**Michael Dickerson**

*Confirming modulator design and manufacturing tolerances requires measuring the amount of inherent chirp. Frequency discrimination and optical-fiber transfer are two accurate techniques.*

Once considered insurmountable problems, transmission impairments such as attenuation, chromatic dispersion, polarization-mode dispersion (PMD), and nonlinear effects have found solutions in optical amplifiers, dispersion compensating fiber, and other mitigating subsystems and devices. Frequency chirping and its effect on span distances and system design is among the transmission phenomena that are now understood.

Whether deliberate or a byproduct of modulation, frequency chirping is a change in the phase of an optical carrier signal that produces an instantaneous frequency change. It is generally encountered during the rising and falling edges of an optical pulse, and it serves to broaden the optical spectrum. When a positively chirped pulse is transmitted through fiber with anomalous dispersion (positive dispersion coefficient, D), the blue-shifted (shorter-wavelength) components of the pulse will travel faster than the red-shifted (longer-wavelength) components (see Fig. 1). As a result, the pulse broadens more quickly than if chirp were not present, reducing the maximum possible transmission distance. If the pulse experienced negative chirp or if the fiber was operated in the normal dispersion region, the pulse would initially narrow before broadening. In this manner, system designers can use chirp to their advantage or deliberately induce it to extend transmission distance.

Chirping is introduced during the process of encoding data on the carrier. The root cause of chirp varies with the modulator technology. When a laser is directly modulated by changing the drive current, the introduction of additional electrical carriers into the material changes its refractive index (n = c/v_{medium}). The light will thus slow down or speed up, resulting in a phase change and chirp. To eliminate the significant amount of chirp introduced by direct modulation, lasers are typically driven in continuous-wave (CW) mode and an external modulator device is used.

One such device is based on the electroabsorption phenomenon in semiconductors. Both the Franz-Keldysh effect and the quantum-confined Stark effect can be exploited to make these devices, but both work on the principle that the application of a small electric field to a semiconductor induces a large change in the absorption of the material at a specific wavelength. In addition, as the electric field moves the absorption edge, it causes a change in the refractive index of the material, which results in aphase change and chirping.

**CHIRP-FREE MODULATION**

Because the chirp in electroabsorption modulators tends to be high, an interferometric configuration can be used to achieve comparatively chirp-free modulation. In a Mach-Zehnder interferometer (MZI), light from a single input is split equally into two separate arms, with each arm traversing different paths. At the far end of the device, the two signals can be recombined in phase and the original source intensity reconstructed with a minor amount of propagation loss. However, if one path is longer (or delayed) by half of the wavelength, the two arms will combine out of phase and no light will emerge at the device output. The most widely used material for fabricating these types of devices is lithium niobate (LiNbO_{3}), in which phase modulation is provided by an applied electric field and the electro-optic effect.

The cause of residual chirp in MZI-based devices is an unbalanced phase change in the two arms of the MZI. At one voltage (v_{peak}), the light from the two arms will be perfectly in phase—they will interfere constructively and device transmission will be peaked. At another voltage (v_{null}), the interference will be destructive and no light will emerge. Chirp arises when the drive voltage is transitioning between v_{null} and v_{peak} and the instantaneous optical frequency of the two optical signals is slightly different. The interference is only partial and the resulting lightwave signal, although diminished in intensity, is at a slightly different frequency than the carrier. However, if the phase changes are equal in magnitude but opposite in sign, there will be no residual phase modulation, only intensity changes.

For X-cut devices, the electric field in the two arms of the structure is equal but opposite in sign. Therefore, even during the voltage transitions, there will be no residual phase change in the carrier. Because current production tolerances have an insignificant contribution to chirp, these devices are typically said to be "zero chirp, by design." In contrast, the electric fields of Z-cut devices are not equal in both arms and the value of chirp can be set by design (it is generally nonzero; see Fig. 2).

**MEASURING CHIRP**

Frequency chirp is commonly expressed in two formats—maximum frequency deviation and linewidth enhancement factor (a parameter). The frequency deviation is simply the deviation in the frequency of the carrier from the nominal center. The α parameter is defined as the phase change in the carrier for a given amplitude change of the carrier:

α _{=}*E**(dØ/dt)(dE/dt)* = 2 I *(dØ/dt)(dI/dt)* (1)

where *E, I, Ø,* and *t* are the electric field, optical intensity, optical phase, and time, respectively. For EA modulators, this expression reduces to:

α = Δ*n*_{r}*/*Δ*n*_{i} (2)

where *n*_{r}is the real index of refraction, responsible for optical-phase changes, and *n*_{i}is the imaginary index, responsible for absorption.

For MZI structures, the above equations become:

α = (Δ*n*_{1} + Δ*n*_{2})*/*(Δ*n*_{1} - Δ*n*_{2}), >(Δ*n*_{1}∝ *E _{1}* (3)

where *Δn _{i}* is the electro-optically induced index change in arm (

*i*), including sign, and is directly proportional to the applied field (

*E*) in that arm.

_{i}^{1}Equation 3 demonstrates how an X-cut device can be designed for zero chirp. For Z-cut devices, the magnitude of the applied electric field is not equal in each arm (larger under the positive electrode). As a result, these devices have a fixed, nonzero chirp.

Confirmation of the device design and manufacturing tolerances require the measurement of these low expected chirp numbers. Two methods are popular for measuring the a parameter of optical sources. The first technique makes use of a device to convert the chirp-induced phase modulation (PM) to amplitude modulation (AM).^{2, 3} The second technique uses a length of optical fiber to determine the α.^{4} Both methods, as well as several other techniques, are very good at measuring moderate- to high-chirp numbers, but the inherent error of some techniques increases significantly as the target chirp measurement decreases.

**FREQUENCY DISCRIMINATOR**

Unlike amplitude changes, it is much more difficult to directly measure phase modulation. Therefore, techniques must be used to convert chirp into something that can be more accurately observed and measured. One commonly used technique employs a frequency-dependent loss/gain element, or frequency discriminator, to convert small frequency modulations into equivalent amplitude modulation, which can then be measured using the usual array of test equipment (see Fig. 3). After the measurement is complete, the resulting AM signal can be used to mathematically determine the chirp.

Any frequency deviations due to chirping are directly converted to amplitude fluctuations as the carrier moves within this passband. To avoid extraneous noise, trace averaging, and therefore a pattern trigger, should be used. The data captured on the digital communications analyzer are composed of both an AM component caused by the pulse pattern generator (PPG) modulation plus an AM component caused by the chirp-induced FM. Once a trace is captured on the digital communications analyzer, the central wavelength of the discriminator is changed to allow a trace to be captured on the other slope of the discriminator passband (see Fig. 4).

The difference between these two traces is the chirp-induced AM (the PPG-induced AM is subtracted out). From this data, the instantaneous frequency deviation can be easily calculated by knowing the slope of the discriminator (m in nm for full-scale, off-to-on), the mean power in the data {[P_{+}(*t*) + P_(*t*)]/2}, the carrier wavelength (λ_{carrier}), and the speed of light (c), where P_{+}(*t*) and P_(*t*) are the optical power at time (*t*) of the waveform taken from the positive and negative discriminator slopes, respectively:

Δƒ(*t*) = [c *m* {P_{+}(*t*) - P_(*t*}]/

[4λ_{carrier}^{2}P_{avg}(*t*)]. (4)

Here, the maximum frequency deviation is 1.1 GHz. To obtain the a parameter, Equation 3 must be used.

**FIBER-TRANSFER FUNCTION**

Another common approach for measuring the α parameter is the vector network analyzer (VNA) technique, which makes use of a dispersive medium (optical fiber) to examine how the carrier and sidebands of a modulated signal move in and out of phase with each other (see Fig. 5). The resulting interference is a function of frequency with nulls described by:

ƒ_{u}^{s}*L*=(c/2DΔ^{2}) [1 + 2*u* - (/2π) rctan(α)] (5)

where ƒ_{u} is the frequency of the u^{th} resonance (or null), L is the length of fiber, *D* is the fiber dispersion coefficient, and a the chirp. Even in the absence of chirp, this interference will occur, but the addition of chirp serves to perturb the frequencies at which the resonances take place.

The function in Equation 5 is a linear function of ƒ*u*^{2}*L* and 2*u*. As a result, any two or more resonant frequencies can be plotted on graph of ƒ*u*^{2}*L* vs. 2*u* and a linear fit applied to determine *D* and α, using Equation 5. Actual measurements of the transfer function can result in significant error in the α parameter due to errors in the determination of the exact frequency of the resonance. The cause of this significant error lies in the tangent relationship used to determine the chirp. A small frequency error can translate into significant chirp variations (see Fig. 6). If the chirp parameter represents a guaranteed specification for a particular device, the use of this characterization technique could itself significantly affect device yield, causing otherwise good devices to be rejected.

**OTHER TECHNIQUES**

A number of additional, novel techniques may be used to measure frequency chirp. One such approach examines the optical spectrum of a modulated signal as passed through yet another modulator. Like the VNA technique, the error is acceptable for modest chirp, but becomes rapidly unacceptable for measurements of extremely low chirp devices.

Chirp in EA modulators can also be determined by measurement of the absorption spectra at the two operating conditions of the device and then examining a set of Kramers-Krönig relations, but this technique is not valid for MZIs.^{5} In addition, chirp should be measured directly from the emitted light since the modulated signal itself contains the chirp.

Finally, some commercially available chirp measuring systems examine the sideband-to-carrier ratio to directly determine the chirp; however, the sign of the chirp is unavailable in these systems.^{6} Additional techniques can be used to determine the sign, such as examination of the pulse broadening through a short section of fiber, but this additional step introduces needless complexity.

**Michael Dickerson** is a member of the technical staff at Codeon, 9112 Guilford Road, Columbia, MD 21046. He can be reached at mdickerson@codeoncorp.com.

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