he amplitude of the symbol transmitted along the jth quadrature in the kth symbol interval. The maximum distance of QAM modulation depends on the adopted pattern and it can be easily evaluated starting from the fact that the power of an optical field is the square of its distance from origin. The minimum distance for a few common QAM patterns is reported in Table 9.2. A general scheme for an optical M-QAM modulator is reported in Figure 9.20: a laser emits a linearly polarized optical field that is divided into two fields with a phase dif- ference equal to by a directional coupler. Each field is independently modulated by an amplitude modulator and then one of them is delayed to introduce a further phase differ- ence of /2. At that point, the two fields are combined again by a directional coupler that introduces a further phase difference of , SO that the total phase difference if 2 + /2 and the two combined fields are exactly the quadratures of the M-QAM signal. Such a scheme is suitable to be integrated in a single optical chip, for example, using lithium-niobate or indium phosphide (InP) technology. Several different modulators have been proposed for particular M-QAM constella- tions. An interesting scheme is that of the hybrid modulator for 64-QAM proposed in [32]. The principal scheme of the modulator is shown in Figure 9.21 besides the way in which the 64-QAM constellation is formed in the different stages of the modulator. The fact that the modulator is composed of a parallel of three dual drivers phase modulators Next Generation Transmission Systems TABLE 9.2 Normalized Constellation Minimum Distance for M-QAM Modulation Format Minimum Distance (Normalized to 4-QAM V2 = 1.14 8-QAM-1 2 =0.632 0.632 8-QAM-2 1- = 0.707 16-QAM = 0.471 32-QAM 2 = 0.263 64-QAM 0.202 128-QAM V335 2 = 0.111 256-QAM 0.094 1024-QAM V==0.046 that has to be driven to produce a four-level phase modulated signal renders the modula- tor control easy and the loss relatively small. The most interesting property of this modulator is, howev