220 COMVONrNTS 3.8.3 The advantage of CGM is that it is conceptually simple. However, there are several drawbacks. The achievable extinction ratio is small (less than 10) since the gain does not really drop to zero when there is an input 1 bit. The input signal power must be high (around 0 dBm) so that the amplifier is saturated enough to produce a good variation in gain. This high-powered signal must be eliminated at the amplifier output by suitable filtering, unless the signal and probe are counterpropagating. Moreover, as the carrier density within the SOA varies, it changes the refractive index as well, which in turn affects the phase of the probe and creates a large amount of pulse distortion. Interferometric Techniques The same phase-change effect that creates pulse distortion in CGM can be used to effect wavelength conversion. As the carrier density in the amplifier varies with the input signal, it produces a change in the refractive index, which in turn mod- ulates the phase of the probe~hence we use the term cross-phase modulation for this approach. This phase modulation can be converted into intensity modulation by using an interferometer such as a Mach-Zehnder interferometer (MZI) (see Sec- tion 3.3.7). Figure 3.78 shows one possible configuration of a wavelength converter using cross-phase modulation. Both arms of the MZI have exactly the same length, with each arm incorporating an SOA. The signal is sent in at one end (A) and the probe at the other end (B). If no signal is present, then the probe signal comes out unmodulated. The couplers in the MZI are designed with an asymmetric cou- pling ratio ?' -r 0.5. When the signal is present, it induces a phase change in each amplifier. The phase change induced by each amplifier on the probe is different be- cause different amounts of signal power are present in the two amplifiers. The MZI translates this relative phase difference between its two arms on the probe into an intensity-modulated signal at the output. This approach has a few interesting properties. The natural state of the MZI (when no input signal is present) can be arranged to produce either destructive or constructive interference on the probe signal. Therefore we can have a choice of whether the data coming out is the same as the input data or is complementary. The advantage of this approach over CGM is that much less signal power is required to achieve a large phase shift compared to a large gain shift. In fact, a low signal power and a high probe power can be used, making this method more attractive than CGM. This method also produces a better extinction ratio because the phase change can be converted into a "digital" amplitude-modulated output signal by the interferometer. So this device provides regeneration with reshaping (2R) of 3.8 Wavelength Converters 221 Figure 3.78 Wavelength conversion by cross-phase modulation using semiconductor optical am- plifiers embedded inside a Mach-Zehnder interferometer. the pulses. Depending on where the MZI is operated, the probe can be modulated with the same polarity as the input signal, or the opposite polarity. Referring to Figure 3.78, where we plot the power coupled out at the probe wavelength versus the power at the signal wavelength, depending on the slope of the demultiplexer, a signal power increase can either decrease or increase the power coupled out at the probe wavelength. Like CGM, the bit rate that can be handled is at most 10 Gb/s and is limited by the carrier lifetime. This approach, however, requires very tight control of the bias current of the SOA, as small changes in the bias current produce refractive index changes that significantly affect the phase of signals passing through the device. We have seen above that the CPM interferometric approach provides regenera- tion with reshaping (2R) of the pulses. As we saw earlier, while 2R cleans up the signal shape, it does not eliminate phase (or equivalently timing) jitter in the signal, which would accumulate with each such 2R stage. In order to completely clean up the signal, including its temporal characteristics, we need regeneration with reshap- ing and retiming (3R). Figure 3.79 shows one proposal for accomplishing this in 222 COMPONENTS Figure 3.79 All-optical regeneration with reshaping and retiming (3R) using a combination of cross-gain modulation and cross-phase modulation in semiconductor optical amplifiers. (After [Chi97].) 3.8 Wavelength Converters 223 3.8.4 the optical domain without resorting to electronic conversion [Chi97, Gui98]. The approach uses a combination of CGM and CPM. We assume that a local clock is available to sample the incoming data. This clock needs to be recovered from the data; we will study ways of doing this in Section 12.2. The regenerator consists of three stages. The first stage samples the signal. It makes use of CGM in an SOA. The incoming signal is probed using two separate signals at different wavelengths. The two probe signals are synchronized and modulated at twice the data rate of the incoming signal. Since the clock is available, the phase of the probe signals is adjusted to sample the input signal in the middle of the bit interval. At the output of the first stage, the two probe signals have reduced power levels when the input signal is present and higher power levels when the input signal is absent. In the second stage, one of the probe signals is delayed by half a bit period with respect to the other. At the output of this stage, the combined signal has a bit rate that matches the bit rate of the input signal and has been regenerated and retimed. This signal is then sent through a CPM-based interferometric converter stage, which then regenerates and reshapes the signal to create an output signal that has been regenerated, retimed, and reshaped. Wave Mixing The four-wave mixing phenomenon that occurs because of nonlinearities in the trans- mission medium (discussed in Section 2.4.8) can also be utilized to realize wavelength conversion. Recall that four-wave mixing causes three waves at frequencies fl, f2, and f3 to produce a fourth wave at the frequency fl + f2 - f3; when fl = f2, we get a wave at the frequency 2fl - f3. What is interesting about four-wave mixing is that the resulting waves can lie in the same band as the interacting waves. As we have seen in Section 2.4.8, in optical fibers, the generated four-wave mixing power is quite small but can lead to crosstalk if present (see Section 5.8.4). For the purposes of wavelength conversion, the four-wave mixing power can be enhanced by using an SOA because of the higher intensities within the device. If we have a signal at frequency fs and a probe at frequency fp, then four-wave mixing will produce signals at frequencies 2fp - fs and 2fs - fp, as long as all these frequencies lie within the amplifier bandwidth (Figure 3.80). The main advantage of four-wave mixing is that it is truly transparent because the effect does not depend on the modulation format (since both amplitude and phase are preserved during the mixing process) and the bit rate. The disadvantages are that the other waves must be filtered out at the SOA output, and the conversion efficiency goes down significantly as the wavelength separation between the signal 224 COMPONENTS Figure 3.80 amplifier. Wavelength conversion by four-wave mixing in a semiconductor optical and probe is increased. We will study the conversion efficiency of four-wave mixing in Section 5.8.4. Summary We have studied many different optical components in this chapter. Couplers, isola- tors, and circulators are all commodity components. Many of the optical filters that we studied are now commercially available, with fiber gratings, thin-film multicavity filters, and arrayed waveguide gratings all competing for use in commercial WDM systems. Erbium-doped fiber amplifiers (EDFAs) are widely deployed and indeed served as a key enabler for WDM. EDFA designs today incorporate multiple stages and gain-flattening filters and provide midstage access between the multiple stages to insert other elements such as dispersion compensating modules and wavelength add/drop multiplexers. A new generation of EDFAs providing amplification in the L-band has recently emerged. We are also now seeing distributed Raman amplifiers used in conjunction with EDFAs in ultra-long-haul systems. Semiconductor DFB lasers are used in most high-speed communication systems today although other single-longitudinal mode laser structures may eventually be- come viable commercially. Compact semicondunctor tunable lasers are now emerg- ing as viable commercial devices. High-speed APDs and pinFET receivers are both available today. Large-scale MEMS-based optical switches for use in wavelength crossconnects are now emerging as commercial devices, and a variety of technologies are avail- able to build smaller-scale switches. All-optical wavelength converters are still in the research laboratories, awaiting significant cost reductions and performance improve- ments before they can become practical. Further Reading 225 Further Reading The book by Green [Gre93] treats many of the optical components considered in this chapter in more detail, particularly tunable filters and lasers. See also [KK97] for more advanced coverage of a number of components. Most of the filters we described are now commercially available. Gratings are described in detail in several textbooks on optics, for example, [KF86, BW99]. The Stimax grating is described in [LL84] and [Gre93]. See [CK94, Ben96, Kas99] for details on fiber grating fabrication and properties, and [Ven96b, Ven96a] for applica- tions of long-period gratings. For a description of how dielectric thin-film multicavity filters work, see [SS96] and [Kni76]. The electromagnetics background necessary to understand their operation is provided, for example, by [RWv93]. Early papers on the arrayed waveguide grating are [DEK91] and [VS91]. The principle behind their operation is described in [McG98, TSN94, TOTI95, TOT96]. The integrated-optics AOTF is described in [SBJC90, KSHS01], and its systems applications are discussed in [Che90]. There is an extensive literature on optical amplifiers. See [BOS99, Des94] for EDFAs, [Flo00] for a summary of L-band EDFAs, and [O'M88] for a tutorial on SOAs. [Tie95, SMB00, FDW01] provide samples of some recent work on gain-clamped SOAs. See [NE01, NE00] and [KK97, Chapter 7] for an overview of Raman amplifiers. There are several textbooks on the subject of lasers alone; see, for example, [AD93]. Laser oscillation and photodetection are covered in detail in [Yar97]. [JQE91] is a good reference for several laser-related topics. Other good tutorials on lasers appear in [BKLW00, LZ89, Lee91, SIA92]. A very readable and up-to-date survey of vertical cavity lasers can be found in [Har00]. See also [MZB97]. Most semiconductor lasers today make use of quantum well structures. See [AY86] for a good introduction to this subject. The mathematical theory behind mode locking is explained in [Yar89] and [Yar65]. There is an extensive discussion of various mode-locking methods for fiber lasers in [Agr95]. Lithium niobate external modula- tors are well described in [Woo00] and [KK97, Chapter 9], and electro-absorption modulators in [BKLW00] and [KK97, Chapter 4]. There is currently significant effort toward realizing commercially viable tunable lasers. We refer the reader to [Col00, Har00, AB98, Gre93, KK97] for more in-depth explorations of this subject. An early review of tunable laser approaches appeared in [KM88]. The VCSEL-based tunable laser is described in [Vak99]. Other types of tunable VCSELs have been demonstrated; see, for instance, [CH00, Har00]. The sampled grating laser structure is explained in [JCC93] and superstructure grating lasers in [Toh93]. See [WMB92, Rig95] for details on the GCSR laser. The arrayed external grating-based laser approaches were proposed in [Soo92, ZJ94, Zir96]. Input 226 COMPONENTS Figure 3.81 A 3 dB coupler with the two outputs connected by a piece of fiber. The tutorial article by Spanke [Spa87] is a good review of large switch architec- tures for optical switches. See also [MS88] for a good collection of papers on optical switching and [Clo53] for the original paper on the Clos switch architecture. The classic book by Bene~ [Ben65] is the authoritative reference for the mathematical theory of large switch architectures developed for telephony applications. A very accessible survey of mechanical switches can be found in [Kas95, Chapter 13]. Several papers [NR01, LGT98, Nei00, Ryf01, Lao99] describe MEMS-based switches. The inkjet-based waveguide switch is described in [Fou00]. See [WL96, PS95] for some early papers on liquid crystal switches. Surveys and comparisons of different types of wavelength converters appear in [Stu00, EM00, NKM98, Yoo96, ISSV96, DMJ+96, Chi97]. 3.1 3.2 Problems Consider the 3 dB 2 x 2 coupler shown in Figure 3.81. Suppose we connect the two outputs with a piece of fiber. Assume that the polarizations are preserved through the device. A light signal is sent in on the first input. What happens? Derive the field transfer function for the device. Assume the coupler used is a reciprocal device so that it works exactly the same way if its inputs and outputs are reversed. Hint: This device is called a loop mirror. Consider a device with three ports where it is desired to send all the energy input at ports 1 and 2 to port 3. We assume, for generality, that all ports can be used as inputs and outputs. The scattering matrix of such a device can be written as 0 0 s13 ) S - 0 0 $23 9 $31 $32 $33 Problems 227 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 Show that a scattering matrix of this form cannot satisfy the conservation of energy condition, (3.4). Thus it is impossible to build a device that combines all the power from two input ports to a third port, without loss. Consider an isolator that is a two-port device where power must be transferred from port 1 to port 2, but no power must be transferred from port 2 to port 1. The scattering matrix of such a device can be written as S_ (Sll s12) 0 $22 " Show that a scattering matrix of this form cannot satisfy the conservation of energy condition, (3.4). Thus the loss occurs in the isolator because the power input at port 2 must be absorbed by it. However, the power input at port 1 can be transferred to port 2 without loss. In Figure 3.10, show that the path length difference between the rays diffracted at angle 0a and traversing through adjacent slits is approximately a[sin(Oi) - sin(0d)] when the grating pitch a is small compared to the distance of the source and the imaging plane from the grating plane. Derive the grating equation for a blazed reflection grating with blaze angle ~, such as the one shown in Figure 3.11. Derive the amplitude distribution of the diffraction pattern of a grating with N narrow slits spaced distance d apart. Show that we obtain diffraction maxima when d sin 0 - m)~. Discuss what happens in the limit as N ~ oo. Show that the resonant frequencies f~ of a Fabry-Perot cavity satisfy f~ - fo + nAf, n integer, for some fixed f0 and Af. Thus these frequencies are spaced equally apart. Note that the corresponding wavelengths are not spaced equally apart. Derive the power transfer function of the Fabry-Perot filter. Derive the expression (3.13) for the finesse of the Fabry-Perot filter. Assume that the mirror reflectivity, R, is close to unity. Show that the fraction of the input power that is transmitted through the Fabry-Perot filter, over all frequencies, is (1 - R)/(1 + R). Note that this fraction is small for high values of R. Thus, when all frequencies are considered, only a small fraction of the input power is transmitted through a cavity with highly reflective facets. Consider a cascade of two Fabry-Perot filters with cavity lengths 11 and 12, respec- tively. Assume the mirror reflectivities of both filters equal R, and the refractive index of their cavities is n. Neglect reflections from the second cavity to the first and vice 228 ' COMPONENTS 3.12 3.13 3.14 3.15 3.16 3.17 versa. What is the power transfer function of the cascade? If 11/12 k/m, where k and m are relatively prime integers, find an expression for the FSR of the cascade. Express this FSR in terms of the FSRs of the individual filters. Show that the transfer function of the dielectric slab filter shown in Figure G.l(b) is identical to that of a Fabry-Perot filter with facet reflectivity ~/~- n2 -nl n2+nl assuming n3 = n 1. Consider a stack of 2k alternating low-index (nL) and high-index (n/4) dielectric films. Let each of these films have a quarter-wave thickness at )~0. In the notation of Section 3.3.6, this stack can be denoted by (HL) ~. Find the reflectivity of this stack as a function of the optical wavelength )~. Thus a single-cavity dielectric thin-film filter can be viewed as a Fabry-Perot filter with wavelength-dependent mirror reflectivities. Derive the power transfer function of the Mach-Zehnder interferometer, assuming only one of its two inputs is active. Consider the Mach-Zehnder interferometer of Section 3.3.7. (a) With the help of a block diagram, show how a 1 x n demultiplexer can be constructed using n - 1 MZIs. Assume n is a power of two. You must specify the path length differences AL that must be used in each of the MZIs. (b) Can you simplify your construction if only a specific one of the signals needs to be separated from the rest of the n - 1 ? Consider the Rowland circle construction shown in Figure 3.26. Show that the differences in path lengths between a fixed-input waveguide and any two successive arrayed waveguides is a constant. Assume that the length of the arc on which the arrayed waveguides are located is much smaller than the diameter of the Rowland circle. Hint: Choose a Cartesian coordinate system whose origin is the point of tangency of the Rowland and grating circles. Now express the Euclidean distance between an arbitrary input (output) waveguide and an arbitrary arrayed waveguide in this coordinate system. Use the assumption stated earlier to simplify your expression. Finally, note that the vertical spacing between the arrayed waveguides is constant. In the notation of the book, this shows that 8i d sinOi, where d is the vertical separation between successive arrayed waveguides, and Oi is the angular separation of input waveguide i from the central input waveguide, as measured from the origin. Derive an expression for the FSR of an AWG for a fixed-input waveguide i and a fixed-output waveguide j. The FSR depends on the input and output waveguides. But show that if the arc length of the Rowland circle on which the input and output Problems 229 3.18 3.19 3.20 3.21 3.22 3.23 waveguides are located (see Figure 3.26) is small, then the FSR is approximately constant. Use the result from Problem 3.16 that ~i = d sin 0i. Consider an AWG that satisfies the condition given in Problem 3.17 for its FSR to be approximately independent of the input and output waveguides. Given the FSR, determine the set of wavelengths that must be selected in order for the AWG to function as the wavelength router depicted in Figure 3.25. Assume that the angular spacing between the input (and output) waveguides is constant. Use the result from Problem 3.16 that c~ i = d sin Oi. Design an AWG that can multiplex/demultiplex 16 WDM signals spaced 100 GHz apart in the 1.55 #m band. Your design must specify, among other things, the spacing between the input/output waveguides, the path length difference between successive arrayed waveguides, the radius R of the grating circle, and the FSR of the AWG. Assume the refractive index of the input/output waveguides and the arrayed wave- guides is 1.5. Note that the design may not be unique, and you may have to make reasonable choices for some of the parameters, which will in turn determine the rest of the parameters. Show that the FWHM bandwidth of the acousto-optic filter is ~ 0.8)~2/1An. Explain how the polarization-independent acousto-optic tunable filter illustrated in Figure 3.28 acts as a two-input, two-output wavelength router when both its inputs are active. Calculate the acousto-optic interaction length that would be required for the AOTF to have a passband width (FWHM) of 1 nm at an operating wavelength of 1.55 #m. Assume An = 0.07. Consider a 16-channel WDM system where the interchannel spacing is nominally 100 GHz. Assume that one of the channels is to be selected by a filter with a 1 dB bandwidth of 2 GHz. We consider three different filter structures for this purpose. Fabry-Perot filter: Assume the center wavelengths of the channels do not drift. What is the required finesse and the corresponding mirror reflectivity of a Fabry-Perot filter that achieves a crosstalk suppression of 30 dB from each adjacent channel? If the center wavelengths of the channels can drift up to 4-20 GHz from their nominal values, what is the required finesse and mirror reflectivity? [] Mach-Zehnder interferometer: Assume a cascade of MZIs, as shown in Fig- ure 3.21(c), is used for this purpose and the same level of crosstalk suppres- sion must be achieved. What is the path length difference AL and the number of stages required, when the channel center wavelengths are fixed and when they can drift by +20 GHz? . receivers are both available today. Large-scale MEMS-based optical switches for use in wavelength crossconnects are now emerging as commercial devices, and a variety of technologies are avail- able. smaller-scale switches. All -optical wavelength converters are still in the research laboratories, awaiting significant cost reductions and performance improve- ments before they can become practical. . diffraction pattern of a grating with N narrow slits spaced distance d apart. Show that we obtain diffraction maxima when d sin 0 - m)~. Discuss what happens in the limit as N ~ oo. Show that