Fiber_Optics_Physics_Technology

index structure. As soon as a fiber is bent, the circular symmetry is broken. And, as may be expected, then additional energy loss arises; bending losses are discussed in Chap. 5.2. Here we can already note this much:

The fiber core may be decisive for waveguiding, but it would be an oversimplification that the light power is guided in the core exclusively. We have seen that the field amplitude decays radially like an exponential function; this implies that there is always a certain fraction of power well outside the core. Di erent modes have di erent geometric field distributions; the fraction outside the core must therefore also be di erent for di erent modes.

The energy transport out of (or into) a certain volume element is described by the Poynting vector

The direction of propagation is perpendicular to the plane containing E and H. Disregarding anisotropic materials, all three vectors are pairwise orthogonal. The reader is reminded that by convention, the direction of polarization designates the direction of the oscillation of E (historically there was once a convention to refer to H, but that has long been obsolete).

In most cases, one describes the energy transport of a wave by giving its irradiance or intensity. By this, one means the temporal average of the instantaneous intensity

Iinst = E(t) H(t).

I gives power per unit area and thus has units of W/m2. Obviously I is the temporal average of the Poynting vector:

I = |S| .

When the wave propagates in nonmagnetic matter with real index n, this becomes

ˆ2

E .

Power P is the integral of intensity over the cross-sectional area s:

After a calculation which we will not show in detail here (see [50, 120]), one finds the energy fraction in the core (PK) and in the cladding (PM) for the fundamental mode:

Chapter 4

Chromatic Dispersion

A light signal propagating in an optical fiber is subject to a variety of ways in which it can get distorted. Many of these are based on di erent propagation velocities for di erent parts of the signal. After such distortion, there is a risk that the signal arrives at the receiver in such a mangled form that it may be impossible to correctly decipher it.

We already encountered one such mechanism: modal dispersion in multimode fibers. Now we will address such distortions as they arise in single-mode fibers.

At the center of explanation is the fact that the refractive index of the fiber glass, just like that of any other material, depends on wavelength (or frequency). No light signal is ever truly monochromatic; rather, it contains Fourier components from a certain spectral interval. In other words, a light pulse of finite duration by necessity has a nonzero spectral width. Di erent frequency components, however, will propagate with di erent velocity. This gives rise to di erential transit time and thus to signal distortion called delay distortion.

The wavelength dependence of the refractive index is behind three di erent contributions to delay distortion. They are collectively called chromatic dispersion. Individually, they are

Material dispersion. Dm: This contribution arises directly from the wavelength dependence of the index. Material dispersion is not specific to fibers but can be found in any bulk glass. It is independent of geometry and (given the material) depends solely on wavelength.

Waveguide dispersion. Dw: In the particular geometry of optical fibers, there is a modification to the di erential propagation time. The reader is reminded that we saw in Chap. 3.13 that the signal power is partitioned between core and cladding; the splitting ratio depends on the wavelength. On the other hand, core and cladding indices are slightly di erent. As the wavelength is varied, we have a crossover from mostly core index to mostly cladding index. The result is a contribution to the wavelength dependence of the e ective index.

Profile dispersion. Dp: Strictly speaking, the index di erence between core and cladding itself, and thus Δ, is also wavelength-dependent. (Core index and

c Springer-Verlag Berlin Heidelberg 2009

cladding index do not vary “in parallel.”) This gives rise to another correction which, however, is often much smaller than material and waveguide dispersion.

Another reason for dispersive distortions in single-mode fibers is related to the state of polarization of the light. As mentioned above, each mode can be decomposed into two mutually orthogonal parts. An ideal fiber has perfect circular symmetry; then both polarization states (polarization modes) propagate with identical velocities. However, real-world glass always has at least some residual birefringence; this implies a slightly di erent e ective index for both states. One can argue that the term “single-mode fiber” is a misnomer: Even when it is true that only a single geometric field amplitude distribution (LP01) can propagate, it still consists of two polarization modes. This is why in real fibers there is polarization mode dispersion. Typically it is a small contribution; we will discuss it further below.

To characterize dispersion, one normally specifies the size of the e ect per distance. For both modal and polarization mode dispersion, one can write the dispersion parameter

where δτ designates the di erence of propagation time after distance L. Units of ps/km are commonly used. For chromatic dispersion, including material, waveguide, and profile dispersion, the following specification is common:

Here, D contains the three parts just mentioned:

This dispersion parameter indicates the propagation time di erence per distance and per wavelength di erence; therefore, units of ps/(nm km) are commonly used.

4.1Material Dispersion

For any glass, the refractive index varies with wavelength. This gives rise to chromatic defects of lenses and the color-separating capability of prisms. For historical reasons, it became common practice (see, e.g., Schott glass catalog [11]) to characterize types of glasses by giving their indices at three wavelengths:

nD, the refractive index at wavelength 589.30 nm (yellow, Fraunhofer’s D line of sodium),

nF at wavelength 486.13 nm (blue–green, Fraunhofer’s F line of hydrogen), and

nC at wavelength 656.27 nm (red, Fraunhofer’s C line of hydrogen).

This choice was motivated by the availability of narrowband emission lamps at these wavelengths. As a further characterization, often the Abbe number is given, defined by

Obviously this is a metric of the wavelength dependence of the index near the central (yellow) wavelength.

Glass catalogs often specify only nD and vD. It is instructive, however, to survey the variation of the index over a wide spectral range. This is schematically shown in Fig. 4.1. There are absorption bands due to electronic transitions in the ultraviolet at wavelengths on the order of 100 nm and molecular vibrations in the infrared around 10 μm. In the vast interval in between, including all of the visible and near-infrared, pure silica glass does not exhibit any resonances. This is the reason, of course, why it appears “crystal-clear” to the eye. The position of the absorption resonances is reflected in the refractive index.

Figure 4.1: Schematic plot of refractive index vs. wavelength. It is dominated by resonances which occur both in the ultraviolet and the infrared.

Pure fused silica (SiO2) has a refractive index of nD = 1.456, decreasing slightly toward longer wavelengths. As long as one stays clear of the resonances, one can empirically describe the wavelength dependence with interpolation formulas. One of the best known such formulas is Sellmeier’s equation

but there are also variants to this. Coe cients Aj denote the resonance strengths and λj the pertaining wavelengths. These coe cients are tabulated in the literature (for various glass types, see [11], and for fibers with various doping materials and concentrations, see e.g., [22]). In most cases three terms in the sum are considered su cient; sometimes, five. Let us emphasize again that a Sellmeier curve is an empirical fit: the coe cients λj must not be construed to indicate the resonance wavelengths in a literal sense.

4.1.1Treatment with Derivatives to Wavelength

We now turn to the actually observed propagation times and the scatter thereof. Consider a plane monochromatic wave with angular frequency ω and wave number β. It is well known that it propagates with phase velocity

whereas the propagation of a signal, like a wave packet, is governed by the group velocity

Since β = nk0 = 2πn/λ, we can rearrange the first fraction in the last line:

or, since λω = 2πc,

We have now expressed the wavelength dependence of the group propagation time as a function of the easily measured quantities n and λ. We emphasize that we here find the bracketed expression taking the role of the usual index n which only appears in the first term of that expression. Therefore we call the bracketed expression the group index:

dn

ngr = n − λ dλ (4.14)

Figure 4.1 shows that throughout the visible and near infrared, n decreases with increasing wavelength; thus, in this range ngr > n. Figure 4.2 shows a comparison of both indices for fused silica.

The scatter of arrival times at the receiver is obtained from

1,48

ngr

n

1,45

Figure 4.2: Refractive index and group index as a function of wavelength, calculated from a three-term Sellmeier’s equation.

Herein, the derivative is

Ld2n

=− c λ dλ2 .

The contribution of the material dispersion Dm to the dispersion coe cient ,

In some cases the signal may occupy a quite broad spectral band. Then one must take into account that D varies with wavelength: Dispersion would not be described to su cient accuracy by D alone. In such case one can additionally specify the dispersion slope:

4.1.2Treatment with Derivatives to Frequency

An alternative terminology to describe dispersion arises when one takes derivatives not with respect to wavelength but with respect to frequency. One starts from a series expansion of the propagation constant β

β(ω) = n(ω) ωc = β0 + β1(ω − ω0) + 12 β2(ω − ω0)2 + ··· (4.19)

with

Let us assess the meaning of βm:

The bracketed expression is the group index ngr as introduced in Eq. (4.14), which implies that

This quantity is called group velocity dispersion parameter or GVD parameter. It is commonly given in units of ps2/km. The GVD parameter is preferred by theorists over the dispersion coe cient D which is widely used by technicians. To convert, one uses

and thus

There are some cases in which higher-order dispersion terms become relevant. Then one can also specify the third-order dispersion (TOD) β3. To convert between dispersion slope S and β3, one can use