identical. This means that the two-point method measures V ( f ) as accurately as the nonlinear response method.
MEASURED RESULTSThe nonlinear response of a LiNbO 3 optical modulator with a DC half-wave voltage of 3 V was measured using the experimental setup labelled B in Figure 1. The laser-source wavelength and modulation frequency are 1.55 m and 6.3 GHz, respectively. Figure 3 shows the measured response of the modulator for input RF powers ranging from Ϫ15 to 15 dBm. The detected output powers were normalized with respect to the input RF power of Ϫ15 dBm. Based on the error analysis presented in the previous section, we know that in order to obtain an accurate result for V ( f ), the RF input power should be set as high as possible; however, in order to avoid modulator thermal damage, the maximum RF input power that we can apply is limited. In this experiment, the maximum and minimum RF input powers used were 15 and Ϫ15 dBm, respectively.The detected RF powers together with Eqs. (8) and (9) were used to measure V ( f ). The values obtained are, respectively, 4.168 and 4.166 V, which are approximately equal. This clearly shows that both the nonlinear response the two-point methods have the same precision. Using Eqs. (10b) and (10c) and the approximation f( x 1 ) Ϸ 1, the difference in the values of V ( f ) using Eqs. (8) and (9) is given byFor the maximum and minimum RF input powers of Ϫ15 and 15 dBm, and V (6.3 GHz) ϭ 4.17 V, the result from this formula is 0.09%, which is consistent with the observed range of the measurement result: (4.168 -4.166)/4.167 ϭ 0.05%.The solid line in Figure 3 denotes the linear response of the modulator plotted using Eq. (8b) and the dashed line denotes the calculated response using Eq. (9) with V ( f ) ϭ 4.17 V, where V 1 in Eq. (9) corresponds to the RF input power of Ϫ15 dBm and V 2 varies from Ϫ15 to 20 dBm. Figure 3 shows the response using the measured RF half-wave voltage of V ( f ) ϭ 4.17 V agrees very well with the experimental result.
CONCLUSIONFour methods for measuring the RF half-wave voltage V ( f ) of an optical modulator have been presented and compared. The error analysis shows that the optical response and two-tone methods can measure V ( f ) with higher precision. However, compared with these two methods, the nonlinear response and the two-point methods are simpler, requiring only a photodetector and an RF spectrum analyzer, rather than an optical spectrum analyzer. If the selected data points are appropriate, the nonlinear response and two-point methods of measuring V ( f ) will provide the same precision as the former two methods.
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