## Abstract

A novel lithium niobate optical modulator of fiber type is proposed. The velocity match and the impedance match can be achieved simultaneously. The modulator is analyzed and designed with the finite element method. The designed modulator operates with a low half-wave voltage of 2.28V, a broad 3dB optical bandwidth of 116.3GHz and a characteristic impedance of 50.67Ω at 1.5µm wavelength.

©2005 Optical Society of America

## 1. Introduction

A LiNbO_{3} electro-optic modulator [1–4] has an important application to future optical communication systems over 10Gb/s due to its chirp-free operation. The bandwidth of such a modulator is mainly restricted by the velocity mismatch between the optical guiding wave and the modulating microwave signal, and the impedance mismatch between the traveling-wave electrodes and the microwave connector. Impedance match can also ensure a maximum power transfer. 50Ω impedance is the current standard for connectors and attenuators, etc. Therefore, the impedance of a practical electro-optic modulator should be matched to 50Ω. The optical guiding wave usually travels slower due to the high dielectric constant of LiNbO_{3}. In order to obtain a broad bandwidth, various modulator structures have been developed [5–8]. However, neither the velocity nor the impedance is matched in these modulators.

In this letter, a novel optical modulator of fiber type is proposed. Fig. 1(a) and 1(b) show the top and cross-sectional views of the proposed modulator, respectively. In the present traveling-wave modulator, a Mach-Zehnder interference structure formed by two LiNbO_{3} fibers (with a rectangular cross-section in the middle region) is placed (through, e.g., adhesive) on a SiO_{2} substrate. To match the impedance of the electrodes to 50Ω, a 3 electrodes of coplanar waveguide (CPW) type are used (see Fig. 1). The positive electrode is in the middle, and the third electrode has a SiO_{2} boss. The finite element method is used to analyze the modulator numerically, and a modulator of broad bandwidth, low half-wave voltage, and matched impedance is designed.

## 2. Theoretical analysis

The performance of the modulator includes half-wave voltage, modulation bandwidth, and characteristic impedance. The half-wave voltage of the present modulator can be calculated by [7]

where *g* is the gap between the electrodes, *λ* is the optical guiding wavelength, *γ*
_{33} is the electro-optic coefficient, *n _{e}* is the refractive index of the extraordinary wave, Γ is the factor of electro-optical overlap-integral, and

*L*is the electrode length.

The characteristic impedance of the modulator can be calculated by

where *C* is the electrode capacitance per unit length, and *C*
_{0} is the electrode capacitance per unit length in free space. In the present paper, we will try to match the characteristic impedance of the modulator to 50Ω.

For a traveling wave modulator, the product of the 3dB optical modulation bandwidth and the electrode length is given by [7]

where *c* is the light speed in vacuum, ${N}_{m}={\left(\frac{C}{{C}_{0}}\right)}^{\frac{1}{2}}$ is the microwave effective refractive index, and *N*
_{0} is the light-wave refractive index.

The electrode capacitance *C* per unit length can be calculated by

where *V*
_{0} is the applied voltage, *φ* is the scalar potential satisfying the Laplace equation ($\left({\epsilon}_{x}\frac{{\partial}^{2}}{\partial {x}^{2}}+{\epsilon}_{y}\frac{{\partial}^{2}}{\partial {y}^{2}}\right)\phi (x,y)=0$),*ε* is the relative dielectric constant along the direction normal to contour *S* (see Fig. 2 below). The potential contour of the present modulator is shown in Fig. 2.

The value of *C* depends on the materials used in the modulator and its structure parameters, such as the electrode thickness (t), the width of the positive electrode (W), and the gap between the electrodes (g). The structure parameters are shown in Fig. 3. In the present modulator, the effective refractive index of microwave can be reduced greatly (to match that of the light wave) since the dielectric constant of the SiO_{2} substrate is much less than that of LiNbO_{3}. Then it is possible to match both the velocity and the impedance of the modulator so that a broadband modulation can be achieved.

## 3. Numerical results

The finite element method (FEM) (see, e.g., Refs.[9, 10]) is used to calculate the electrode capacitance per unit length and the performance of the electro-optic modulator.

The thickness H_{0} of the LiNbO_{3} rectangular fiber (with a refractive index of 2.181) and the thickness H of the SiO_{2} substrate are assumed to be 10µm and 200µm, respectively. The numerical results for the microwave effective refractive index Nm and the characteristic impedance Z of the modulator are shown in Fig. 4 as the gap (g) between the electrodes varies when the width W=4,5,6,8µm (with fixed t=4µm). From this figure one sees that the microwave effective refractive index is too small (as compared to the effective refractive index for the light wave; note that the refractive index of optical-wave equals 2.138 at 1.5µm wavelength) and decreases as the width W increases (thus we choose W=4µm in the our design). The characteristic impedance Z increases as the gap (g) between the electrodes increases. Z is about 50Ω when g=5µm (with W=4µm and t=4µm). Fig. 5 shows Nm and *Z* when the electrodes thickness (t) varies from 0.1µm to 8µm (with fixed W=4µm and g=5µm).

Obviously, t=3.3µm gives the best microwave effective refractive index (Nm=1.974) and characteristic impedance (*Z*=50.67Ω). The designed modulator has a broad 3dB bandwidth of 116.3GHz, a characteristic impedance of 50.67Ω (very close to 50Ω) a low half-wave voltage of 2.28V (when the electrode length *L*=1cm).

## 4. Discussion and conclusion

In the present paper we have proposed a novel lithium niobate optical modulator of fiber type. The rectangular LiNbO_{3} fibers need to be placed on a silica substrate by using e.g. some special adhesive (the charges trapped around the interfaces between LiNbO_{3} and silica should be minimal). Since Silica and LiNbO_{3} have different thermal expansion coefficients, temperature control should be used for the present modulator. The performance of the novel modulator has been analyzed by using the finite element method. The numerical results have shown that an optimally designed modulator can give a broad bandwidth, low half-wave voltage, and good impedance match. The present modulator has a potential application to high speed and broadband optical communication systems.

## Acknowledgments

One of the authors (J.Sh.Li) would like to thank Prof. Z.W.Bao for helpful discussion. The work was supported by the National Basic Research Program of China (2004CB719800) and the National Natural Science Foundation of China under a key project (90101024).

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