# Techniques take differing measures of chromatic dispersion

**Kevin Lefebvre**

**Measurement of chromatic dispersion precedes any compensation process. Methods for measuring chromatic dispersion include the phase method, time-of-flight technique, soliton method, and interferometer.**

Increasing demand for bandwidth within optical networks requires higher data rates. This means that optical pulses are becoming closer together and optical fibers need to be able to address the high data rates. But as the pulses come closer together, the amount at which a pulse can spread before overlapping into an adjacent pulse becomes smaller.

A significant cause of intersymbol interference in high-data-rate optical networks is chromatic dispersion. Compensating for chromatic dispersion within an optical fiber is possible by several means. Before compensation is possible, however, chromatic dispersion measurement of the existing system is necessary.

In an ideal world, a laser light source at the input of an optical fiber would generate a monochromatic wave or pulse for transmitting data. A monochromatic wave is described by *B*exp[*j*(ωt - β*z*)] where *B* is the amplitude, ω (= 2π*f*) is the angular frequency and β is the propagation constant. The wave travels down the fiber with a velocity of *c/n* where *n* is the refractive index of the fiber and *c* is the speed of light. The propagation constant for a monochromatic wave is given by *n*ω*/c* where *n* is the refractive index of the optical fiber.

Real light sources, however, are not monochromatic but consist of a narrow spectrum of light. Furthermore, the interaction between an electromagnetic wave and the bound electrons of a material results in a frequency-dependent dielectric constant and a frequency-dependent refractive index. The combination of a finite-width source and the frequency dependence of the refractive index causes each spectral component of an optical pulse to travel at a different velocity down the fiber. This results in the narrow pulse spreading in time as the pulse travels down the fiber with the propagation constant, β(ω), of *n*(ω)ω*/c*. This spreading is defined as chromatic dispersion.

The effects of chromatic dispersion can be accounted for by expanding the propagation constant in a Taylor series around the center of the pulse spectrum as follows: β(ω) = β_{o} + β_{1}(ω - ω_{o}) + 0.5β_{2}(ω - ω_{o})^{2} + ... where β_{n} = (δ^{n}β/δω^{n})ω = ω_{o}. In the expansion, β1 is the inverse of the group velocity of the pulse and β_{2} is the group velocity dispersion, which is responsible for the spreading of the pulse.

Chromatic dispersion is usually defined as the dispersion parameter *D.* The dispersion parameter is related to the group velocity dispersion by *D* = δβ_{1}/δλ = -2π*c*β_{2}/λ^{2}. As mentioned, β_{1} is the inverse of the group velocity, which translates the dispersion parameter to *D* = δβ_{1}/δλ = *L*^{-1}δτ/δλ; essentially the chromatic dispersion is the rate of change of the group delay with wavelength. Therefore, measurement of the time delay of an optical pulse as a function of wavelength will yield the dispersion of the optical fiber (see Fig. 1).

To determine the overall effects of chromatic dispersion, the nonlinear Schrödinger equation should be solved for the system, including both the fiber and the source. The nonlinear Schrödinger equation accounts for nonlinear effects, chromatic dispersion, and attenuation of the fiber, as well as the shape of the input pulse.

**MEASUREMENT TECHNIQUES**

There are several methods for measuring chromatic dispersion. Each method has a particular fiber range and operating conditions in which the measurements will yield good results. For instance, in the phase-shift method, changes in the phase of a modulated pulse are measured and the accuracy of the phase method is determined by the design of the test equipment. The time-of-flight method is used to characterize fibers of lengths greater than several kilometers so that the small differences in the pulse delay at different frequencies can be measured. The soliton technique can be used to measure intermediate lengths but is questionable on the accuracy of measuring the dispersion slope. The interferometer method has excellent resolution but only for fiber shorter than 5 m.

**FIGURE 1. Plot of the relative group delay, along with the dispersion, demonstrates how different wavelengths of the same pulse travel at different velocities where wavelengths away from λ _{o} travel slower than λo.**

The phase of a wave traversing a fiber is related to the propagation constant as demonstrated previously. The change in the propagation constant as a function of wavelength will change the phase of the pulse where a phase shift in the frequency domain is equivalent to a time delay in the time domain. Therefore, measurement of a phase change of an optical pulse that has traveled through the fiber will result in a measurement of the time delay, as well as the dispersion of the fiber.

The two phase-shift methods for measuring chromatic dispersion are the modulated phase-shift method and the differential phase-shift method. Both of these methods measure the phase shift through the fiber as a function of frequency, and both methods also take advantage of the fiber delay causing a phase shift as the pulse traverses the fiber. Modulation of the intensity for both techniques is required.

Chromatic dispersion measured by the modulated phase-shift method (see Fig. 2, top) requires the amplitude of the input signal to be modulated by a reference signal and applied to the fiber under test. The transmitted signal is detected at the output of the fiber and the phase of the transmitted signal is compared to the reference signal used to modulate the input signal. The measured value for the modulated phase-shift method is the group delay corresponding to a wavelength interval. The group delay can then be calculated by equation set 1 (see Table 1). The chromatic dispersion can be calculated by taking the derivative of the group delay with respect to wavelength and dividing it by the wavelength. This measurement is repeated at different wavelengths.

The differential phase-shift method (see Fig. 2, bottom) differs from the modulated phase method in that it measures the dispersion directly instead of the group delay. Just as in the modulated phase technique, the amplitude is modulated. In the differential phase method, however, the wavelength is also modulated around a central wavelength where the group delay is to be measured. The detected signal not only has a phase difference but also varies with frequency. At the output end of the fiber, the transmitted signal is compared with the reference signal yielding the phase difference. To obtain the dispersion curve, the measurement is repeated for several wavelengths. The dispersion coefficient can be calculated from equation set 2 (see Table 1).

Accuracy of the phase method is dependent upon the system design. The phase of the pulse is sensitive to temperature, dispersion, modulation frequency, and the wavelength step. Care must be taken to ensure that the combination of the modulation frequency, dispersion, and the wavelength step does not result in a phase change that exceeds the capability of the phase detection equipment. Furthermore, dispersion measurements of narrow-band devices, such as Bragg gratings and multiplexers, require high wavelength-resolution to accurately measure the relative group delay in these fine structures.

The time-of-flight technique measures the time delay between two pulses at the end of the fiber. This technique requires a source that is tunable and has a narrow pulse width. The pulses are captured by a high-speed sampling scope and the time delay between the pulses is measured.

Calculating chromatic dispersion from the time-of-flight measurement utilizes a fitting equation for the time delay. A fitting equation requires several measurements of the time delay at different wavelengths. The number of wavelengths or measurements required is determined by the fitting equation chosen: the number of wavelengths equals the number of unknown constants (see Table 2). The coefficients of the time delays are calculated and the derivative of each time delay, with respect to wavelength, yields chromatic dispersion.

The accuracy of the time-of-flight method is not only determined by the number of wavelengths and fitting equation but is also a function of the dispersion, distance, and pulse width. A time delay between the pulses requires distances greater than several kilometers. Furthermore, temporal and spectral resolutions are limited by the source resolution. The accuracy of the measurement is also limited by the use of Raman systems for the generation of the pulse, which adds jitter and optical triggering instabilities.

The soliton method utilizes the competing effects of dispersion and self-phase modulation. When a pulse is transmitted down a fiber with sufficient energy, self-phase modulation is induced. This effect, combined with dispersion, causes wavelengths that are longer than λ_{o} to shift to longer wavelengths and wavelengths shorter than λ_{o} to shift to shorter wavelengths. This shift creates a soliton, provided that the fiber has a negative dispersion. The measured power spectrum results in a notch at λ_{o} that can be confirmed by solving the modified nonlinear Schrödinger equation. The power measurement can be performed from both ends of the fiber, yielding more accurate results.

The soliton measurement technique has been shown to be independent of the input wavelength, source power, and fiber length. For shorter fiber lengths, a higher input power is required to induce self-phase modulation sooner within the fiber. Measurement of the dispersion slope can be extracted from the width of the notch at lo; however, the width is a function of input power and could yield inaccurate results.

The interferometric technique for measuring dispersion utilizes the short coherence times that exist with partially coherent light. The resolution of this technique can be several orders of magnitude smaller than the time-of-flight technique. The experimental setup for this technique incorporates a Mach-Zehnder interferometer to split the incoming filtered white light into two beams. One beam is directed into the fiber under test while the other is directed into a reference fiber. The two beams are recombined at the output with a beamsplitter. The cross-correlation between the two beams depends on the time-delay difference of the reference and test beams.

The main drawback with the interferometric technique is the inability to measure larger fiber lengths.

**REFERENCES**

- G. P. Agrawal, Nonlinear Fiber Optics, Academic Press (San Diego, 1995).
- L. G. Cohen, IEEE J. Light. Tech. 5, 958 (1995).
- M. Stern et al., IEEE J. Light. Tech. 10, 1777 (1992).

**Kevin Lefebvre** is the director of engineering research for optical networks testing at GN Nettest, Optical Division, 6 Rhoads Rd., Center Green Building 4, Utica, NY 13502; e-mail kevin.lefebvre@gnnettest.com