# Measurement of Modulator Chirp Factor and Fiber Dispersion

## Overview

The chirp factor of a modulator is defined as the ratio between the modulator output frequency modulation and amplitude modulation [1]:

where θ and Eout are phase and amplitude of electric field of the optical signal, respectively.

In an ideal Mach-Zehnder modulator with perfect power split between arms and driven with opposite voltages the chirp factor can be assumed equal to zero. There can be other sources of chirp such as asymmetry of construction between the upper and lower arms and applied reverse bias at the phase-shifters.

## Chirp Factor Measurement

The chirp factor as defined above is used in the industry to characterize spurious frequency modulation at the output of the modulator and can be measured directly, see for instance [2].

Modulator’s chirp factor and chromatic dispersion of the fiber are two of the parameters responsible for setting an upper limit of the fiber-optic system performance. Interplay between fiber nonlinearities and dispersion also impact the system performance. Both – modulator chirp factor and fiber dispersion - are input parameters for the modulator and nonlinear fiber models in Synopsys OptSim. These parameters are typically supplied by the manufacturers and easily available from the datasheets.

However, there are times when a user may want to measure chirp factor and fiber dispersion in response to design changes or operating conditions, such as temperature, reverse bias or mismatch in modulator arm-lengths. In this application note, we demonstrate a measurement method [2] that is applicable for measuring for both the chirp factor and the dispersion.

## Simulation Setup in Synopsys OptSim

The simulation setup for the measurement of modulator chirp (if dispersion is known) and fiber dispersion (if chirp factor is known) is shown in Figure 1.

Figure 1:  Schematic of the measurement setup for modulator chirp factor and fiber dispersion

The setup measures small-signal frequency response of the modulator, fiber and detector as described in Ref[2]. The detector is an ideal, noise-less photodiode with responsivity of 1 A/W. The fiber is 100-km long, and the nonlinearity in the fiber is ignored. A C-band laser (1547-nm wavelength) is used in the setup although the method is also applicable to O- and L-bands. The fiber dispersion is 16.8 ps/nm.km.

## Operating Principle of the Measurement Setup

The small-signal analysis of the setup reveals that the resonance frequencies fu in detected spectrum follow the following law [2]:

where:
α is modulator chirp factor, fu is frequency of the uth null, L is length of the fiber, D is the dispersion parameter of the fiber, λ is center wavelength of the laser, and c is the speed of light.

The modulator chirp factor α and fiber dispersion D are related to the first null (i.e., u=0) of the transfer function:

The above expressions can be used to calculate modulator chirp factor if dispersion parameter is known, and dispersion if the modulator chirp factor is known.

## Discussion and Results

In this application note, we (i) perform a parameter scan over modulator chirp factor α, (ii) observe the first resonance frequency of the transfer function, (iii) use it to calculate the values of the chirp factor, and (iv) compare them with the scanned values.

Click on the “Scan” button to run the simulation. After the simulation is over, double-click the spectrum analyzer to view RF spectra for different values of modulator chirp factor (Fig. 2).

Figure 2:  Detected signal spectra for different values of modulator chirp factor

As can be seen in the spectra of Fig. 2, the first resonance frequencies for chirp factors 0, 2 and 4 respectively are 6.12-GHz, 7.98-GHz, and 8.3-GHz. Using these values to calculate chirp factor gives us Table 1.

 Observed Null Scanned Value of α Calculated Value of α 6.12-GHz 0 0.0073 7.98-GHz 2 2.02 8.3-GHz 4 4.1

Table 1:  Calculating modulator chirp factor from observed first frequency-nulls

As we can see, values of the modulator chirp factor used in the simulation match very well with the calculated values of the chirp factor using observed resonance nulls in the analytical expression [B].

In this application note, we used a pre-supplied library model of the Mach-Zehnder modulator (MZM) in the simulation setup. Instead, an MZM can also be constructed from two phase shifter models, each using effective index vs. applied voltage data files in order to observe the effect of bias voltage on the modulator chirp factor. A slight mismatch in the MZM arm-lengths can also cause chirping.

The same simulation setup and measurement method can be used to measure dispersion of the fiber using equation [C] when the modulator chirp factor is known.

## References

1. F. Koyoma, and K. Iga, “Frequency Chirping in External Modulators,” J. Lightwave Technol., vol. 6, pp. 87-92, 1988.
2. F. Devaux, Y. Sorel, and J. F. Kerdiles, “Simple Measurement of Fiber Dispersion and Chirp Parameter of Intensity Modulated Light Emitter,” J. Lightwave Technol., vol. 11, pp. 1937-40, 1993.