11}\cdot{e}^{-\alpha_{11}(\lambda)\cdot{L}}\]

where \(\alpha_{11}(\lambda)\) is the LP_{11} attenuation as a function of wavelength. If we consider wavelengths short relative to LP_{11} mode cutoff, then the LP_{11} mode is well confined and the excess attenuation is low, so that \(P_\text{out}(\lambda)\approx{P}_{01}+P_{11}\).

Since the LP_{01} mode is a singly degenerate set of modes and the LP_{11} mode is a doubly degenerate set of modes, we assume that \(P_{11}=2*P_{01}\) and then \(P_\text{out}(\lambda)\sim3*P_{01}\) at short wavelength.

At long wavelengths relative to LP_{11} mode cutoff, the attenuation of the LP_{11} mode is very high and \(P_\text{out}(\lambda)\sim{P_{01}}\).

At wavelengths between the two extremes, the power level at the output of the fiber transitions between \(3*P_{01}\) and \(P_{01}\), as the fiber transitions from two-moded to single-moded behavior.

It is important to note that because the length and layout of the fiber sample will determine the level of excess LP_{01} attenuation, the location of the transition from two-moded to single-moded behavior will also vary with fiber length and layout.

Notice that the wavelength where only LP_{01} power is observed at the output of the fiber is considerably below \(\lambda_\text{c}^\text{th}\). In other words, at wavelengths considerably below \(\lambda_\text{c}^\text{th}\), the LP_{11} mode has become effectively cutoff. By convention, the effective cutoff wavelength has been defined as the wavelength where \(P_\text{out}(\lambda)\) has risen by 0.1 dB above \(P_{01}\). It can be shown from Eq. (2.17) that the attenuation of the LP_{11} mode at \(\lambda_\text{c}^\text{eff}\) is 19.2 dB.

The fiber effective cutoff wavelength has been defined by international standards groups to be measured on a 2-m length of fiber that is deployed in a ‘‘nominally’’ straight configuration except for a single 28-cm diameter loop.

This fiber configuration was defined so that fiber manufacturers could easily implement the procedure in their factories on readily available spectral attenuation test benches.

Because this factory-friendly measurement configuration may not represent field-deployed conditions, a need arose to relate the fiber effective cutoff wavelength to the effective cutoff wavelength of the fiber when it was deployed as a cable section or jumper in an operating transmission system.

Many groups studied how the LP_{11} mode cutoff scales in wavelength as the length and bending configuration of fiber under test is varied. The studies showed that length and bending scaling of cutoff varied significantly across fiber designs.

For example, the change in the cutoff wavelength with variation in length of the fiber under test was significantly different for the matched-cladding, depressed-cladding and dispersion shifted fiber designs manufactured during the late 1980s.

In an effort to ensure that fibers are effectively single moded in the various configurations that they are likely to be deployed in, the cable effective cutoff wavelength has been defined.

The cabled fiber deployment configurations viewed as worst-case scenarios for outside plant, building, and interconnection cables were defined. For outside plant cables, the concern is that a short section of restoration cable, as short as 20 m in length, may be spliced into the transmission path to replace a damaged section of cable.

Typically when a telecommunications cable is damaged, for example, by excavation at a construction site near a cable right of way, then the damaged section is removed and a short section of cable is spliced in place to bridge the gap.

If the fiber in the short restoration cable is not effectively single moded at the system operating wavelength, then there is the potential for the paired splices to generate excess additive noise, which is referred to as modal noise.

Much of the energy lost by the incoming LP_{01} mode at the first splice will be coupled into the LP_{11} mode of the restoration fiber. If the transmission loss of the LP_{11} mode in the restoration fiber is low enough, then energy in both the LP_{01} and LP_{11} mode will reach the second splice and will add coherently if the optical path length difference for the two modes is less than the coherence length of the optical source.

The energy coupled into LP_{01} of the output fiber of the second splice depends on the electric field shape at the input to the splice, which is a function of the coherent interference of the two modes entering the splice.

Because the modal interference at the splice can be time dependent (because of laser wavelength variations, environmental variations that change the optical path length of the restoration fiber, etc.), the LP_{11} mode splice loss of the second splice can be time dependent, which generates modal noise.

With modal noise in mind, the outside plant cable cutoff wavelength deployment configuration is designed to mimic a 20-m restoration cable and the associated splice closures at its ends.

The configuration is defined as a 22-m length of fiber, coiled with minimum bend diameter of 28 cm, with one 75-mm diameter loop at each end of the fiber.

Many suppliers of fiber for use in the outside plant specify that the cable effective cutoff wavelength of their fiber is 1260 nm or less. Although it depends on the specifics of the fiber design and, therefore, varies considerably, typically the fiber effective cutoff wavelength is roughly 100 nm below the theoretical cutoff wavelength for many standard single-mode fibers.

Likewise, the cable effective cutoff wavelength is typically an additional 60–80 nm below the fiber effective cutoff wavelength for typical standard single-mode fibers.

The next tutorial introduces ** atomic energy levels and spontaneous emission**.