Optical Waveguides: Numerical Modeling
 

Dispersive Properties

Dispersion refers to any phenomenon in which the velocity of propagation of an electromagnetic wave is wavelength dependent. Therefore, dispersion plays a critical role in many applications that involve short pulse propagation in optical fibers. Owing to dispersion, different frequencies within the pulse spectrum propagate at slightly different velocities along the optical fiber. In the so-called normal dispersion regime, red frequency components propagate faster than blue components. In the anomalous dispersion case, blue components propagate faster. As a result, different frequencies arrive with different delays, leading to broadening and chirping of the output pulse.

Dispersion-induced broadening can be detrimental for fiber-optic transmission systems. As was schematically illustrated in Fig. 4 in the Introduction, at certain point, two neighboring pulses may overlap and their amplitudes may reduce to a level when they cannot be detected by a receiver at the end of the transmission system. On the other hand, in many cases, dispersive effects can be advantageous. For example, in conjunction with a nonlinear effect of self-phase modulation, fiber dispersion gives rise to a pulse that propagates without any temporal and spectral changes, called soliton.

There are different types of dispersion taking place in optical fibers, including a group-velocity and higher-order fiber dispersion (also called chromatic dispersion or intramodal dispersion), waveguide dispersion, and modal dispersion in multimode fibers.

On a fundamental level, the response of any dielectric medium is related to the resonance frequencies at which the medium absorbs the incoming electromagnetic wave and, as a result, the response is frequency dependent. Resulting chromatic dispersion reveals itself in the frequency dependence of the refractive index .

The effect of group-velocity dispersion (GVD) can be described by expanding the mode propagation constant in a Taylor series (11)

where m=1, 2, … (12)

In Eqs. (11)-(12), is inversely proportional to the group velocity, a velocity of the pulse envelope, and  describes the GVD responsible for pulse broadening. These terms can expressed as (13)

Higher order terms in Eqs. (11)-(12) may also be important, especially as the pulse width decreases; however, in many practical cases, the effect of the GVD dominates. In silica fibers, the GVD vanishes at around 1.3 μm. This wavelength is often referred to as a zero-dispersion wavelength. Therefore, it seems logical to design the transmission systems around this wavelength. Unfortunately, fiber losses are not minimized at 1.3 μm; however, they minimize at 1.55 μm. As a result, modern systems operate in the 1.55 μm transmission window.

Waveguide dispersion is usually relatively small in comparison to material dispersion in standard telecom fibers (except near the zero-dispersion wavelength) and its effect is to shift the zero-dispersion wavelength to slightly longer wavelengths. However, a significant advantage of the waveguide dispersion is that it depends on fiber design and, therefore, can be controlled at least to some degree. In particular, it was used to shift the zero-dispersion wavelength from 1.3 μm to 1.55 μm in so-called dispersion-shifted fibers. In addition, emergence of microstructured optical fibers opened new opportunities in waveguide dispersion design.

Finally, there is another dispersive effect that results from the difference in propagation velocities of light in the orthogonal principal polarization states of the fiber, called polarization mode dispersion (PMD). As a result of the PMD, different polarization components within the optical pulse arrive at different times, degrading the received optical signal.

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