Abstract:
An optical modulator, including: a substrate; an optical waveguide embedded in the substrate; a traveling wave electrode mounted on the substrate and having a traveling wave applied thereon so that a light wave is modulated by the traveling wave with an electro-optic effect. The traveling wave electrode includes a center electrode and ground electrodes; in which the optical waveguide has a plurality of interaction optical waveguides that form a Mach-Zehnder optical waveguide that modulates the light wave in a phase modulation manner when the traveling wave is applied to the traveling wave electrode, the interaction optical waveguides form a region where respective widths of the interaction optical waveguides are different from each other, and the center electrode and the ground electrodes are positioned such that interaction efficiencies between the high frequency electric signal and the light wave guided in the respective interaction optical waveguides are substantially equal to each other.

Description:
TECHNICAL FIELD 
     The present invention relates to an optical modulator with high modulation velocity, low alpha parameter, high extinction ratio, low drive voltage, and low DC bias voltage. 
     BACKGROUND ART 
     Up until now, there has been developed an optical modulator such as a traveling wave electrode type of lithium niobate optical modulator (hereinafter simply referred to as an LN optical modulator) comprising a substrate (hereinafter simply referred to as an LN substrate) made of a material such as lithium niobate (LiNbO 3 ) having an electro-optic effect to cause a refractive index of an incident light to be varied in response to an electric field applied to the substrate, thereby making it possible to form an optical waveguide and a traveling wave electrode in and on the substrate. The LN optical modulator can be applied to a large volume optical transmission system having a capacity in the range of 2.5 Gbit/s to 10 Gbit/s due to the excellent chirping characteristics. In recent years, the LN optical modulator thus constructed is under review to be applied to the optical transmission system having a super large capacity of 40 Gbit/s and therefore expected as a key device in this technological field. 
     (First Prior Art) 
     There are two types of LN optical modulators with respect to the states of the substrate, one substrate having a z-cut state, and the other having an x-cut state (or a y-cut state). Here, x-cut LN substrate type LN optical modulator will be described as the first prior art, wherein the LN optical modulator comprises an x-cut LN substrate and a coplanar waveguide (CPW) forming a traveling wave electrode.  FIG. 13  is a perspective view showing the x-cut substrate type LN optical modulator.  FIG. 14  is a sectional view taken along the line A-A′ of  FIG. 13 . 
     The conventional LN modulation device comprises an x-cut LN substrate  1 , an SiO 2  buffer layer  2 , and an optical waveguide  3  formed to be flush with an upper surface of the x-cut LN substrate  1  wherein the SiO 2  buffer layer  2  has a thickness of 200 nm to 1 μm and transparent to incident light having optical wavelength typically utilized for optical communications such as for example 1.3 μm and 1.55 μm. The optical waveguide  3  is formed with a process of evaporating a metal Ti (titanium) on the x-cut LN substrate  1  and a process of thermal diffusing at a temperature of 1050° C. for approximately 10 hours, the optical waveguide  3  forming a Mach-Zehnder interferometer (a Mach-Zehnder optical waveguide). The optical waveguide  3  includes two interaction optical waveguides, that is, two arms  3   a  and  3   b  at the position where an electric signal and an incident light are interacted with each other (the position will be referred to as an interaction portion). The conventional LN optical modulator further comprises a traveling wave electrode  4  having two ground electrodes  4   b ,  4   c  and a center electrode  4   a  placed between the two ground electrodes  4   b ,  4   c.    
     As shown in  FIG. 14 , the interaction optical waveguides  3   a  and  3   b  have widths represented by W a  and W b , respectively. In this first prior art, the interaction optical waveguides  3   a  and  3   b  have same widths with each other, that is, W a =W b , where W a  and W b  both represent the same value exemplified by 9 μm. The legend G wg  represents the distance between the interaction optical waveguides  3   a  and  3   b  (or a gap between the waveguides), the distance being set at, such as, 16 μm. The legend Δ represents a distance in the horizontal direction between one edge of the center electrode  4   a  and the center of the interaction optical waveguide  3   b  (that is, the center line), the edge of the center electrode  4   a  facing the ground electrode  4   c . In general, the interaction optical waveguides  3   a  and  3   b  are symmetrically positioned with respect to the center electrode  4   a  and the ground electrodes  4   b  and  4   c , which leads to the fact that the distance in the horizontal direction between another edge of the center electrode  4   a  facing the ground electrode  4   b  and the center of the interaction optical waveguide  3   b  is represented by Δ. 
     In  FIG. 14 , the center of the center electrode  4   a  in the width direction is represented by a center line  18 . The centers of the interaction optical waveguides  3   a  and  3   b  in the width direction are represented by center lines  19   a  and  19   b , respectively. 
       FIG. 15  is a top view showing the optical waveguide  3 . The length of the interaction optical waveguides  3   a  and  3   b  is represented by the legend L. The position of the line A-A′ in  FIG. 15  corresponds to the position of the line A-A′ in the perspective view shown in  FIG. 13 , though only the optical waveguide  3  is shown in  FIG. 15 . 
     In this first prior art, both a bias voltage (generally a DC bias voltage) and a high frequency electric signal (an RF electric signal) are superimposed and applied between the center electrode  4   a  and the ground electrodes  4   b  and  4   c , thereby resulting in the fact that the incident light is phase modulated not only by the RF electric signal but also by the DC bias voltage at each of the interaction optical waveguides. The buffer layer  2  is important in that it functions to expand a modulation bandwidth of the optical modulator by reducing a microwave equivalent refractive index n m  of the electric signal traveling through the traveling wave electrode  4  to be close to an effective refractive index n 0  of the incident lights traveling through the respective interaction optical waveguides  3   a  and  3   b.    
     The operation of the LN optical modulator thus constructed will be described hereinafter. Firstly, the DC bias voltage and the RF electric signal are necessary to be applied between the center electrode  4   a  and the ground electrodes  4   b  and  4   c  to realize the operation of the LN optical modulator. 
       FIG. 16  is a graph showing the relationship between the applied voltage and the output light power of the LN optical modulator under a certain condition with the DC bias voltage set at “Vb”. As shown in  FIG. 16 , the DC bias voltage “Vb” is generally set such that the output light power becomes middle value of the peak-to-peak value. 
       FIG. 17  is a graph showing the relationship between the distance A and the product of the half wavelength voltage Vπ and the length L (that is, Vπ·L, which is utilized as a barometer of the driving voltage), where the legend Δ represents the distance in the horizontal direction between one edge of the center electrode  4   a  and the center line  19   b  of the interaction optical waveguide  3   b , and the legend L represents the length of the interaction optical waveguides. In this calculation, the value Δ is varied by changing the gap G wg  representing the distance between the interaction optical waveguides  3   a  and  3   b . As shown in  FIG. 17 , the distance Δ in the horizontal direction between one edge of the center electrode  4   a  and the center line  19   b  of the interaction optical waveguide  3   b  should be small to a certain extent, and there exists an optimum value. 
     When the value Δ, the distance in the horizontal direction between the edge of the center electrode  4   a  and the center line  19   b  ( 19   a ) of the interaction optical waveguide  3   b  ( 3   a ), is set to be smaller in order to lower the driving voltage, the gap G wg , the distance between the interaction optical waveguides  3   a  and  3   b , also becomes smaller. The optical modulator, however, encounters such a problem that the optical power ratio between the condition of the optical output is ON state and OFF state, that is, extinction ratio, is deteriorated under the condition that the gap G wg  between the interaction optical waveguides  3   a  and  3   b  becomes smaller. This results from the fact that the degree of coupling between the interaction optical waveguides  3   a  and  3   b  becomes severely large. 
     (Second Prior Art) 
     There has been two methods to make the degree of coupling between the interaction optical waveguides  3   a  and  3   b  smaller, one being achieved by setting the gap G wg  wider so that the incident lights in the interaction optical waveguides  3   a  and  3   b  are transmitted to be away from each other, the other being achieved by setting the widths of the interaction optical waveguides different from each other so that the incident lights respectively traveling the incident optical waveguides have the respective effective refractive indexes (propagation constants) different from each other. 
     However, the value Vπ·L becomes larger as the gap G wg  between the interaction optical waveguides  3   a  and  3   b  becomes larger as shown in  FIG. 17 , which results in the fact that the driving voltage is required to be higher. To avoid this problem, the widths W a ′and W b ′ of the interaction optical waveguides  5   a  and  5   b  are set to be different from each other according to the second prior art as shown in  FIG. 19 .  FIG. 20  is a top view showing the optical waveguide  5  according to the second prior art.  FIG. 19  is a sectional view taken along the line B-B′ of  FIG. 20 , shown with the x-cut LN substrate  1 , the center electrode  4   a , the ground electrodes  4   b  and  4   c , and the buffer layer  2 . 
     The legend Δ′ represents the distance in the horizontal direction between one edge of the center electrode  4   a  and the center line of the interaction optical waveguide  5   b , the edge facing the ground electrode  4   c . In  FIG. 19 , the centers of the interaction optical waveguides  5   a  and  5   b  in the width direction are represented by center lines  20   a  and  20   b , respectively. 
     The second prior art as described above, however, encounters some problems. Firstly, each of the widths W a ′ and W b ′ of the interaction optical waveguides  5   a  and  5   b  is set to be partially varied as shown in  FIG. 20 . Here, the interaction optical waveguides  5   a  and  5   b  form a first region having a length of L 1  and a second region having a length of L 2 . 
     As shown in  FIG. 20 , the interaction optical waveguides  5   a  and  5   b  have taper portions  6 ,  7 ,  8 ,  9 ,  10  and  11  to have the widths of the interaction optical waveguides varied, each of the interaction optical waveguides  5   a  and  5   b  being required to have three taper portions. It is well known that radiation loss is caused at the portion where the width of the optical waveguide is varied. Furthermore, the radiation loss at the taper portion where the optical waveguide is widened and the radiation loss at the taper portion where the optical waveguide is narrowed are different from each other. Therefore, the incident lights respectively traveling the interaction optical waveguides  5   a  and  5   b  have powers different from each other, which results in the deterioration of the extinction ratio. 
     The major problem, that is, the chirping problem encountered by the second prior art will be described hereinafter. The degree of chirping can be represented by an alpha parameter (i.e., “a” parameter) as described by the formula (1), wherein the alpha parameter is calculated by a phase “φ” of the optical signal pulse outputted from the optical modulator and an intensity (amplitude) “E” of the optical signal pulse (disclosed in non-patent document 1).
 
α=[ dφ/dt ]/[(1 /E )( dE/dt )]  (1)
 
     As can be seen in the above, the “α” parameter is calculated with an amount of phase shift and an amount of intensity variation of the optical signal pulse outputted from the optical modulator. 
     The “α” parameter can be represented by a formula (2), which is further developed from the formula (1).
 
α=(Γ1−Γ2)/(Γ1+Γ2)  (2)
 
     “Γ1”: An interaction efficiency normalized by the numerical number  1  in the form of overlap integration between the amplitude of the electric signal and the power of the incident light passing through the interaction optical waveguide  5   a.    
     “Γ2”: An interaction efficiency normalized by the numerical number  1  in the form of overlap integration between the amplitude of the electric signal and the power of the incident light passing through the interaction optical waveguide  5   b . The value “Γ1” for the interaction optical waveguide  5   a  at the first region becomes equal to the value “Γ2” for the interaction optical waveguide  5   b  at the second region while the value “Γ2” for the interaction optical waveguide  5   b  at the first region becomes equal to the value “Γ1” for the interaction optical waveguide  5   a  at the second region, under the condition that the width W a ′ of the interaction optical waveguide  5   a  at the first region having a length of L 1  is set to be equal to the width W b ′ of the interaction optical waveguide  5   b  at the second region having a length of L 2  while the width W b ′ of the interaction optical waveguide  5   b  at the first region is set to be equal to the width W a ′ of the interaction optical waveguide  5   a  at the second region. However, this does not mean that the chirping becomes zero, that is, α=0, by setting the length L 1  of the first region equal to the length L 2  of the second region. 
     This stems from the fact that the traveling wave electrode  4  constituted by the center electrode  4   a  and ground electrodes  4   b ,  4   c  not shown in  FIG. 20  causes high propagation loss to the high frequency electric signal traveling therethrough, which results in the high frequency electric signal attenuated as the high frequency electric signal propagates the traveling wave electrode  4 . In order to make the alpha parameter “α” in the formula (2) to be zero, it is required to fulfill the following condition due to the attenuation of the high frequency electric signal.
 
L1&lt;L2  (3)
 
     The condition to be imposed on the length L 1  of the first region and the length L 2  of the second region shown in  FIG. 20  to make the chirping zero, that is, α=0 will be described hereinafter in detail. The microwave propagation loss at the frequency “f” under the condition that the high frequency electric signal is imposed on the traveling wave electrode  4  formed by the center electrode  4   a  and ground electrodes  4   b ,  4   c  is represented by β m (f). The integration values calculated by the interaction efficiency between the incident lights in the respective interaction optical waveguides  5   a ,  5   b  and the electric signal integrated by the length at the first region with the length of L 1  and the second region with the length of L 2  are represented by I 1 (f) and I 2 (f), respectively (the integration value simply be referred to as a modulation efficiency). 
     Each of the modulation efficiency I 1 (f) and I 2 (f) depends on the frequency “f” and can be described as the formulas (4) and (5) where the incident light and the electric signal propagate in the “z” direction. 
     
       
         
           
             
               
                 
                   
                     
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     The chirping can be zero at any frequency “f” by setting the length L 1  of the first region and the length L 2  of the second region in such a way that the modulation efficiencies I 1 (f) and I 2 (f) fulfill the following condition.
 
 I   1 ( f )= I   2 ( f )  (6)
 
     In other words, the alpha parameter becomes zero when the condition of the formula (6) is fulfilled. 
     According to the above described calculation, there is a relationship between the length L 1  of the first region and the length L 2  of the second region as follows.
 
 L 1/ L 2≈0.9  (7)
 
     However, manufacturing variations generally occurs typically in the width and the thickness of the center electrode  4   a , the shape of the trapezoid and the inverted trapezoid, and the gaps between the center electrode  4   a  and ground electrodes  4   b  and  4   c , due to the fact that the traveling wave electrode  4  is formed with a thick gold plating having a thickness of 20 μm or more. This results in the fact that the microwave propagation loss β m (f) at the frequency “f” of the high frequency electric signal propagating the traveling wave electrode  4  varies within the z-cut LN substrate not shown in  FIG. 20 , and varies from run-to-run of the manufacturing. Therefore, the process yield achieving the condition of α=0 can not be expected. 
     (Non-patent Document 1) 
     Nadege Courjal et al “Modeling and Optimization of Low Chirp LiNbO 3  Mach-Zehnder Modulators With an Inverted Ferroelectric Domain Section “Journal of Lightwave Technology vol. 22 No. 5 May 2004 
     DISCLOSURE OF THE INVENTION 
     Problems to be Solved by the Invention 
     As described above, the distance between two interaction optical waveguides becomes short when the interaction optical waveguides are positioned to be close to the center electrode in order to reduce the driving voltage, which results in the deterioration in the extinction ratio, the pulse shape, and the high frequency modulation characteristics such as a chirping, due to the coupling between the incident lights according to the first prior art. The process yield is worsened and the extinction ratio is deteriorated to achieve the condition of chirping-zero under the condition that the widths of the interaction optical waveguides are set to be different from each other due to the propagation loss of the high frequency electric signal according to the second prior art. 
     Means for Solving the Problems 
     It is, therefore, an object of the present invention to provide an optical modulator to solve the problems in accordance with the prior arts as described above. According to a first aspect of the present invention, there is provided an optical modulator, comprising: a substrate having an electro-optic effect; an optical waveguide embedded in the substrate to have a light wave guided therein; and a traveling wave electrode mounted on the substrate to have a high frequency electric signal applied thereon so that the light wave is modulated by the high frequency electric signal with the electro-optic effect, the traveling wave electrode being constituted by a center electrode and ground electrodes; in which the optical waveguide has a plurality of interaction optical waveguides to collectively form a Mach-Zehnder optical waveguide operative to modulate the light wave in a phase modulation manner under the condition that the high frequency electric signal is applied to the traveling wave electrode, the interaction optical waveguides collectively form a region where respective widths of the interaction optical waveguides are different from each other, and the center electrode and the ground electrodes are positioned such that interaction efficiencies between the high frequency electric signal and the light wave guided in the respective interaction optical waveguides are substantially equal to each other. 
     According to a second aspect of the present invention, there is provided an optical modulator as set forth in claim  1 , in which the interaction optical waveguides are formed such that magnitude relationship between the widths of the interaction optical waveguides is unchanged. 
     According to a third aspect of the present invention, there is provided an optical modulator as set forth in claim  1 , in which the interaction optical waveguides form additional region such that magnitude relationship between the widths of the interaction optical waveguides at the region and the additional region is reciprocal. 
     According to a fourth aspect of the present invention, there is provided an optical modulator as set forth in claim  3 , in which the two regions have longitudinal lengths equal to each other, where the widths of the interaction optical waveguides at each of the two regions are different from each other. 
     According to a fifth aspect of the present invention, there is provided an optical modulator as set forth in claims  1  to  4 , in which center of the center electrode is away from centers of the interaction optical waveguides with respective distances different from each other. 
     According to a sixth aspect of the present invention, there is provided an optical modulator as set forth in claims  1  to  4 , in which the interaction optical waveguides collectively form a gap, and center of the gap is positioned away from center of the center electrode in a direction parallel to the surface of the substrate. According to a seventh aspect of the present invention, there is provided an optical modulator as set forth in claims  1  to  6 , in which the center electrode and the ground electrodes each form gaps, the gaps having respective sizes different from each other. 
     Advantageous Effect of the Invention 
     The optical modulators according to the first to third, fifth, and sixth aspects of the present invention make it possible to improve the extinction ratio by setting the widths of the interaction optical waveguides constituting the Mach-Zehnder optical waveguide different from each other, due to the fact that the optical coupling between the two interaction optical waveguides is suppressed. Furthermore, it is possible to make the chirping to be zero by optimally setting the positions of the center electrode and ground electrodes so that the interaction efficiencies between the high frequency electric signals and the incident lights in the respective interaction optical waveguides become equal with each other. 
     The optical modulator according to the fourth aspect of the present invention makes it possible to suppress filter characteristics against the optical wavelength due to the fact that the optical path lengths of the interaction optical waveguides forming the Mach-Zehnder interferometer are equal with each other. 
     The optical modulator according to the seventh aspect of the present invention makes it possible to effectively equalize the interaction efficiencies between the high frequency electric signal and respective incident lights in the interaction optical waveguides by setting the gaps between the center electrode and the ground electrodes different from each other, due to the fact that the electric field intensity between the center electrode and the ground electrode becomes larger as the gap between the center electrode and the ground electrode becomes narrower. 
     The optical modulator according to the seventh aspect of the present invention can additionally prevent any other high frequency modulation characteristics from being deteriorated caused by the above mentioned optical coupling. This comes from the fact that the optical coupling is suppressed even when the gap between the two interaction optical waveguides is small. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a sectional view taken along the line C-C′ of  FIG. 2  showing the optical modulator according to the first embodiment of the present invention; 
         FIG. 2  is a top view showing the optical waveguide according to the first embodiment of the present invention; 
         FIG. 3  is a graph showing the relationship between the width of the optical waveguide and the spot-size of the incident light; 
         FIG. 4  is a graph showing the relationship between the modulation efficiency and the distance between the center electrode and the optical waveguide; 
         FIG. 5  is a graph showing the relationship between the width of the optical waveguide and the spot-size of the incident light; 
         FIG. 6  is a graph showing the relationship between the modulation efficiency and the distance between the center electrode and the optical waveguide; 
         FIG. 7  is a sectional view showing the optical modulator according to the third embodiment of the present invention; 
         FIG. 8  is a top view showing the optical waveguide according to the fourth embodiment of the present invention; 
         FIG. 9  is a sectional view taken along the line D-D′ of  FIG. 8  showing the optical modulator according to the fourth embodiment of the present invention; 
         FIG. 10  is a sectional view taken along the line E-E′ of  FIG. 8  showing the optical modulator according to the fourth embodiment of the present invention; 
         FIG. 11  is a sectional view taken along the line D-D′ of  FIG. 8  showing the optical modulator according to the fifth embodiment of the present invention; 
         FIG. 12  is a sectional view taken along the line E-E′ of  FIG. 8  showing the optical modulator according to the fifth embodiment of the present invention; 
         FIG. 13  is a perspective view showing the optical modulator according to the first prior art; 
         FIG. 14  is a sectional view taken along the line A-A′ of  FIG. 13  showing the optical modulator according to the first prior art; 
         FIG. 15  is a top view showing the optical waveguide according to the first prior art; 
         FIG. 16  is a graph to explain the operation of the optical modulator according to the first prior art; 
         FIG. 17  is a graph showing the relationship between Vπ·L and Δ; 
         FIG. 18  is a graph showing the relationship between the degree of coupling and G wg ; 
         FIG. 19  is a sectional view showing the optical modulator according to the second prior art; and 
         FIG. 20  is a top view showing the optical waveguide according to the second prior art. 
     
    
    
     DESCRIPTION OF THE REFERENCE NUMERALS 
       1 : x-cut LN substrate (substrate) 
       2 : SiO 2  buffer layer (buffer layer) 
       3 ,  5 ,  12 ,  23 : optical waveguide 
       3   a ,  3   b ,  5   a ,  5   b ,  12   a ,  12   b ,  23   a ,  23   b : interaction optical waveguide 
       4 ,  17 ,  24 : traveling wave electrode (CPW traveling wave electrode) 
       4   a ,  17   a ,  24   a ,  27   a : center electrode 
       4   b ,  4   c ,  17   b ,  17   c ,  24   b ,  24   c ,  27   b ,  27   c : ground electrode 
       6 ,  7 ,  8 ,  9 ,  10 ,  11 ,  13 ,  14 ,  15 ,  16 : taper portion 
       18 ,  22 ,  26 ,  28 : center of the center electrode (center line) 
       19   a ,  19   b ,  20   a ,  20   b ,  21   a ,  21   b ,  25   a ,  25   b : center line of the interaction optical waveguide (center line, center) 
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     The embodiments of the optical modulator according to the present invention will now be described in detail. The constitutional elements of the embodiments the same as those of the prior art shown in  FIGS. 13 to 20  will not be described but bear the same reference numerals and legends as those of the prior art. 
     First Embodiment 
       FIG. 1  is a sectional view showing the optical modulator according to the first embodiment of the present invention.  FIG. 2  is a top view showing the optical waveguide  12 . The optical waveguide  12  is constituted by interaction optical waveguides  12   a  and  12   b . The legend W a ″ represents a width of the interaction optical waveguide  12   a  while the legend W b ″ represents a width of the interaction optical waveguide  12   b . The legend G wg ″ represents a distance (a gap) between the edges of the interaction optical waveguides  12   a  and  12   b , the edges facing each other. Here,  FIG. 1  is a sectional view taken along the line C-C′ of  FIG. 2 , shown with the center electrode  4   a , the ground electrodes  4   b  and  4   c , the x-cut LN substrate  1 , and the SiO 2  buffer layer  2 . Centers (center lines)  21   a  and  21   b  of the interaction optical waveguides  12   a  and  12   b  in the width direction are also shown in  FIG. 1 . 
     The legend Δ 1 ″ represents the distance in the horizontal direction between one edge of the center electrode  4   a  and the center  21   a  of the interaction optical waveguide  12   a . The legend Δ 2 ″ represents the distance in the horizontal direction between the other edge of the center electrode  4   a  and the center  21   b  of the interaction optical waveguide  12   b.    
     As shown in  FIG. 2 , the interaction optical waveguides  12   a  and  12   b  include taper portions  13 ,  14 ,  15  and  16  to have the widths of the interaction optical waveguides varied, each of the interaction optical waveguides  12   a  and  12   b  having two taper portions. In this embodiment, the optical modulator has an advantage of improved extinction ratio due to the fact that the total number of taper portions is less than the optical modulator according to the second prior art, and the number of taper portions to widen the interaction optical waveguides  12   a ,  12   b  and the number of taper portions to narrow the interaction optical waveguides  12   a ,  12   b  are equal to each other. 
     The chirping characteristic, which is the most important characteristic of the optical modulator, will now be described hereinafter.  FIG. 3  is a graph showing the spot-sizes of the incident lights traveling the interaction optical waveguides  12   a  and  12   b  with respect to the widths W a ″, W b ″ of the interaction optical waveguides  12   a  and  12   b . As shown in  FIG. 3 , the incident light traveling the interaction optical waveguide  12   a  has a spot-size larger than that of the incident light traveling the interaction optical waveguide  12   b , resulting from the fact that the interaction optical waveguide  12   a  has a remarkably narrow width of, for example, 6 μm compared to the width (for example 11 μm) of the interaction optical waveguide  12   b  according to the first embodiment. 
     Though there has been described that the widths W a ″ and W b ″ of the interaction optical waveguides  12   a  and  12   b  are respectively set at 6 μm and 11 μm, these widths are only examples, and it goes without saying that each of the widths can be set at any size in the range of approximately 3 μm to 15 μm in this invention. 
     The fact that the spot-size of the incident light traveling the interaction optical waveguide  12   a  is larger than that of the incident light traveling the interaction optical waveguide  12   b  leads to the fact that the normalized interaction efficiency Γ1″ in the form of overlap integration between the electric signal (amplitude) and the incident light (power) passing through the interaction optical waveguide  12   a  is smaller than the normalized interaction efficiency Γ2″ in the form of overlap integration between the electric signal (amplitude) and the incident light (power) passing through the interaction optical waveguide  12   b.    
     This means that the interaction efficiency Γ1″ of the interaction optical waveguide  12   a  and the interaction efficiency Γ2″ of the interaction optical waveguide  12   b  with respect to the distance Δ 1 ″ between one edge of the center electrode  4   a  and the center  21   a  of the interaction optical waveguide  12   a  and with respect to the distance Δ 2 ″ between another edge of the center electrode  4   a  and the center  21   b  of the interaction optical waveguide  12   b  have a relationship that the interaction efficiency Γ1″ is smaller than the interaction efficiency Γ2″ within the whole range, as shown in  FIG. 4 . 
     In this invention, the chirping zero, that is, α=0 is achieved by adjusting the distance Δ 1 ″ between one edge of the center electrode  4   a  and the center  21   a  of the interaction optical waveguide  12   a  and the distance Δ 2 ″ between another edge of the center electrode  4   a  and the center  21   b  of the interaction optical waveguide  12   b  to ensure that the interaction efficiency Γ1″ of the interaction optical waveguide  12   a  and the interaction efficiency Γ2″ of the interaction optical waveguide  12   b  become equal with each other. Here, the center line  18  of the center electrode  4   a  is positioned at the middle of the edges. 
     This results in the fact that the distance between one edge of the center electrode  4   a  and the center  21   a  of the interaction optical waveguide  12   a  is shorter than the distance between another edge of the center electrode  4   a  and the center  21   b  of the interaction optical waveguide  12   b  (Δ 1 ″&lt;Δ 2 ″). 
     The chirping zero (α=0) is, therefore, achieved by setting the widths of the interaction optical waveguides  12   a  and  12   b  different from each other to ensure that the optical coupling between the interaction optical waveguides  12   a  and  12   b  is suppressed while setting the positional relationship to be asymmetric against the center electrode  4   a  and ground electrodes  4   b ,  4   c.    
     The fact that the distance Δ 1 ″ between one edge of the center electrode  4   a  and the center  21   a  of the interaction optical waveguide  12   a  and the distance Δ 2 ″ between another edge of the center electrode  4   a  and the center  21   b  of the interaction optical waveguide  12   b  have a relationship of Δ 1 ″≠Δ 2 ″, as above mentioned, can be translated to the fact that the middle point of the center  21   a  and the center  21   b  is positioned away from the center  18  of the center electrode  4   a . In this case, the middle point of the gap, the gap being defined by the interaction optical waveguides  12   a  and  12   b , may be positioned away from the center  18  of the center electrode  4   a , or may be overlapped with the center  18  of the center electrode  4   a.    
     The spot-size of the incident light in the direction parallel to the surface of the substrate is considered to be designed in the above description to simplify the explanation. However, the designing with higher accuracy can be achieved when the spot-size of the incident light in the direction perpendicular to the surface of the substrate is considered to be designed. Here, the spot-size in the direction perpendicular to the surface of the substrate becomes larger as the widths of the interaction optical waveguides  12   a  and  12   b  become narrower. 
     It has been described about the magnitude relationship between the distance Δ 1 ″ from one edge of the center electrode  4   a  to the center  21   a  of the interaction optical waveguide  12   a  and the distance Δ 2 ″ from another edge of the center electrode  4   a  to the center  21   b  of the interaction optical waveguide  12   b . Here, the magnitude relationship between Δ 1 ″ and Δ 2 ″ is the same as the magnitude relationship between the distance from the center  18  of the center electrode  4   a  to the center  21   a  of the interaction optical waveguide  12   a  and the distance from the center  18  of the center electrode  4   a  to the center  21   b  of the interaction optical waveguide  12   b . The chirping can be suppressed more by setting the lengths of the taper portions  13 ,  14 ,  15  and  16  as short as possible unless optical loss is caused. These constitutions can be applied to all embodiments of the present invention. 
     The optical modulator according to the embodiments of the present invention can realize the zero chirping with high process yield resulting from the fact that the degree of chirping is independent of the factors relating to the lengths, which is different from the second prior art. Therefore, the chirping characteristic can be suppressed independently of the variation of the propagation loss of the high frequency electric signal. 
     Second Embodiment 
     As shown in  FIG. 3 , the spot-size of the interaction optical waveguide  12   a  is larger than that of the interaction optical waveguide  12   b  due to the fact that the width of the interaction optical waveguide  12   a  is set to be tremendously narrow. This results in the interaction efficiency Γ1″ of the interaction optical waveguide  12   a  smaller than the interaction efficiency Γ 2 ″ of the interaction optical waveguide  12   b  according to the first embodiment of the present invention. 
     In the second embodiment of the present invention, the width of the interaction optical waveguide  12   a  is set at, for example, 7 μm to ensure that the spot-size of the incident light in the interaction optical waveguide  12   a  becomes smaller than the spot-size of the incident light in the interaction optical waveguide  12   b.    
     In this case, the interaction efficiency Γ1″ of the interaction optical waveguide  12   a  and the interaction efficiency Γ2″ of the interaction optical waveguide  12   b  with respect to the distance Δ 1 ″ between one edge of the center electrode  4   a  and the center  21   a  of the interaction optical waveguide  12   a  and with respect to the distance Δ 2 ″ between another edge of the center electrode  4   a  and the center  21   b  of the interaction optical waveguide  12   b  have a relationship that the interaction efficiency Γ1″ is larger than the interaction efficiency Γ2″ within the whole range, as shown in  FIG. 6 . 
     The relationship Γ1″=Γ2″, that is, chirping zero can be achieved in this second embodiment by setting the distance Δ 2 ″ between another edge of the center electrode  4   a  and the center  21   b  of the interaction optical waveguide  12   b  shorter than the distance Δ 1 ″ between one edge of the center electrode  4   a  and the center  21   a  of the interaction optical waveguide  12   a.    
     The optical modulator according to the embodiments of this invention has such a characteristic that the interaction optical waveguides  12   a  and  12   b  have positional relationship to be asymmetric against the center electrode  4   a  and ground electrodes  4   b ,  4   c . Therefore, the distance Δ 1 ″ between one edge of the center electrode  4   a  and the center  21   a  of the interaction optical waveguide  12   a  and the distance Δ 2 ″ between another edge of the center electrode  4   a  and the center  21   b  of the interaction optical waveguide  12   b  have a relationship of Δ 1 ″≠Δ 2 ″, as described in the first embodiment. 
     In this case, the middle point of the gap defined by the interaction optical waveguides  12   a ,  12   b  may be positioned away from the center  18  of the center electrode  4   a , or may be overlapped with the center  18  of the center electrode  4   a.    
     The optical modulator may be formed to be Δ 1 ″=Δ 2 ″ under a certain condition of the widths of the interaction optical waveguides  12   a  and  12   b  or a certain condition of these constructions, while the center of the gap between the interaction optical waveguides is positioned away from the center  18  of the center electrode  4   a , which makes it possible to have the interaction efficiencies between the high frequency electric signal and the incident lights at the two interaction optical waveguides  12   a  and  12   b  equal to each other. These constitutions can be applied not only to the first and the second embodiments but also to all embodiments of the present invention. 
     Third Embodiment 
       FIG. 7  is a sectional view showing the optical modulator according to the third embodiment of the present invention. In this embodiment, the constitution is developed compared to the constitution of the first embodiment shown in  FIGS. 1 to 4  by utilizing the fact that the electric field intensity between the center electrode  17   a  and the ground electrodes  17   b ,  17   c  of the CPW traveling wave electrode  17  becomes stronger as the gap between the center electrode and the ground electrodes becomes narrower. 
     Here, the center (or the center line)  22  of the center electrode  17   a  in the width direction is shown in  FIG. 7 . 
     In this embodiment, the relationship Γ1″=Γ2″, that is, the chirping zero is effectively achieved by setting the gaps G 1  and G 2  between the center electrode  17   a  and the respective ground electrodes  17   b ,  17   c  of the CPW traveling wave electrode to be different from each other (G 1 ≠G 2 ), while the interaction optical waveguides  12   a  and  12   b  are positioned to be asymmetrical with the center electrode  17   a  of the CPW traveling wave electrode  17  (the center  22  of the center electrode  17   a  is positioned away from the center of the interaction optical waveguides  12   a ,  12   b ) (G 1 &lt;G 2  in this  FIG. 7 ). 
     Here, it is possible that the interaction optical waveguides  12   a  and  12   b  are positioned symmetrically with the center electrode  17   a  of the CPW traveling wave electrode  17  (Δ 1 ″=Δ 2 ″, or the center electrode may be positioned at the center of the gap between the two interaction optical waveguides  12   a  and  12   b ) as long as the relationship Γ1″=Γ2″ can be maintained by setting the gap to be G 1 ≠G 2 . The constitution to set the gaps to be G 1 ≠G 2  can be applied to any embodiments of this invention including the first and the second embodiments. 
     Fourth Embodiment 
       FIG. 8  is a top view showing the optical waveguide  23  of the optical modulator according to the fourth embodiment of the present invention. The optical waveguide  23  is constituted by interaction optical waveguides  23   a  and  23   b . The legend W a ″ represents a width of the interaction optical waveguide  23   a  while the legend W b ″ represents a width of the interaction optical waveguide  23   b . The widths W a ″ and W b ″ are set to be different from each other to ensure that the optical coupling between the interaction optical waveguides  23   a  and  23   b  are suppressed. The legend G wg ″ represents a distance (a gap) between the edges of the interaction optical waveguides  23   a  and  23   b.    
     As shown in  FIG. 8 , the magnitude relationship between the widths W a ″ and W b ″ of the interaction optical waveguides  23   a  and  23   b  at a first region having a length of L 1 ″ is reciprocal to the magnitude relationship between the widths Wa″ and Wb″ of the interaction optical waveguides  23   a  and  23   b  at a second region having a length of L 2 ″. 
       FIGS. 9 and 10  are sectional views respectively taken along the line D-D′ and E-E′ of  FIG. 8  showing the optical modulator. Here,  FIGS. 9 and 10  are sectional views respectively taken along the line D-D′ and E-E′ of  FIG. 8 , shown with the center electrode  24   a  and the ground electrodes  24   b ,  24   c  of the traveling wave electrode  24 , the x-cut LN substrate  1 , and the buffer layer  2 . Center (center line)  26  of the center electrode  24   a  in the width direction and centers (center lines)  25   a  and  25   b  of the respective interaction optical waveguides  23   a  and  23   b  in the width direction are also shown in  FIGS. 9 and 10 . 
     The legend Δ 1 ″ represents the distance in the horizontal direction between one edge of the center electrode  24   a  and the center  25   a  of the interaction optical waveguide  23   a . The legend Δ 2 ″ represents the distance in the horizontal direction between another edge of the center electrode  24   a  and the center  25   b  of the interaction optical waveguide  23   b.    
     The positions of the center  26  of the center electrode  24   a  in  FIG. 9  and in  FIG. 10  are away from each other in a direction parallel to the surface of the x-cut LN substrate  1 . The amount of shift is, however, small enough to have an order of micron or sub-micron. Therefore, deterioration of the electrical characteristic can be prevented in the case that the first region and the second region are separated with each other with a predetermined length (for example 50 μm) to ensure that the center electrode  24   a  and the ground electrodes  24   b  and  24   c  at the first region can be connected linearly or gently with those of the second region, respectively. 
     As shown in  FIG. 8 , the widths Wa″ and Wb″ of the interaction optical waveguides  23   a  and  23   b  at the first region having a length of L 1 ″ have a relationship of Wa″&lt;Wb″ in the fourth embodiment. The optical modulator is, therefore, constructed to have a relationship of Δ 1 ″&lt;Δ 2 ″ as shown in  FIG. 9  to ensure that the interaction efficiencies (aforementioned Γ1″ and Γ2″) between the high frequency electric signals and the incident lights passing through the interaction optical waveguides  23   a  and  23   b  become equal with each other, which is in a similar manner with the first embodiment shown in  FIG. 1 . 
     On the other hand, the widths Wa″ and Wb″ have a relationship of Wa″&gt;Wb″ at the second region having a length of L 2 ″. Therefore, the optical modulator is constructed to have a relationship of Δ 1 ″&gt;Δ 2 ″ to ensure that the interaction efficiencies (Γ1″ and Γ2″) between the high frequency electric signals and the incident lights passing through the interaction optical waveguides  23   a  and  23   b  become equal with each other. 
     The middle point of the gap defined by the interaction optical waveguides  23   a ,  23   b  may be positioned away from the center  18  of the center electrode  24   a , or may be overlapped with the center  18  of the center electrode  24   a , in a similar manner with the first embodiment and the second embodiment. 
     In this invention, the optical modulator is characterized in that the positional relationship between the interaction optical waveguides  23   a ,  23   b  and the traveling wave electrode  24  is shifted from the symmetry position, the traveling wave electrode  24  being constituted by a center electrode  24   a  and ground electrodes  24   b ,  24   c.    
     As described in the second embodiment, the distance Δ 1 ″ between one edge of the center electrode  24   a  and the center  25   a  of the interaction optical waveguide  23   a  and the distance Δ 2 ″ between another edge of the center electrode  24   a  and the center  25   b  of the interaction optical waveguide  23   b  have a relationship of Δ 1 ″≠Δ 2 ″ in this fourth embodiment. However, the distance Δ 1 ″ and Δ 2 ″ may have a relationship of Δ 1 ″=Δ 2 ″ under the condition that the interaction optical waveguides  23   a  and  23   b  have a certain widths or formed under a certain condition, while the middle point of the gap defined by the interaction optical waveguides  23   a ,  23   b  is positioned away from the center  26  of the center electrode  24   a.    
     In this invention, the interaction efficiencies between the high frequency electric signals and the incident lights respectively passing through the interaction optical waveguides  23   a  and  23   b  become equal with each other. This means that there is a relationship Γ1″=Γ2″ at each of the first and the second regions, where the interaction efficiency at the interaction optical waveguide  23   a  is represented by the legend Γ1″ and the interaction efficiency at the interaction optical waveguide  23   b  is represented by the legend Γ2″. Therefore, the length L 1 ″ of the first region and the length L 2 ″ of the second region can be set without any restriction due to the fact that it is unnecessary to consider the amount of phase variation between the first region and the second region caused by the propagation loss of the electrode. This results in the fact that it is possible to set the length L 1 ″ of the first region and the length L 2 ″ of the second region to be equal with each other (L 1 ″=L 2 ″). 
     In general, effective refractive index of optical waveguide having large width is higher than that of the optical waveguide having small width. Therefore, the fact that the length L 1 ″ of the first region and the length L 2 ″ of the second region are equal with each other (L 1 ″=L 2 ″) results in the fact that the optical path lengths of the interaction optical waveguides  23   a  and  23   b  forming the Mach-Zehnder interferometer are equal with each other. This results in the fact that the DC bias voltage is not necessary to be changed against the optical wavelengths due to the fact that the optical modulator can suppress filter characteristics against the optical wavelength. The optical modulator is, therefore, advantageous to be used with the optical communication methods using a wide band of optical wavelength such as WDM (Wavelength Division Multiplexing) and DWDM (Dense Wavelength Division Multiplexing). 
     Fifth Embodiment 
     The interaction optical waveguides  23   a  and  23   b  according to the fourth embodiment of the present invention shown in  FIG. 8  may be formed with a CPW traveling wave electrode  17  having gaps respectively have sizes different from each other, in a similar manner with the third embodiment, this constitution being disclosed as a fifth embodiment.  FIGS. 11 and 12  are sectional views respectively taken along the line D-D′ and E-E′ of  FIG. 8  showing the optical modulator. 
       FIGS. 11 and 12  additionally show a center (a center line)  28  of the center electrode  27   a  partly forming the CPW traveling wave electrode. The legend Δ 1 ″ represents the distance in the horizontal direction between one edge of the center electrode  27   a  and the center  25   a  of the interaction optical waveguide  23   a . The legend Δ 2 ″ represents the distance in the horizontal direction between another edge of the center electrode  27   a  and the center  25   b  of the interaction optical waveguide  23   b.    
     As shown in  FIGS. 11 and 12 , the magnitude relationship between the gaps G 1  and G 2  formed by the center electrode  27   a  and ground electrodes  27   b ,  27   c  of the CPW traveling wave electrode  27  at the first region having a length of L 1 ″ is reciprocal to the magnitude relationship at the second region having a length of L 2 ″. (G 1 &lt;G 2  and Δ 1 ″&lt;Δ 2 ″ at the first region shown in  FIG. 11 . G 1 &gt;G 2  and Δ 1 ″&gt;Δ 2 ″ at the second region shown in  FIG. 12 .) In this embodiment, it is important to have a relationship of G 1 ≠G 2 . Therefore, Δ 1 ″ and Δ 2 ″ may have aforementioned relationship as an example, but not limited thereto. The middle point of the gap defined by the interaction optical waveguides  23   a ,  23   b  may be positioned away from the center  28  of the center electrode  27   a , or may be overlapped with the center  28  of the center electrode  27   a.    
     In this embodiment, the length L 1 ″ of the first region and the length L 2 ″ of the second region can be set without any restriction due to the fact that the interaction efficiencies between the high frequency electric signals and the incident lights passing through the respective interaction optical waveguides  23   a  and  23   b  become equal with each other at each of the first region and the second region. This constitution makes it possible to suppress filter characteristics against the optical wavelength under the condition that the lengths L 1 ″ and L 2 ″ are set to be L 1 ″=L 2 ″, in a similar manner with the fourth embodiment of this invention. 
     There may be three or more regions in the case that the optical modulator has a region to ensure that the magnitude relationship between the interaction optical waveguides  23   a  and  23   b  are interchanged, in a similar manner with the fourth and fifth embodiments of this invention. In this case, the filter characteristics can be suppressed by making the length where the interaction optical waveguide is narrow and the length where the interaction optical waveguide is wide to be equal to each other. 
     While particular embodiments have been described, it will be appreciated by those in the art that changes and modifications may be made thereto without departing from the invention in its broader aspects. 
     Each Embodiment 
     There has been described about the fact that the traveling wave electrode is constituted by the CPW electrode, the traveling wave electrode may be constituted by any types of traveling wave electrode, or may be replaced by a lumped parameter electrode. 
     It is within the scope of this invention that the optical modulator has a portion where the magnitude relationship between the interaction optical waveguides  12   a  and  12   b  becomes reciprocal as long as the relationship of the lengths between the portion and other portion is different from the relationship described by the formula (7), resulting from the fact that the method described in the second prior art is not utilized. 
     There has been described about the fact that the two interaction optical waveguides are formed to have widths different from each other from the aspect of spot-size. However, the fact that the widths of the two interaction optical waveguides are varied with each other is tantamount to the fact that the effective refractive indexes are varied with each other. This leads to the fact that the difference of the effective refractive indexes can be large by leaving a buffer layer formed by the SiO 2  or SiO x  over the interaction optical waveguide having higher effective refractive index while removing the buffer layer over the interaction optical waveguide having lower effective refractive index. This results in the two interaction optical waveguides difficult to be coupled with each other. This constitution can be easily applied to any embodiments of this invention. 
     Furthermore, in each embodiment, the LN substrate may have an x-cut state, a y-cut state, or a z-cut state. In other words, the LN substrate may have a surface direction such that x-axis, y-axis, or z-axis of the crystal is perpendicular to the surface (cut surface) of the LN substrate. The LN substrate may be formed such that the main surface direction, the direction as described above, is mixed with a sub surface direction different from the main surface direction. The LN substrate may be formed with another material having the electro-optic effect such as lithium tantalite and a semiconductor. 
     Industrial Applicability of the Present Invention 
     In accordance with the present invention, there is provided an optical modulator which is available for an optical modulator with high modulation speed, low alpha parameter, high extinction ratio, low drive voltage, and low DC bias voltage.