# Dispersion Compensation with Dispersion-Equalizing Filters

Fiber gratings constitute an example a whole class of optical filters that can be employed for compensating dispersion in long-haul systems. In this tutorial, we consider several other dispersion-equalizing filters that may be made using fibers or planar waveguides. Such a compact optical filter can be combined with an amplifier module such that both the loss and the dispersion of optical fibers are compensated simultaneously in a periodic fashion. Moreover, an optical filter can also reduce the amplifier noise if its bandwidth is much smaller than the amplifier bandwidth.

#### 1. Gires-Tournois Filters

Any interferometer acts as an optical filter because it is sensitive to the frequency of input light by its very nature and exhibits frequency-dependent transmission and reflection characteristics. A simple example is provided by the Fabry-Perot interferometer. The only problem from the standpoint of dispersion compensation is that the transfer function of a Fabry-Perot filter affects both the amplitude and phase of passing light. A dispersion-equalizing filter should affect the phase of light but not its amplitude.

The problem is easily solved by using a Gires-Tournois (GT) interferometer, which is simply a Fabry-Perot interferometer whose back mirror has been made 100% reflective. The transfer function of a GT filter is obtained by considering multiple round trips inside its cavity and is given by

where the constant H0 takes into account losses, |r|2 is the front-mirror reflectivity, and Tr is the round-trip time within the filter cavity. If losses are constant over the signal bandwidth, |HGT(ω)| is frequency-independent, and only the special phase is modified by such a filter.

However, the phase φ(ω) of HGT(ω) is far from ideal. It is a periodic function, peaking at frequencies that correspond to longitudinal modes of the cavity. In the vicinity of each peak, a spectral region exists in which phase variations are nearly quadratic in ω. The group delay, tg = dφ(ω)/dω, is also a periodic function. The quantity φ2 ≡ dtg/dω, related to the slope of the group delay, represents the total dispersion of the GT filter. At frequencies corresponding to the longitudinal modes, φ2 is given by

As an example, for a 2-cm-thick GT filter designed with r = 0.8, φ2 ≈ 2,200 ps2. Such a device can compensate the GVD acquired over 110 km of standard fiber.

Several experiments have shown the potential of GT filters as a compact dispersion compensator. In a 1991 experiment, such a device was used to transmit a 8-Gb/s signal over 130 km of standard fiber. The GT filter had a 1-mm-long cavity with 70% reflectivity for the front mirror. The relatively high insertion loss of 8 dB was compensated by using an optical amplifier. However, 6-dB losses were due to the 3-dB fiber coupler that was used to separate the reflected signal from the incident signal. This amount can be reduced to below 1 dB using an optical circulator. The micro-electro-mechanical system (MEMS) technology has also been employed for fabricating a GT filter whose cavity length can be adjusted electronically.

In another approach, two mirrors of the GI filter are replaced with two fiber gratings, one of which is made nearly 100% reflective. The two gratings can even overlap physically, resulting in the so-called distributed GT filter. The figure below shows schematically the basic idea behind such a device together with the measured reflectivity, group delay, and dispersion as a function of wavelength for a device consisting of a 1-cm-long grating with 98% reflectivity and another 6-mm-long grating with 11% reflectivity. Whereas the reflectivity is nearly constant over the 20-nm spectral window, group delay and dispersion exhibit a periodic pattern. The 50-GHz channel spacing results from a 2-nm shift in the Bragg wavelength of the two gratings.

A GT filter can compensate dispersion for multiple channels simultaneously because, as seen in the first equation, it exhibits a periodic response at frequencies that correspond to the longitudinal modes of the underlying Fabry-Perot cavity. However, the periodic nature of the transfer function also indicates that φ2 is the same for all channels. In other words, a GT filter cannot compensate for the dispersion slope of the transmission fiber without suitable design modifications. Several schemes have been proposed for dispersion slope compensation. In one approach, two or more cavities are coupled such that the entire device acts as a composite GT filter. In another approach, GT filters are cascaded in series. In a 2004 experiment, cascaded GT filters were used to compensate dispersion of 40 channels (each operating at 10 Gb/s) over a length of 3,200 km.

Another approach employs two grating-based distributed GT filters, which are cascaded using a circulator as shown in the figure below.

This figure also shows schematically the basic idea behind the dispersion slope compensation. A four-port circulator forces the input WDM signal to pass through the two filters in a sequential fashion. Two GT filters have different device parameters, resulting in group-delay profiles whose peaks are slightly shifted and have different amplitudes. This combination results in a composite group-delay profile that exhibits different slopes (and hence a different dispersion parameter D) near each peak. Changes in D occurring from one peak to the next can be designed to satisfy the slope condition in the first equation by choosing the filter parameters appropriately.

#### 2. Mach-Zehnder Filters

A Mach-Zehnder (MZ) interferometer can also act as an optical filter. Such a fiber-based device can be constructed by connecting two fiber couplers in series. The first coupler splits the input signal into two parts, which acquire different phase shifts if optical path lengths are different, before they interfere at the second coupler. The signal may exit from either of the two output ports depending on its frequency and the arm lengths. In the case of 3-dB couplers, the transfer function for the cross port is given by

where τ is the extra delay in the longer arm of the MZ interferometer.

A single MZ interferometer is not suitable for dispersion compensation. However, it turned out that a cascaded chain of several MZ interferometers acts as an excellent dispersion-equalizing filter. Such filters have been fabricated in the form of a planar lightwave circuit (PLC) using silica waveguides on a silicon substrate. Figure (a) below shows schematically a specific circuit design.

This device was 52 x 71 mm2 in size and exhibited a chip loss of 8 dB. It consists of 12 couplers with asymmetric arm lengths that are cascaded in series. A chromium heater is deposited on one arm of each MZ interferometer to provide thermo-optic control of the optical phase. The main advantage of such a device is that its dispersion-equalization characteristics can be controlled by changing the arm lengths and the number of MZ interferometers.

The operation of the MZ filter can be understood from the unfolded view shown in figure (b) above. The device is designed such that the higher-frequency components propagate in the longer arm of the MZ interferometers. As a result, they experience more delay than the lower-frequency components taking the shorter route. The relative delay introduce by such a device is just eh opposite of that introduced by a standard fiber exhibiting anomalous dispersion near 1.55 μm. The transfer function H(ω) can be obtained analytically and is used to optimize the device design and performance. In a 1994 implementation, a planar lightwave circuit with only five MZ interferometers provided a relatively delay of 836 ps/nm. Such a device is only a few centimeters long, but it is capable of compensating dispersion acquired over 50 km of fiber. Its main limitations are a relatively narrow bandwidth (~10 GHz) and sensitivity to input polarization. However, it acts as a programmable optical filter whose GVD as well as the operating wavelength can be adjusted. In one device, the GVD could be varied from -1,006 to 834 ps/nm.

It is not easy to compensate for the dispersion slope of the fiber with a single MZ chain. A simple solution is to demultiplex the WDM signal, employ a MZ chain designed suitably for each channel, and then multiplex the WDM channels back. Although this process sounds too complicated to be practical, all components can be integrated on a single chip using the silica-on-silicon technology. The following figure shows the schematic of such a planar lightwave circuit. The use of a separate MZ chain for each channel allows the flexibility that the device can be tuned to match dispersion experienced by each channel. By 2008, the use of a lattice-form filer allowed tuning over a range of ±500 ps/nm.

#### 3. Other All-Pass Filters

It is possible to design several other types of filters that affect the signal phase but leave the signal amplitude intact. Such filters are known as all-pass filters (as they pass all of the optical power incident on them) and have attracted considerable attention within the context of dispersion compensation. A ring resonator constitutes a simple example of an all-pass filter. Indeed, ring resonators have been employed for this purpose since 1998.

The following figure shows schematically three designs that use directional couplers and phase shifters to form a ring resonator. Although a single ring can be employed for dispersion compensation, cascading of multiple rings increases the amount of dispersion. More complicated designs combine a MZ interferometer with a ring. The resulting device can compensate even the dispersion slope of a fiber. Such devices have been fabricated using the silica-on-silicon technology. With this technology, the phase shifters are incorporated using thin-film chromium heaters. One such device exhibited dispersion that ranged from -378 to -3,026 ps/nm depending on the channel wavelength.

In general, all-pass filters such as those shown in the figure above suffer from a narrow bandwidth over which dispersion can be compensated. The amount of dispersion can be increased by using multiple stages but the bandwidth is reduced even further. A solution is provided by the filter architectures shown in the figure below in which the WDM signal is split into individual channels using a demultiplexer. In figure (a) below, an array of dispersive elements, following by delay lines and phase shifters, is used to compensate for the dispersion of each channel. Individual channels are then multiplexed back. Configurations (b) and (c) simply the design using a single mirror or an array of movable mirrors. Such designs, although more complicated, provide the most flexibility.

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