Optical Networks: A Practical Perspective - Part 78 potx

740 PULSE PROPAGATION IN OPTICAL FIBER ~ /2 ~/8 ~/ 0 Figure E.1 The magnitude of the pulse envelope of the second-order soliton. [Agr97] G.P. Agrawal. Fiber-Optic Communication Systems. John Wiley, New York, 1997. [Mar80] D. Marcuse. Pulse distortion in single-mode fibers. Applied Optics, 19:1653-1660, 1980. [Mar81] D. Marcuse. Pulse distortion in single-mode fibers. 3: Chirped pulses. Applied Optics, 20:3573-3579, 1981. Nonlinear Polarization T HE LINEAR EQUATION (2.7) for the relationship between the induced polarization P and the applied electric field E holds when the power levels and/or bit rates are moderate. When this is not the case, this must be generalized to include higher powers of E(r, t). For an isotropic medium and an electric field polarized along one direction so that it has a single component E(r, t), this relationship can be written as follows: 79(r, t) / ~0 x (x) (1)(r, t - tl)E(r, tl)dtl + ~0 X (x) oo (2) (t - tl, t - t2)E(r, tl)E(r, t2)dtl dt2 /// + ~0 X oo oo oo (3) (t tl, t - t2, t - t3)E(r, tl)E(r, t2)E(r, t3) dtl dt2 dt3 + (El) Now X(1)(r, t) is called the linear susceptibility to distinguish it from x(i)(r, t), i - 2, 3 which are termed the higher-order nonlinear susceptibilities. Owing to certain symmetry properties of the silica molecule, X(2)(r, t) - 0. The effect of the higher-order susceptibilities x (4)(, ), x (5)(, ) is negligible in comparison with that of X (3)(, ). Thus we can write (E 1) as P(r, t) - ~)L (r, t) 4- ~)NL (r, t). 741 742 NONLINEAR POLARIZATION Here 7)L(r, t) is the linear polarization given by (2.18). The nonlinear polarization 7)NL (r, t) is given by 79NL (r, t) G0 X (9O OO OO (3)(t - tl, t - t2, t - t3) E(r, q)E(r, t2)E(r, t3) dtl dt2 dt3. (F.2) The nonlinear response of the medium occurs on a very narrow time scale of less than 100 fsmmuch smaller than the time scale of the linear response and thus can be assumed to be instantaneous for pulse widths greater than 1 ps. Note that even if the pulse occupies only a tenth of the bit interval, this assumption is satisfied for bit rates greater than 100 Gb/s. We will consider only this instantaneous nonlinear response case in this book. When this assumption is satisfied, X(3)(t-tl, t-t2, t-t3) =X (3)S(t tl)S(t t2)g(t t3), where X (3) on the right-hand side is now a constant, independent of t. This assump- tion enables us to simplify (E2) considerably. It now reduces to T)NL(r, t) e0X(3)E3(r, t), which is equation (2.19). Multilayer Thin-Film Filters T O UNDERSTAND THE PRINCIPLE of operation of dielectric thin-film multicavity filters, we need to digress and discuss some results from electromagnetic theory. G.1 Wave Propagation at Dielectric Interfaces A plane electromagnetic wave is one whose electric and magnetic fields vary only in the spatial coordinate along the direction of propagation. In other words, along any plane perpendicular to the direction of propagation, the electric and magnetic fields are constant. The ratio of the amplitude of the electric field to that of the magnetic field at any such plane is called the impedance at that plane. In a medium that supports only one propagating wave (so there is no reflected wave), this impedance is called the intrinsic impedance of the medium and is denoted by 7. If ~ is the dielectric permittivity of the medium and # is its magnetic permeability, r/ - ~,/-~-7~. If we denote the intrinsic impedance of vacuum by rl0, for a nonmagnetic dielectric medium with refractive index n, the intrinsic impedance , - rlo/n. (A nonmagnetic dielectric material has the same permeability as that of a vacuum. Since most commonly used dielectrics are nonmagnetic, in the rest of the discussion, we assume that the dielectrics considered are nonmagnetic.) Consider the interface between two dielectrics with refractive indices nl and n2, illustrated in Figure G.l(a). Assume that a plane electromagnetic wave is incident normal to this interface. The reflection coefficient at this interface is the ratio of the amplitude of the electric field in the reflected wave to that in the incident wave. From 743 744 MULTILAYER THIN-FILM FILTERS Figure G.1 (a) The interface between two dielectric media. (b) A dielectric slab or film placed between two other dielectric media. (c) Multiple dielectric slabs or films stacked together. the principles of electromagnetics [RWv93, Section 6.7], it can be shown that the reflection coefficient at this interface (for normal incidence) is r/2 r/1 n 1 n2 p -~ = . (G.1) r]2 q- 771 n l -+- n2 Thus the fraction of power transmitted through this interface is 2 1 -[p[2 = 1 - nl - n2 . nl +n2 Here, as in the rest of the discussion, we assume that the dielectrics are lossless so that no power is absorbed by them. Now consider a slab of a dielectric material of thickness I and refractive index n2 (dielectric 2) that is placed between two dielectrics with refractive indices nl and n3 (dielectrics 1 and 3, respectively). Assume that dielectrics 1 and 3 have very large, essentially infinite, thicknesses. This is illustrated in Figure G.l(b). A part of any signal incident from dielectric 1 will be reflected at the 1-2 interface and a part transmitted. Of the transmitted part, a fraction will be reflected at the 2-3 interface. Of this reflected signal, another fraction will be reflected at the 2-1 interface and the remainder transmitted to dielectric 1 and added to the first reflected signal, and so on. In principle, the net signal reflected at the 1-2 interface can be calculated by adding all the reflected signals calculated using the reflection coefficients given by G.1 Wave Propagation at Dielectric Interfaces 745 (G.1), with the proper phases. But the whole process can be simplified by using the concept of impedances and the following result concerning them. If the impedance at some plane in a dielectric is ZL, called the load impedance, the impedance at distance I in front of it, called the input impedance, is given, as a function of the wavelength ~., by Zi = 77 ( ZL cos(2rcnl/)O + irl sin(27rnl/)O ) . (G.2) rl cos(2rcnl/)~) + i ZL sin(2rcnl/)O Here, r/is the intrinsic impedance of the dielectric, and n is its refractive index. Note that in a single dielectric medium, ZL = r/, and (G.2) yields Zi = r/as well. This agrees with our earlier statement that the impedance at all planes in a single dielectric medium is ~. The reason that the concept of impedance is useful for us is that the reflection and transmission coefficients may be expressed in terms of impedances. Specifically, the reflection coefficient at an interface with load impedance Z L, in a dielectric with intrinsic impedance r/, is given by ZL I 7 P- ZL + ~" (G.3) The transmission coefficient at the same interface is given by 2ZL r = 1 - p = ~. (G.4) Z L [- I 7 Note that (G.1) is a special case of (G.3) obtained by setting rl = ~1 and ZL = ~2. Now consider again the case of a single dielectric slab, placed between two other dielectrics, illustrated in Figure G. l(b). The impedance at the 2-3 interface is r13. Thus the impedance at the 1-2 interface may be calculated using (G.2) as Z12 ~72 ( ~73 c~ + i~72 sin(2rcnl/)O ) 172 cos(2rcnl/)~) + it13 sin(2rcnl/)O " Using this, the reflection coefficient at the 1-2 interface can be obtained from (G.3) as p Z12- 171 Z12 + r]l If the slab of a dielectric of thickness 1 shown in Figure G.1 (b) is viewed as a filter, its power transfer functionmthe fraction of power transmitted by it is given by T(~) 1 - IPl 2. 746 MULTILAYER THIN-FILM FILTERS .o o -3 0'5 i 1'5 )~0/)~ Figure G.2 Transfer function of the filter shown in Figure G.l(b) for nl - n3 - 1.52, n2 - 2.3, and I - )~0/2n2. Let )~0 - 2nl so that the optical path length in the slab is a half wavelength. Note that T(k0) 1. In Figure G.2, T(k) is plotted as a function of ko/)~, assuming n l = n3 1.5 and n2 = 2.3. Note that for the case nl - n3, this filter becomes a Fabry-Perot filter (see Problem 3.12). This result can be generalized to an arbitrary number of dielectric slabs as follows. Consider a series of k dielectrics with refractive indices nl, n2 nk (not necessarily distinct) and thicknesses l l, 12 lk, which are stacked together as shown in Fig- ure G.1. We also assume that l l and lk are very large, essentially infinite. This can be viewed as a filter of which a special case is the DTME We assume that the input signal is incident normal to the 1-2 interface. If we find the reflection coefficient, p, at the 1-2 interface, we can determine the power transfer function, T 00, of the filter, using T()Q - 1 -[p[2. Using the impedance machinery, this is quite easy to do. If rig is the intrinsic impedance of dielectric i, i - 1 k, ~7i - ~7o/ni. We start at the right end of the filter, at the (k- 1)-(k) interface. The impedance at this plane is just the intrinsic impedance of medium k, namely, ~k. The intrinsic impedance at the (k - 2)-(k- 1) interface can be calculated using (G.2) with ZL Ok, ~ Ok-l, n n~-l, and 1 - lk-1. Continuing in the same manner, we can recursively calculate the input G.2 Filter Design 747 impedances at the interfaces (k - 3)-(k - 2) 1-2. From this, the reflection co- efficient at the 1-2 interface can be calculated using (G.3), and the power transfer function of the filter can be determined. G.2 Filter Design Although the power transfer function of any given stack of dielectrics can be deter- mined using the preceding procedure, designing a filter of this type to meet a given filter requirement is a more typical problem encountered in practice. The multiple dielectric slab structure exemplified by Figure G.l(c) is quite versatile, and a number of well-known filter transfer functions, such as the Butterworth and the Chebyshev, may be synthesized using it [Kni76]. However, the synthesis of these filters calls for a variety of dielectric materials with different refractive indices. This may be a difficult requirement to meet in practice. It turns out, however, that very useful filter transfer functions can be synthesized using just two different dielectric materials, a low-index dielectric with refractive in- dex nL and a high-index dielectric with refractive index nH [Kni76]. Assume we want to synthesize a bandpass filter with center wavelength ~.0. Then, a general structure for doing this is to use alternate layers of high-index and low-index dielectrics with thicknesses equivalent to a quarter or a half wavelength at )~0. (A quarter-wavelength slab of the dielectric with refractive index nL would have a thickness ~.0/4nL.) Since these thicknesses at optical wavelengths are quite small, the term thin film is more appropriately used instead of slab. The dielectric thin films that are a half-wavelength thick at )~0 are called the cavities of the filter. A particularly useful filter structure consists of a few cavities separated by several quarter-wavelength films. If H and L denote quarter-wavelength films (at ~.0) of the high- and low-index dielectrics, re- spectively, then we can represent any such filter by a sequence of H s and Ls. Two Ls or two H s in succession would represent a half-wavelength film. For example, if the lightly shaded dielectrics are of low index and the darker shaded are of high index, the filter consisting of the multiple dielectric films 2-8 shown in Figure G.l(c) can be represented by the sequence H L H L L H L/-/. If the surrounding dielectrics, 1 and 9, are denoted by G (for glass), the entire structure in Figure G.l(c) can be represented by the sequence GHLHLLHLHG. If we know the refractive indices no, nL, and nH of the G, L, and H dielectrics, respectively, the transfer function of the filter can be calculated using the procedure outlined. For no = 1.52, a typical value for the cover glass, nL = 1.46, which is the refractive index of SiO2 (a low-index dielectric), and nH= 2.3, which is the refractive index of TiO2 (a high-index dielectric), this transfer function is plotted in Figure G.3. From this figure, we see that the main lobe is quite 748 MULTILAYER THIN-FILM FILTERS Figure G.3 Transfer function of the filter shown in Figure G.l(c) for nG = 1.52, nL = 1.46, and n/-/= 2.3. wide compared to the center wavelength, and the side lobe suppression is less than 10 dB. Clearly, a better transfer function is needed if the filter is to be useful. A narrower passband and greater side lobe suppression can be achieved by the use of more quarter-wavelength films than just three. For example, the filter described by the sequence G(HL)9HLL(HL)9HG has the transfer function shown in Figure G.4. The notation (HL) 1` denotes the sequence HL. HL HL (k times). Note that this filter is a single-cavity filter since it uses just one half-wave film. However, it uses 38 quarter-wave films, 19 on each side of the cavity. The transfer function of a dielectric thin-film filter is periodic in frequency or in )~0/)~, just like the Fabry-Perot filter. In Figure G.4(a), the transfer function of the filter for one complete period is shown. However, this figure hides the passband structure of the filter. Therefore, the transfer function of the filter is shown in Figure G.4(b) for a narrow spectral range around the center wavelength )~0. The passband structure of the filter can now be clearly seen. The resemblance to the Fabry-Perot filter transfer function (Figure 3.17) is no accident (see Problem 3.12). The use of multiple cavities leads to a flatter passband and a sharper transition from the passband to the stop band. Both effects are illustrated in Figure 3.19, where the filter transfer function, around the center wavelength )~0, is plotted for a G.2 Filter Design 749 -40 ~) 015 1 115 89 ~" -10 o E -20 -30 (a) 0 - ko/k -40 ~" -10 o r~ ., , ~ -20 -30 9 9 | 9 9 9 I , , , I , | , 0.996 0.998 1 1.002 1.004 (b) ko/)~ Figure G.4 Transfer function of a single-cavity dielectric thin-film filter. The sequence structure is G(HL)9HLL(HL)9HG. nG= 1.52, nL = 1.46, and nH= 2.3. single-cavity, two-cavity, and three-cavity dielectric thin-film filter. The single-cavity filter is the same as the one considered here. The two-cavity filter is described by the sequence G(HL)6HLL(HL)12HLL(HL)6H G. . interface and a part transmitted. Of the transmitted part, a fraction will be reflected at the 2-3 interface. Of this reflected signal, another fraction will be reflected at the 2-1 interface and. equivalent to a quarter or a half wavelength at )~0. (A quarter-wavelength slab of the dielectric with refractive index nL would have a thickness ~.0/4nL.) Since these thicknesses at optical wavelengths. filter. A particularly useful filter structure consists of a few cavities separated by several quarter-wavelength films. If H and L denote quarter-wavelength films (at ~.0) of the high- and low-index