BACKGROUND 
       [0001]    1. Technical Field 
         [0002]    The present disclosure relates to integrated optics and, particularly, to an electro-optic modulator having a high extinction ratio when functioning as a switch. 
         [0003]    2. Description of Related Art 
         [0004]    Electro-optic modulators, such as Mach-Zehner electro-optic modulators, change a refractive index of a branch of a Y-shaped waveguide (hereinafter the first branch) using a modulating electric field, utilizing electro-optic effect. Thus, the modulator can alter a phase of lightwaves traversing the first branch. As a result, the lightwaves traversing the first branch have a phase shift and thus interfere with lightwaves traversing another branch of the Y-shaped waveguide (hereinafter the second branch). An output of the Y-shaped waveguide is modulated as the output depends on the phase shift, which in turn depends on the modulating electric field. However, being limited by manufacturing imprecision, all the properties of the lightwaves traversing the first and second branches are not the same. As such, when the modulator is used as a switch, the output is often larger than zero in an off state (i.e., the phase shift is π) or less than a desired maximum value in an on state (i.e., the phase shift is zero). An extinction ratio of the switch is thus less than satisfactory. 
         [0005]    Therefore, it is desirable to provide an electro-optic modulator that can overcome the above-mentioned problems. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0006]    Many aspects of the present disclosure can be better understood with reference to the following drawings. The components in the drawings are not necessarily drawn to scale, the emphasis instead being placed upon clearly illustrating the principles of the present disclosure. 
           [0007]      FIG. 1  is an isometric view of an electro-optic modulator, according to an embodiment. 
           [0008]      FIG. 2  is a cross-sectional view taken along a line II-II of  FIG. 1 . 
       
    
    
     DETAILED DESCRIPTION 
       [0009]    Embodiments of the present disclosure will be described with reference to the drawings. 
         [0010]      FIGS. 1 and 2  show an electro-optic modulator  10 , according to an embodiment. The modulator  10  includes a substrate  110 , a waveguide  120 , a first modulating electrode  130 , a second modulating electrode  140 , and three ground electrodes  150 . 
         [0011]    The substrate  110  is made of lithium niobate (LiNbO 3 ) crystal to increase a bandwidth of the modulator  10 , as LiNbO 3  crystal has a high response speed. In this embodiment, the substrate  110  is substantially rectangular and includes a top surface  111 . 
         [0012]    The waveguide  120  is formed by applying a layer of titanium as a coating on a shape corresponding to the waveguide  120  and diffusing the titanium into the substrate  110  by, for example, a high temperature diffusion technology. In this embodiment, the waveguide  120  is formed in the top surface  111 . 
         [0013]    The waveguide  120  is Y-shaped and formed in the substrate  110 . The waveguide  120  includes a first section  121  and a second section  122 . The first section  121  is Y-shaped and includes a first branch  124  and a second branch  125 . The second section  122  is Y-shaped and includes a third branch  127  and a fourth branch  128 . 
         [0014]    The first to fourth branches  124 ,  125 ,  127 ,  128  are substantially parallel with each other and the second and fourth branches  125 ,  128  are located at two opposite sides of the first and third branches  124 ,  127 . 
         [0015]    In addition to the first section  121  and the second section  122 , the waveguide  120  includes an input section  129  and an output section  12   a . The first and second sections  121 ,  122  diverge from the input section  129  and are converged into the output section  12   a.    
         [0016]    In addition to the first branch  124  and the second branch  122 , the first section  121  includes a first input branch  12   b  and a first output branch  12   c.  The first and second branches  124 ,  125  diverge from the first input branch  12   b  and are converged into the first output branch  12   c.    
         [0017]    In addition to the third branch  127  and the fourth branch  128 , the second section  122  includes a second input branch  12   d  and a second output branch  12   e.  The third and fourth branches  127 ,  128  diverge from the second input branch  12   d  and are converged into the second output branch  12   e.    
         [0018]    The substrate  110  defines first to fifth recesses  112 - 116 , all of which are substantially rectangular and arranged to be parallel with the first to fourth branches  124 ,  125 ,  127 ,  128 , in the top surface  111 . A depth of each of the first to fifth recesses  112 - 116  is larger than a thickness of the waveguide  120 . The first and second recesses  112 ,  113  are located at two opposite sides of the second branch  125  and have the same length as, and are aligned with, the second branch  125 . The first recess  112  is located between the first and second branches  124 ,  125 . The third and fourth recesses  114 ,  115  are at two opposite sides of the fourth branch  128  and have the same length as, and are aligned with, the fourth branch  128 . The third recess  114  is located between the third and fourth branches  127 ,  128 . The fifth recess  116  is located between the first and third branches  124 ,  127 . Orthogonal projections of the first to fourth recesses  112 - 115  on the fifth recess  116  fall within the fifth recess  116 . 
         [0019]    The first and second modulating electrodes  130 ,  140  are fully filled within the first and third recesses  112 ,  114 , respectively. The ground electrodes  150  are fully filled within the second, fourth, and fifth recesses  113 ,  115 ,  116 . 
         [0020]    The first and second modulating electrodes  130 ,  140  and the ground electrodes  150  receive voltages and modulate the first and second sections  121 ,  122  such that optical outputs of the first and second sections  121 ,  122  are equal to each other. 
         [0021]    In principle, the output of the output section  12   a  can be calculated by the following equation: 
         [0000]      ae i(α-wt) =a 1 e i(φ-wt) +a 2 e i(β-wt) , 
         [0022]    wherein, a, a 1 , a 2  are amplitudes of lightwaves traversing the output section  12   a , the first output branch  12   c , and the second output branch  12   e  respectively, α, φ, β are phases of lightwaves traversing the output section  12   a , the first output branch  12   c , and the second output branch  12   e  respectively, and where e is the natural exponent, i is the imaginary unit, ω is an angular velocity, and t is a time variable. 
         [0023]    The output of the output section  12   a  can be calculated by the following equation: 
         [0000]        S=a   2   =a   1   2   +a   2   2 30 2 a   1   a   2  cos(φ−β),
 
         [0024]    wherein S is the output of the output section  12   a.    
         [0025]    Similarly, the outputs of the first and second output branches  12   c ,  12   e  can be calculated by the following equations: 
         [0000]        a   1   e   i(φ-wt)   =a   11   e   i(φ     1     -wt)   +a   12   e   i(φ     2     -wt) , 
         [0000]        Q   1   =a   1   2   =a   11   2   +a   12   2 +2 a   11   a   12  cos(φ 1 −φ 2 ),
 
         [0000]        a   2   e   i(φ-wt)   =a   21   e   i(β     1     -wt)   +a   22   e   i(β     2     -wt) , and 
         [0000]        Q   2   =a   2   2   =a   21   2   +a   22   2 +2 a   21   a   22  cos(β 1 −β 2 ),
 
         [0026]    wherein a 11 , a 12 , a 22 , a 22  are amplitudes of lightwaves traversing the first to fourth branches  124 ,  125 ,  127 ,  128  respectively, φ 1 , φ 2 , β 1 , β 2 , are phases of lightwaves traversing the first to fourth branches  124 ,  125 ,  127 ,  128  respectively, and Q 1 , Q 2  are the respective outputs of the first and second output branches  12   c ,  12   e.    
         [0000]        a   1   e   i(φ-wt)   =a   11   e   i(φ     1     -wt)   +a   12   e   i(φ     2     -wt) , 
         [0000]        Q   1   =a   1   2   =a   11   2   +a   12   2 +2 a   11   a   12  cos(φ 1 −φ 2 ),
 
         [0000]        a   2   e   i(φ-wt)   =a   21   e   i(β     1     -wt)   +a   22   e   i(β     2     -wt) , and 
         [0000]        Q   2   =a   2   2   =a   21   2   +a   22   2 +2 a   21   a   22  cos(β 1 −β 2 ),
 
         [0027]    The lightwaves have transverse electric waves (hereinafter the TE mode) and transverse magnetic waves (hereinafter the TM mode). In a coordinate system xyz (see  FIG. 1 ), wherein x axis is a vertical height of the substrate  110  (i.e., perpendicular to the top surface  111 ), y axis is a horizontal width of substrate  110  (parallel with the top surface  111  and perpendicular to the first to fourth branches  124 ,  125 ,  127 ,  128 ), and z axis is a length of the substrate  110  (i.e., along a direction that is parallel with the first to fourth branches  124 ,  125 ,  127 ,  128 ), the TE mode has an electric field component {right arrow over (Ey)} vibrating along the y axis only. The TM mode has an electric field component {right arrow over (Ex)} vibrating along the x axis and a {right arrow over (Ez)} vibrating along the z axis. 
         [0028]    By constructing the first to fifth recesses  112 - 116 , the first and second modulating electrodes  130 ,  140 , and the ground electrodes  150 , as described above, modulating electric fields            1 ,            2  generated by the first and second modulating electrodes  130 ,  140  respectively and the corresponding ground electrodes  150  traverse the first to fourth branches  124 ,  125 ,  127 ,  128 . A portion of the electric field            1  interacting with the first second secondary branches  124 ,  125  is substantially parallel with the y axis, and thus efficiently modulates the TE mode (i.e. Ey) and alters the phases φ 1 , φ 2 . Similarly, a portion of the electric field            2  interacting with the fourth branch  128  is substantially parallel with the y axis, and thus efficiently modulates the TE mode (i.e. Ey) and alters the phases β 1 , β 2 . 
         [0029]    By changing the phases φ 1 , φ 2 , β 1 , β 2 , the equations Q 1 =Q 2 , and φ−β=0 (or φ−β=π) can be applied. As such, when the modulator  10  is used as a switch, the output of the waveguide  120  can be exactly zero in an off-state and can be substantially at a desired maximum value in an on state, and thus an extinction ratio of the modulator  10  is increased. 
         [0030]    It will be understood that the above particular embodiments are shown and described by way of illustration only. The principles and the features of the present disclosure may be employed in various and numerous embodiments thereof without departing from the scope of the disclosure. The above-described embodiments illustrate the possible scope of the disclosure but do not restrict the scope of the disclosure.
An electro-optic modulator includes a substrate, an end-to-end Y-shaped waveguide for optical divergence and convergence, and electrodes. The waveguide is formed in the substrate and the electrodes are formed in the substrate and received voltages act to modulate first and second sections of the waveguide such that the optical output by the first and second sections are equal or opposite to each other in all necessary respects regarding phase and amplitude.