## Abstract

We analyzed optical-signal processing based on time-space conversion in an arrayed-waveguide grating (AWG). General expressions for the electric fields needed to design frequency filters were obtained. We took into account the effects of the waveguides and clearly distinguished the temporal frequency axis from the spatial axis at the focal plane, at which frequency filters were placed. Using the analytical results, we identified the factors limiting the input-pulse width and clarified the windowing effect and the effect ofphase fluctuation in the arrayed waveguide.

© 2000 Optical Society of America

## 1. Introduction

High-speed signal processing is needed to cope with the continuing rapid growth in Internet traffic. However, signal processing at data rates greater than 100 Gbit/s is difficult to provide using conventional electronic processing. Thus, all-optical signal processing at a wavelength of 1.55 µm will be necessary.

The time-space-conversion method proposed by Weiner is one of the most promising optical-signal processing methods because it is not based on time-domain (real-time) operation but on frequency-domain operation [1]. Pulse-pattern generation, waveform reshaping, and pattern recognition using diffraction-grating (DG) pairs and lenses in free-space optics have been demonstrated, mainly at visible wavelengths [1–5]. These achievements have led to femtosecond optical-signal processing.

To apply time-space-conversion method to optical-fiber communication systems, we previously proposed an optical-signal processing system that uses an arrayed-waveguide grating (AWG) and demonstrated optical-pulse-train generation [6], differential processing [7], 2nd- and 3rd-order dispersion compensation [8], optical phase-shift-keying direct-detection [9], and optical code-division multiplexing encoding and decoding [10]. This signal-processing AWG system is compatible with fiber optics and is compact. Moreover, many parameters, for example, the temporal and frequency resolutions of AWG devices can be flexibly designed because such devices provide a flexible choice of diffraction orders and the number of waveguides in the array.

The signal processing in the DG system has been well analyzed [11–13]; however, the findings cannot be applied directly to the signal-processing AWG system because the AWG is composed of waveguides. To incorporate the findings into optical communication systems, precise analysis of the waveforms processed by AWG systems is needed.

We have analyzed optical-signal processing based on the time-space-conversion scheme in AWGs and developed general expressions for time-space-conversion optical-signal processing using AWG devices. We also investigated the limitations on performance of a signal-processing AWG system identifying the factors limiting the input-pulse width and clarified the window effect and the effect of imperfections in the arrayed waveguide.

## 2. General expressions for optical-signal processing using an AWG

#### 2.1 Overview of signal processing based on time-space conversion

Figure 1 shows schematic diagrams of time-space-conversion optical-signal processing systems using (a) DGs and (b) AWGs. In both systems, the temporal input waveform is converted into a spatial waveform by a dispersive element, then spatially decomposed into temporal frequency components at the focal plane. The amplitude and/or phase of these components can be manipulated with the spatial filter array placed at the focal plane. The modulated frequency components are reassembled by reversing the process, and a temporal output waveform is obtained as a convolution of the input temporal waveform and the impulse response of the spatial filter.

In AWG processing, two slab waveguides whose input and output planes are concave
function as Fourier-transform lenses. The waveguide array functions as the
dispersive element. An advantage of the AWG is that the diffraction order,
*m*, is determined by the difference in path lengths between
adjacent waveguides [14]. The analytical results of the DG system cannot be
directly applied to the AWG system because the AWG system consists of
waveguides. In the following sections we describe general expressions we
developed for signal-processing AWG systems.

## 2.2. Spatial field profiles before filtering

In the following analysis, we treat the temporal input signal as its Fourier
coefficient so that we can easily distinguish the *temporal* and
*spatial* frequencies at the focal plane, as discussed below.
This will also make it easy to take some temporal-frequency-dependent effects, for
example, chromatic dispersion, into account. We do not distinguish between the
center optical frequency of the input waveform and the designed center frequency of
the AWG because these two frequencies are often the same in practice.

The model and definition of the axes used for the analysis are shown in Fig. 2. The model is based on a reflection-type AWG. The eigen modes of the channel waveguides in the array are approximated by a Gaussian function. By using slowly varying envelope approximation, we can represent the temporal input waveform as

where *u*(*t*) is the complex amplitude of
*f*(*t*). Its temporal Fourier coefficient is
expressed as

Distribution function

We define the normalized eigen mode of the input waveform as
*e*(*x*
_{0}) and its mode field radius as
*w _{IO}*. At the interface between the input/output
(I/O) waveguide and the first slab waveguide, the spatial electric field with
temporal frequency

*ν*is expressed as

The field,
*F*
_{0,ν}(*x*
_{0})
is diffracted in the first slab waveguide and illuminates the arrayed waveguide. The
distribution function of the first slab waveguide can thus be derived as

with

where *n _{s}* is the effective index of the slab waveguides
and

*L*is the focal length of the slab waveguides. Parameter

_{f}*α*is important because it relates the space domain to the spatial-frequency domain, as discussed below. Generally,

*α*depends on the temporal frequency; however, we use the approximation

*ν*≅

*ν*

_{0}in the following discussion because

*ν*-

*ν*

_{0}≪

*ν*

_{0}.

Interface between first slab waveguide and arrayed waveguide

By using distribution function
*β*(*x*
_{1}), we can express the
electric field at the interface between the first slab waveguide and the arrayed
waveguide as

where *N* is the number of waveguides in the array, *d*
is the spacing between waveguides along the *x*
_{1} and
*x*
_{2} axes, *w _{AW}* is the mode
field radius in the channel waveguides in the array,

*δ*(

_{S}*x*) is defined as

rect(*x*) is a rectangular function defined as

and * denotes convolution. The function
*δ*
_{S}(*x*) represents the
discreteness of the arrayed waveguide for repetitions of *d*. In Eq. (6), the amplitude of the field can be treated
approximately as constant within the width of each waveguide in the array. This is
because the spatial width of the eigen mode of the waveguides is much smaller than
the expansion of the distribution function.

Interface between arrayed waveguide and second slab waveguide

The lengths of the adjacent waveguides differ by a constant value of
Δ*L*=*mc*/*n _{c}ν*

_{0}, where

*n*is the effective index of the channel waveguides in the array and

_{c}*m*is the diffraction order of the arrayed waveguide. This structure produces a temporal-frequency-dependent phase shift written as

The electric field at the output plane of the arrayed waveguide can thus be expressed as

$$={U}_{\nu}\xb7[\left\{\beta \left({x}_{2}\right)\xb7\mathrm{rect}\left(\frac{{x}_{2}}{\mathit{Nd}}\right)\xb7\mathrm{exp}\left(i2\pi m\frac{\nu}{{\nu}_{0}d}{x}_{2}\right)\right\}*{\delta}_{\mathrm{S}}\left({x}_{2}\right)]$$

$$*\mathrm{exp}(-\pi \frac{{{x}_{2}}^{2}}{{{w}_{\mathit{AW}}}^{2}}).$$

Focal plane of second slab waveguide

By considering the one-dimensional diffraction in the second slab waveguide, which
acts as a focusing lens, we can derive the field pattern at the focal plane of the
second slab waveguide by using the spatial Fourier transform off
*f*
_{2,ν}(*x*
_{2}):

where *ξ* represents the spatial frequency and is defined
using Eq. (5),

and Δ_{S}(*ξ*) and
B(*ξ*) are spatial Fourier transforms of
*δ*
_{S}(*x*) and
*β*(*x*) expressed as

and

The sinc function sinc(*x*) is defined as
sinc(*x*)≡sin(*πx*)/*πx*.
In Eq. (11),
B(*ξ*)*sinc(*Ndξ*)
determines the spot size at the focal plane, and
*δ*(*ξ*-*mν*/*ν*
_{0}
*d*)
gives the propagation direction of a beam with a temporal frequency of
*ν*. In other words, each temporal frequency component
is centered at a spatial frequency of

with a beam profile of
B(*ξ*)*sinc(*Ndξ*).

## 2.3 Derivation of basic parameters

Spatial dispersion at focal plane and free spectral range of AWG

The spatial dispersion, *γ*, at the focal plane of the
second slab waveguide is one of the most important parameters of the AWG because it
relates the temporal frequency spectrum of the input pulse spread over the focal
plane to the spatial filters (represented in the space domain). From Eqs. (12) and (15), we get

The focal plane of the AWG is illuminated not only by the *m*th-order
beam but also by the other order beams (i.e.,
the(*m*±1)th-order beams, etc.). Therefore, the frequency
range in which the AWG can process is limited. The free spectral range (FSR) of the
AWG is defined as the frequency range corresponding to the spatial span between the
*m*th-order beam and the
(*m*+1)th-order beam with frequency
*ν*
_{0}. From the coincidence of the direction
of the (*m*+1)th-order diffraction of a beam with
frequency *ν*
_{0} and that of the
*m*th-order beam with frequency
*ν*
_{0}+*ν _{FSR}*,
the FSR is derived by using Eq (15) as follows:

The FSR is also a crucial parameter because its inverse limits the temporal resolution of the AWG.

Temporal frequency resolution of AWG

The temporal frequency resolution, Δ*ν*, is
defined by the spatial field profile at the output plane of the arrayed waveguide.
From Eq. (11), the shape of the spot at the focal plane is expressed
as B(*ξ*)*sinc(*Ndξ*) because
the envelope of the field at the output of the arrayed waveguide is written as
*β*(*x*
_{2})·rect(*x*
_{2}/*Nd*).
The B(*ξ*)*sinc(*Ndξ*) means
that the frequency resolution depends greatly on the envelope of the field at the
output of the arrayed waveguide. In the following, we consider the two extremes. If
we assume that the distribution from the I/O waveguide to the array is sufficiently
uniform (i.e.,
*β*(*x*
_{2})≈1/*N*),
Δ*ν* is mainly determined by the sinc
function. In this case, it is reasonable that
Δ*ν* is determined as
*ξ*=1/*Nd* because the main lobe of
sinc(*Ndξ*) drops to zero at
*ξ*=±1/*Nd*. From Eq. (15), we get

If *β*(*x*
_{2}) is not uniform,
B(*ξ*), which is approximately expressed as Gaussian,
is the major limiting factor of Δ*ν*. When we
denote the spot size of B(*ξ*) as
Δ*ξ* in the spatial frequency,
Δ*ν* is expressed as

where *N _{eff}* is defined as 1/(

*d*Δ

*ξ*) and means the effective number of illuminated waveguides.

From Eqs. (15) and (16), the three parameters (*ξ*,
*x*, and *ν*) are not independent.
However, there is not a one-to-one correspondence between
*ν* and *ξ* or between
*ν* and *x* because a beam spot with
temporal frequency *ν* has finite size. Therefore,
frequency and space are coupled in signal processing using an AWG.

## 2.4 Spatial field profiles after filtering

If a spatial filter with frequency response function
*H*(*ξ*) is placed at the focal plane,
the input signal is modulated by the filter. The spatial field of the processed
signal can be derived by reversing the previous discussion. The electric fields at
the planes after filtering are expressed as follows:

$$\phantom{\rule{.2em}{0ex}}\times {\Delta}_{\mathrm{S}}\left(\xi \right)\xb7\mathrm{exp}(-{\pi}^{2}{{w}_{\mathit{AW}}}^{2}{\xi}^{2}),$$

at interface between second slab waveguide and arrayed waveguide

$$*\mathrm{exp}(-\pi \frac{{{x}_{2}}^{2}}{{w}^{2}})$$

at interface between arrayed waveguide and first slab waveguide

$$\phantom{\rule{.2em}{0ex}}\times \mathrm{exp}(-i2\pi m\frac{\nu}{{\nu}_{0}d}{x}_{1})]*\mathrm{exp}(-\pi \frac{{{x}_{1}}^{2}}{{w}^{2}}).$$

Output temporal signal

The spatial field at the interface between the first slab waveguide and the I/O
waveguide is given as the spatial Fourier transform of
*g*
_{2,ν}(*x*
_{2})
:

$$\phantom{\rule{.2em}{0ex}}\times \mathrm{exp}(-{\pi}^{2}{{w}_{\mathit{AW}}}^{2}{\xi}^{2}).$$

If we assume that only one I/O waveguide is placed in the direction of
*ξ*=0, the temporal frequency component of the output
signal, *V _{ν}*, can be derived by considering the
coupling between field

*G*

_{0,ν}(

*ξ*) and the eigen-mode function of the I/O waveguide. Because

*x*

_{0}=

*αξ*at the interface, the eigen-mode function,

*e*(

*x*

_{0}), is expressed as

*e*(

*αξ*). Therefore,

*V*is expressed as

_{ν}$$=\frac{\pi {{w}_{\mathit{AW}}}^{2}}{\sqrt{i\alpha}}\xb7{U}_{\nu}\int e\left(\alpha \xi \right)\xb7H\left(\xi +m\frac{\nu}{{\nu}_{0}d}\right)\xb7(B\left(\xi \right)*\mathrm{sinc}\left(\mathit{Nd}\xi \right))\xb7{\Delta}_{\mathrm{S}}\left(\xi +m\frac{\nu}{{\nu}_{0}d}\right)$$

$$\phantom{\rule{.2em}{0ex}}\times \mathrm{exp}(-{\pi}^{2}{{w}_{\mathit{AW}}}^{2}{\xi}^{2})d\xi .$$

Output signal *g*(*t*) can be obtained as the temporal
inverse Fourier transform of
*V _{ν}*=

*V*(

*ν*-

*ν*

_{0}).

## 3. Performance of signal-processing AWG

In this section, we discuss the relation between the parameters of the
signal-processing AWG and its performance. The most important design parameters are
the number of waveguides in the array, *N*, diffraction order
*m*, and distribution function
*β*(*x*
_{1}). The phase
fluctuation in the array is another important factor because the shape and loss of
the output signal are greatly affected by the phase characteristics.

The following discussion is based mainly on the simulated electric fields in the AWG. Table 1 shows the parameters we used. They are the same as those previously reported for AWG devices [8–10,14]. The AWG was silica-based and fabricated using planer-lightwave-circuit technology. The relative refractive index difference of the waveguide was 0.75%. [15]

Factors limiting width of input pulse

The number of waveguides and diffraction order are basic parameters in this type of
signal processing. As shown in Eqs. (17), (18), and (19), these parameters define
*ν _{FSR}* and
Δ

*ν*. Therefore, there are some limitations on the temporal width of the input pulses to be processed. The minimum width is determined by

*ν*, and the temporal resolution is given by the inverse of

_{FSR}*ν*:

_{FSR}Because frequency components that exceed the FSR will not contribute to formation of
the output pulse, the output pulse is not the same as the input one. On the other
hand, the maximum temporal width of the input pulse is restricted by
Δ*ν*. The inverse of
Δ*ν* gives the maximum time span,
*T*
_{0}, in which signals can be processed:

Transform-limited pulses wider than *T*
_{0} have frequency
components smaller than Δ*ν*. Therefore, they
cannot be spatially decomposed into their frequency components.

The parameters, *N* and *m*, limit the width of the
input pulse, as shown by Eqs. (25) and (26). In designing signal-processing AWGs, the smaller the
*m*, the higher the temporal resolution. Moreover, a large
*N* is needed to obtain a large *T*
_{0}.
In designing AWG devices, the dimension is determined by parameter,
*Nm*, because the maximum path-length difference of the arrayed
waveguide is expressed by
*Nm*·*λ*/*n _{c}*.

Previously reported AWG devices [8–10,14] were fabricated on a 4-inch Si wafer. Their maximum
*Nm* was estimated to be around 3×10^{4}. If
AWGs whose relative refractive index difference is 1.5% or higher are fabricated on
a 6-inch wafer, higher-performance
(*Nm*⋍1×10^{5}) AWGs can be made.
This means that, for example, we can design AWGs with *N*=2000 and
*m*=50 (i.e., *T*
_{0}=500 ps and
Δ*t*=0.25 ps at a wavelength of 1.55 µm).
Such performance should be sufficient for future high-speed communications.

Effect of windowing in arrayed waveguide

In Sec. 2, we explained why the beam profile at the focal plane depends on the
distribution function. The shape of the distribution function depends on the spot
size of the I/O waveguide. To estimate the effect of the distribution on the output
signal, we examined the output by using an amplitude filter with a narrow-stripe
mirror. We assumed that the width of the stripe corresponds to the temporal
frequency resolution, shown in Fig. 3(a). If the temporal frequency is ideally resolved at
the focal plane, the narrow-stripe mirror reflects only one temporal frequency
component, and the output is constant over *T*
_{0}. In
practice, however, the stripe mirror is illuminated by a few (or more than a few)
temporal frequency components because each component has a finite spot size, as
shown in Fig. 3(b). Therefore, the temporal frequency spectrum
reflected by the filter is like that shown in Fig. 3(c), and the temporal shape of the output is
restricted, as shown in Fig. 3(d). Therefore, it is reasonable to define the figure
of merit, *F*, as the ratio of the maximum point of the output
waveform to the minimum point. In the following discussion, we use parameter
*a*/*Nd*, where *a* is the spot
size of the Gaussian beam, *A*
exp(-*x*
^{2}/*a*
^{2}),
illuminating the arrayed waveguide.

Figure 4 shows the figure of merit versus
*a*/*Nd*. The larger the
*a*/*Nd*, the more the beam profile of the focal
plane resembles a sinc function and the more it spreads spatially. However, the
effect of crosstalk becomes weaker because the crosstalk components in the electric
fields barely couple to the single-mode I/O waveguide and become zero at the center
of the stripe. The smaller the *a*/*Nd*, the larger
the loss. Because there is a trade-off between *F* and loss, the
previously reported AWG was designed with an *a*/*Nd*
of 0.57, as shown in Fig. 4 [6–10,14].

The shape of the distribution function of the AWG for wavelength-division-multiplexing (WDM) applications should be Gaussian because this type of AWG must minimize crosstalk between the adjacent output ports in the frequency plane. On the other hand, in signal-processing AWGs, a non-Gaussian shape is better. This is a key difference between signal-processing AWGs and AWGs for WDM applications.

Phase errors in arrayed waveguide

The phase fluctuation in an arrayed waveguide causes crosstalk between its output
channels for WDM applications [16–17]. It also degrades performance of signal-processing AWGs.
The fluctuation is caused by core-size errors, refractive-index errors of the core
and cladding, and waveguide-length errors. A fluctuating phase front disturbs
spectral filtering at the focal plane, reducing the efficiency of the coupling to
the I/O waveguide. When the signal propagates from the arrayed waveguide to the
focal plane of the second slab waveguide, each temporal frequency component spreads
over the focal plane because of the disturbed phase front. Therefore, each temporal
frequency component is modulated not only by an appropriate component of the spatial
filter, but also by other components. When it propagates from the arrayed waveguide
to the I/O waveguide, the coupling efficiency is degraded if there is phase
fluctuation because the spectrum components become difficult to couple to the I/O
waveguide due to the distortion of the focused image. The larger the
*m* and *N*, the larger the total phase error in the
waveguide because longer waveguides are needed to obtain a larger maximum
path-length difference. The standard deviation of the phase error, σ, in
a silica-based waveguide with a relative refractive index difference of 0.75% is
typically 0.8×10^{-2} rad/mm [18].

To estimate the effects of phase fluctuation, we simulated output waveforms by using
a step-like filter. This is the simplest type phase filter; it shifts the phase of
one sideband by π [7, 9, 15]. Figure 5 shows the number-of-waveguides dependence on the
coefficient of determination (*R*
^{2}) calculated from the
output pulse shape for σ=0.8 and 0.2×10^{-2} rad/mm,
where the diffraction order is fixed at *m*=72 and the input pulse
width was 1 ps. For σ=0.8×10^{-2} (typical value), the
output waveform was distorted and the excess loss increased when *N*
exceeded 500 (*mN*>3.6×10^{4}). A
Smaller phase fluctuation is needed for higher-performance signal-processing AWGs.
If *N*⋍2000
(*mN*⋍1.5×10^{5}), for example,
optical waveguides with a standard deviation of the phase error of
0.2×10^{-2} is needed to obtain non-distorted output
waveforms.

## 4. Conclusion

We analyzed optical-signal processing in AWG devices by considering the effects of the waveguides. We clearly distinguished the temporal frequency axis from the spatial axis at the focal plane of the AWG. Using simulation based on this analytical method, we showed that the maximum and minimum temporal widths of the input pulse were restricted by the number of waveguides in the array and by the diffraction order. We found that the distribution function played an important role in determining the shape of the envelope in the time window.

We also estimated the relation between the output waveform and phase fluctuation in the arrayed waveguide and found that the phase error must be reduced to achieve the performance that will be needed for signal-processing AWGs in the near future.

## Acknowledgments

We would like to thank Dr. Katsunari Okamoto, Dr. Kenji Kawano, Dr. Kazunori Naganuma, Dr. Hiroyuki Suzuki, Dr. Hiroki Itoh and Dr. Chikara Amano of NTT Photonics Laboratories for their helpful discussions, and Dr. Seiko Mitachi and Dr. Hidetoshi Iwamura of NTT Photonics Laboratories for their continuous encouragement.

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