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
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 . 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 , differential processing , 2nd- and 3rd-order dispersion compensation , optical phase-shift-keying direct-detection , and optical code-division multiplexing encoding and decoding . 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 . 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
We define the normalized eigen mode of the input waveform as e(x 0) and its mode field radius as wIO. 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
where ns is the effective index of the slab waveguides and Lf is the focal length of the slab waveguides. Parameter 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, wAW is the mode field radius in the channel waveguides in the array, δS(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 ΔL=mc/ncν 0, where nc is the effective index of the channel waveguides in the array and 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
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 ΔS(ξ) and B(ξ) are spatial Fourier transforms of δ S(x) and β(x) expressed as
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 mth-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 mth-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 mth-order beam with frequency ν 0+νFSR, 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, Δν, 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), Δν is mainly determined by the sinc function. In this case, it is reasonable that Δν 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 Δν. When we denote the spot size of B(ξ) as Δξ in the spatial frequency, Δν is expressed as
where Neff is defined as 1/(d Δξ) 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:
at interface between second slab waveguide and arrayed waveguide
at interface between arrayed waveguide and first slab waveguide
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) :
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, Vν, 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, Vν is expressed as
Output signal g(t) can be obtained as the temporal inverse Fourier transform of Vν=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%. 
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 νFSR and Δν. Therefore, there are some limitations on the temporal width of the input pulses to be processed. The minimum width is determined by νFSR, and the temporal resolution is given by the inverse of νFSR :
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 Δν. The inverse of Δν 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 Δν. 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/nc.
Previously reported AWG devices [8–10,14] were fabricated on a 4-inch Si wafer. Their maximum Nm was estimated to be around 3×104. If AWGs whose relative refractive index difference is 1.5% or higher are fabricated on a 6-inch wafer, higher-performance (Nm⋍1×105) 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 Δt=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 .
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>3.6×104). A Smaller phase fluctuation is needed for higher-performance signal-processing AWGs. If N⋍2000 (mN⋍1.5×105), 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.
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.
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.
References and links
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