Living with Dispersion in Fiber Optic Systems

Modern telecommunication systems transmit information by modulating the intensity of a light source; information is conveyed as a series of pulses representing binary encoded data. As long as these pulses travel through the fiber without changing their shape, the data can be transmitted with few errors. But if the pulses broaden as they travel along the fiber and start to spread and overlap each other, the data can become seriously corrupted.

Dispersion is the general term applied to this problem. Dispersion has several causes, and there are different ways to mitigate its effects.

Monochromatic light travels through a fiber with propagation constant, b. The propagation constant is fundamental to understanding how the optical fiber transmits light and is given by:

In equation 1, neff is the effective index of refraction and l is the wavelength of the light in a vacuum. The effective index depends on such characteristics as the fiber's material properties and the size and shape of the core/cladding regions, as well as any stress-causing birefringence.

(Equation 1)

When you think of a traveling wave, you probably associate the velocity of the wave with the rate at which wavefronts of a constant phase travel along the fiber (see Figure 1). The velocity of these wavefronts is called the phase velocity, and is given by:

In equation 2, v is the optical frequency in radians per second.

(Equation 2)

Another important concept is the velocity with which a pulse (which, according to the Fourier theory, is composed of many different sine waves of various amplitudes and phases) travels through the fiber (see Figure 1). The velocity at which a pulse (or a group of sine waves) travels along the fiber is called the group velocity. In non-dispersive media, the phase velocity equals the group velocity. However, in dispersive media (like optical fibers) the group velocity differs from the phase velocity and is given by:

In optical fibers both the propagation constant and the group velocity depend on the wavelength of the light and the state of polarization. When the group velocity depends on the wavelength of light, the fiber has chromatic dispersion (CD). When the group velocity depends on the state of polarization, the fiber has polarization mode dispersion (PMD).

Imagine pulses of light traveling along an optical fiber, and suppose these pulses are composed of different wavelengths. This happens with real-world transmitters, which always have finite optical bandwidths. The very act of modulating a lightwave introduces additional frequencies (wavelengths). Because of dispersion, these different wavelengths travel through the fiber with different group velocities. Consequently, after traveling through the fiber the pulses become spread out due to CD.

Figure 2 shows square pulses with a range of wavelengths entering an optical fiber. After traveling through the fiber, these pulses spread out and become attenuated, with the shorter-wavelength components at the leading edge of the pulse and the longer-wavelength components at the trailing edge. Depending on the wavelength of the transmitter, and its relationship to the wavelength of zero dispersion, it's also possible for the red wavelengths to lead the blue wavelengths. Either case results in dispersion.

CD in single-mode fibers results from two things. First, there is material dispersion, due to the wavelength-dependent index of refraction in the bulk material, which makes up the fiber. It is this property that Newton exploited when he conducted his experiments with glass prisms, showing that white sunlight is actually made up of many different colors.

The other source of CD is more complicated. The ray approximation typically shows little squiggly lines bouncing off the core-cladding interface due to total internal reflection. In single-mode waveguides, this over simplified view is incorrect because a significant amount of the optical power is actually carried in the cladding, as well as in the core.

Furthermore, solution of Maxwell's equations shows that the diameter of this mode increases with increasing wavelength (see Figure 3). Thus, the fundamental mode in standard single-mode fiber is slightly larger at 1550 nm, for example, than at 1310 nm, and this causes it to see a slightly different effective group index. Thus, the total effective group index is a function of material dispersion, as well as the geometry of the core/cladding region.

No differentiation

Usually, the CD specification for fiber doesn't distinguish between waveguide CD and material CD. Instead, both effects are included in a combined specification that describes how the group velocity of light varies with wavelength in the fiber.

The impact of CD depends on the bandwidth of the transmitter, with wider-bandwidth sources being affected by CD more than sources with narrower optical bandwidths. It also depends on the length of the fiber. Fiber manufacturers typically specify CD in units of:

To determine the amount of pulse broadening, multiply the specification by the spectral bandwidth of the source, and the length of the optical fiber. Obviously, one approach to mitigating the effects of CD is to use narrow-bandwidth transmitters, such as distributed feedback (DFB) lasers.

Typical measurements of CD are made in the time domain, either by measuring the relative group delay for pulses of different wavelengths directly, or by measuring the phase delay of an intensity-modulated source.¹ Often, these tests measure the group delay at several different wavelengths, and then interpolate to estimate the CD over a continuous span of optical frequencies.

CD is a bulk property of optical fiber, and can be considered stable over time. This means that if the total CD of a particular section of optical fiber is too large, a passive component with an equal but negative amount of CD can be used to compensate. Alternate techniques for compensating CD involve chirping the pulses, so that the wavelengths that travel fastest start out at the end of the pulse. Then, because of CD, the faster wavelengths overtake the slower ones, so at the end of the fiber the chirp is reversed, but the pulse shape remains intact.

The stable nature of CD makes it straightforward to deal with, if not exactly easy to compensate. When the fiber has birefringence, however, the resulting PMD is not so easily dealt with. PMD is something that happens in birefringent fibers, where the group velocity is a function of the state of polarization (see Figure 4).

Birefringence and DGD

Birefringence results from several possible factors, all of which somehow introduce stress into the fiber. Sometimes this stress is deliberate, as in the case of polarization-maintaining fibers. More often, the effect is deleterious and results from unwanted asymmetries in the fiber and/or external stresses.

Two axes, called the principal states of polarization, characterize a fiber with PMD. These two states of polarization (which are orthogonal) define the endpoints of an axis through the center of the Poincare sphere. Light polarized along either of the principal states of polarization will emerge from the other end of the fiber in the same state of polarization in which it was launched. The polarization dispersion vector (PDV) is defined as a vector originating at the center of the Poincare sphere and pointing toward one of the principal states of polarization. The vectors length is equal to the differential group delay (DGD). The principal axis toward which the PDV points is the one for which the PDV rotates in a counterclockwise direction with increasing optical frequency.

CD is typically a much larger impairment than PMD, and can be a significant effect even at relatively low data rates on long fibers (depending on the transmitter type). For most fibers, PMD is not a serious problem until data rates exceed 10 Gbps. Unlike CD, however, PMD on long fibers cannot be corrected with passive devices because the PMD changes randomly over time. This means that both the amount of DGD changes, and the orientation of the principal axes. It's this dynamic characteristic of PMD that makes it such a difficult problem for high-speed optical networks.

One way of dealing with PMD is to actively compensate it. Another approach is to use soliton pulses. Recent demonstrations of both these approaches have been impressive. Still, these approaches add expense, and even with active compensation there are limits to the amount of DGD that can be corrected. So, it's still important in many circumstances to measure the DGD in components and fiber, to ensure that the combined effect will be within system specifications.

Time and frequency

Techniques for measuring PMD are broadly classified into two categories: Time domain and frequency domain. In addition to these, there are other techniques based on measuring the polarization DGD as a function of the amount of optical power in nonlinear mixing products, and some techniques that measure the changing state of polarization in the backscatter of optical time-domain reflectometers.

Time-domain techniques measure the DGD directly. Figure 4 illustrates one way of measuring DGD. Short pulses of polarized light are injected into the device under test (DUT) along the two principal axes, and the DGD is measured with fast optical detectors. While straightforward, this technique requires measurements in the time domain with picosecond accuracy, techniques for finding the principal axes, and the generation of short optical pulses.

Another common time-domain technique measures the phase shift in an RF-modulated polarized lightwave. This modulated light is input to the device along the principal axes, and the DGD is measured as a function of the phase difference. Variants of this technique allow CD to be measured at the same time, and both types of dispersion to be characterized over wavelength.

Yet another type of time-domain technique uses a broadband light source in an interferometric configuration. The instrument physically scans a mirror in one of the legs in the interferometer and is able to measure the DGD by applying numerical algorithms to the resulting interferogram.

The third type of measurement technique involves swept or tunable lasers. These are called frequency-domain techniques, and they all involve three things:

• Transmitting polarized light through the DUT at a given wavelength

• Measuring some physical parameter that is a function of the state of polarization of the light transmitted through the DUT

• Changing the optical wavelength by a known amount, and repeating steps 1 and 2.

Post-test processing analyzes the data acquired, and calculates the DGD (as a function of wavelength) based on how the measurements in step 2 change when the wavelength changes. The different frequency-domain techniques differ in the type of physical parameter(s) they measure (in step 2) and the mathematical algorithms they use.

Imagine light passing through a device with PMD. Upon leaving the DUT the light has a specific state of polarization. If we change the wavelength of this light (but not the input state of polarization), the output state of polarization changes. Now imagine tracing the movement of the state of polarization on the Poincare sphere as the wavelength changes. For stable devices we find that the state of polarization traces out an arc on the sphere as it rotates around the principal axes. For non-stable occurrences, like long lengths of fiber, the state of polarization moves over the Poincare sphere in a squiggly, random manner.

More methods

There are several different methods of measuring the DGD by measuring quantities related to changes in polarization as a function of wavelength. The most common of these is the fixed-analyzer method. Two other examples of frequency-domain measurements are the Jones-Matrix-Eigenanalysis method and the Poincare-arc method.

The Jones-Matrix-Eigenanalysis method uses three defined input states of polarization to the DUT (linear polarization at 0, 45, and 90 degrees) and a polarimeter at the output.^2,3 From this setup the system is able to measure changes in the Jones matrix of the DUT as a function of wavelength. It then determines the DGD from the following equation:

(Equation 4)

In equation 4, r1 and r2 are the eigenmodes of the measured Jones matrices (Equation 4.5). The two-element Jones vector describes the geometric and phase information in polarized light. The Jones matrix is a means of representing the polarization characteristics of different optical components as two-by-two matrices. The Jones Calculus is a means of using Jones vectors and matrices in mathematical equations to predict the state of polarization of light after passing through various types of polarizing components. ^4,5

The Poincare arc method uses a polarimeter to measure directly the changing state of polarization transmitted through the DUT as a function of the wavelength. It then calculates the DGD from the following formula:

In equation 5, is the rotation of the Stokes vector around the principal states axis, and is the change in optical frequency (in radians) responsible for this rotation.

Advantageous approaches

An important advantage of frequency domain techniques is that they can readily provide measurements of DGD over wavelength, especially in swept-wavelength test systems used to measure wavelength-dependent transmission and reflectivity characteristics of optical components. They offer the potential for full device characterization in a single test setup. However, since frequency-domain techniques involve division by they require extreme accuracy in the wavelength measurements of the tunable or swept lasers used in the test configuration.

For example, suppose the DGD of a device is nominally 10 ps, and that we wish to measure the DGD at wavelength intervals of every 50 picometers (pm). Given this amount of DGD, the Stokes vector will nominally move 22.5 degrees as the wavelength changes by 50 pm. If the wavelength error is plus/minus 10 pm, however, the resulting measurement of the DGD calculation (assuming no error in measuring the angle of the Stokes vector) will be 4.2 ps, or roughly 42% of the nominal DGD. Reducing the wavelength error to 1 pm would reduce the DGD error to less than 0.5 ps.

A designer can obtain picometer accuracy in stepped systems by using existing wavelength meters, though swept systems (in which the laser sweeps at thousands of pm per second) require a more specialized swept-wavelength meter. ⁶

(Equation 5)

From equation 5 errors in the DGD measurement are directly proportional to errors in measuring the change in the Stokes vector over a given change in the wavelength. For the Jones matrix method, it becomes equally important that the input states of polarization are truly linear and spaced 45 degrees apart. Assuming that these requirements can be met, frequency domain measurements of the DGD can provide sufficient accuracy and resolution to meet PDL measurement requirements for data rates up to 40 Gbps and higher. The advantage of having transmission and reflectivity measurements in a single compact test configuration are significant.

As data rates continue to increase, dispersion management will become an even more important requirement for system designers. Active PMD compensation and channel-specific CD compensation will become the norm for high-speed WDM systems. Test equipment will be necessary, not only for verifying system performance, but also for verifying the performance of individual components that go into these high-speed photonic networks.

1. EIA/TIA FOTP-169. 1992. Chromatic Dispersion Measurement of Singlemode Optical Fibers by the Phase Shift Method. Washington, DC: Telecommunications Industry Association.

2. TIA/EIA FOTP-122, 1996. Polarization-Mode Dispersion Measurement for Singlemode Optical Fibers by Jones Matrix Eigenanalysis, Washington, DC: Telecommunications Industry Association.

3. Heffner, B.L., "Automated Measurement of Polarization Mode Dispersion Using Jones Matrix Eigenanalysis," IEEE, Photonics Technology Letters, No. 9, Vol. 4 (September 1992), pp. 1066-1069.

4. Jones, R. C., A New Calculus for the Treatment of Optical Systems, J. Opt. Soc. Am., 31, 488-503, July 1941.

5. Jones, R. C., A New Calculus for the Treatment of Optical Systems, J. Opt. Soc. Am., 32, 486-493, Aug. 1941.

6. . Anderson, D., "Real-Time Wavelength Calibration with Picometer Accuracy in Swept-Laser Systems," National Fiber Optics Engineers Conference, Baltimore, MD., July 2001.