## Abstract

The present paper aims to describe other functionalities for an arrayed waveguide grating (AWG)-based device, showing that this widely used configuration can be designed not only to frequency multiplex/demultiplex wavelength division multiplexing (WDM) signals, but also to perform the discrete Fourier transform (DFT) and the discrete fractional Fourier transform (DFrFT) of a signal, in all-optical orthogonal frequency division multiplexing (OFDM) systems. In addition 1 *× N* and *N × N* phased array switches architectures are described, as well as a new configuration to perform polarization diversity demultiplexing. Finally, a general approach, based on an analogy with the finite impulse response (FIR) filter approach, is presented to design optical modulators for any modulation format, using either phase modulators (PM) or electro-absorption modulators (EAM).

©2012 Optical Society of America

## 1. Introduction

Flexible optical transceivers are becoming a hot topic in optical communication research, as they are viewed as a possible solution to upgrade current system towards Tb/s capacities, while optimizing the use of physical resources. Many research and development efforts have been directed toward the design and fabrication of multi-carrier transceivers to generate and process optical ultra-high capacity channels (superchannels) that can be adapted for different wavelengths and modulation order, according to the traffic demand and required system performance [1].

OFDM and Nyquist WDM are seen as the most suitable technologies to introduce bandwidth allocation flexibility into core, metro and access networks, by using orthogonal sub-carriers that have rectangular shapes in time or frequency domains, respectively [2]. However, in the most part of actual implementations, signal processing is performed in the electronic domain, using expensive and power consuming analog-to-digital converters (ADC).

The present paper demonstrates that is possible to implement signal processing directly in the optical domain, and describes new AWG configurations, that integrate different functionalities for multiplexing/demultiplexing OFDM sub-channels, vector modulating and switching optical signals, onto a single optical planar lightwave circuit (PLC), significantly improving the size and cost, with respect to more traditional optical assemblies.

A novel OFDM technique is described, based on fractional Fourier sub-carriers that are orthogonal over a symbol duration, if the corresponding parameters are adequately chosen; in addition, their chirped feature can compensate chromatic dispersion.

In time-frequency reference frame, a signal *s(t)* is represented along the time axis, and the corresponding Fourier transform (FT)

*F*can be viewed as a change in the representation of a signal corresponding to a π/2 counterclockwise axis rotation.

The fractional Fourier transform (FrFT) is a generalization of the FT, firstly introduced by V. Namias in 1980 [3]

*s(t)*on an axis that forms an angle

*p*π/2, with 0<

*p*<2, so that the FrFT operator performs a rotation in the time-frequency plane of an angle

*p*π/2, as shown in Fig. 1 [4].

The inverse transformation is obtained by changing the sign of the parameter *p*, $s\left(t\right)={F}^{-p}\left\{{S}^{p}\right\}\left(t\right)$ and the FrFT has the additivity property

*p*= 1, the FrFT coincides with the canonical FT of the signal

*s(t)*of Eq. (1), evaluated at

*u = f*.

The simplest optical interpretation of the FrFT is based on the light propagation through graded index (GRIN) media [5], that exhibit imaging and Fourier transforming properties at a given distance *L*. The medium can be regarded as consisting of infinitesimal layers uniformly distributed along the axial direction, and the FrFT can be physically defined as the functional form of field distribution measured after a distance *pL*. On the other hand, the FrFT can be also obtained in a bulk two-lens optical system, with a proper choice of the focal length and propagation distance [6–8].

In the present paper, new planar architectures for optical modulators are presented, that are wavelength and bit-rate transparent, and can be easily adapted to enlarge or decrease the constellation size. In particular, a general method to design PLC digital modulators of any order and modulation format is described, using an AWG configuration and a reduced number of straight PMs or EAMs, with respect to conventional schemes. Detailed design guidelines for quadrature amplitude modulation (QAM) and phase shift keying (PSK) of any order are given. In addition, the same PLC device used for QAM modulators can be designed to implement a 1 × *N* and a *N* × *N* phased array switch.

Finally, an alternative scheme for polarization multiplexing is proposed, that has an AWG configuration combined with a polarization grating structure.

## 2. Discrete Fourier transform

The AWG device is one the earliest PLC devices, designed by M. K. Smit in 1988, that replaced bulk gratings and filters to multiplex and demultiplex WDM channels [9]. A standard AWG architecture is composed of two slab couplers, connected through a set of delay lines with increasing lengths, as shown in Fig. 2
. A slab coupler can be designed in a confocal configuration, so that the radius of curvature *R* equates the slab length *l*, i.e. the distance between the two surfaces (Fig. 3(a)
). From a functional point of view, this system is equivalent to a bulk-optic configuration composed of two lenses with focal length *R,* separated of a distance *l* (Fig. 3(b)); in this case, the field amplitude distribution *b(x)* at the output plane is proportional to the analog FT of the amplitude *a(x)* on the input plane, evaluated at the spatial frequency *x/(λl)*, where *λ* is the central wavelength [10]

The input and output waveguide arrays sample the field distribution (Fig. 3(c)), so that the optical signal at the *m*-th output waveguide has the expression

*d*and

_{i}*d*are the pitches of the input and output waveguide gratings. In the previous expression, I have set

_{o}*N = λl/d*, to obtain the DFT. It is worth to observe that the design guidelines of an AWG to perform the DFT are similar to those corresponding to a conventional frequency demultiplexer [11]; the main difference is that the DFT device has

_{i}d_{o}*N*input/output ports, being

*N*the number of the waveguides in the grating. Although the condition

*N = λl/d*is exactly satisfied only at a single wavelength, the device can be used for broadband application [12]. In addition, the angular dispersion in the slab, due to the frequency dependence of the refractive index, causes a linear phase variation across the output of the grating array that must be carefully considered in the device design [13, 14]. For sake of simplicity, in the following discussion, the input slab coupler in the AWG is replaced by a standard splitter.

_{i}d_{o}It is important to observe that the slab coupler of Fig. 3(c), when the layout parameters satisfy the condition *N = λl/d _{i}d_{o}*, is a 360°/

*N*hybrid, and there is a fixed phase relation between the input and the output fields. For instance, by setting

*N*= 4, the slab coupler is a standard 90° hybrid.

Introducing delays multiple of τ in the arrayed waveguides, the impulse response and the transfer function for the *m*-th output (*m = 0,1,2..N-1*) of the AWG device are

*δ(t)*is the Dirac delta function, and

*T = Nτ*the symbol period. Alternatively, by using the sampling property of the delta function, Eq. (6) can be written as

*rect*for

_{T}(t) = 1*–T/2<t<T/2*and zero otherwise. According to the convolution theorem, the transfer function can be also written as

Figure 4 (a) shows the time waveform of the output signal from an AWG device that implements the DFT, when the input signal is a 2-ps Gaussian laser pulse, and the transfer function is shown in Fig. 4 (b).

This device configuration was first proposed in 2004 [15] and some prototypes with 8, 15, 16, 32 and 50 ports have been fabricated and used in many optical code division multiplexing (OCDM) experiments [16]. Recently, the same architecture has been further theoretically investigated [17] and used in all-optical OFDM implementations [18].

## 3. Discrete fractional Fourier transform

In a conventional OFDM system, the *m*-th sub-channel waveform is

*T*

*δ*is the Kronecker symbol. To move from an analog to a discrete (digital) representation and use the DFT, the temporal waveform ${\overline{h}}_{m}\left(t\right)$is sampled at

_{mm’}*t = nτ,*to obtain Eqs. (6) and (8).

In the case of FrFT, the sub-carrier waveforms

*m*= 0 and

*p*= 1/8. The FTs (spectra) of the FrFT sub-carriers are

*u*is selected asthen the sub-carriers satisfy the orthogonality condition of Eq. (11). It is evident that the FrFT waveforms coincide with conventional OFDM sub-carriers of Eq. (10) when

_{m}*p*= 1. Finally, sampling at

*t = nτ,*I obtain

The same AWG device of Fig. 2 can implement the DFrFT if the slab parameters are selected to satisfy the conditions [20]

From an inspection of Eq. (16), it is evident that the subcarriers are chirped signals, and present the same spectrum, shifted of *m/T* and multiplied by a complex factor ${e}^{-j\pi \frac{{m}_{}^{2}}{2}\mathrm{sin}\left(p\pi \right)}$. The chirp feature of the subcarriers can be used to compensate chromatic dispersion in the fiber link, by selecting

*D*is the dispersion parameter,

*L*the fiber link length.

## 4. QAM modulator

There are two main general approaches to multilevel modulate in-phase (I) and quadrature-phase (Q) optical signals: the first one requires two pairs of complementary *N*-level electrical waveforms, that drive a conventional IQ modulator. In this case, the high-speed electrical devices required are quite expensive and power hungry, and it is difficult to achieve high linearity and a wide dynamic range for the electrical signals. The second method requires only complementary two-level electrical signals, but a more complex PLC device composed of a nested configuration of Mach Zehnder interferometers (MZI) and a set of phase shifters: in general, *N* MZIs are required for 2^{N}–QAM modulation format, along with two trees of asymmetric splitters and a set of phase shifters [21]. For instance, the modulator of Fig. 5(a)
generate 16-QAM optical signals, respectively, if the MZIs are driven by *N* = 4 complementary two-level electrical signals ± *V _{n}* (

*n*= 0,1,..

*N*-1).

To simplify the planar layout of an optical modulator, I observe that its transfer function is similar to that one corresponding to a FIR filter

where the coefficients*a*depend on the electric voltages applied to either PMs or EAMs; the parameter

_{n}*z = exp(j2π/M)*is a constant phase shift that, opposite to the FIR filter case, does not depends on time delays or frequency. In addition,

*M*is a suitable parameter, related to the modulation format; for QAM modulators, I set

*M*= 4, so that Eq. (19) becomes

The *z ^{−1}* blocks of a FIR filter structure can be optically implemented by a passive 90° hybrid (or slab coupler with suitable parameters), and the filter coefficients a

_{0}, a

_{1}= ± 1; a

_{2}, a

_{3}= ± 2, …, a

_{N-2},a

_{N-1}= ± 2

^{N/2-1}by an asymmetric star coupler, as shown in Fig. 5(b). The sign of the coefficients depends on the voltages

*V*(

_{n}*n*= 0,1,2,..

*N-1*) applied to the PM electrodes. The output signal is always taken at port

*m = 1*, so that the phase shifts introduced by the 90° hybrid are

*j*(

^{n}*n = 1,2,3,4*). It is important to observe also that these modulator layouts are completely flexible, so that a 16-QAM configuration can also generate 4-QAM modulation format, if only two ports are connected to the light source. It is also possible to replace PMs with EAMs, making the device layout more compact [22]. Finally, I observe that, with respect to other optical modulator configurations presented in literature, the novel architecture does not require MZIs but only straight PMs or EAMs (therefore the number of electrodes is halved), and the phase shifters are completely eliminated. The proposed architecture does not suffer for chirp effects, and its design is similar to the that one an AWG device to perform the DFT; also in this case, the angular dispersion in the slab, due to the frequency dependence of the refractive index, causes a linear phase variation across the output of the grating array.

## 5. Phased array switch

If the splitters of Fig. 5(b) are symmetric, the PLC device becomes a phased array switch, because the phase changes at the hybrid ports are compensated by the PMs (or phase shifters), as shown in Fig. 6(a)
. In fact, if the phases of the coefficients are changed by applying *N*-level electrical voltages *V _{n}* (

*n*= 0,1,..

*N-1*), so that they are complex conjugate of the phase factors introduced by the slab coupler

*m*-th port. A phased array switch was firstly designed and fabricated introduced by C. Doerr

*et al*. in 1999 [23] and many prototypes have been fabricated by T. Tanemura

*et al*. [24]. The innovation of the proposed approach is that a diffraction slab coupler is replaced by a 360°/

*N*hybrid, and this is a great simplification of the PLC layout. In addition, thanks to the periodic behavior of the hybrid, with respect to the input port number, it is also possible to design a

*N*x

*N*switch configuration, as shown in Fig. 6(b).

## 6. PSK modulator

To design a PSK modulator, it is more advantageous to refer to the FIR filter transfer function

whose zeros are*z*In this case, I set

_{k}/(1-z_{k-1}).*z*or

_{1}, z_{2},…z_{N-1}= 0*1*; and the device layout is composed of a set of electro-optical (EO) switches and some 360°/

*N*hybrids. The 8-PSK constellation can be generated by the architecture of Fig. 7 , which is composed of three EO switches driven by the electrical voltages

*V*(

_{n}*n*= 0,1,2) and a 180°, 90° and a 45° hybrid. Also in this case, the schematic can be further generalized adding more hybrids and switches to increase the order of the PSK modulation or to reduce the number of switches. It is also possible to consider alternative architectures that integrate PMs [25].

The QAM and PSK modulators presented in this paper are only examples of a general approach to design an optical digital modulator using the same approach as FIR filter method. The design parameters are the choice of the filter coefficients (or zeros) and the parameter *z = exp(j2π/M)*, that should be adapted to different advanced modulation formats.

## 7. Polarization multiplexer

The AWG architecture of Fig. 2 can be also designed to implement a polarization demultiplexer, if the waveguides in the arrayed grating are designed to be alternatively single-mode transverse magnetic (TM) and transverse electric (TE) [26]. Alternatively, planar polarization rotators can be placed at the end of the odd (or even) grating waveguides, to rotate the polarization state of 90°. If all the grating waveguides are uniformly illuminated by the input slab, the impulse response and the transfer function at *m-*th output can be written as

*x*and

*y*axes. It is evident that the polarization state of the signal depends on the output port number

*m*and that the output ports

*m*and

*m*+

*N/2*have orthogonal polarization states. Therefore, this device can be used to perform simultaneously the sub-channel decomposition of an OFDM signal, and dual polarization demultiplexing, as shown in Fig. 8 .

## 4. Conclusions

I have demonstrated that it is possible to add new functionalities to an AWG configuration, so that a suitable choice of the device parameters allows us to all-optically implement the DFT and the DFrFT. In the latter case, the chirped feature of the orthogonal waveform can be used to compensate chromatic dispersion. In addition, adding PMs in the grating arms, it is possible to implement a 1 × *N* and *N* × *N* phased array switch.

A general approach to implement coherent modulators is also provided, based on an analogy with the standard FIR filter design. It is possible to generate any constellation map, and detailed design guidelines for QAM- and PSK-modulators are also provided. It is important to observe that the new configurations use half number of PMs or EAMs, with respect to conventional vector modulators based on MZIs.

Finally, a novel AWG architecture has been presented, that can be used simultaneously as OFDM and polarization diversity multiplexer.

However, the main innovation of the present work is the flexibility, because all these configurations can be easily combined together in a single multi-function PLC component.

## Acknowledgment

This work was supported by the European Commission through ICT-ASTRON (Contract No. 318714) project funded under the 7th Framework Programme.The Author is grateful to an anonymous Reviewer whose comments have enhanced the technical quality of the paper.

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