## Abstract

An incoherent photonic microwave filter implementing multiple positive and negative coefficients based on two sets of optical carriers and dispersive media is proposed and demonstrated. Positive and negative coefficients are obtained thanks to the π phase inversion in a single electro-optic Mach-Zehnder modulator as well as the modulator V_{π} dependence with wavelength. To show the feasibility of this technique to implement practical filter transfer functions, the Parks-McClellan algorithm has been used to design a 5-tap flat bandpass filter. Experimental results show an excellent agreement with theory.

© 2006 Optical Society of America

## 1. Introduction

The use of optical techniques to process microwave and millimeter-wave signals has attracted great interest because of advantages such as large time-bandwidth products, immunity to electromagnetic interference, reduced size and weight and low and constant electrical loss. Among the applications of optical signal processing of microwave signals, the implementation of microwave filters in the optical domain has been a very active area of research Refs. [1–4].

Most of the proposed photonic microwave filter architectures work under incoherent regime since incoherent optical processing is insensitive to environmental conditions. However, in incoherent systems the filter coefficients (taps) are always positive. Since the number of transfer functions arising when using positive coefficients is quite limited Ref [1], negative coefficients are required to obtain the synthesis of practical passband transfer functions. To overcome this limitation, several schemes which obtain the equivalent to negative coefficients have been proposed Ref. [5–9].

Recently the authors have proposed an all-optical photonic microwave notch filter with negative coefficients based on the π phase inversion in a single electro-optic Mach-Zehnder modulator and the modulator V_{π} dependence with wavelength Ref. [9].

In this paper, a multi-tap photonic microwave filter is implemented using a similar all-optical technique which avoids the limitations of microwave components and the fully exploitation of the advantages of photonic technology. In addition, the availability of multi-tap filters allows the implementation of high performance transfer functions based on conventional discrete filter design theory.

## 2. Principle of operation

The principle of the technique is based on the π phase inversion which happens in Mach-Zehnder Modulators (MZM) when two transfer function slopes of different sign are used. As can be seen in Fig. 1, there are two linear modulation regions with opposite slopes centred at
*V _{bias}*

^{+}or

*V*

_{bias}^{-}. If a modulating microwave signal is applied to the MZM at each one of the former bias points, an optical modulated output is obtained with the same average power and with the same phase or in counterphase (π radians shift) depending on the bias point chosen. Therefore, photonic microwave filters with negative coefficients can be implemented if the input microwave signal is modulated over both slopes. This can be done, as in Ref. [8], using two MZMs each one biased in a different slope. However, the synchronization of both modulating signals can be difficult when high frequency modulating signals are used. In addition, both MZMs have to have the same transfer function to avoid the amplitude dependence of the coefficients with frequency.

A different approach that avoids these problems is possible. It is based on the MZM half-wave voltage (V_{π}) dependence on wavelength as given by Eq. (1) (Ref [10]).

where *λ* is the wavelength, *n _{0}* is the optical index of the active layer at zero applied voltage,

*r*is the relevant electro-optical coefficient,

_{ij}*d*is the spatial gap between the electrodes across which the voltage is applied,

*γ*is the optical confinement factor and

*L*is the modulation length.

This dependence causes a shift in the MZM transfer function if different wavelengths are used. This shift is slight for the usual wavelength spacing used in WDM systems but if optical carriers around 1300 nm and 1550 nm are used simultaneously, this effect has to be taken into account, since the wavelength spacing (around 250 nm) is considerable.

Figure 2 shows that biasing the MZM at -0.6 V, the microwave signal modulated over the optical carrier around 1300 nm will suffer a π phase inversion (negative slope) over the same microwave signal which is modulated over the optical carrier around 1550 nm (positive slope). It allows the all-optical implementation of a photonic microwave filter with negative coefficients using a single modulator. In Ref. [9], this principle of operation was proposed for the implementation of a photonic notch filter. Notch filters can be suitable for some applications but, in general, multi-tap filters are by far more interesting.

## 3. Multi-tap photonic microwave filter architecture

A multi-tap photonic microwave filter can be implemented using the technique described in section II and using two sets of optical carriers centered around 1300 and 1550 nm (Fig. 3). One of the sets will correspond with the filter positive taps whereas the other set will implement the negative ones, using each one different slopes of the modulator transfer function. The time delay between the filter coefficients (optical carriers equally spaced, Δλ) will be obtained by means of dispersive elements. In particular, a standard single mode fiber (SSMF) can be used as dispersive element for the set of optical carriers around 1550 nm and dispersion shifted fiber (DSF) as dispersive element for the carriers around 1300 nm.

Since there is a large wavelength spacing between the two sets of optical carriers (Δλ_{a}), dispersion introduces a large time delay between the two sets of optical carriers. This time delay has to be compensated to obtain a multi-tap photonic microwave filter. Therefore, the two sets of optical carriers have to be split using a low cost coarse demultiplexer (CWDM demux) and a time delay has to be introduced to one branch to compensate the dispersive time delay between sets of optical carriers.

The time delay caused by dispersion between two optical carriers (λ_{1},λ_{2}) have to be calculated using Eq. (2) (Ref. [9]) due to the large wavelength spacing between optical carriers and the dependence of the dispersion parameter, *D*, with wavelength :

where L is the fiber coil length, λ_{0} is the zero-dispersion wavelength and S_{0} is the dispersion slope at λ_{0}.

The multi-tap photonic microwave filter architecture is shown in Fig. 4. The two sets of optical carriers are amplitude modulated in a single Mach-Zehnder modulator. The number of optical carriers of each set (N,K) determines the number of positive and negative coefficients.

Once the signals have been amplitude modulated by the RF signal, the two subsets of carriers are demultiplexed and launched to a dispersive medium. One of the branches is time delayed using a non-dispersive optical delay line to compensate the time delay between the two sets of optical carriers. Then, both sets are combined and photodetected.

## 4. Experimental results

To show that the technique described here can be used to obtain practical filter transfer functions, the Parks-McClellan algorithm (Ref [11]) has been used to design a photonic microwave filter of five coefficients [-0.08 -0.3 0.61 -0.3 -0.08]. This coefficient distribution translates on a single optical carrier at second window (1307.6 nm) and four at the third window ([1548.3 1549.1 1550.7 1551.5] nm). The output power of the lasers has been windowed accordingly to the amplitude weights obtained theoretically. This coefficient distribution does not require the use of DSF since there is just one positive coefficient.

The experimental configuration used to show the feasibility of the architecture is shown in Fig. 5.

Five optical carriers are combined and amplitude modulated by a LiNbO_{3} Mach-Zehnder modulator. The dispersive medium is a coil of 10 km of standard single mode fiber. Once the optical carriers have been delayed by chromatic dispersion the two sets of optical carriers (around 1300 and 1550 nm) are demultiplexed using a coarse WDM demultiplexer. The time delay due to dispersion between the two sets of optical carriers (*ΔT _{a}*) is compensated by means of an absolute time delay between both sets of carriers. In order to introduce this delay, a fiber patch cord and two optical delay lines have been used. The time delay needed is calculated using Eq. (2). The λ

_{0}and S

_{0}parameters of the fiber have been measured using an optical network analyser. The parameters are λ

_{0}=1314.23 nm y S

_{0}=91.05 fs/(nm

^{2}∙km) for the particular fiber used in the setup. Using Eq. (2) with these parameters gives a time delay equal to 21.565 μs. Using a fiber patchcord of 4.44 meters and doing the fine time delay adjusting with the optical delay lines the absolute time delay can be obtained.

The filter response measured with the experimental setup is shown in Fig. 6.

As can be seen, the measurement has an excellent agreement with the expected theoretical values, showing the feasibility of implementing all-optically microwave filters with practical passband transfer functions.

Changes in the laser output power, instabilities in laser wavelength, etc, may result in a degradation of the filter transfer function. Random changes of these parameters create a residual sidelobe level in the filter transfer function (Ref [12]). However, the errors using off-the-self devices are usually small and an excellent agreement can be obtained between measurements and theory as shown in Fig. 6.

## 7. Conclusion

A multi-tap photonic microwave filter with negative coefficients based on multiple optical carriers and dispersive media has been proposed and experimentally demonstrated. Negative coefficients are obtained by means of the π phase inversion in a single electro-optic Mach-Zehnder modulator and the modulator V_{π} dependence with wavelength, in this way negative coefficients using a single MZM can be obtained. Since the technique does not require electrical devices to implement negative coefficients, the filter operation bandwidth is not limited by electric components and fully exploits the benefits of optical technology to obtain practical filter responses.

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