| With the recent developments in multi-media services, computer networks and wireless access, the need for external modulation in optical fiber transmission systems has been firmly established. In high-speed direct optical modulation of laser diodes, the frequency chirp of the modulated signal cannot be eliminated. Then the optical external modulators play more and more important role in large capacity optical transmission systems, and have become a subject of intense study. At present, the practical modulators include M-Z waveguide LiNbO3 modulator and electro-absorb type semiconductor modulator. Especially, LiNbO3 optical modulators, using M-Z waveguides and traveling-wave electrodes, are essential for high-speed and long-haul optical fiber transmission systems since the wavelength chirp is very small and the effect of the fiber dispersion is minimized.Firstly, we describe the fundamental theory and types of the electrooptical modulators in detail, and make an introduction to the means used to analyze the modulators, especially, to the Schwartz-Christoffel(SC) conformal mapping(CM) technique. Then a quasi-static analysis using the conventional conformal mapping approaches based on elliptic integrals has been used in evaluation of the microwave characteristics of the conventional coplanar waveguide modulator, the effective index, the characteristic impedance and the conductor loss. It has been shown by our calculations that conductor loss is greatly reduced under velocity matching with relatively thick coplanar waveguide electrodes and thick buffer layer, but the characteristic impedance can not match with that of the external circuit at the same time, and the modulation bandwidth is confined in this case.The cross section of modulator electrodes we study is trapezoidal but not rectangular in common, so it is necessary to make an analysis of the effects of trapezoidal electrodes on the characteristics of coplanar waveguide. The calculation results have shown that when the angles of the center electrode and the grounds decrease, the effective index, the characteristic impedance, and the conductor loss all decrease. But, the characteristic impedance is much smaller than the signal resourceimpedance 50 Ω, we just can realize broadband modulation by adjusting the height of the center electrode and the grounds. The modulator with a ridge structure is a hotspot of modulator investigation now. To evaluate the characteristics of ridged structure modulator,the hyperelliptic integrals are cumbersome due to the complex polygon in the physical domain of a ridged structure, and the conventional conformal mapping approaches based on elliptic integrals can not be used in the calculation. In some papers, a quasi-static analysis by means of the finite element method(FEM) has been used for numerical simulations on ridged structure modulator in general, but in this paper we use the SC toolbox to perform these computations, and the finite element method combined with conformal mapping(FEM-CM) has been used in our work at the first time to evaluate the characteristics of the ridged modulator. After making an analysis of the dependence of the characteristics of the ridged structure on the ridge height and other parameters, we can obtain that as the ridge height increases, the effective index and the conductor loss decrease, but the characteristic impedance increases. By optimizing the modulator dimension parameters, impedance matching and small conductor loss can be realized under the conditions of velocity matching, therefore the optimum design device with the ridge structure is suitable for ultra broadband modulation. At last, we use a quasi-static finite element method(QS FEM) to analyze the electric field and potential distributions of the ridged structure and those of conventional coplanar waveguide modulators, and a comparison between these two cases is made to show that half-wave voltage of the ridge structure is lower than that of the conventional coplanar waveguides. As a result, we have obtained...
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