Patent Publication Number: US-9835798-B2

Title: Planar optical waveguide device, polarization multiplexing 4-value phase modulator, coherent receiver, and polarization diversity

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation application based on a PCT Patent Application No. PCT/JP2015/070548, filed Jul. 17, 2015, whose priority is claimed on Japanese Patent Application No. 2014-165190, filed on Aug. 14, 2014, the entire content of which are hereby incorporated by reference. 
    
    
     BACKGROUND OF THE INVENTION 
     Field of the Invention 
     The present invention relates to a planar optical waveguide device, a polarization multiplexing 4-value phase modulator, a coherent receiver, and a polarization diversity used in optical fiber communication. 
     Description of the Related Art 
     In recent years, the amount of information transmitted through optical communications has been steadily increasing. To cope with the increase in the amount of information, measures such as increasing a signal speed or increasing the number of channels based on wavelength multiplexing communication have been developed. Particularly, in the next generation 100 Gbps digital coherent transmission technology for high-speed information communication, in order to double the amount of information per unit time, a polarization multiplexing method for carrying information in two polarizations where electric fields are orthogonal to each other is used. 
     However, in a modulation method for high-speed communication including polarization multiplexing, an optical modulator having a complicated structure is necessary, which can cause a problem in that the size of an apparatus becomes large and the manufacturing cost increases. In order to solve these problems, research regarding an optical modulator based on a planar optical waveguide using silicon having merits such as easy processing, size reduction due to integration, cost reduction due to mass production, or the like has been performed. 
     However, polarization multiplexing in such a planar optical waveguide can have the following problems. In general, a planar optical waveguide has an asymmetric shape in a width direction parallel to a substrate and in a height direction perpendicular to the substrate. Thus, a characteristic such as an effective refractive index varies between two types of polarization modes of a mode (hereinafter, referred to as a TE mode) in which a width-directional electric field component is a main component and a mode (hereinafter, referred to as a TM mode) in which a height-directional electric field component is a main component. In many cases, TE 0  and TM 0  among the two polarization modes are frequently used. Here, TE 0  is a mode in which an effective refractive index is largest in the TE mode, and TM 0  is a mode in which an effective refractive index is largest in the TM mode. 
     In a case where an optical modulation operation is performed with respect to these polarization modes in which characteristics are different from each other, it is difficult to perform the optical modulation operation by only using a single planar optical waveguide device. Thus, it is necessary to provide a planar optical waveguide device optimally designed for each polarization mode, which causes a problem in that a large amount of effort is necessary for development of a planar optical waveguide device. 
     In order to solve the problems, a method for using TE 0  as input light to a planar optical waveguide device optimally designed for TE 0  and polarization-converting output light thereof into TM 0  may be used. Here, the “polarization conversion” refers to conversion from TE 0  to TM 0  or from TM 0  to TE 0 . In order to perform the optical modulation operation, it is necessary to provide a planar optical waveguide device that performs polarization conversion on a substrate. 
     In order to perform polarization conversion on a substrate, a technique that combines a conversion between TE 0  and TE 1  and a conversion between TE 1  and TM 0  may be used. The invention pays attention to the conversion between TE 0  and TE 1  among these conversions. Here, TE 1  represents a mode having a second largest effective refractive index in the TE mode. 
     As a related art relating to an optical waveguide device having a function for the conversion between TE 0  and TE 1 , there is an optical waveguide device disclosed in Daoxin Dai and John E. Bowers, “Novel concept for ultracompact polarization splitter-rotator based on silicon nanowires.” Optics Express, Vol. 19, Issue. 11, pp. 10940-10949 (2011) (hereinafter, referred to as Non-Patent Document 1). 
       FIGS. 69A and 69B  are diagrams illustrating an optical waveguide device which is a model of a structure disclosed in Non-Patent Document 1.  FIG. 69A  is a plan view thereof, and  FIG. 69B  is a sectional view thereof. 
     The optical waveguide device includes core portions  81  and  82 , and a cladding  15 . The cladding  15  includes a lower cladding  7  and an upper cladding  6 . 
     The core portions  81  and  82  are linear waveguides, and are disposed in parallel to form a directional coupler. In the directional coupler, TE 0  of the core portion  81  and TE 1  of the core portion  82  are coupled to perform mode conversion. 
     In order to efficiently perform mode conversion in the directional coupler, it is necessary to maintain effective refractive indexes in TE 0  and TE 1  at the same level. Thus, a waveguide structure is adjusted according to each mode. 
     In this optical waveguide device, in order to maintain the effective refractive indexes in TE 0  and TE 1  at the same level, the widths of the core portions  81  and  82  are adjusted. Since the widths of the core portions  81  and  82  are different from each other, such a directional coupler is referred to as an “asymmetric directional coupler”. 
     However, the above-described optical waveguide device combines different modes. Thus, even if the condition for “maintaining effective refractive indexes in TE 0  and TE 1  at the same level” with respect to a specific wavelength is satisfied by adjustment of a waveguide structure (adjustment of the width of a core portion, or the like), a wavelength may deviate from the specific wavelength. Further, in a case where a waveguide structure is changed due to a manufacturing error, deviation occurs between effective refractive indexes of the two modes. Accordingly, conversion efficiency may be lowered. 
     Accordingly, in the related art, there are problems in that a wavelength band for allowing highly efficient conversion is narrow and stability against a manufacturing error is weak. 
     Hereinafter, the problems will be described using an asymmetric directional coupler in the related art shown in  FIGS. 69A and 69B  as an example. 
     In this example, core portions  81  and  82  are formed of Si (having a refractive index of 3.48), and an upper cladding  6  and a lower cladding  7  are formed of SiO 2  (having a refractive index of 1.44). The heights of the core portions  81  and  82  are 220 nm. A gap between the core portions  81  and  82  is 200 nm. 
     A waveguide which guides light in TE 0  which is a mode conversion target and has the core portion  81  having a smaller width is referred to as “waveguide 1”, and a waveguide which guides light in TE 1  and has the core portion  82  having a larger width is referred to as “waveguide 2”. 
     The width of the core portion  81  is 400 nm. Here, at a wavelength of 1580 nm, the width of the core portion  82  is set to 838 nm so that effective refractive indexes in TE 0  of the core portion  81  and TE 1  of the core portion  82  are at the same level. Calculation results of the effective refractive indexes are shown in Table 1. A finite element method (FEM) is used for the calculation. 
     
       
         
           
               
               
               
             
               
                   
                 TABLE 1 
               
               
                   
                   
               
               
                   
                 TE 0  of waveguide 1 
                 TE 1  of waveguide 2 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
            
               
                 Effective refractive index 
                 2.178818 
                 2.178940 
               
               
                   
               
            
           
         
       
     
     A conversion efficiency of the asymmetric directional coupler is as follows. Here, a conversion efficiency T is a ratio of power of output TE 1  to power of input TE 0 .
 
[Expression 1]
 
 T=F  sin 2 ( qL )  (1)
 
     Here, F and q are expressed as the following expressions, respectively. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   F 
                   = 
                   
                     1 
                     
                       1 
                       + 
                       
                         
                           ( 
                           
                             δ 
                             χ 
                           
                           ) 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
             
               
                 
                   [ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
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                     3 
                   
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                   q 
                   = 
                   
                     
                       
                         χ 
                         2 
                       
                       + 
                       
                         δ 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     Here, δ is expressed as the following expression. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     4 
                   
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                   δ 
                   = 
                   
                     
                       π 
                       λ 
                     
                     ⁢ 
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     N 
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     Here, L represents the length of an asymmetric directional coupler in a light propagation direction, ΔN represents a difference between effective refractive indexes (difference between effective refractive indexes in Table 1) in TE 0  of the waveguide 1 and TE 1  of the waveguide 2 in a case where two waveguides are independently present, and λ represents a wavelength. Further, χ represents the strength of coupling of two waveguides, and is referred to as a coupling coefficient. 
     In the asymmetric directional coupler, even if effective refractive indexes of two modes which are coupling targets match each other by adjusting a waveguide structure such as the width of a core portion or the like at a certain wavelength (1580 nm in this example), if the wavelength is changed, deviation occurs in the effective refractive indexes. 
     This problem does not occur in a symmetric directional coupler that has the same heights and widths in two cores and handles coupling of the same modes but occurs in an asymmetric directional coupler that handles coupling of different modes. 
       FIG. 70  is a diagram illustrating a relationship between a wavelength and an absolute value of ΔN in an optical waveguide device in this example. It can be understood from  FIG. 70  that the absolute value of ΔN becomes larger as the wavelength deviates farther from 1580 nm. 
     Since the conversion efficiency T is lowered according to the deviation of the wavelength, from Expression (1), (2), and (4), highly efficient conversion is not preferable in a wide wavelength band. 
     Then, the conversion efficiency with respect to the wavelength (1520 nm to 1640 nm) is calculated based on Expression (1) to Expression (4). The result is shown in  FIG. 71 . Here, L in Expression (1) is a value in which a minimum value of the conversion efficiency in the wavelength band of 1520 nm to 1640 nm becomes a maximum, and in this case, L is 16.1 μm. 
     Referring to  FIG. 71 , the conversion efficiency becomes lower as the wavelength becomes more distant from the vicinity of 1580 nm, and becomes equal to or greater than approximately −0.94 dB in the wavelength band of 1520 nm to 1640 nm. This is because the absolute value of ΔN increases with respect to the above-described wavelength. 
     Subsequently, a relationship between a manufacturing error and conversion efficiency will be described. If a waveguide structure is changed, the level of light confinement is changed, and an effective refractive index associated therewith is changed. Thus, even if a waveguide structure is designed so that effective refractive indexes of two modes which are coupling targets are at the same level at a certain wavelength, the waveguide structure is changed due to a manufacturing error, and the effective refractive indexes of two modes deviate from each other. 
     Thus, the conversion efficiency is lowered as in the above description regarding the wavelength dependency. 
     In order to confirm this problem, a manufacturing error of the width of a core portion generated due to lithography or etching will be described as an example. 
     Normally, a manufacturing error locally occurs in two core portions  81  and  82  by the same amount (δ), as shown in  FIG. 72 , with respect to design values of the widths of the core portions (the widths of core portions regulated by a mask, for example, W 81  and W 82  in  FIG. 72 ). In this example, it is assumed that that positions on both side edges of respective cores are changed inward or outward by δ/2, respectively. 
     Hereinafter, a case where a manufacturing error δ (=−30 nm) occurs with respect to the core portion  81  (design value: width of 400 nm) and the core portion  82  (design value: width of 838 nm) of the optical waveguide device shown in  FIGS. 69A and 69B  is considered.  FIG. 73  is a diagram illustrating a relationship between a wavelength and an absolute value of ΔN. 
     It can be understood from  FIG. 73  that effective refractive indexes in TE 0  of the core portion  81  and TE 1  of the core portion  82  significantly deviate from each other, and thus, the absolute value of ΔN becomes large. The conversion efficiency is calculated based on this result. L employs the above-described value (L=16.1 μm). A result thereof is shown in  FIG. 74 . 
     It can be understood from  FIG. 74  that since the absolute value of ΔN becomes large due to the manufacturing error, the conversion efficiency is significantly lowered. Specifically, the conversion efficiency becomes equal to or greater than about −5.16 dB at 1580 nm, and becomes equal to or greater than −7.32 dB in a range of 1520 nm to 1640 nm. In this view, it can be said that an asymmetric directional coupler is weak against a manufacturing error. 
     In this way, in an optical waveguide device including an asymmetric directional coupler in the related art, there are problems in that a wavelength band in mode conversion is narrow and stability against a manufacturing error is weak. 
     In consideration of the above-described problems, an object of the invention is to provide a planar optical waveguide device capable of securing high conversion efficiency in a wide wavelength band, and securing efficient mode conversion even in a case where a waveguide structure is changed due to a manufacturing error. 
     SUMMARY 
     According to a first aspect of the invention, a planar optical waveguide device includes: a substrate; and an optical waveguide that includes a core and a cladding, the core including a first core portion and a second core portion that are disposed in parallel on the substrate, the cladding having a refractive index smaller than that of the core. The core forms a preceding-stage mode conversion section and a subsequent-stage mode conversion section, the preceding-stage mode conversion section being configured to convert a mode of input light, the subsequent-stage mode conversion section being configured to convert a mode of light output from the preceding-stage mode conversion section. Sectional shapes of the first core portion and the second core portion are not congruent with each other at an input end of the preceding-stage mode conversion section, the sectional shape or size of at least one core is continuously changed along a light waveguide direction, and sectional shapes of the first core portion and the second core portion are congruent with each other at an output end of the preceding-stage mode conversion section. The subsequent-stage mode conversion section includes an output portion to which the first core portion and the second core portion are connected with a gap being provided therebetween in a width direction. At a connection end at which the first core portion and the second core portion are connected to the output portion, the center of the output portion in the width direction and the center of a width directional range including the first core portion, the second core portion, and a gap between the first core portion and the second core portion match each other. 
     At the connection end, the width of the output portion may be larger than a sum of the width of the first core portion, the width of the second core portion, and the gap between the first core portion and the second core portion. 
     The first core portion, the second core portion and the output portion may have rectangular sections vertical to the light waveguide direction. 
     In the preceding-stage mode conversion section, the heights of the first core portion and the second core portion may be same, and the width of the first core portion having a larger section than that of the second core portion at the input end continuously may decrease along the light waveguide direction so that the sectional shapes of the first core portion and the second core portion are congruent with each other at the output end. 
     The core may include a slab portion that extends in the width direction of the first core portion and the second core portion, and the slab portion may have a height dimension smaller than those of the first core portion and the second core portion, may be provided at least between the first core portion and the second core portion, and may be formed to connect the first core portion and the second core portion. 
     The slab portion may have an outer extension region that is formed to extend outward in the width direction from each of the first core portion and the second core portion. 
     The slab portion may have an outer extension region that is formed to extend outward in the width direction from the output portion. 
     The subsequent-stage mode conversion section may be configured so that an output-side core portion having a width smaller than that of the output portion is connected to a rear end of the output portion to form a multi-mode interferometer. 
     The preceding-stage mode conversion section may be capable of converting TE 0  into an odd mode which is a super mode of TE 0 , and the subsequent-stage mode conversion section may be capable of converting the odd mode which is the super mode into TE1. 
     The core may include a bent waveguide formed by bending at least one of the first core portion and the second core portion in a planar view on an input side of the preceding-stage mode conversion section, and in the bent waveguide, the first core portion and the second core portion may become closer to each other as a distance to the preceding-stage mode conversion section becomes shorter. 
     The planar optical waveguide device may further include: an intermediate core portion that is provided between the preceding-stage mode conversion section and the subsequent-stage mode conversion section and connects the preceding-stage mode conversion section and the subsequent-stage mode conversion section. 
     The core may be formed of Si, and the cladding may be formed of SiO 2 . 
     The planar optical waveguide device may further include: a high-order polarization-converting section that is connected to an output-side of the subsequent-stage mode conversion section and is capable of converting TE 1  obtained in the subsequent-stage mode conversion section into TM 0 . 
     According to a second aspect of the invention, a polarization multiplexing 4-value phase modulator is provided including the planar optical waveguide device. 
     According to a third aspect of the invention, a coherent receiver is provided including the planar optical waveguide device. 
     According to a fourth aspect of the invention, a polarization diversity is provided including the planar optical waveguide device. 
     The planar optical waveguide device according to the above-described aspect has a configuration in which a preceding-stage mode conversion section (super mode-generating element) and a subsequent-stage mode conversion section (matching coupling element) are combined. 
     In the super mode-generating element having a structure (for example, a tapered waveguide) in which a waveguide structure is changed in a light waveguide direction, input TE 0  is converted into an odd mode which is a super mode of TE 0 . In the matching coupling element, the odd mode is converted into TE 1  by using similarity of electric field distributions of the odd mode which is the super mode of TE 0  and TE 1  of a rectangular waveguide. 
     Since the super mode-generating element has a configuration in which shapes and sizes of sections of two core portions at an output end are the same (congruent with each other), the super mode-generating element is not easily affected by a manufacturing error and also has small wavelength dependency. 
     Since even in a case where a wavelength changes or a waveguide structure is changed due to a manufacturing error, electric field distributions of both of the odd mode and TE 1  are changed, the matching coupling element is not easily affected by wavelength change or a manufacturing error. 
     Accordingly, it is possible to perform conversion over a wide wavelength band with high efficiency, and to secure efficient mode conversion even in a case where a waveguide structure is changed due to a manufacturing error. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1A  is a plan view illustrating a planar optical waveguide device according to a first embodiment of the invention. 
         FIG. 1B  is a sectional view at a sectional position (a) of the planar optical waveguide device according to the first embodiment of the invention. 
         FIG. 2A  is a plan view illustrating an example of an optical waveguide device. 
         FIG. 2B  is a sectional view illustrating an example of the optical waveguide device. 
         FIG. 3A  is a diagram illustrating a simulation result of an electric field distribution (E x  component) in an even mode in the optical waveguide device shown in  FIGS. 2A and 2B . 
         FIG. 3B  is a diagram illustrating a simulation result of an electric field distribution (E x  component) in an odd mode in the optical waveguide device shown in  FIGS. 2A and 2B . 
         FIG. 3C  is a graph of the electric field distribution (E x  component) in the even mode in the optical waveguide device shown in  FIGS. 2A and 2B . 
         FIG. 3D  is a graph of the electric field distribution (E x  component) in the odd mode in the optical waveguide device shown in  FIGS. 2A and 2B . 
         FIG. 4A  is a plan view illustrating a structure of a preceding-stage mode conversion section. 
         FIG. 4B  is a sectional view at a sectional position (c), illustrating the structure of the preceding-stage mode conversion section. 
         FIG. 4C  is a sectional view at a sectional position (b), illustrating the structure of the preceding-stage mode conversion section. 
         FIG. 4D  is a sectional view at a sectional position (a), illustrating the structure of the preceding-stage mode conversion section. 
         FIG. 5  is a diagram illustrating effective refractive indexes in a case where two waveguides are independently present. 
         FIG. 6  is a diagram illustrating effective refractive indexes in a case where two waveguides are contiguous to each other. 
         FIG. 7A  is a diagram illustrating a simulation result of an electric field distribution (E x  component) in a mode #0 at the sectional position (a) in  FIG. 4A . 
         FIG. 7B  is a diagram illustrating a simulation result of an electric field distribution (E x  component) in a mode #1 at the sectional position (a) in  FIG. 4A . 
         FIG. 7C  is a diagram illustrating a simulation result of an electric field distribution (E x  component) in the mode #0 at the sectional position (b) in  FIG. 4A . 
         FIG. 7D  is a diagram illustrating a simulation result of an electric field distribution (E x  component) in the mode #1 at the sectional position (b) in  FIG. 4A . 
         FIG. 7E  is a diagram illustrating a simulation result of an electric field distribution (E x  component) in the mode #0 at the sectional position (c) in  FIG. 4A . 
         FIG. 7F  is a diagram illustrating a simulation result of an electric field distribution (E x  component) in the mode #1 at the sectional position (c) in  FIG. 4A . 
         FIG. 8A  is a plan view illustrating a structure of a subsequent-stage mode conversion section. 
         FIG. 8B  is a sectional view of core portions, illustrating the structure of the subsequent-stage mode conversion section. 
         FIG. 8C  is a sectional view of an output portion, illustrating the structure of the subsequent-stage mode conversion section. 
         FIG. 9A  is a diagram illustrating a simulation result showing an electric field distribution (E x  component) in the output portion of the subsequent-stage mode conversion section. 
         FIG. 9B  is a graph illustrating the electric field distribution (E x  component) in the output portion of the subsequent-stage mode conversion section. 
         FIG. 10A  is a diagram illustrating a simulation result showing an electric field distribution (E x  component) in the core portions of the subsequent-stage mode conversion section. 
         FIG. 10B  is a graph illustrating the electric field distribution (E x  component) in the core portions of the subsequent-stage mode conversion section. 
         FIG. 11  is a graph illustrating a relationship between the width of the output portion of the subsequent-stage mode conversion section and conversion efficiency. 
         FIG. 12A  is a diagram illustrating a simulation result showing an electric field distribution (E x  component) in the output portion of the subsequent-stage mode conversion section. 
         FIG. 12B  is a graph illustrating the electric field distribution (E x  component) in the output portion of the subsequent-stage mode conversion section. 
         FIG. 13  is a graph illustrating a relationship between a wavelength of light in the subsequent-stage mode conversion section and conversion efficiency. 
         FIG. 14  is a graph illustrating a relationship between a variation of a core width (waveguide width) and conversion efficiency. 
         FIG. 15A  is a plan view illustrating an example of a planar optical waveguide device having a bent waveguide. 
         FIG. 15B  is a sectional view at a sectional position (a), illustrating the example of the planar optical waveguide device having the bent waveguide. 
         FIG. 16  is a graph illustrating a relationship between the length of the preceding-stage mode conversion section and conversion efficiency. 
         FIG. 17  is a diagram illustrating a simulation result showing an electric field distribution (E x  component). 
         FIG. 18  is a graph illustrating a relationship between a wavelength of light and conversion efficiency. 
         FIG. 19  is a graph illustrating a relationship between a wavelength and conversion efficiency in a case where the width of a core portion (and an output portion) is changed. 
         FIG. 20  is a diagram illustrating a simulation result showing an electric field distribution (E x  component). 
         FIG. 21  is a graph illustrating a relationship between a wavelength of light and conversion efficiency. 
         FIG. 22  is a graph illustrating a relationship between a wavelength and conversion efficiency in a case where the width of a core portion (and an output portion) is changed. 
         FIG. 23  is a plan view illustrating a first example of a planar optical waveguide device having a structure in which an intermediate core portion is provided between the preceding-stage mode conversion section and the subsequent-stage mode conversion section. 
         FIG. 24  is a plan view illustrating a second example of the planar optical waveguide device having the structure in which the intermediate core portion is provided between the preceding-stage mode conversion section and the subsequent-stage mode conversion section. 
         FIG. 25  is a plan view illustrating a third example of the planar optical waveguide device having the structure in which the intermediate core portion is provided between the preceding-stage mode conversion section and the subsequent-stage mode conversion section. 
         FIG. 26  is a plan view illustrating a planar optical waveguide device using a modification example of a matching coupling element. 
         FIG. 27  is a plan view illustrating an example of a planar optical waveguide device using a high-order polarization conversion element. 
         FIG. 28A  is a plan view schematically illustrating an example of the high-order polarization conversion element shown in  FIG. 27 . 
         FIG. 28B  is a sectional view schematically illustrating an example of the high-order polarization conversion element shown in  FIG. 27 . 
         FIG. 29A  is a plan view illustrating another example of the high-order polarization conversion element. 
         FIG. 29B  is a sectional view at a sectional position (h), illustrating another example of the high-order polarization conversion element. 
         FIG. 29C  is a sectional view at a sectional position (g), illustrating another example of the high-order polarization conversion element. 
         FIG. 29D  is a sectional view at a sectional position (f), illustrating another example of the high-order polarization conversion element. 
         FIG. 30A  is a plan view illustrating a planar optical waveguide device according to a second embodiment of the invention. 
         FIG. 30B  is a sectional view at a sectional position (a), illustrating the planar optical waveguide device according to the second embodiment of the invention. 
         FIG. 31A  is a plan view illustrating an example of an optical waveguide device. 
         FIG. 31B  is a sectional view illustrating the example of the optical waveguide device. 
         FIG. 32A  is a diagram illustrating a simulation result of an electric field distribution (E x  component) in an even mode in the optical waveguide device shown in  FIGS. 31A and 31B . 
         FIG. 32B  is a diagram illustrating a simulation result of an electric field distribution (E x  component) in an odd mode in the optical waveguide device shown in  FIGS. 31A and 31B . 
         FIG. 32C  is a graph of the electric field distribution (E x  component) in the even mode in the optical waveguide device shown in  FIGS. 31A and 31B . 
         FIG. 32D  is a graph of the electric field distribution (E x  component) in the odd mode in the optical waveguide device shown in  FIGS. 31A and 31B . 
         FIG. 33A  is a plan view illustrating a structure of a preceding-stage mode conversion section. 
         FIG. 33B  is a sectional view at a sectional position (c), illustrating the structure of the preceding-stage mode conversion section. 
         FIG. 33C  is a sectional view at a sectional position (b), illustrating the structure of the preceding-stage mode conversion section. 
         FIG. 33D  is a sectional view at a sectional position (a), illustrating the structure of the preceding-stage mode conversion section. 
         FIG. 34A  is a diagram illustrating effective refractive indexes in a case where two waveguides are independently present. 
         FIG. 34B  is a sectional view illustrating a structure of one waveguide in a case where two waveguides are independently present. 
         FIG. 34C  is a sectional view illustrating a structure of the other waveguide in a case where two waveguides are independently present. 
         FIG. 35  is a diagram illustrating effective refractive indexes in a case where two waveguides are contiguous to each other. 
         FIG. 36A  is a diagram illustrating a simulation result of an electric field distribution (E x  component) in a mode #0 at a sectional position (a) in  FIG. 33A . 
         FIG. 36B  is a diagram illustrating a simulation result of an electric field distribution (E x  component) in a mode #1 at the sectional position (a) in  FIG. 33A . 
         FIG. 36C  is a diagram illustrating a simulation result of an electric field distribution (E x  component) in the mode #0 at a sectional position (b) in  FIG. 33A . 
         FIG. 36D  is a diagram illustrating a simulation result of an electric field distribution (E x  component) in the mode #1 at the sectional position (b) in  FIG. 33A . 
         FIG. 36E  is a diagram illustrating a simulation result of an electric field distribution (E x  component) in the mode #0 at the sectional position (c) in  FIG. 33A . 
         FIG. 36F  is a diagram illustrating a simulation result of an electric field distribution (E x  component) in the mode #1 at the sectional position (c) in  FIG. 33A . 
         FIG. 37A  is a plan view illustrating a structure of a subsequent-stage mode conversion section. 
         FIG. 37B  is a sectional view of core portions, illustrating the structure of the subsequent-stage mode conversion section. 
         FIG. 37C  is a sectional view of an output portion, illustrating the structure of the subsequent-stage mode conversion section. 
         FIG. 38A  is a diagram illustrating a simulation result showing an electric field distribution (E x  component) in the output portion of the subsequent-stage mode conversion section. 
         FIG. 38B  is graph illustrating the electric field distribution (E x  component) in the output portion of the subsequent-stage mode conversion section. 
         FIG. 39A  is a diagram illustrating a simulation result showing an electric field distribution (E x  component) in the core portions of the subsequent-stage mode conversion section. 
         FIG. 39B  is a graph illustrating the electric field distribution (E x  component) in the core portions of the subsequent-stage mode conversion section. 
         FIG. 40  is a graph illustrating a relationship between the width of the output portion and conversion efficiency. 
         FIG. 41A  is a diagram illustrating a simulation result showing an electric field distribution (E x  component) in the output portion of the subsequent-stage mode conversion section. 
         FIG. 41B  is a graph illustrating the electric field distribution (E x  component) in the output portion of the subsequent-stage mode conversion section. 
         FIG. 42  is a graph illustrating a relationship between a wavelength of light and conversion efficiency. 
         FIG. 43  is a graph illustrating a relationship between a wavelength and conversion efficiency in a case where the width of a core portion (and an output portion) is changed. 
         FIG. 44A  is a plan view illustrating an example of a planar optical waveguide device having a bent waveguide. 
         FIG. 44B  is a sectional view at a sectional position (a), illustrating the example of the planar optical waveguide device having the bent waveguide. 
         FIG. 45  is a graph illustrating a relationship between the length of the preceding-stage mode conversion section and conversion efficiency. 
         FIG. 46  is a diagram illustrating a simulation result showing an electric field distribution (E x  component). 
         FIG. 47  is a graph illustrating a relationship between a wavelength of light and conversion efficiency. 
         FIG. 48  is a graph illustrating a relationship between a wavelength and conversion efficiency in a case where the width of a core portion (and an output portion) is changed. 
         FIG. 49  is a diagram illustrating a simulation result showing an electric field distribution (E x  component). 
         FIG. 50  is a graph illustrating a relationship between a wavelength of light and conversion efficiency. 
         FIG. 51  is a graph illustrating a relationship between a wavelength and conversion efficiency in a case where the width of a core portion (and an output portion) is changed. 
         FIG. 52  is a graph illustrating a relationship between a wavelength of light and loss. 
         FIG. 53A  is a plan view illustrating a first example of a planar optical waveguide device that employs a rib waveguide structure. 
         FIG. 53B  is a sectional view of an output portion, illustrating the first example of the planar optical waveguide device that employs the rib waveguide structure. 
         FIG. 53C  is a sectional view of core portions, illustrating the first example of the planar optical waveguide device that employs the rib waveguide structure. 
         FIG. 54A  is a plan view illustrating a second example of the planar optical waveguide device that employs the rib waveguide structure. 
         FIG. 54B  is a sectional view of an output portion, illustrating the second example of the planar optical waveguide device that employs the rib waveguide structure. 
         FIG. 54C  is a sectional view of core portions, illustrating the second example of the planar optical waveguide device that employs the rib waveguide structure. 
         FIG. 55A  is a plan view illustrating a third example of the planar optical waveguide device that employs the rib waveguide structure. 
         FIG. 55B  is a sectional view of an output portion, illustrating the third example of the planar optical waveguide device that employs the rib waveguide structure. 
         FIG. 55C  is a sectional view of core portions, illustrating the third example of the planar optical waveguide device that employs the rib waveguide structure. 
         FIG. 56  is a plan view illustrating an example of a planar optical waveguide device having a tapered waveguide. 
         FIG. 57A  is a diagram illustrating an example of the tapered waveguide, which is a sectional view of one end thereof. 
         FIG. 57B  is a plan view illustrating the example of the tapered waveguide. 
         FIG. 57C  is a diagram illustrating the example of the tapered waveguide, which is a sectional view of the other end thereof. 
         FIG. 58A  is a diagram illustrating another example of the tapered waveguide, which is a sectional view of one end thereof. 
         FIG. 58B  is a plan view illustrating another example of the tapered waveguide. 
         FIG. 58C  is a diagram illustrating another example of the tapered waveguide, which is a sectional view of the other end thereof. 
         FIG. 59  is a plan view illustrating a fourth example of the planar optical waveguide device having the structure in which the intermediate core portion is provided between the preceding-stage mode conversion section and the subsequent-stage mode conversion section. 
         FIG. 60  is a plan view illustrating a fifth example of the planar optical waveguide device having the structure in which the intermediate core portion is provided between the preceding-stage mode conversion section and the subsequent-stage mode conversion section. 
         FIG. 61  is a plan view illustrating a sixth example of the planar optical waveguide device having the structure in which the intermediate core portion is provided between the preceding-stage mode conversion section and the subsequent-stage mode conversion section. 
         FIG. 62  is a plan view illustrating a planar optical waveguide device using a modification example of a matching coupling element. 
         FIG. 63  is a plan view illustrating an example of a planar optical waveguide device (polarization conversion element) using a high-order polarization conversion element. 
         FIG. 64A  is an overall plan view illustrating an example of the planar optical waveguide device using the high-order polarization conversion element. 
         FIG. 64B  is a plan view of the high-order polarization conversion element, illustrating the example of the planar optical waveguide device using the high-order polarization conversion element. 
         FIG. 64C  is a sectional view of an ending portion of the high-order polarization conversion element, illustrating the example of the planar optical waveguide device using the high-order polarization conversion element. 
         FIG. 64D  is a sectional view of a start portion of the high-order polarization conversion element, illustrating the example of the planar optical waveguide device using the high-order polarization conversion element. 
         FIG. 65  is a plan view illustrating an example of the planar optical waveguide device using the high-order polarization conversion element. 
         FIG. 66  is a schematic view illustrating an example of a DP-QPSK modulator. 
         FIG. 67  is a schematic view illustrating an example of a polarization diversity coherent receiver. 
         FIG. 68  is a schematic view illustrating an example of a polarization diversity technique. 
         FIG. 69A  is a plan view illustrating an example of a planar optical waveguide device in the related art. 
         FIG. 69B  is a sectional view illustrating the example of the planar optical waveguide device in the related art. 
         FIG. 70  is a graph illustrating a relationship between a wavelength of light and an absolute value of ΔN. 
         FIG. 71  is a graph illustrating a relationship between a wavelength of light and conversion efficiency. 
         FIG. 72  is a diagram illustrating a manufacturing error of the width of a core portion. 
         FIG. 73  is a graph illustrating a relationship between a wavelength and an absolute value of ΔN in a case where the width of a core portion is changed. 
         FIG. 74  is a graph illustrating a relationship between a wavelength and conversion efficiency in a case where the width of a core portion is changed. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     &lt;Overview of Present Embodiment&gt; 
     A planar optical waveguide device according to an embodiment of the invention has a configuration in which a preceding-stage mode conversion section (super mode-generating element) and a subsequent-stage mode conversion section (matching coupling element) are combined. 
     In the super mode-generating element having a structure (for example, a tapered waveguide) in which a waveguide structure changes in a light waveguide direction, TE 0  which is input is converted into an odd mode which is a super mode of TE 0 . In the matching coupling element, the odd mode is converted into TE 1 . 
     The super mode-generating element converts the mode of TE 0  into the odd mode by continuously changing the waveguide structure in the light waveguide direction, using a so-called adiabatic change phenomenon. Thus, if a waveguide (for example, a tapered waveguide) of such a structure is set to be sufficiently long, it is possible to enhance conversion efficiency to the odd mode. 
     The matching coupling element uses electric field distribution similarity between the odd mode which is the super mode of TE 0  and TE 1  of a rectangular waveguide, to thereby make it possible to enhance conversion efficiency to TE 1  from the odd mode. 
     Since the super mode-generating element has a configuration in which shapes and sizes of sections of two core portions at an output end are the same (congruent), the super mode-generating element is not easily affected by a manufacturing error and also has small wavelength dependency. 
     Since even in a case where a wavelength changes or a waveguide structure is changed by a manufacturing error, electric field distributions of both of the odd mode and TE 1  are changed, the matching coupling element is not easily affected by wavelength change or a manufacturing error. 
     Accordingly, in the present embodiment, it is possible to perform conversion over a wide wavelength band with high efficiency, and to secure efficient mode conversion even in a case where a waveguide structure is changed due to a manufacturing error. 
     Hereinafter, the planar optical waveguide device according to the present embodiment will be described in detail. 
     First, a specific example of the planar optical waveguide device according to the present embodiment will be presented. Then, a conversion principle between TE 0  and the odd mode in the super mode-generating element will be described with reference to the specific example. Thereafter, a conversion principle between the odd mode and TE 1  in the matching coupling element will be described, and then, effects of the invention will be described. 
     &lt;Planar Optical Waveguide Device&gt; 
     A structure of a planar optical waveguide device  10  according to a first embodiment of the invention will be described with reference to  FIGS. 1A and 1B .  FIG. 1A  is a plan view illustrating the planar optical waveguide device  10 , and  FIG. 1B  is a sectional view at a sectional position (a) in  FIG. 1A , which shows a cross section perpendicular to a light waveguide direction. 
     In the cross section, a dimension in a direction where a core portion  1  and a core portion  2  face each other (a direction perpendicular to the light waveguide direction) is referred to as a width, and a dimension in a direction perpendicular to the width direction (a direction vertical to a substrate S) is referred to as a height. 
     As shown in  FIGS. 1A and 1B , the planar optical waveguide device  10  (mode conversion element) includes an optical waveguide  4  that includes a core  5  and a cladding  15 . The optical waveguide  4  is formed on the substrate S. 
     The cladding  15  is formed of a material having a refractive index which is lower than that of the core  5 , and is formed to cover the core  5 . The cladding  15  includes an upper cladding  6  and a lower cladding  7 . The upper cladding  6  is provided on the core  5  and the lower cladding  7 . 
     The lower cladding  7  is formed of SiO 2 , for example. The upper cladding  6  is formed of SiO 2  or an air layer, for example. 
     The core  5  is formed of a material having a refractive index which is higher than that of the cladding  15 , and includes a pair of core portions  1  and  2  which are disposed in parallel with each other, and an output portion  3  provided in a subsequent-stage (output-side) of the core portions  1  and  2 . 
     The core portions  1  and  2  and the output portion  3  are preferably formed of Si (silicon). 
     Hereinafter, the core portion  1  may be referred to as a first core portion  1 , and the core portion  2  may be referred to as a second core portion  2 . 
     The invention is not limited to a silicon waveguide in which a core is formed of Si, and may be applied to an optical waveguide using a core formed of SiO2, for example, an optical waveguide such as a planar lightwave circuit (PLC). 
     As shown in  FIG. 1B , sectional shapes of the core portions  1  and  2  may be rectangular. The sectional shapes of the core portions  1  and  2  are not limiting. 
     It is preferable that heights H 1  and H 2  of the core portions  1  and  2  be equal to each other. If the heights of the core portions  1  and  2  are equal to each other, it is possible to reduce the number of times of etching to the minimum when forming the core. 
     As shown in  FIG. 1A , the core  5  includes a preceding-stage mode conversion section  8  (super mode-generating element) that converts the mode of light that propagates through the core portions  1  and  2 , and a subsequent-stage mode conversion section  9  (matching coupling element) that converts the mode of light passed through the preceding-stage mode conversion section  8 . 
     The planar optical waveguide device  10  may be manufactured by processing a silicon-on-insulator (SOI) substrate. For example, an SiO 2  layer of the SOI substrate may be formed as a lower cladding, and an Si layer thereof may be formed as a core through a lithography/etching process. After formation of the core, an SiO 2  layer may be provided as an upper cladding. 
     &lt;Principle of Super Mode-Generating Element&gt; 
     A basic principle of the super mode-generating element will be described. 
     In the super mode-generating element, TE 0  of one waveguide among two contiguous waveguides is gradually mode-coupled with TE 0  of the other waveguide, and thus, TE 0  is converted into the odd mode which is the super mode of TE 0 . 
     The two waveguides in the planar optical waveguide devices are respectively referred to as a waveguide 1 and a waveguide 2. The waveguide 1 is formed so that the width of a core portion is smaller than the width of a core portion of the waveguide 2 at an input end. 
     The waveguide refers to a path which guides light and is formed by a core and a cladding. 
     In the preceding-stage mode conversion section  8  of the planar optical waveguide device  10  shown in  FIGS. 1A and 1B , the waveguide 1 is formed by a core portion  11  and a cladding portion that covers the core portion  11 . The waveguide 2 is formed by a core portion  12  and a cladding portion that covers the core portion  12 . 
     The “mode coupling” refers to a phenomenon that with respect to light that propagates through a core portion of one waveguide in a certain mode, a part of the light leaks out from the core portion through which the light propagates and moves to the other waveguide. 
     In order to efficiently perform the mode coupling, it is necessary that effective refractive indexes of respective coupling target modes in waveguides are at the same level. The “same level” means that a difference between effective refractive indexes is smaller than χ×wavelength/π using a coupling coefficient χ (which will be described later). A state where this condition is satisfied is referred to as “phase matching”. Here, in a case where the shapes and sizes of the cores are the same (congruent), since effective refractive indexes of light components in the same mode that propagate through the core portions become the same, phase-matching is constantly achieved. The constant phase matching is established since even when a wavelength changes, the shape of the core is not changed. In addition, even in a case where manufacturing errors of the same amount occur in two core portions, such as a case where the same variation (variation in width or height) occurs in two core portions, since a congruent relationship of the core is not broken, phase matching is not broken. 
       FIGS. 2A and 2B  show an optical waveguide device having core portions  21  and  22  having the same width. 
     As shown in  FIGS. 2A and 2B , when both TE 0  and TE 0  are mode-coupled between contiguous waveguides, in a case where heights of core portions are equal to each other, since the shapes of the cores become the same by setting the widths of the core portions to be the same, the phase-matching condition is satisfied. This condition is constantly satisfied even when a wavelength changes. 
     Furthermore, in a structure in which the widths of the core portions are equal to each other, even in a case where manufacturing errors of the same amount occur in two core portions, such as a case where the same variation (variation in width or height) occurs in two core portions, phase matching is not broken. 
     Here, both TE 0  and TE 0  in waveguides which are contiguous to each other are mode-coupled. Thus, modes in which light is guided through cross sections in which two core portions are disposed in parallel are divided into a mode (referred to as an even mode) in which TE 0  and TE 0  of respective waveguides are coupled and electric field components are symmetric to each other in a width direction, as shown in  FIGS. 3A and 3C , and a mode (referred to as an odd mode) in which TE 0  and TE 0  of respective waveguides are coupled and electric field components are asymmetric to each other in a width direction, as shown in  FIGS. 3B and 3D . These modes are collectively referred to as a super mode of TE 0  (or simply a super mode). 
     When phase matching of respective modes which are coupling targets in contiguous waveguides is established, the lengths of waveguides necessary for movement of light leaked out from one waveguide to the other waveguide to form a super mode depend on the coupling coefficient χ which represents the strength of mode coupling. Here, χ is expressed as follows.
 
[Expression 5]
 
χ∝∫ −∞   ∞ ∫ −∞   ∞ ( N   2   −N   1   2 ) E   1   *·E   2   dxdy  or ∫ −∞   ∞ ∫ −∞   ∞ ( N   2   −N   2   2 ) E   1   ·E   2   *dxdy   (5)
 
     Here, E i  (i=1, 2) represents electric field vectors of modes of coupling targets that are guided by two contiguous waveguides i (i=1, 2), N represents a refractive index distribution when two waveguides are contiguous, N i  represents a refractive index distribution when a waveguide i is independently present, and coordinates x and y represent a width direction and a height direction, respectively. 
     It can be understood from Expression (5) that since inner products of electric fields of both modes are integrated in the waveguide 1 or the waveguide 2, coupling between the waveguides becomes stronger as light leaked out from a core portion becomes larger. As the coupling coefficient χ becomes larger, it is possible to generate the super mode at a shorter distance. 
     &lt;Specific Example of Super Mode-Generating Element&gt; 
     The preceding-stage mode conversion section  8  which is a specific example of a super mode-generating element will be described with reference to  FIGS. 1A and 1B . 
     Ranges of the core portions  1  and  2  where the preceding-stage mode conversion section  8  is configured are respectively referred to as a preceding-stage first core portion  11  and a preceding-stage second core portion  12 . 
     Input ends  11   a  and  12   a  (preceding-stage input ends) of the core portions  11  and  12  are end portions at which light is input to the core portions  11  and  12 , respectively. Output ends  11   b  and  12   b  (preceding-stage output ends) are end portions at which light is output from the core portions  11  and  12 . 
     It is preferable that the preceding-stage first core portion  11  linearly extend in a planar view and the width and the height thereof be uniform in a length direction (light waveguide direction). In the example shown in the figure, since the width and the height are uniform in the length direction, in the preceding-stage first core portion  11 , a sectional shape (the shape of a cross section vertical to the light waveguide direction) is also uniform over an entire length thereof. 
     An inner edge  11   c  (side edge on the side of the core portion  12  among both side edges of the core portion  11 ) and an outer edge  11   d  (side edge opposite to the inner edge  11   c ) of the preceding-stage first core portion  11  are linearly formed, respectively. 
     It is preferable that the sectional shapes of the core portions  11  and  12  be rectangular. 
     A waveguide structure of the preceding-stage second core portion  12  is continuously changed from the input end  12   a  to the output end  12   b.    
     In order to change the waveguide structure in the light waveguide direction, it is preferable that the width of the core portion be changed along the light waveguide direction. 
     Since the width of the core portion is related to light confinement to the core portion, it is possible to arbitrarily adjust an effective refractive index in a mode of light that is guided by the core portion by changing the width. 
     A method for changing the waveguide structure may include a method for changing the height of the core portion. On the other hand, a method for changing the width of the core portion in the length direction of the core portion while maintaining the height of the core portion to be uniform is preferable since the core portion can be manufactured by one time of etching in processing of an SOI substrate. 
     An inner edge  12   c  of the preceding-stage second core portion  12  (side edge on the side of the core portion  11  among both side edges of the core portion  12 ) is formed in a linear shape which is parallel to the inner edge  11   c  of the preceding-stage first core portion  11 . 
     An outer edge  12   d  of the preceding-stage second core portion  12  (side edge opposite to the inner edge  12   c ) is inclined and formed in a linear shape to gradually approach the inner edge  12   c  from the input end  12   a  to the output end  12   b.    
     Thus, the preceding-stage second core portion  12  is formed in a tapered shape in which the width (width W 2  in  FIG. 1B ) continuously decreases from the input end  12   a  to the output end  12   b  at a constant rate. 
     Since the width of the preceding-stage second core portion  12  gradually decreases, the size of a cross section thereof continuously decreases from the input end  12   a  to the output end  12   b  at a constant rate. 
     In the present embodiment, in the input ends  11   a  and  12   a , since the width of the second core portion  12  (width W 2  in  FIG. 1B ) is larger than the width (width W 1  in  FIG. 1B ) of the first core portion  11 , the cross section of the core portion  12  is larger than the cross section of the core portion  11 . That is, a sectional area of the core portion  12  is larger than a sectional area of the core portion  11 . 
     At the output ends  11   b  and  12   b , the widths of the core portion  11  and  12  are equal to each other as the second core portion  12  is tapered. Thus, at the output ends  11   b  and  12   b , the shapes and sizes of the cross sections of the core portions  11  and  12  are equal to each other. 
     The core portions  11  and  12  are separated from each other. A gap between the core portions  11  and  12  may be set to be uniform from the input ends  11   a  and  12   a  to the output ends  11   b  and  12   b.    
     As long as the sectional shapes at the input ends  11   a  and  12   a  are not congruent with each other and the shape or size of a cross section of at least one core continuously changes along the light waveguide direction so that the sectional shapes of the core portions  11  and  12  are congruent with each other at the output ends  11   b  and  12   b , the structure of the core portions  11  and  12  is not limited to the example shown in the figure. 
     For example, a structure in which the size of a core portion having a small cross section at an input end continuously increases in the light waveguide direction so that sectional shapes of two core portions become congruent with each other at output ends may be used. 
     Here, the “continuous change” means that the cross sections of the core portions  11  and  12  are changed to such a degree that there is no rapid change in the structures of the core portions  11  and  12  and, for example, such a non-continuous uneven portion as to form a step difference portion is not generated on any outer surface of the core portions. 
     Hereinafter, the preceding-stage mode conversion section  8  will be more specifically described with reference to  FIGS. 4A to 4D . 
       FIGS. 4A to 4D  are diagrams illustrating the preceding-stage mode conversion section  8 .  FIG. 4A  is a plan view thereof,  FIG. 4B  is a sectional view at a sectional position (c) in  FIG. 4A ,  FIG. 4C  is a sectional view at a sectional position (b), and  FIG. 4D  is a sectional view at a sectional position (a). 
     The core portion  11  (core portion  1 ) and the core portion  12  (core portion  2 ) are formed of Si (having a refractive index of 3.48 (at a wavelength of 1580 nm)), and the upper cladding  6  and the lower cladding  7  are formed of SiO 2  (having a refractive index of 1.44 (at the wavelength of 1580 nm)). Further, the heights of the core portions  11  and  12  (core portions  1  and  2 ) are 220 nm. The gap between the core portions  11  and  12  (core portions  1  and  2 ) is 200 nm. 
     As shown in  FIG. 4D , the width of the core portion  11  (core portion  1 ) is 400 [nm], and the width of the core portion  12  (core portion  2 ) is 400−X [nm] (−200≦X≦0). X is linearly changed from −200 to 0, from the input end  12   a  to the output end  12   b.    
     Thus, the core portion  12  (core portion  2 ) is formed in a tapered shape so that the width gradually decreases from the input end  12   a  (X=−200) to the output end  12   b  (X=0). 
       FIG. 4C  shows a cross section at an intermediate position (X=−20) between the input ends  11   a  and  12   a  and the output ends  11   b  and  12   b.    
     In the super mode-generating element, based on the above-described phase-matching condition, core shapes of contiguous waveguides are set not to be congruent with each other at input ends. Thus, the phase-matching condition is broken. On the other hand, at output ends, the phase-matching condition is satisfied since the core shapes of contiguous waveguides are congruent with each other. 
     Further, by continuously changing the shape or size of the core portion along the light waveguide direction (that is, by forming the core portion in a tapered shape), phase matching is continuously performed from the input ends to the output ends. 
     In the example illustrated in  FIGS. 4A and 4D , at the input ends  11   a  and  12   a , the width (width W 12a  in  FIG. 4D ) of the core portion  12  (core portion  2 ) is larger than the width (width W 11a  in  FIG. 4D ) of the core portion  11  (core portion  1 ). Thus, the cross section of the core portion  12  is formed to be larger than the cross section of the core portion  11 . Thus, phase matching is not established, and mode coupling is not nearly performed. 
     On the other hand, at the output ends  11   b  and  12   b , the widths (widths W 11b  and W 12b  in  FIG. 4B ) of the core portions  11  and  12  (core portions  1  and  2 ) are equal to each other, and thus, the shapes and sizes of cross sections of the core portions  11  and  12  are equal to each other. Thus, phase matching is established. 
     Since the core portion  12  is formed in a tapered shape, phase matching is gradually performed along the light waveguide direction from the input end to the output end, and as a result, mode coupling is progressed. Thus, by sufficiently increasing the length (taper length) of the tapered waveguide (core portion  12 ), it is possible to convert TE 0  input to the waveguide 1 into the odd mode with almost no loss. 
     As described above, the lengths of waveguides necessary for movement of light leaked out from one waveguide to the other waveguide to form a super mode depend on the coupling coefficient χ. Thus, as the coupling coefficient χ becomes larger, mode conversion may be performed with higher accuracy using shorter waveguides (shorter device length). 
     A method for adjusting an effective refractive index by changing the width of a waveguide employs a phenomenon that as the size of the waveguide becomes larger, confinement of light in a core portion becomes larger, and thus, the influence of a refractive index of the core portion more strongly acts and an effective refractive index becomes larger. 
     This principle will be more specifically described with reference to the above-described specific example. 
     In order to check that it is possible to break the phase-matching condition at input ends by changing the width of a core portion (that is, by tapering a waveguide) in a light waveguide direction, mode effective refractive indexes in a case where waveguides 1 and 2 are independently present were calculated. A result thereof is illustrated in  FIG. 5 . A wavelength was set to 1580 nm. 
     It can be understood from  FIG. 5  that effective refractive indexes of TE 0  of the waveguide 1 and TE 0  of the waveguide 2 are the same and phase matching is established when X=0. 
     As X is separated from 0, deviation occurs in the effective refractive indexes in TE 0  and TE 0  of the waveguides 1 and 2, the phase-matching condition is broken. 
     The reason why the effective refractive index in TE 0  of the waveguide 1 is smaller than the effective refractive index in TE 0  of the waveguide 2 in the range of −200≦X&lt;0 is that the width of the core portion in the waveguide 1 is smaller than that of the waveguide 2. 
     Subsequently,  FIG. 6  illustrates effective refractive indexes in modes in a case where the waveguides 1 and 2 are contiguous to each other. 
     #0 and #1 represent effective refractive indexes in modes in which effective refractive indexes are largest and second largest, respectively, among modes on cross sections of two waveguides. 
     Compared with  FIG. 5  illustrating the effective refractive indexes in a case where the waveguides are independently present, in  FIG. 6 , #0 and #1 do not match and are separated from each other when X=0. 
     This is because two modes mutually act due to mode coupling to form a mixed mode (super mode) since the phase-matching condition is satisfied between TE 0  of the waveguide 1 and TE 0  of the waveguide 2. 
     If X is separated from 0, since the phase-matching condition is not satisfied, such a mutual action does not occur, and the same mode distribution as in a case where the waveguides are independently present is obtained. As a result, effective refractive indexes do not greatly change compared with a case where the waveguides are independently present. When X=0, the mode #0 becomes an even mode, and the mode #1 becomes an odd mode. 
     In a structure in which a structure of a waveguide is gradually changed in a light waveguide direction, such as a tapered waveguide, it is known that mode conversion is performed so as to change on a curve of one curve effective refractive index (referred to as an adiabatic change). 
     Thus, in  FIG. 6 , by inputting TE 0  to the waveguide 1 when X=−200 (input end) and gradually changing X from −200 to 0 in the length direction of the waveguide, it is possible to convert TE 0  into the odd mode which is the super mode of TE 0  when X=0. 
     In order to confirm this mode conversion,  FIGS. 7A to 7F  illustrate electric field distributions in modes #0 and #1 at sectional positions (a) to (c) (see  FIG. 4A ). 
       FIGS. 7A and 7B  are diagrams illustrating simulation results (mode #0 in  FIG. 7A  and mode #1 in  FIG. 7B ) showing electric field distributions (E x  components) at the sectional position (a).  FIGS. 7C and 7D  are diagrams illustrating simulation results (mode #0 in  FIG. 7C  and mode #1 in  FIG. 7D ) showing electric field distributions (E x  components) at the sectional position (b).  FIGS. 7E and 7F  are diagrams illustrating simulation results (mode #0 in  FIG. 7E  and mode #1 in  FIG. 7F ) showing electric field distributions (E x  components) at the sectional position (c). 
     Here, x and y represent a width direction and a height direction, respectively. The electric field distributions in  FIGS. 7E and 7F  are the same as in  FIGS. 3A and 3B , respectively. 
     Referring to the mode #1, at the sectional position (a) (X=−200) illustrated in  FIG. 7B , TE 0  is present in the waveguide 1. 
     At the sectional position (b) (X=−20) illustrated in  FIG. 7D , it can be understood that mode coupling to TE 0  of the waveguide 2 starts. 
     At the sectional position (c) (X=0) illustrated in  FIG. 7F , since the phase-matching condition is satisfied, an odd mode which is a super mode in which TE 0  of the waveguide 1 and TE 0  of the waveguide 2 are mixed can be viewed. 
     In this way, it can be understood that it is possible to change TE 0  input to the waveguide 1 to an odd mode by gradually changing a waveguide structure in a light waveguide direction. 
     Hereinbefore, the principle of the super mode-generating element has been described. 
     In the illustrated example, since the width continuously decreases from the input end  12   a  to the output end  12   b , the preceding-stage second core portion  12  is formed in a tapered shape over the entire length. However, the shape of the preceding-stage second core portion  12  is not limited thereto, and only a part thereof in the length direction may be formed in a tapered shape in which the width continuously decreases. 
     In  FIGS. 4A to 4D , the gap between the waveguides (gap between the core portion  11  and  12 ) of the super mode-generating element (preceding-stage mode conversion section) is uniform. However, the gap is not limited thereto, and may vary in the length direction of the waveguides. 
     Furthermore, as long as a condition that in two core portions which are contiguous to each other, the sizes of cross sections thereof at output ends are equal to each other and an effective refractive index in TE 0  of a first core portion is smaller than an effective refractive index in TE 0  of a second core portion in a range other than the output ends is satisfied, a configuration in which only one of two core portions is tapered may be used, or a configuration in which both of two core portions are tapered may be used. 
     For example, the first core portion  11  may be formed in a tapered shape in which the width continuously decreases from the input end  11   a  to the output end  11   b , similar to the second core portion  12 . 
     Furthermore, in the above-described specific example, the core portion  2  is a tapered waveguide in which the width linearly changes in the length direction, but the tapered waveguide is not limited thereto, and may be a curve-tapered waveguide. 
     A method for changing the waveguide structure may include a method for changing the height of a core portion in a light waveguide direction, instead of a method for changing the width of the core portion. By changing the height of the core portion, it is possible to arbitrarily adjust an effective refractive index in a mode of light that is guided by the core portion. 
     Further, in the specific example, the input ends and the output ends of the core portions are formed to be vertical to the light waveguide direction, but may be inclined with respect to the vertical direction. 
     &lt;Principle of Matching Coupling Element&gt; 
     Subsequently, a principle of converting an odd mode into TE 1  by a matching coupling element will be described. 
     The odd mode between TE 0  and TE 0  is asymmetric in an E x  component which is a main electric field distribution in a TE mode, and shows an electric field distribution having two peaks. 
     On the other hand, TE 1  also has an asymmetric E x  component, and has an electric field distribution having two peaks. 
     Thus, TE 1  and the odd mode have high similarity. Thus, even when two waveguides through which the odd mode propagates and one waveguide by which TE 1  is guided are discontinuously connected, it is possible to convert the odd mode into TE 1  with high coupling efficiency. 
     &lt;Specific Example of Matching Coupling Element&gt; 
     The subsequent-stage mode conversion section  9  which is a specific example of a matching coupling element will be described with reference to  FIGS. 1A and 1B . 
     The subsequent-stage mode conversion section  9  includes a subsequent-stage first core portion  13  formed to be connected to an output-side of the core portion  11 , a subsequent-stage second core portion  14  formed to be connected to an output-side of the core portion  12 , and an output portion  3  to which output ends  13   b  and  14   b  of the core portions  13  and  14  are connected. 
     The input ends  13   a  and  14   a  (subsequent-stage input ends) of the core portions  13  and  14  are end portions at which light is input to the core portions  13  and  14 , respectively, and the output ends  13   b  and  14   d  (subsequent-stage output ends) are end portions at which light is output from the core portions  13  and  14 . 
     Since the subsequent-stage mode conversion section  9  is formed to be connected to the output-side of the preceding-stage mode conversion section  8 , the input end  13   a  is disposed at the same position as that of the output end  11   b , and the input end  14   a  is disposed at the same position as that of the output end  12   b.    
     It is preferable that the subsequent-stage first core portion  13  linearly extend and the width and the height thereof be uniform in the length direction (light waveguide direction). It is preferable that the width of the subsequent-stage first core portion  13  be the same as the width of the preceding-stage first core portion  11 . 
     An inner edge  13   c  (side edge on the side of the core portion  14  among both side edges of the core portion  13 ) and an outer edge  13   d  (side edge opposite to the inner edge  13   c ) of the subsequent-stage first core portion  13  are linearly formed, respectively. 
     It is preferable that the subsequent-stage second core portion  14  linearly extend and the width and the height thereof be uniform in the length direction (light waveguide direction). It is preferable that the width of the subsequent-stage second core portion  14  be the same as the width on the output end  12   b  of the preceding-stage second core portion  12 . 
     An inner edge  14   c  (side edge on the side of the core portion  13  among both side edges of the core portion  14 ) and an outer edge  14   d  (side edge opposite to the inner edge  14   c ) of the subsequent-stage second core portion  14  are linearly formed, respectively. 
     It is preferable that the subsequent-stage first core portion  13  and the subsequent-stage second core portion  14  be parallel with each other in a planar view. 
     The core portions  13  and  14  are linear waveguides that extend in the same direction as that of the preceding-stage second core portion  12 . 
     It is preferable that the core portions  13  and  14  have the same width. It is preferable that sectional shapes of the core portions  13  and  14  be rectangular. 
     The core portions  13  and  14  are separated from each other, and a gap therebetween is uniform from the input ends  13   a  and  14   a  to the output ends  13   b  and  14   b.    
     Since the widths and heights of the core portions  13  and  14  illustrated in the figures are uniform in the length direction, the sectional shapes (shapes of cross sections vertical to the light waveguide direction) are uniform over the entire length. 
     The output ends  13   b  and  14   b  of the core portions  13  and  14  are commonly connected to an input end  3   g  (connection end) of the output portion  3 . 
     It is preferable that the output portion  3  be a linear waveguide that linearly extends and the width and the height thereof be uniform in the length direction (light waveguide direction). The output portion  3  may be formed in a rectangular sectional shape. 
     It is preferable that the width of the output portion  3  on the input end  3   g  be equal to or greater than a sum (W 13 +W 14 +gap) of the widths of the core portions  13  and  14  and the gap between the core portions  13  and  14  at the output ends  13   b  and  14   b , and it is more preferable that the width be greater than the sum (W 13 +W 14 +gap). 
     In the output portion  3  in the example illustrated in the figures, one side edge  3   c  is disposed outwardly with reference to the outer edge  13   d . Thus, the one side portion of the output portion  3  is formed to protrude outward from the outer edge  13   d  of the core portion  13 . The portion that protrudes outward (to the left in  FIG. 1A ) from the outer edge  13   d  is referred to as a protrusion  3   e.    
     The other side edge  3   d  of the output portion  3  is disposed outwardly with reference to the outer edge  14   d . Thus, the other side portion of the output portion  3  is formed to protrude outward from the outer edge  14   d  of the core portion  14 . The portion that protrudes outward (to the right in  FIG. 1A ) from the outer edge  14   d  is referred to as a protrusion  3   f.    
     The output portion  3  is a linear waveguide that extends in the same direction as those of the core portions  13  and  14 . Thus, the side edges  3   c  and  3   d  of the output portion  3  are parallel with the outer edges  13   d  and  14   d.    
     It is preferable that the input end  3   g  to which the core portions  13  and  14  are connected be vertical to the extension directions of the core portions  13  and  14  in a planar view. 
     It is preferable that the height of the output portion  3  be equal to the heights of the core portions  1  and  2 . 
     Hereinafter, the subsequent-stage mode conversion section  9  will be more specifically described with reference to  FIGS. 8A to 8C . 
       FIGS. 8A to 8C  are diagrams illustrating the subsequent-stage mode conversion section  9 , in which  FIG. 8A  is a plan view thereof,  FIG. 8B  is a sectional view of core portions, and  FIG. 8C  is a sectional view of an output portion. 
     In the subsequent-stage mode conversion section  9 , “waveguide 1” corresponds to a waveguide having the core portion  13 , and “waveguide 2” corresponds to a waveguide having a core portion  14 . “Waveguide 3” corresponds to a waveguide having the output portion  3 . 
     The core portion  13  (core portion  1 ) and the core portion  14  (core portion  2 ) are formed of Si (having a refractive index of 3.48 (at a wavelength of 1580 nm)), and the upper cladding  6  and the lower cladding  7  are formed of SiO 2  (having a refractive index of 1.44 (at the wavelength of 1580 nm)). Further, the heights of the core portions  13  and  14  (core portions  1  and  2 ) are 220 nm. 
     The widths W 13  and W 14  of the core portions  13  and  14  (core portions  1  and  2 ) are set to 400 [nm], respectively. 
     The gap between the core portions  13  and  14  (core portions  1  and  2 ) is set to “gap” [nm] (“gap”=200). 
     As shown in  FIG. 8A , at the input end  3   g , a central line C 2  of the output portion  3  in the width direction and a central line C 1  between the core portions  13  and  14  in the width direction match each other. 
     The central line C 1  between the core portions  13  and  14  is a line that passes through the center of a width directional range (range from the outer edge  13   d  of the core portion  13  to the outer edge  14   d  of the core portion  14 ) including the core portions  13  and  14  and the gap therebetween, at the output ends  13   b  and  14   b  (input end  3   g ), and extends along the direction where the core portions  13  and  14  extend. Distances from the central line C 1  to the outer edges  13   d  and  14   d  are respectively (W 13 +W 14 +gap)/2. 
     The central line C 2  of the output portion  3  is a line that extends along the direction where the output portion  3  extends in a planar view. At the input end  3   g , distances from the central line C 2  to the side edges  3   c  and  3   d  are respectively W 3 /2. 
     The central lines C 1  and C 2  may be set to be parallel with each other. 
     First, a case where W which represents the width of the waveguide 3 is equal to the sum of the widths W 13  and W 14  of the waveguides 1 and 2 and the gap between the waveguides 1 and 2, that is, a case where W=W 13 +W 14 +gap (=1000 nm) is satisfied may be considered. In this case, a “protrusion” is not formed in the waveguide 3. 
       FIGS. 9A and 9B  are diagrams showing an Ex component (y=−0.00730942 μm) of TE 1  in the waveguide 3 (output portion  3 ) under the condition that W=1000.  FIG. 9A  is a diagram illustrating a simulation result showing an electric field distribution (E x  component), and  FIG. 9B  is a graph illustrating the E x  component. 
       FIGS. 10A and 10B  are diagrams showing an odd mode in the waveguides 1 and 2 (core portions  13  and  14 ) under the same condition as in  FIGS. 9A and 9B .  FIG. 10A  is a diagram illustrating a simulation result showing an electric field distribution (E x  component), and  FIG. 10B  is a graph illustrating the E x  component. Here, the electric field distributions in  FIGS. 10A and 10B  are the same as in  FIGS. 3B and 3D , respectively. 
     When comparing  FIGS. 9A and 9B  (electric field distributions of TE 1  in the output portion  3 ) with  FIGS. 10A and 10B  (electric field distributions in the odd mode in the core portions  13  and  14 ), it can be understood that they are similar to each other. 
     In a case where the waveguides 1 and 2 and the waveguide 3 are discontinuously connected to each other, a conversion efficiency based on matching coupling therebetween is expressed as the following expression (here, since the E x  component is a main component in the TE mode, contribution of other components is ignored).
 
[Expression 6]
 
 T=K∫∫E   x   odd mode of waveguides 1 and 2 ×( E   x   TE1 of waveguide 3 )* dxdy   (6)
 
     Here, signs are determined as follows. Integration is performed over an overall cross section of a connecting portion of the waveguides 1 and 2 and the waveguide 3.
 
[Expression 7]
 
 T=K∫∫E   x   odd mode of waveguides 1 and 2 ×( E   x   TE1 pf waveguide 3 )* dxdy    6
 
E x   odd mode of waveguides 1 and 2 : E x  component of electric field in odd mode of waveguides 1 and 2
 
E x   TE1 of waveguide 3 : E x  component in TE 1  of waveguide 3
 
K: other constant
 
     It can be understood from Expression (6) that as the electric field distributions in the odd mode of the waveguides 1 and 2 and TE 1  of the waveguide 3 become more similar to each other, the conversion efficiency becomes higher. In reality, in the matching coupling element shown in  FIG. 8A , a conversion efficiency at W=1000 becomes a high value (about −0.294 dB) (at a wavelength of 1580 nm). 
     Then, a case where the width W of the waveguide 3 is larger than the sum of the widths W 13  and W 14  of the waveguides 1 and 2 and the gap between the waveguides 1 and 2, that is, a case where W&gt;W 13 +W 14 +gap is satisfied may be considered. 
     The output portion  3  includes the protrusion  3   e  that protrudes toward one side (to the left in  FIG. 8A ) from the outer edge  13   d  of the core portion  13 , and the protrusion  3   f  that protrudes toward the other side (to the right in  FIG. 8A ) from the outer edge  14   d  of the core portion  14 . 
     By employing such a structure, it is possible to make the electric field distributions in the odd mode of the waveguides 1 and 2 and TE 1  of the waveguide 3 to be closer to each other, which will be described as follows. 
       FIG. 11  illustrates a relationship between the width of the waveguide 3 and the conversion efficiency between the odd mode and TE 1  (at a wavelength of 1580 nm). 
     As shown in  FIG. 11 , the conversion efficiency becomes a maximum value (−0.022 dB) in the vicinity of W=1250. 
     This is because peak positions with respect to the odd mode are aligned by increasing W (compared with a case where W is 1000) and an integrated value of Expression (6) is increased. 
       FIGS. 12A and 12B  are diagrams illustrating electric field distributions in TE 1  and an Ex component (y=0.00729512 μm) in a case where W=1250, that is, W(=W 3 )&gt;W 13 +W 14 +gap. 
     When comparing  FIG. 12B  with  FIG. 9B  (W=1000), peak positions move outward in the width direction in  FIG. 12B , compared with  FIG. 9B . That is, it can be understood that the peak positions in  FIG. 12B  approach the peak positions in the odd mode in  FIG. 10B . Thus, overlapping of the electric field distributions in the odd mode of the waveguides 1 and 2 and TE 1  of the waveguide 3 becomes large. 
     In this way, by forming the output portion of the matching coupling element to satisfy “W 3 &gt;W 13 +W 14 +gap”, it is possible to convert the odd mode into TE 1  using similarity in electric field distributions of the odd mode and TE 1 . 
     TE 0  input to the super mode-generating element is converted into the odd mode, and then, is input to the matching coupling element and is converted into TE 1 . 
     In  FIGS. 1A and 1B  and  FIGS. 8A to 8C , the matching coupling element (subsequent-stage mode conversion section  9 ) includes the output portion  3  and the core portions  13  and  14 , but the matching coupling element may be configured by only the output portion  3 . 
     Effects of the Present Embodiment 
     [First Effect 1] 
     As a first effect, by applying the present embodiment, it is possible to achieve conversion with high accuracy over a long wavelength band, and to reduce the influence of a manufacturing error. The reason is as follows. 
     In an asymmetric directional coupler in the related art, it is necessary to maintain a phase-matching condition for maintaining high conversion efficiency. 
     However, in a case where a wavelength changes or a waveguide structure is changed due to a manufacturing error, the condition is not satisfied, and the conversion efficiency decreases. 
     On the other hand, in the super mode-generating element used in the present embodiment, since the shapes and sizes of the cross sections of the core portions are equal to each other at the output ends, even when a wavelength changes, deviation does not occur in effective refractive indexes, and phase matching is not broken. 
     Accordingly, it is possible to secure high conversion efficiency at a wide wavelength band. 
     Further, since a general manufacturing error due to lithography, etching or the like occurs in two core portions by the same amount, phase matching is not broken even when the widths or heights of the core portions vary due to a manufacturing error. In addition, even in a case where there is variation in the heights of layers in a wafer (SOI substrate or the like) to be used, its influence equally affects two core portions, which does not cause an obstacle in establishment of phase matching. 
     Accordingly, it is possible to secure efficient mode conversion even in a case where a waveguide structure is changed due to a manufacturing error. 
     Further, in the super mode-generating element, since a waveguide structure is gradually changed in a light waveguide direction, it is possible to sufficiently reduce decrease energy loss due to a so-called adiabatic change. 
     Thus, by setting the length (taper length) of a portion (for example, tapered waveguide) in which a waveguide structure is changed to be sufficiently long, it is possible to perform mode conversion with low loss. 
     In the matching coupling element used in the present embodiment, it is also possible to perform high efficiency conversion over a wide wavelength band, and the influence of a manufacturing error is small. The reason is as follows. 
     In a case where a wavelength changes, an electric field distribution in a mode accordingly spreads (in a case where the wavelength increases) or shrinks (in a case where the wavelength decreases) with respect to a core. 
     Since the change is the same in an arbitrary mode, even in the case of the odd mode and TE 1 , the same change of the electric field distribution occurs according to the change of the wavelength. Thus, the conversion efficiency of the matching coupling becomes a high value. 
     In order to confirm this,  FIG. 13  illustrates a relationship between a wavelength and conversion efficiency when W is 1250. 
     It can be understood from  FIG. 13  that high conversion efficiency is maintained even though the wavelength changes. 
     Further, in a case where there is a manufacturing error due to lithography or etching or variation in the heights of layers of a wafer (SOI substrate or the like), its influence (change or the like in the width or height of a core) locally occurs in respective portions of waveguides by the same amount. Thus, the waveguides 1 and 2 and the waveguide 3 are changed in the same manner (for example, the widths increase or decrease together). Accordingly, electric field distributions in modes of the respective waveguides also spread or shrink in the same manner. 
     Thus, in the matching coupling element, reduction in conversion efficiency does not occur, and thus, it is possible to secure high conversion efficiency. 
     As a specific example,  FIG. 14  illustrates a relationship between a variation and conversion efficiency in a case where core widths (in the figure, written as waveguide widths) (for example, W 3 , W 13 , and W 14  in  FIG. 8A ) of the waveguides 1 to 3 of the matching coupling element (subsequent-stage mode conversion section) are all changed by −30 nm. In  FIG. 14 , a portion in which the conversion efficiency exceeds 100% is present, as a matter of calculation accuracy. 
     It can be understood from  FIG. 14  that even when the core widths of the waveguides 1 to 3 are changed, high conversion efficiency is maintained. 
     As described above, the super mode-generating element (preceding-stage mode conversion section) and the matching coupling element (subsequent-stage mode conversion section) are all excellent in terms of wavelength dependency and characteristics against a manufacturing error. 
     Accordingly, in the present embodiment, by using the super mode-generating element (preceding-stage mode conversion section) and the matching coupling element (subsequent-stage mode conversion section), it is possible to perform conversion with high accuracy over a wide wavelength band, and to secure efficient mode conversion even in a case where a waveguide structure is changed due to a manufacturing error. 
     Further, TE 1  is obtained by inputting the odd mode of TE 0  to the matching coupling element, but in the present embodiment, it is possible to use a new super mode-generating element using a part of the principle of the tapered directional coupler (see MICHAEL G. F. WILSON, et. al., “Tapered Optical Directional Coupler,” IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-23, NO. 1, JANUARY 1975) known in the related art in order to generate the odd mode. 
     Further, in the present embodiment, regardless of handling of conversion to a different mode, in at least an output end of the super mode-generating element and the matching coupling element, it is possible to use a symmetric waveguide structure. As described above, the symmetric waveguide structure is excellent in terms of wavelength dependency and characteristic when a manufacturing error occurs. 
     In addition, it is possible to use the adiabatic change phenomenon by using the super mode-generating element, and to use similarity of electric field distributions of the odd mode and TE 1  in the matching coupling element. 
     Accordingly, in the present embodiment, there are remarkable effects that it is possible to secure high conversion efficiency in a wide wavelength band, and it is possible to secure efficient mode conversion even in a case where a waveguide structure is changed due to a manufacturing error. 
     [Second Effect] 
     As a second effect, it is possible to perform mode multiplexing between TE 0  and TE 1 . 
     In the present embodiment, if TE 0  is input to the waveguide 2 of the super mode-generating element, TE 0  is output from the matching coupling element as it is. 
     This is because TE 0  input to the waveguide 2 becomes the even mode in the super mode-generating element in the effective refractive index curve shown in  FIG. 6 , and then, is coupled with TE 0  of the waveguide 3 in the matching coupling element. 
     Accordingly, by inputting TE 0  to the waveguides 1 and 2, respectively, it is possible to output light in which TE 0  and TE 1  are multiplexed. 
     The electric field distribution in the even mode is symmetric in the width direction, and thus is not coupled with TE 1  of the waveguide 3 which is asymmetric in the width direction. Thus, mixture with TE 1  does not occur. 
     Here, the “mode multiplexing” represents a phenomenon that light in a mode (referred to as a mode B) which is different from a mode (referred to as a mode A) generated by mode conversion from one waveguide to the other waveguide is input to the other waveguide, so that light in the mode A and light in the mode B are simultaneously output from the other waveguide. 
     [Third Effect] 
     As a third effect, it is possible to manufacture a core by one lithography/etching process, to thereby easily perform manufacturing. 
     For example, it is possible to form the Si layer of the SOT substrate as the core  5  (the core portions  1  and  2 , and the output portion  3 ) illustrated in  FIGS. 1A, 1B  or the like by one lithography/etching process. 
     Further, since there is no particular request for the height of the core or the like, and since it is sufficient if a general condition for optical waveguides is satisfied, it is possible to easily perform integration with other optical waveguide devices. 
     [Fourth Effect] 
     As a fourth effect, it is possible to easily perform manufacturing without necessity of a fine process. 
     For example, as a conversion structure TE 0  and TE 1  in the related art, there is an asymmetric Y branch, but in the Y branch, it is not easy to manufacture a base portion with high accuracy, and low formation accuracy leads to low performance. Particularly, this problem is noticeable in silicon wire waveguides in which the dimensions of respective portions are in the order of sub-micrometers. 
     On the other hand, in the present embodiment, a gap between core portions is uniform in the length direction. Since it is not necessary to gradually bring two waveguides closer to each other until they are in contact with each other, differently from the base portion of the Y portion, it is possible to easily perform manufacturing. 
     [Fifth Effect] 
     As a fifth effect, it is possible to reduce a device length. 
     As a conversion structure between the odd mode and TE 1 , a symmetric Y branch structure may be considered, but in this case, since it is necessary to gradually bring waveguides closer to each other, a device length tends to be long. 
     On the other hand, in the present embodiment, the matching coupling element is used for conversion between the odd mode and TE 1 . Since the matching coupling can be performed in a very short distance, it is possible to reduce the device length, compared with the Y branch structure. 
     In the planar optical waveguide device according to the present embodiment, in a case where input and output directions are opposite to the above-described directions, an operation opposite to the above-described operation can be performed. For example, in the planar optical waveguide device  10  illustrated in  FIGS. 1A and 1B , if TE 1  is input to the output portion  3 , TE 0  is output from the input end  11   a  of the core portion  11 . On the other hand, if TE 0  is input to the output portion  3 , TE 0  is output from the input end  12   a  of the core portion  12 . In this way, since a core portion for output is changed according to an input mode, the planar optical waveguide device  10  may be applied to a mode splitter. 
     Example 1 
     &lt;Planar Optical Waveguide Device&gt; 
       FIGS. 15A and 15B  are diagrams illustrating a planar optical waveguide device (mode conversion element)  20  according to Example 1 of the invention.  FIG. 15A  is a plan view thereof, and  FIG. 15B  is a sectional view at a sectional position (a) in  FIG. 15A . The same reference numerals are given to the same configurations as those of the above-described planar optical waveguide device, and description thereof will not be repeated. 
     The width of the core portion  11  (core portion  1 ) is 400 [nm], and the width of the core portion  12  (core portion  2 ) is 400−X [nm] (−200≦X≦0). Here, X is linearly changed from −200 to 0, from the input end  12   a  to the output end  12   b . The widths of the core portions  13  and  14  are respectively 400 [nm]. The gap between the core portions  1  and  2  is 200 [nm]. The width of the output portion  3  is 1250 [nm]. The heights of the core portions  1  and  2  and the output portion  3  are 220 nm. 
     In the planar optical waveguide device  20 , the first core portion  1  includes a linear waveguide  23 , and the second core portion  2  includes a bent waveguide  24 . The linear waveguide  23  is formed on an input side of the core portion  11 , and the bent waveguide  24  is formed on an input side of the core portion  12 . 
     The bent waveguide  24  is formed to be bent in a planar view. 
     Thus, the core portion  1  (linear waveguide  23 ) and the core portion  2  (bent waveguide  24 ) become closer to each other as a distance to the preceding-stage mode conversion section  8  becomes shorter in a length range including at least output ends  23   b  and  24   b.    
     The curved shape of the bent waveguide  24  may be an arc shape, for example. The shape of the bent waveguide  24  is not limited thereto, and may be an arbitrary shape. For example, a high-order curve shape (for example, a quadratic curve shape) such as an elliptical arc shape, a parabolic shape, or a hyperbolic shape may be employed. The bent waveguide  24  illustrated in the figure has an S shape formed by combining two approximately circular arcs. 
     In the planar optical waveguide device  20 , on an input side of the preceding-stage mode conversion section  8 , the first core portion  1  (linear waveguide  23 ) and the second core portion  2  (bent waveguide  24 ) are formed to become closer to each other as the distance to the preceding-stage mode conversion section  8  becomes shorter. Thus, it is possible to reduce unnecessary reflection of light. 
     As described above, in the preceding-stage mode conversion section  8 , mode coupling on an input side is reduced by tapering the core portion  12 , but since the linear waveguide  23  and the bent waveguide  24  become more distant from each other as the distance to the preceding-stage mode conversion section  8  becomes longer, it is possible to reliably reduce mode coupling compared with the tapered structure. Thus, it is possible to enhance mode conversion efficiency in the preceding-stage mode conversion section  8 . 
     This is because a structure in which core portions become distant from each other has a higher effect of weakening coupling compared with a tapered structure. 
     In the planar optical waveguide device  20 , the core portion  1  (core portion  11 ) includes the linear waveguide  23 , and the core portion  2  (core portion  12 ) that has a width larger than that of the core portion  1  includes the bent waveguide  24 . This is because light confinement is weak in the core portion  1  having a narrow width, and if curved, loss becomes large. 
     The planar optical waveguide device  20  has no problem of functioning as a planar optical waveguide device even when the linear waveguide  23  and the bent waveguide  24  are not provided. However, as described above, since there are advantages such as reduction of reflection and reduction of mode coupling, it is preferable that they (the linear waveguide  23  and the bent waveguide  24 ) be employed. 
     In the planar optical waveguide device  20  illustrated in  FIGS. 15A and 15B , only the core portion  2  among the core portions  1  and  2  includes the bent waveguide  24 . However, as long as the core portions  1  and  2  become closer to each other as the distance to the preceding-stage mode conversion section  8  becomes shorter in a bent waveguide, a position where the bent waveguide is formed is not limited to the example illustrated in the figures. 
     For example, a structure in which the core portion  2  includes a linear waveguide and the core portion  1  includes a bent waveguide may be employed, and a structure in which both of the core portions  1  and  2  include bent waveguides may be employed. 
     The planar optical waveguide device  20  may be manufactured by processing an SOI substrate. An intermediate SiO 2  layer (having a refractive index of 1.44) of the SOI substrate is used as a lower cladding, and an Si layer (having a refractive index of 3.47) is used as a core. After the core is formed, an SiO 2  layer is provided as an upper cladding. 
     In order to show that mode conversion is possible according to this example, a conversion efficiency (ratio of power of output TE 1  to power of input TE 0 ) in TE 1  output when TE 0  was input to the core portion  1  was calculated using a finite-difference time-domain (FDTD). 
     The length L 2  of the matching coupling element (the subsequent-stage mode conversion section  9  in  FIGS. 15A and 15B ) was set to 1 μm. A wavelength was set to 1550 nm. A calculation result is shown in  FIG. 16 . 
       FIG. 16  is a diagram illustrating a relationship between a taper length L 1  (the length of tapered waveguide (core portion  12 )) and conversion efficiency in the super mode-generating element. 
     It can be understood from  FIG. 16  that as the taper length L 1  of the super mode-generating element becomes longer, the width of a core portion in the light waveguide direction becomes smoother, so that an adiabatic change condition is more easily satisfied and the conversion efficiency becomes higher. 
       FIG. 17  is a diagram illustrating an electric field distribution when the taper length L 1  (the length of the core portion  12 ) of the super mode-generating element (preceding-stage mode conversion section  8 ) is 100 μm.  FIG. 17  is a diagram illustrating an E x  component when y is 0.1 μm in a case where TE 0  is input to the core portion  1  from the input end (lower end). The wavelength was set to 1550 nm. 
     It can be understood from  FIG. 17  that light is coupled in the super mode-generating element and TE 0  is converted into an odd mode in which TE 0  is distributed in both core portions at an output end. Further, it can be also confirmed that the odd mode is changed to TE 1  by the matching coupling element. 
       FIG. 18  is a graph illustrating a result obtained by simulating wavelength dependency (a relationship between a wavelength and conversion efficiency) in this example using FDTD. The taper length L 1  of the super mode-generating element was set to 100 μm. 
     It can be confirmed from  FIG. 18  that a high conversion efficiency which is equal to or greater than −0.56 dB from 1520 nm to 1640 nm is achieved in this example. 
     Since an electric field distribution more greatly spreads outside a core portion and coupling to a contiguous waveguide becomes stronger as a wavelength becomes longer, the conversion efficiency of the super mode-generating element is enhanced at a long wavelength, and thus, the overall conversion efficiency is enhanced. 
     Next, in order to confirm the influence of a manufacturing error in this example, a relationship between a wavelength and conversion efficiency when the widths of overall core portions (and an output portion) were changed by −30 nm was simulated using FDTD. The taper length L1 of the super mode-generating element was set to 100 μm. 
     A calculation result is shown in  FIG. 19 . 
     When comparing  FIG. 19  with  FIG. 18 , fluctuation in the conversion efficiency in a case where the widths of the core portions (and the output portion) are changed by −30 nm is within 0.24 dB at each wavelength, and high conversion efficiency is maintained. 
     It can be confirmed from  FIG. 19  that this structure is less affected by a manufacturing error. 
     In a case where a manufacturing error is present ( FIG. 19 ), the conversion efficiency is enhanced compared with a case where there is no manufacturing error ( FIG. 18 ). The reason is as follows. 
     If the width of a core portion is changed, the level of light confinement into the core is changed. In a case where the width becomes small, since the confinement becomes weak, leakage from the core increases, and thus, coupling to a contiguous waveguide in the super mode-generating element becomes strong. Thus, the conversion efficiency increases in a case where there is a manufacturing error. 
     Here, there is a case where the confinement becomes strong according to a manufacturing error, but since the width of fluctuation in conversion efficiency is approximately at the same level between waveguides, there is no change in the fact that this structure is less affected by a manufacturing error. 
     Next, in this example ( FIGS. 15A and 15B ), mode multiplexing of TE 0  of the core portion  2  and TE 1  (mode which is converted from TE 0  input to the core portion  1 ) is possible, which will be described. 
     To this end, a transmittance in TE 0 ′ output from the matching coupling element when TE 0  (written as TE 0 ′ for distinction) was inputted to the core portion  2  from the input side (ratio of power in TE 0 ′ output from the matching coupling element to power in TE 0 ′ input to the core portion  2 ) was simulated using FDTD. 
       FIG. 20  is a diagram illustrating an electric field distribution calculated using FDTD when the taper length of the super mode-generating element is 100 μm. The wavelength was set to 1550 nm.  FIG. 20  is a diagram illustrating an E x  component when y is 0.1 μm in a case where TE 0  is input to the core portion  2  from the input end (lower end). 
     Here, it can be understood that the transmittance becomes −0.86 dB and a large amount of power is transmitted. As described above, mode coupling is possible in this example. 
     &lt;Comparison with Related Art&gt; 
     This example will be compared with performance of an asymmetric directional coupler which is a technique in the related art. Specifically, Example 1 will be compared with Comparative Example 1 having a structure shown in  FIGS. 69A and 69B . First, validity of the comparison will be considered from the following viewpoints. 
     Both the super mode-generating element used in this example and the asymmetric directional coupler in the related art use the mode coupling principle. 
     In the mode coupling, as leakage of light into a contiguous waveguide becomes larger, the coupling becomes stronger, so that the efficiency becomes higher. For this purpose, it is sufficient if the width of a core portion is reduced and light confinement is weakened. 
     However, in consideration of actual manufacturing, if the width of the core portion is too narrow, there are problems such that reproducibility is lowered or a waveguide as mask design cannot be manufactured depending on the accuracy of lithography. Thus, the width of the core portion is set to have a minimum value capable of manufacturing a waveguide. 
     Accordingly, it is possible to perform the comparison between Example 1 and Comparative Example 1 by setting a minimum width of a core portion as the same condition. Since the coupling is also strengthened by decreasing a gap between core portions, the gaps between the core portions in Example 1 and Comparative Example 1 are set to be the same. 
     In Example 1, in a state where the width of an output end (a portion which was necessary for narrowing a core to the minimum) of the super mode-generating element that used the principle of mode coupling was set to 400 nm, the widths of core portions other than the output end were determined. The gap between the core portions was set to 200 nm. 
     In Comparative Example 1 (the asymmetric directional coupler shown in  FIGS. 69A and 69B ), the width of the core portion  1  (a portion which was necessary for narrowing a core to the minimum) that guided light in TE 0  which was a coupling target was set to 400 nm, and the width of the core portion  2  was determined so that phase matching was established. 
     Since the minimum width of the core portion and the gap between the core portions are under the same condition, it is possible to perform the comparison between Example 1 and Comparative Example 1. 
       FIG. 21  shows comparison results of the influences of wavelengths on conversion efficiencies in Example 1 and Comparative Example 1. The results of Example 1 and Comparative Example 1 are written as Example 1-1 and Comparative Example 1-1, respectively. These results are the same as in graphs illustrated in  FIGS. 18 and 71 . 
     Referring to  FIG. 21 , in Comparative Example 1 (Comparative Example 1-1), loss is low compared with Example 1 in the vicinity of a wavelength of 1580 nm, but as the wavelength is changed, the conversion efficiency is greatly reduced. Thus, a loss change due to the wavelength is large. 
     On the other hand, Example 1 (Example 1-1) is inferior to Comparative Example 1 (Comparative Example 1-1) in the vicinity of 1580 nm, but a loss change depending on a wavelength is small at 1520 nm to 1640 nm (a wavelength range which covers a C+L band in optical communication). 
     Further, when comparing the minimum conversion efficiency in this wavelength range, it can be understood that Example 1 (Example 1-1) has a higher minimum conversion efficiency. 
     As described above, in Example 1 (Example 1-1), the conversion can be performed with high efficiency over a wide wavelength range compared with Comparative Example 1 (Comparative Example 1-1). 
     In Example 1 (Example 1-1), since the super mode-generating element uses adiabatic change, by increasing a taper length, it is possible to further lower loss. 
     On the other hand, in the asymmetric directional coupler of Comparative Example 1 (Comparative Example 1-1), since it is difficult to remarkably change the length, no further improvement in conversion efficiency can be expected. 
     Subsequently,  FIG. 22  shows comparison results of the influences of manufacturing errors on conversion efficiencies in Example 1 and Comparative Example 1.  FIG. 22  shows a conversion efficiency when the width of a core portion (and an output portion) is changed by −30 nm. The results in Example 1 and Comparative Example 1 are written as Example 1-2 and Comparative Example 1-2. These results are the same as in graphs illustrated in  FIG. 19  and  FIG. 74 . 
     Referring to  FIG. 22 , in Comparative Example 1 (Comparative Example 1-2), phase matching is not established and the conversion efficiency is reduced, but in Example 1 (Example 1-2), high conversion efficiency is maintained. Accordingly, Example 1 (Example 1-2) has a small influence due to a manufacturing error compared with Comparative Example 1 (Comparative Example 1-2). 
     It can be understood from these results that Example 1 has higher conversion efficiency in a wide wavelength band and is less influenced by a manufacturing error compared with the related art technique. 
     Example 2 
     &lt;Planar Optical Waveguide Device&gt; 
       FIG. 23  is a plan view illustrating a planar optical waveguide device  30  (mode conversion element) according to Example 2. Example 2 corresponds to a first example of a structure in which an intermediate core portion is provided between a preceding-stage mode conversion section and a subsequent-stage mode conversion section, in which the intermediate core portion connects the preceding-stage mode conversion section and the subsequent-stage mode conversion section. 
     The planar optical waveguide device  30  includes a pair of core portions  31  and  32  which are disposed in parallel with each other, and an output portion  33  (subsequent-stage mode conversion section  39 ) provided on a subsequent-stage side (output-side) of the core portions  31  and  32 . It is preferable that the heights of the core portions  31  and  32  and the output portion  33  be equal to each other. 
     The planar optical waveguide device  30  is different from the planar optical waveguide device  10  illustrated in  FIGS. 1A and 1B  in that an intermediate core portion  34  including tapered core portions  31 B and  32 B is interposed between a preceding-stage mode conversion section  38  and a subsequent-stage mode conversion section  39 . 
     Among core portions  31 A and  32 A that form the preceding-stage mode conversion section  38 , the core portion  31 A linearly extends, and the width thereof is uniform in a length direction (light waveguide direction). 
     The core portion  32 A is formed in a tapered shape in which the width thereof continuously decreases from an input end  32 Aa to an output end  32 Ab. A gap between the core portion  31 A and  32 A is uniform along the length direction. 
     In the preceding-stage mode conversion section  38 , since the width of the core portion  32 A is larger than the width of the core portion  31 A at the input ends  31 Aa and  32 Aa, a cross section of the core portion  32 A is larger than a cross section of the core portion  31 A. 
     At the output ends  31 Ab and  32 Ab, since the widths of the core portions  31 A and  32 A are equal to each other, the shapes and sizes of the cross sections of the core portions  31 A and  32 A are equal to each other. 
     In the preceding-stage mode conversion section  38 , phase matching is not established at the input ends  31 Aa and  32 Aa, and phase matching is established at the output ends  31 Ab and  32 Ab. 
     The core portion  31 B among the core portions  31 B and  32 B that form the intermediate core portion  34  is formed to be connected to the core portion  31 A, and is formed in a tapered shape in which the width continuously decreases from an input end  31 Ba to an output end  31 Bb. 
     The width of the input end  31 Ba of the core portion  31 B is the same as the width of the output end  31 Ab of the core portion  31 A. 
     The core portion  32 B is formed to be connected to the core portion  32 A, and is formed in a tapered shape in which the width continuously decreases from an input end  32 Ba to an output end  32 Bb. 
     The width of the input end  32 Ba of the core portion  32 B is the same as the width of the output end  32 Ab of the core portion  32 A. Thus, the shapes and sizes of the cross sections of the core portions  31 B and  32 B at the input ends  31 Ba and  32 Ba are equal to each other. 
     The widths of the core portions  31 B and  32 B at the output ends  31 Bb and  32 Bb are equal to each other. Thus, the shapes and sizes of the cross sections of the core portions  31 B and  32 B at the output ends  31 Bb and  32 Bb are equal to each other. 
     A gap between the core portions  31 B and  32 B are uniform in the length direction. 
     The output portion  33  may have a rectangular cross section. 
     The output portion  33  is a linear waveguide that linearly extends, and it is preferable that the width and the height be uniform in the length direction (light waveguide direction). 
     The width of the output portion  33  at an input end  33   g  is larger than the sum of the widths of the core portions  31 B and  32 B and the gap between the core portions  31 B and  32 B at the output ends  31 Bb and  32 Bb. Thus, the output portion  33  includes a protrusion  33   e  that protrudes outward from an outer edge of the core  31 B, and a protrusion  33   f  that protrudes outward from an outer edge of the core  32 B. 
     At a connection end (input end of the output portion  33 ), a central line of the output portion  33  in the width direction and a central line between the core portions  31 B and  32 B in the width direction match each other. 
     The central line between the core portions  31 B and  32 B is a line that passes through the center of a width directional range including the core portions  31 B and  32 B and the gap therebetween. 
     The central line of the output portion  33  is a line that extends along the direction where the output portion  33  extends, and passes through the center of the output portion  33  in the width direction. 
     In the planar optical waveguide device  30 , since the intermediate core portion  34  having a tapered structure is provided, it is possible to smoothly change the width of the core portion  31  from the preceding-stage mode conversion section  38  to the subsequent-stage mode conversion section  39 . 
     Thus, it is possible to connect the preceding-stage mode conversion section  38  and the subsequent-stage mode conversion section  39  with low loss, although the widths of the core portion  31 A and the core portion  31 C are different from each other. 
     Example 3 
     &lt;Planar Optical Waveguide Device&gt; 
       FIG. 24  is a plan view illustrating a planar optical waveguide device  40  (mode conversion element) according to Example 3. Example 3 corresponds to a second example of a structure in which an intermediate core portion is provided between a preceding-stage mode conversion section and a subsequent-stage mode conversion section. 
     The planar optical waveguide device  40  includes a pair of core portions  41  and  42  which are disposed in parallel with each other, and an output portion  43  provided on a subsequent-stage side (output-side) of the core portions  41  and  42 . It is preferable that the heights of the core portions  41  and  42  and the output portion  43  be equal to each other. 
     The planar optical waveguide device  40  is different from the planar optical waveguide device  10  illustrated in  FIGS. 1A and 1B  in that an intermediate core portion  44  including tapered core portions  41 B and  42 B is interposed between a preceding-stage mode conversion section  48  and a subsequent-stage mode conversion section  49 . 
     Among core portions  41 A and  42 A that form the preceding-stage mode conversion section  48 , the core portion  41 A linearly extends, and the width thereof is uniform in a length direction (light waveguide direction). The core portion  42 A is formed in a tapered shape in which the width continuously decreases from an input end to an output end. 
     In the preceding-stage mode conversion section  48 , since the width of the core portion  42 A is larger than the width of the core portion  41 A at input ends, a cross section of the core portion  42 A is larger than a cross section of the core portion  41 A. 
     Since the widths of the core portions  41 A and  42 A are equal to each other at output ends, the shapes and sizes of the cross sections of the core portions  41 A and  42 A are equal to each other. 
     In the preceding-stage mode conversion section  48 , phase matching is not established at the input ends, and phase matching is established at the output ends. 
     The core portion  41 B among the core portions  41 B and  42 B that form the intermediate core portion  44  is formed to be connected to the core portion  41 A, and the width is uniform in the length direction (light waveguide direction). The core portion  41 B linearly extends, and the width at the input end is the same as the width at the output end of the core portion  41 A. 
     The core portion  42 B is formed to be connected to the core portion  42 A, and the width is uniform in the length direction (light waveguide direction). The core portion  42 B linearly extends, and the width at the input end is the same as the width at the output end of the core portion  42 A. 
     The widths of the core portions  41 B and  42 B are equal to each other. Thus, the shapes of cross sections of the core portions  41 B and  42 B are equal to each other over the entire length. 
     A gap between the core portions  41 B and  42 B is uniform. 
     The output ends of the core portions  41 B and  42 B are connected to the output portion  43 . 
     The output portion  43  that forms the subsequent-stage mode conversion section  49  has the same structure as in the output portion  33  in the previous figure. 
     That is, the width of the output portion  43  at the input end is larger than the sum of the widths of the core portions  41 B and  42 B at the output ends and the gap between the core portions  41 B and  42 B. Thus, the output portion  43  is formed to protrude outward from outer edges of the cores  41 B and  42 B, respectively. 
     A central line of the output portion  43  in the width direction and a central line between the cores  41 B and  42 B in the width direction match each other, at a connection end (an input end of the output portion  43 ). 
     The central line between the core portions  41 B and  42 B is a line that passes through the center of a width directional range including the core portions  41 B and  42 B and the gap therebetween. 
     The central line of the output portion  43  is a line that extends in the direction where the output portion  43  extends, and passes through the center of the output portion  43  in the width direction. 
     In the planar optical waveguide device  40 , since the linear intermediate core portion  44  is provided between the preceding-stage mode conversion section  48  and the subsequent-stage mode conversion section  49 , it is possible to increase the degree of freedom in disposition of the subsequent-stage mode conversion section  49 . 
     Example 4 
     &lt;Planar Optical Waveguide Device&gt; 
       FIG. 25  is a plan view illustrating a planar optical waveguide device  50  (mode conversion element) according to Example 4. Example 4 corresponds to a third example of a structure in which an intermediate core portion is provided between a preceding-stage mode conversion section and a subsequent-stage mode conversion section. 
     The planar optical waveguide device  50  includes a pair of core portions  51  and  52  which are disposed in parallel with each other, and an output portion  53  provided on a subsequent-stage side (output-side) of the core portions  51  and  52 . It is preferable that the heights of the core portions  51  and  52  and the output portion  53  be equal to each other. 
     The planar optical waveguide device  50  is different from the planar optical waveguide device  10  illustrated in  FIGS. 1A and 1B  in that an intermediate core portion  54  including curved core portions  51 B and  52 B is interposed between a preceding-stage mode conversion section  58  and a subsequent-stage mode conversion section  59 . 
     Among core portions  51 A and  52 A that form the preceding-stage mode conversion section  58 , the core portion  51 A linearly extends, and the width thereof is uniform in a length direction (light waveguide direction). The core portion  52 A is formed in a tapered shape in which the width continuously decreases from an input end to an output end. 
     In the preceding-stage mode conversion section  58 , since the width of the core portion  52 A is larger than the width of the core portion  51 A at input ends, a cross section of the core portion  52 A is larger than a cross section of the core portion  51 A. 
     Since the widths of the core portions  51 A and  52 A are equal to each other at output ends, the shapes and sizes of the cross sections of the core portions  51 A and  52 A are equal to each other. 
     In the preceding-stage mode conversion section  58 , phase matching is not established at the input ends, and phase matching is established at the output ends. 
     The core portion  51 B among the core portions  51 B and  52 B that form the intermediate core portion  54  is formed to be connected to the core portion  51 A, and the width is uniform in the length direction (light waveguide direction). The width of the input end of the core portion  51 B is the same as the width at the output end of the core portion  51 A. 
     The core portion  52 B is formed to be connected to the core portion  52 A, and the width is uniform in the length direction (light waveguide direction). The width of the input end of the core portion  52 B is the same as the width at the output end of the core portion  52 A. 
     It is preferable that planar shapes of the core portions  51 B and  52 B be arc shapes, but the planar shapes are not limited thereto, and may be arbitrary curve shapes. For example, a high-order curve shape (for example, a quadratic curve shape) such as an elliptical arc shape, a parabolic shape, or a hyperbolic shape may be employed. 
     It is preferable that the widths of the core portions  51 B and  52 B be equal to each other. 
     A gap between the core portions  51 B and  52 B is uniform in the length direction. 
     The output ends of the core portions  51 B and  52 B are connected to the output portion  53 . 
     The output portion  53  that forms the subsequent-stage mode conversion section  59  has the same structure as in the output portion  43  in the previous figure. 
     That is, the width of the output portion  53  at the input end is larger than the sum of the widths of the core portions  51 B and  52 B at the output ends and the gap between the core portions  51 B and  52 B. Thus, the output portion  53  is formed to protrude outward from outer edges of the cores  51 B and  52 B, respectively. 
     A central line of the output portion  53  in the width direction and a central line between the cores  51 B and  52 B in the width direction match each other, at a connection end (an input end of the output portion  53 ). 
     The central line between the core portions  51 B and  52 B is a line that passes through the center of a width directional range including the core portions  51 B and  52 B and the gap therebetween. 
     The central line of the output portion  53  is a line that extends in the direction where the output portion  53  extends, and passes through the center of the output portion  53  in the width direction. 
     In the planar optical waveguide devices  30  to  50 , the intermediate core portion may be included in the preceding-stage mode conversion section. 
     Example 5 
     &lt;Planar Optical Waveguide Device&gt; 
       FIG. 26  is a plan view illustrating a planar optical waveguide device  80  (mode conversion element) according to Example 5. 
     The planar optical waveguide device  80  has the same configuration as that of the planar optical waveguide device  20  illustrated in  FIGS. 15A and 15B  except that a core  85  instead of the core  5  is provided. 
     The core  85  includes a preceding-stage mode conversion section  8  and a subsequent-stage mode conversion section  89 . 
     The subsequent-stage mode conversion section  89  includes core portions  13  and  14 , an output portion  3  provided on an output-side of the core portions  13  and  14 , and a single output-side core portion  86  that extends from an output-side of the output portion  3 . 
     The width of the output-side core portion  86  is smaller than the width of the output portion  3 . It is preferable that the output-side core portion  86  have a rectangular cross section. 
     The subsequent-stage mode conversion section  89  is configured so that two core portions  13  and  14  are connected to the input side of the output portion  3  and one output-side core portion  86  is connected to the output-side thereof. Thus, the subsequent-stage mode conversion section  89  may be used as a 1×2 MMI (multi-mode interferometer). 
     Example 6 
     &lt;Planar Optical Waveguide Device&gt; 
       FIG. 27  is a plan view illustrating a planar optical waveguide device  60  (polarization conversion element) according to Example 6. 
     The planar optical waveguide device  60  has a configuration in which a high-order polarization conversion element  101  (high-order polarization-converting section) is provided on the output-side of the planar optical waveguide device  10  (mode conversion element) illustrated in  FIGS. 1A and 1B . The high-order polarization conversion refers to conversion between TE 1  and TM 0 . 
       FIGS. 28A and 28B  are diagrams illustrating examples of a structure of the high-order polarization conversion element  101  (see Non-Patent Document 1), in which  FIG. 28A  is a plan view thereof, and  FIG. 28B  is a sectional view thereof. The high-order polarization conversion element  101  includes an optical waveguide that includes a core  102 , a lower cladding  103  having a refractive index lower than that of the core, and an upper cladding  104  having a refractive index lower than that of the core  102 . 
     The core  102  is formed of Si, for example. The lower cladding  103  is formed of SiO 2 , for example. The upper cladding  104  is formed of air, for example. 
     In order to perform the high-order polarization conversion, it is necessary that the upper cladding  104  and the lower cladding  103  have different refractive index. 
     The planar optical waveguide device  60  illustrated in  FIG. 27  is capable of converting TE 0  into TE 1  by the planar optical waveguide device  10 , and is capable of converting TE 1  into TM 0  by the high-order polarization conversion element  101 . 
     In the high-order polarization conversion element  101 , since TE 0  is not converted into a different mode, TE 0  (written as TE 0 ′ for distinction) which is input to the core portion  2  and is output from the output portion  3  is not converted. 
     Thus, an output obtained by combining TM 0  and TE 0 ′ is output from an output-side of the high-order polarization conversion element  101 . Accordingly, the planar optical waveguide device  60  can be used as an element for performing polarization multiplexing. 
     Example 7 
     &lt;Planar Optical Waveguide Device (Polarization Conversion Element)&gt; 
     In the planar optical waveguide device  60  in  FIG. 27 , a high-order polarization conversion element  61  illustrated in  FIGS. 29A to 29D , instead of the high-order polarization conversion element  101 , may be used. 
       FIG. 29A  is a plan view of a core  62 , and  FIGS. 29B to 29D  are sectional views of an ending portion, an intermediate portion, and a starting portion of the core  62 , respectively. A cladding (not shown) is provided in the vicinity of the core  62 . In  FIG. 29A , shading is given to a lower core  64 . 
     In the high-order polarization conversion element  61 , the core  62  includes the lower core  64  which has a rectangular cross section, and an upper core  63  which is formed on the lower core  64  and has a rectangular cross section. In a starting portion  68  and an ending portion  69 , since both side edges of the upper core  63  are disposed at positions that overlap both side edges of the lower core  64 , respectively, the core  62  is formed to have a rectangular cross section. 
     The width W 68  of the starting portion  68  is larger than the width W69 of the ending portion  69 . Both the heights of the starting portion  68  and the ending portion  69  are H 62 , and the height H 64  of the lower core  64  is lower than the core height H 62 . 
     In a section L 61  from the starting portion  68  to the intermediate portion  70 , the width of the lower core  64  is uniform, and the width of the upper core  13  gradually decreases from the starting portion  68  to the intermediate portion  70 . 
     In a section L 62  from the intermediate portion  70  to the ending portion  69 , the width of the lower core  64  gradually decreases from the intermediate portion  70  to the ending portion  69 , and the width of the upper core  63  is uniform. 
     In the high-order polarization conversion element  61 , the core  62  has an asymmetric structure in a height direction, and the width of a part of each of the upper core  63  and the lower core  64  is smoothly changed. Thus, it is possible to convert TE 1  into TM 0 . 
     &lt;Planar Optical Waveguide Device&gt; 
     A structure of the planar optical waveguide device  110  according to a second embodiment of the invention will be described with reference to  FIGS. 30A and 30B .  FIG. 30A  is a plan view illustrating the planar optical waveguide device  110 , and  FIG. 30B  is a sectional view at a sectional position (a) in  FIG. 30A . The same reference numerals are given to the same configurations as those of the above-described planar optical waveguide device  10  shown in  FIGS. 1A and 1B , and description thereof will not be repeated. 
     As shown in  FIGS. 30A and 30B , the planar optical waveguide device  110  (mode conversion element) has the same configuration as that of the planar optical waveguide device  10  illustrated in  FIGS. 1A and 1B  except that a core  105  is provided, instead of the core  5 . In  FIG. 30A , shading is given to a slab portion  16 . 
     The core  105  includes a pair of core portions  1  and  2  (rib portions) which are disposed in parallel with each other, a slab portion  16  which is formed at least between the core portions  1  and  2 , and an output portion  3  provided on a subsequent-stage side (output-side) thereof. 
     The slab portion  16  is formed to have a height lower than those of the core portions  1  and  2 . That is, as shown in  FIG. 30B , the height H 16  of the slab portion  16  is lower than the heights H 1  and H 2  of the core portions  1  and  2 . 
     As shown in  FIG. 30A , the slab portion  16  is formed to connect the core portions  1  and  2  in at least a part of the core portions  1  and  2  in a length direction. 
     The slab portion  16  is formed between opposite inner surfaces of the core portions  1  and  2  (that is, between side surfaces of inner edges  11   c  and  13   c  of the core portion  1  and side surfaces of inner edges  12   c  and  14   c  of the core portion  2 ). 
     In the example shown in the figures, the slab portion  16  is formed over the entire length of the core portions  1  and  2 . Here, the slab portion  16  may be formed only over a partial length of the core portions  1  and  2 . 
     The slab portion  16  is formed of the same material (preferably, Si) as those of the core portions  1  and  2 , and is integrally formed with the core portions  1  and  2 . 
     As shown in  FIG. 30B , the slab portion  16  is formed to extend from lower parts of the inner surfaces of the core portions  1  and  2 , and a lower surface of the slab portion  16  is continuous with lower surfaces of the core portions  1  and  2 . The core portions  1  and  2  protrude upward from an upper surface of the slab portion  16 . 
     The core  105  forms a so-called half-rib waveguide since the slab portion  16  is provided on only one side (inner side) of each of the core portions  1  and  2  in the width direction. 
     As shown in  FIG. 30A , the core  105  includes a preceding-stage mode conversion section  108  (super mode-generating element) that converts a mode of light that propagates through the core portions  1  and  2 , and a subsequent-stage mode conversion section  109  (matching coupling element) that converts a mode of light passed through the preceding-stage mode conversion section  108 . 
     The preceding-stage mode conversion section  108  includes core portions  11  and  12  (rib portions), and the slab portion  16  provided therebetween. 
     The subsequent-stage mode conversion section  109  includes core portions  13  and  14  (rib portions), and the slab portion  16  provided therebetween. The subsequent-stage mode conversion section  109  is formed to be connected to a rear end (output-side) of the preceding-stage mode conversion section  108 . 
     The planar optical waveguide device  110  may be manufactured by processing an SOI substrate. For example, an SiO 2  layer of the SOI substrate may be formed as a lower cladding, and an Si layer thereof may be formed as a core through a lithography/etching process. 
     The core may be formed by performing a lithography/etching process two times. That is, first, a core having a predetermined thickness is manufactured by a lithography/etching process. Then, a part thereof is thinned by a lithography/etching process to be formed as a slab portion, to thereby form a core having core portions and a slab portion. 
     &lt;Principle of Super Mode-Generating Element&gt; 
       FIGS. 31A and 31B  illustrate an optical waveguide device that includes core portions  21  and  22  of which the widths are equal to each other and a slab portion  16  formed between the core portions  21  and  22 . 
     As shown in  FIGS. 31 and 31B , modes when TE 0  and TE 0  are mode-coupled between contiguous waveguides are divided into an even mode as shown in  FIGS. 32A and 32C , and an odd mode as shown in  FIGS. 32B and 32D . These modes are collectively referred to as a super mode of TE 0  (or simply referred to as a super mode). 
     As described above, when phase matching is established, the lengths of waveguides necessary for movement of light leaked out from one waveguide to the other waveguide to form a super mode depend on a coupling coefficient χ which represents the strength of mode coupling. 
     &lt;Specific Example of Super Mode-Generating Element&gt; 
     The preceding-stage mode conversion section  108  which is a specific example of a super mode-generating element will be described with reference to  FIGS. 30A and 301B , and  FIGS. 33A to 33D . 
       FIG. 33A  is a plan view thereof,  FIG. 33B  is a sectional view at a sectional position (c) in  FIG. 33A ,  FIG. 33C  is a sectional view at a sectional position (b), and  FIG. 33D  is a sectional view at a sectional position (a). 
     “Waveguide 1” corresponds to a waveguide having the core portion  11 , and “waveguide 2” corresponds to the waveguide having the core portion  12 . 
     The core portion  11  (core portion  1 ) and the core portion  12  (core portion  2 ) are formed of Si (having a refractive index of 3.48 (at a wavelength of 1580 nm)), and the upper cladding  6  and the lower cladding  7  are formed of SiO 2  (having a refractive index of 1.44 (at the wavelength of 1580 nm)). Further, the heights of the core portions  11  and  12  (core portions  1  and  2 ) are 220 nm. The height of the slab portion  16  is 95 nm. A gap between the core portions  11  and  12  (core portions  1  and  2 ) is 200 nm. 
     As shown in  FIG. 33D , the width of the core portion  11  (core portion  1 ) is set to 400 [nm], and the width of the core portion  12  (core portion  2 ) is set to 400−X [nm](−200≦X≦0). Here, X is linearly changed from −200 to 0, from an input end  12   a  to an output end  12   b . Thus, the core portion  12  (core portion  2 ) is formed in a tapered shape so that the width gradually decreases from the input end  12   a  (X=−200) to the output end  12   b  (X=0). 
       FIG. 33C  shows a cross section at an intermediate position (X=−20) between the input ends  11   a  and  12   a  and the output ends  11   b  and  12   b.    
     In the example illustrated in  FIGS. 33A to 33D , at the input ends  11   a  and  12   a , the width (width W 12a  in  FIG. 33D ) of the core portion  12  is larger than the width (width W 11a  in  FIG. 33D ) of the core portion  11 . Thus, a cross section of the core portion  12  is formed to be larger than a cross section of the core portion  11 . Thus, phase matching is not established, and mode coupling is not nearly performed. 
     On the other hand, at the output ends  11   b  and  12   b , the widths (widths W 11b  and W 12b  in  FIG. 33B ) of the core portions  11  and  12  are equal to each other, and thus, the shapes and sizes of cross sections of the core portions  11  and  12  are equal to each other. Thus, phase matching is established. 
     Since the core portion  12  is formed in a tapered shape, phase matching is gradually performed along the light waveguide direction from the input end to the output end, and as a result, mode coupling is progressed. Thus, by sufficiently increasing the length (taper length) of the tapered waveguide (core portion  12 ), it is possible to convert TE 0  input to the waveguide 1 into the odd mode which is the super mode of TE 0  with almost no loss. 
     As described above, the lengths of waveguides necessary for movement of light leaked out from one waveguide to the other waveguide to form a super mode depend on the coupling coefficient χ. Thus, as the coupling coefficient χ becomes larger, mode conversion may be performed with higher accuracy using shorter waveguides (shorter device length). 
     This principle will be described with reference to the above-described specific example. 
     In order to confirm that a phase-matching condition is broken at an input end by changing the width of a core portion in the light waveguide direction (that is, by tapering a waveguide), an effective refractive index in a mode in a case where the waveguides 1 and 2 are independently present are illustrated in  FIG. 34A . Here, a wavelength was set to be 1580 nm (this is similarly applied hereinafter). 
       FIG. 34B  is a sectional view of the waveguide 1 in a case where the waveguide 1 having the core portion  1  is independently present. A core of the independent waveguide 1 includes a core portion  11  and a slab portion  16 A (having a height of 95 nm) that extends from the core portion  11  in a width direction, and the entire width is the same as the width (400+200+(400−X) [nm]) of the core  105 . 
       FIG. 34C  is a sectional view illustrating the waveguide 2 in a case where the waveguide 2 having the core portion  12  is independently present. A core of the independent waveguide 2 includes a core portion  12  and a slab portion  16 B (height of 95 nm) that extends from the core portion  12  in the width direction, and the entire width is the same as the width (400+200+(400−X) [nm]) of the core  105 . 
     It can be understood from  FIGS. 34A to 34C  that effective refractive indexes in TE 0  of the waveguide 1 and TE 0  of the waveguide 2 are the same when X=0 and phase matching is established. 
     As X is separated from 0, deviation occurs in the effective refractive indexes in TE 0  and TE 0  of the waveguides 1 and 2, the phase-matching condition is broken. 
       FIG. 35  is a diagram illustrating effective refractive indexes in a case where waveguides 1 and 2 are contiguous to each other. 
     Compared with  FIG. 34A  illustrating the effective refractive index in a case where the waveguides are independently present, in  FIG. 35 , #0 and #1 do not match and are separated from each other when X=0. 
     This is because two modes mutually act due to mode coupling to form a mixed mode (super mode) since the phase-matching condition is satisfied between TE 0  of the waveguide 1 and TE 0  of the waveguide 2. 
     If X is separated from 0, since the phase-matching condition is not satisfied, such a mutual action does not occur, and the same mode distribution as in a case where the waveguides are independently present is obtained. As a result, the effective refractive indexes do not greatly change compared with a case where the waveguides are independently present. 
     In a structure in which a structure of a waveguide is gradually changed in a light waveguide direction, such as a tapered waveguide, it is known that mode conversion is performed so as to change on a curve of one effective refractive index (referred to as an adiabatic change). 
     Thus, in  FIG. 35 , by inputting TE 0  to the waveguide 1 when X=−200 (input end) and gradually changing X from −200 to 0 in the length direction of the waveguide, it is possible to convert TE 0  into the odd mode which is the super mode of TE 0  when X=0. 
     In order to confirm this mode conversion,  FIGS. 36A to 36F  illustrate electric field distributions in modes #0 and #1 at sectional positions (a) to (c) (see  FIG. 33A ). 
       FIGS. 36A and 36B  are diagrams illustrating simulation results (mode #0 in  FIG. 36A  and mode #1 in  FIG. 36B ) showing electric field distributions (E x  components) at the sectional position (a).  FIGS. 36C and 36D  are diagrams illustrating simulation results (mode #0 in  FIG. 36C  and mode #1 in  FIG. 36D ) showing electric field distributions (E x  components) at the sectional position (b).  FIGS. 36E and 36F  are diagrams illustrating simulation results (mode #0 in  FIG. 36E  and mode #1 in  FIG. 36F ) showing electric field distributions (E x  components) at the sectional position (c). 
     Here, x and y represent a width direction and a height direction, respectively. The electric field distributions in  FIGS. 36E and 36F  are the same as in  FIGS. 32A and 32B , respectively. 
     Referring to the mode #1, at the sectional position (a) (X=−200) illustrated in FIG.  36 B, TE 0  is present in the waveguide 1. 
     At the sectional position (b) (X=−20) illustrated in  FIG. 36D , it can be understood that mode coupling to TE 0  of the waveguide 2 starts. 
     At the sectional position (c) (X=0) illustrated in  FIG. 36F , since the phase-matching condition is satisfied, an odd mode which is a super mode in which TE 0  of the waveguide 1 and TE 0  of the waveguide 2 are mixed can be viewed. 
     In this way, by gradually changing a waveguide structure in a light waveguide direction, it is possible to change TE 0  input to the waveguide 1 to an odd mode which is a super mode of TE 0 . 
     &lt;Specific Example of Matching Coupling Element&gt; 
     The subsequent-stage mode conversion section  109  which is a specific example of a matching coupling element will be described with reference to  FIGS. 37A to 37C . 
       FIGS. 37A to 37C  illustrate the subsequent-stage mode conversion section  109 , in which  FIG. 37A  is a plan view thereof,  FIG. 37B  is a sectional view of core portions, and  FIG. 37C  is a sectional view of an output portion. 
     In the subsequent-stage mode conversion section  109 , “waveguide 1” corresponds to a waveguide having the core portion  13 , and “waveguide 2” corresponds to a waveguide having a core portion  14 . “Waveguide 3” corresponds to a waveguide having the output portion  3 . 
     The core portion  13  (core portion  1 ) and the core portion  14  (core portion  2 ) are formed of Si (having a refractive index of 3.48 (at a wavelength of 1580 nm)), and the upper cladding  6  and the lower cladding  7  are formed of SiO 2  (having a refractive index of 1.44 (at the wavelength of 1580 nm)). Further, the heights of the core portions  13  and  14  (core portions  1  and  2 ) are 220 nm. The height of the slab portion  16  is 95 nm. 
     The widths W 13  and W 14  of the core portions  13  and  14  (core portions  1  and  2 ) are set to 400 [nm], respectively. 
     The gap between the core portions  13  and  14  (core portions  1  and  2 ) is set to “gap” [nm] (“gap”=200). 
     As shown in  FIG. 37A , at the input end  3   g , a central line C 2  of the output portion  3  in the width direction and a central line C 1  between the core portions  13  and  14  in the width direction match each other. 
     The central line C 1  between the core portions  13  and  14  is a line that passes through the center of a width directional range (range from the outer edge  13   d  of the core portion  13  to the outer edge  14   d  of the core portion  14 ) including the core portions  13  and  14  and the gap therebetween, at the output ends  13   b  and  14   b  (input end  3   g ), and extends along the direction where the core portions  13  and  14  extend. 
     The central line C 2  of the output portion  3  is a line that extends along the direction where the output portion  3  extends in a planar view. 
     First, a case where W which represents the width of the waveguide 3 is equal to the sum of the widths W 13  and W 14  of the waveguides 1 and 2 and the gap between the waveguides 1 and 2, that is, a case where W=W 13 +W 14 +gap (=1000 nm) is satisfied may be considered. 
       FIGS. 38A and 38B  are diagrams showing an Ex component (y=0.00730942 μm) in TE 1  in the waveguide 3 (output portion  3 ) under the condition that W=1000.  FIG. 38A  is a diagram illustrating a simulation result showing an electric field distribution (E x  component), and  FIG. 38A  is a graph illustrating the E x  component. 
       FIGS. 39A and 39B  are diagrams showing an odd mode in the waveguides 1 and 2 (core portions  13  and  14 ) under the same condition as in  FIGS. 38A and 38B .  FIG. 39A  is a diagram illustrating a simulation result showing an electric field distribution (E x  component), and  FIG. 39B  is a graph illustrating the E x  component. Here, the electric field distributions in  FIGS. 39A and 39B  are the same as in  FIGS. 32B and 32D , respectively. 
     When comparing  FIGS. 38A and 38B  (electric field distributions in TE 1  in the output portion  3 ) with  FIGS. 39A and 39B  (electric field distributions in the odd mode in the core portions  13  and  14 ), it can be understood that they are similar to each other. 
     In a case where the waveguides 1 and 2 and the waveguide 3 are discontinuously connected to each other, a conversion efficiency based on matching coupling therebetween is expressed as the above-described Expression (6) (here, since the E x  component is a main component in the TE mode, contribution of other components is ignored). 
     It can be understood from Expression (6) that as the electric field distributions in the odd mode of the waveguides 1 and 2 and TE 1  of the waveguide 3 become more similar to each other, the conversion efficiency becomes higher. In reality, in the matching coupling element shown in  FIG. 37A , a conversion efficiency at W=1000 becomes a high value (about −0.134 dB) (at a wavelength of 1550 nm). 
     Next, a case where the width W of the waveguide 3 is larger than the sum of the widths W 13  and W 14  of the waveguides 1 and 2 and the gap between the waveguides 1 and 2, that is, a case where W&gt;W 13 +W 14 +gap is satisfied may be considered. 
     By employing such a structure, it is possible to make the electric field distributions in the odd mode of the waveguides 1 and 2 and TE 1  of the waveguide 3 to be closer to each other, which will be described as follows. 
       FIG. 40  illustrates a relationship between the width of the waveguide 3 and a conversion efficiency between the odd mode and TE 1  (at a wavelength of 1550 nm). 
     As shown in  FIG. 40 , the conversion efficiency becomes a maximum value (−0.087 dB) in the vicinity of W=1200. 
     This is because peak positions with respect to the odd mode are aligned by increasing W and an integrated value of Expression (6) is increased. 
       FIGS. 41A and 41B  are diagrams illustrating electric field distributions in TE 1  and an E x  component in a case where W=1200, that is, W(=W 3 )&gt;W 13 +W 14 +gap. 
     When comparing  FIG. 41B  with  FIG. 38B  (W=1000), peak positions move outward in the width direction in  FIG. 41B , compared with  FIG. 38B . That is, it can be understood that the peak positions in  FIG. 41B  approach the peak positions in the odd mode in  FIG. 39B . Thus, overlapping of the electric field distributions in the odd mode of the waveguides 1 and 2 and TE 1  of the waveguide 3 becomes large. 
     In this way, by forming the output portion of the matching coupling element to satisfy “W 3 &gt;W 13 +W 14 +gap”, it is possible to convert the odd mode into TE 1  using similarity in electric field distributions of the odd mode and TE 1 . 
     According to the present embodiment, the super mode-generating element (preceding-stage mode conversion section) converts TE 0  into an odd mode which is a super mode of TE 0  at output ends of first and second cores. 
     The matching coupling element (subsequent-stage mode conversion section) converts the odd mode into TE 1 . 
     Thus, TE 0  input to the super mode-generating element is converted into the odd mode, and then, is input to the matching coupling element and is converted into TE 1 . 
     Effects of the Present Embodiment 
     [Sixth Effect] 
     In the present embodiment, similar to the first embodiment, it is possible to secure high conversion efficiency in a wide wavelength band, and to secure mode conversion efficiency even in a case where a waveguide structure is changed due to a manufacturing error. 
     Further, in the present embodiment, by forming a slab portion between a pair of core portions, compared with a case where there is no slab portion, a refractive index difference between a core and a cladding becomes substantially smaller between two core portions, and leakage of light increases. Thus, a coupling coefficient χ becomes larger. As the coupling coefficient χ becomes larger, coupling of light between contiguous waveguides becomes stronger. 
     Accordingly, it is possible to perform mode conversion in a short distance, and thus, it is possible to reduce the device length. 
     [Seventh Effect] 
     In the present embodiment, since a structure in which a slab portion is formed between core portions is provided, it is possible to integrally form the core portions and the slab portion by performing a lithography/etching process two times. 
     That is, first, a core having a predetermined thickness is manufactured by a lithography/etching process. Then, a part thereof is thinned by a lithography/etching process to form a slab portion, to thereby form a core. 
     Since the height of the core and the height of the slab portion are not particularly limited, and since it is sufficient if a general condition for optical waveguides is satisfied, it is possible to easily perform integration with other optical waveguide devices (an optical modulator or the like including a rib-type phase modulator) having the slab portion. 
     [Eighth Effect] 
     In the present embodiment, similar to the planar optical waveguide device  10  according to the first embodiment illustrated in  FIGS. 1A and 1B , it is possible to perform conversion over a wide wavelength band with high efficiency by using the super mode-generating element, and to secure efficient mode conversion even in a case where a waveguide structure is changed due to a manufacturing error. 
     Further, it is possible to perform conversion over a wide wavelength band with high efficiency using a matching coupling element, and to secure efficient mode conversion even in a case where a waveguide structure is changed due to a manufacturing error. 
     The reason is as follows. 
     In a case where a wavelength changes, an electric field distribution in a mode accordingly spreads (in a case where the wavelength increases) or shrinks (in a case where the wavelength decreases) with respect to a core. 
     Since the change is the same in an arbitrary mode, even in the case of the odd mode and TE 1 , the same change of the electric field distribution occurs according to the change of the wavelength. Thus, the conversion efficiency of the matching coupling becomes a high value. 
     In order to confirm this,  FIG. 42  illustrates a relationship between a wavelength and conversion efficiency when W is 1200. 
     It can be understood from  FIG. 42  that high conversion efficiency is maintained even though the wavelength changes. 
     Further, in a case where there is a manufacturing error due to lithography or etching or variation in the heights of layers of a wafer (SOI substrate or the like), its influence (change or the like in the width or height of a core) locally occurs in respective portions of waveguides by the same amount. Thus, the waveguides 1 and 2 and the waveguide 3 are changed in the same manner (for example, the widths increase or decrease together). Accordingly, electric field distributions in modes of the respective waveguides also spread or shrink in the same manner. 
     Thus, in the matching coupling element, reduction in conversion efficiency does not occur, and thus, it is possible to secure high conversion efficiency. 
     As a specific example,  FIG. 43  is a graph illustrating a relationship between a wavelength and conversion efficiency in a case where core widths (in the figure, written as waveguide widths in  FIG. 43 ) (for example, W 3 , W 13 , and W 14  in  FIG. 37A ) of the waveguides 1 to 3 of the matching coupling element (subsequent-stage mode conversion section) are all changed by −30 nm. 
     It can be understood from  FIG. 43  that even when the core widths of the waveguides 1 to 3 are changed, high conversion efficiency is maintained. 
     Accordingly, in the present embodiment, by using the super mode-generating element (preceding-stage mode conversion section) and the matching coupling element (subsequent-stage mode conversion section), it is possible to perform conversion with high accuracy over a wide wavelength band, and to secure efficient mode conversion even in a case where a waveguide structure is changed due to a manufacturing error. 
     [Ninth Effect] 
     In the planar optical waveguide device  110 , since the slab portion  16  is provided, in the subsequent-stage mode conversion section  9 , conversion efficiency from an odd mode to TE 1  is enhanced, and conversion efficiency from an even mode to TE 0  is also enhanced. 
     Thus, the planar optical waveguide device  110  is advantageous in performing mode multiplexing between TE 0  and TE 1 . 
     Example 8 
     &lt;Planar Optical Waveguide Device&gt; 
       FIGS. 44A and 44B  are diagrams illustrating a planar optical waveguide device  120  (mode conversion element) according to Example 8 of the invention.  FIG. 44A  is a plan view thereof, and  FIG. 44B  is a sectional view at a sectional position (a) in  FIG. 44A . The same reference numerals are given to the same configurations as those of the planar optical waveguide device  110  shown in  FIGS. 30A and 30B , and description thereof will not be repeated. 
     The width of the core portion  11  (core portion  1 ) is set to 400 [nm], and the width of the core portion  12  (core portion  2 ) is set to 400−X [nm] (−200≦X≦0). Here, X is linearly changed from −200 to 0, from an input end  12   a  to an output end  12   b . The widths of the core portions  13  and  14  are respectively 400 [nm]. A gap between the core portions  1  and  2  is 200 [nm]. The width of the output portion  3  is 1250 [nm]. The heights of the core portions  1  and  2  and the output portion  3  are 220 nm. The height of the slab portion  16  is 95 nm. 
     In the planar optical waveguide device  120 , the first core portion  1  includes a linear waveguide  23 , and the second core portion  2  includes a bent waveguide  24 . The linear waveguide  23  is formed on an input side of the core portion  11 , and the bent waveguide  24  is formed on an input side of the core portion  12 . 
     In the planar optical waveguide device  120 , the first core portion  1  (linear waveguide  23 ) and the second core portion  2  (bent waveguide  24 ) are formed to become closer to each other as a distance to the preceding-stage mode conversion section  108  becomes shorter. Thus, it is possible to reduce unnecessary reflection of light. 
     Since the linear waveguide  23  and the bent waveguide  24  become more distant from each other as the distance to the preceding-stage mode conversion section  108  becomes longer, it is possible to reliably reduce mode coupling compared with a tapered structure. Thus, it is possible to enhance mode conversion efficiency in the preceding-stage mode conversion section  108 . 
     In order to show that mode conversion is possible according to this example, a conversion efficiency (ratio of power of output TE 1  to power of input TE 0 ) in TE 1  output when TE 0  was input to the core portion  1  was calculated using a finite-difference time-domain (FDTD). 
     The length L 2  of the matching coupling element (the subsequent-stage mode conversion section  109  in  FIG. 44A ) was set to 1 μm. A wavelength was set to 1550 nm. A calculation result is shown in  FIG. 45 . 
       FIG. 45  is a diagram illustrating a relationship between a taper length L 1  (the length of a tapered waveguide (core portion  12 )) and conversion efficiency in the super mode-generating element. 
     It can be understood from  FIG. 45  that as the taper length L 1  of the super mode-generating element becomes longer, the width of a core portion in the light waveguide direction becomes smoother, so that an adiabatic change condition is more easily satisfied and the conversion efficiency becomes higher. 
       FIG. 46  is a diagram illustrating an electric field distribution when the taper length L 1  (the length of the core portion  12 ) of the super mode-generating element (preceding-stage mode conversion section  108 ) is 12 μm.  FIG. 46  is a diagram illustrating an E x  component when y is 0.1 μm in a case where TE 0  is input to the core portion  1  from the input end (lower end). The wavelength was set to 1550 nm. 
     It can be understood from  FIG. 46  that light is coupled in the super mode-generating element and TE 0  is converted into an odd mode in which TE 0  is distributed in both core portions. Further, it can also be confirmed that the odd mode is changed to TE 1  by the matching coupling element. 
       FIG. 47  is a graph illustrating a result obtained by simulating wavelength dependency (relationship between a wavelength and conversion efficiency) in this example using FDTD. The taper length L 1  of the super mode-generating element was set to 12 μm. 
     It can be confirmed from  FIG. 47  that a high conversion efficiency which is equal to or greater than −0.27 dB from 1520 nm to 1640 nm is achieved in this example. 
     Since an electric field distribution more greatly spreads outside a core portion and coupling to a contiguous waveguide becomes stronger as a wavelength becomes longer, the conversion efficiency of the super mode-generating element is enhanced at a long wavelength, and thus, the overall conversion efficiency is enhanced. 
     Next, in order to confirm the influence of a manufacturing error in this example, a relationship between a wavelength and conversion efficiency when the widths of overall core portions (and an output portion) were changed by −30 nm was simulated using FDTD. The taper length L 1  of the super mode-generating element was set to 12 μm. 
     A calculation result is shown in  FIG. 48 . 
     When comparing  FIG. 48  with  FIG. 47 , fluctuation in the conversion efficiency in a case where the widths of the core portions (and the output portion) are changed by −30 nm is within 0.15 dB at each wavelength, and high conversion efficiency is maintained. 
     It can be confirmed from  FIG. 48  that this structure is less affected by a manufacturing error. 
     Next, in this example, mode multiplexing of TE 0  of the core portion  2  and TE 1  (mode which is converted from TE 0  input to the core portion  1 ) is possible, which will be described. 
     To this end, a transmittance in TE 0 ′ output from the matching coupling element when TE 0  (written as TE 0 ′ for distinction) was inputted to the core portion  2  from the input side (ratio of power in TE 0 ′ output from the matching coupling element to power in TE 0 ′ input to the core portion  2 ) was simulated using FDTD. 
       FIG. 49  is a diagram illustrating an electric field distribution calculated using FDTD when the taper length of the super mode-generating element is 12 μm. The wavelength was set to 1550 nm.  FIG. 49  is a diagram illustrating an E x  component when y is 0.1 μm in a case where TE 0  is input to the core portion  1  from the input end (lower end). 
     Here, it can be understood that the transmittance becomes −0.17 dB and a large amount of power is transmitted. As described above, mode coupling is possible in this example. 
     &lt;Comparison with Related Art&gt; 
     This example will be compared with performance of an asymmetric directional coupler which is a technique in the related art. Specifically, Example 8 will be compared with Comparative Example 1 having a structure shown in  FIGS. 69A and 69B . First, validity of the comparison will be considered from the following viewpoints. 
     Both the super mode-generating element used in this example and the asymmetric directional coupler in the related art use the mode coupling principle. 
     In the mode coupling, as leakage of light into a contiguous waveguide becomes larger, the coupling becomes stronger, so that the efficiency becomes higher. For this purpose, it is sufficient if the width of a core portion is reduced and light confinement is weakened. 
     However, in consideration of actual manufacturing, if the width of the core portion is too narrow, there are problems such that reproducibility is lowered or a waveguide as mask design cannot be manufactured depending on the accuracy of lithography. Thus, the width of the core portion is set to have a minimum value capable of manufacturing a waveguide. 
     Accordingly, it is possible to perform the comparison between Example 8 and Comparative Example 1 by setting a minimum width of a core portion as the same condition. Since the coupling is also strengthened by decreasing a gap between core portions, the gaps between the core portions in Example 8 and Comparative Example 1 are set to be the same. 
     In Example 8, in a state where the width of an output end (a portion which was necessary for narrowing a core to the minimum) of the super mode-generating element that used the principle of mode coupling was set to 400 nm, the widths of core portions other than the output end were determined. The gap between the core portions was set to 200 nm. 
     In Comparative Example 1 (the asymmetric directional coupler shown in  FIGS. 69A and 69B ), the width of the core portion  1  (a portion which was necessary for narrowing a core to the minimum) that guided light in TE 0  which was a coupling target was set to 400 nm, and the width of the core portion  2  was determined so that phase matching was established. 
     Since the minimum width of the core portion and the gap between the core portions are under the same condition, it is possible to perform the comparison between Example 8 and Comparative Example 1. 
       FIG. 50  shows comparison results of the influences of wavelengths on conversion efficiencies in Example 8 and Comparative Example 1. The results of Example 8 and Comparative Example 1 are written as Example 8-1 and Comparative Example 1-1, respectively. These results are the same as in graphs illustrated in  FIGS. 47 and 71 . 
     Referring to  FIG. 50 , in Comparative Example 1 (Comparative Example 1-1), loss is low compared with Example 8 in the vicinity of a wavelength of 1580 nm, but as the wavelength is changed, the conversion efficiency is greatly reduced. Thus, a loss change due to the wavelength is large. 
     On the other hand, Example 8 (Example 8-1) is inferior to Comparative Example 1 (Comparative Example 1-1) in the vicinity of 1580 nm, but a loss change depending on a wavelength is small at 1520 nm to 1640 nm (a wavelength range which covers a C+L band in optical communication). 
     Further, when comparing the minimum conversion efficiency in this wavelength range, it can be understood that Example 8 (Example 8-1) is a higher minimum conversion efficiency. 
     As described above, in Example 8 (Example 8-1), the conversion can be performed with high efficiency over a wide wavelength range compared with Comparative Example 1 (Comparative Example 1-1). 
     In Example 8 (Example 8-1), since the super mode-generating element uses adiabatic change, by increasing a taper length, it is possible to further lower loss. 
     On the other hand, in the asymmetric directional coupler of Comparative Example 1 (Comparative Example 1-1), since it is difficult to remarkably change the length, no further improvement in conversion efficiency can be expected. 
     Subsequently,  FIG. 51  shows comparison results of the influences of manufacturing errors on conversion efficiencies in Example 8 and Comparative Example 1.  FIG. 51  shows a conversion efficiency when the width of a core portion (and an output portion) is changed by −30 nm. The results in Example 8 and Comparative Example 1 are written as Example 8-2 and Comparative Example 1-2. These results are the same as in graphs illustrated in  FIG. 48  and  FIG. 74 . 
     Referring to  FIG. 51 , in Comparative Example 1 (Comparative Example 1-2), phase matching is not established and the conversion efficiency is reduced, but in Example 8 (Example 8-2), high conversion efficiency is maintained. 
     Accordingly, Example 8 (Example 8-2) has a small influence due to a manufacturing error compared with Comparative Example 1 (Comparative Example 1-2). 
     The planar optical waveguide device  120  having the structure of Example 8 was actually manufactured. 
       FIG. 52  shows a measurement result of a loss (conversion efficiency (dB) which is assigned a negative (−) sign) when TE 0  is converted into TE 1 , with respect to a wavelength. It can be understood from  FIG. 52  that conversion of TE 0  to TE 1  can be sufficiently performed. 
     A conversion loss in this case is equal to or smaller than 0.4 dB in a wide wavelength band from 1430 nm to 1630 nm, which is extremely low. This is because wavelength dependency of the planar optical waveguide device is small and the influence due to a manufacturing error is also small. 
     It can be understood from the results shown in  FIGS. 50 to 52  that Example 8 has higher conversion efficiency in a wide wavelength band and is strong against a manufacturing error compared with the related art technique. 
     Example 9 
     &lt;Planar Optical Waveguide Device&gt; 
       FIGS. 53A to 53C  illustrate a planar optical waveguide device  170  (mode conversion element) according to Example 9 of the invention. Example 9 corresponds to a first example of a planar optical waveguide device that employs a rib waveguide structure.  FIG. 53A  is a plan view thereof,  FIG. 53B  is a sectional view of an output portion  3 , and  FIG. 53C  is a sectional view of core portions  1  and  2 . 
     The planar optical waveguide device  170  has the same configuration as that of the planar optical waveguide device  120  illustrated in  FIGS. 44A and 44B  except that a core  175  instead of the core  105  is provided. 
     The core  175  includes a pair of core portions  1  and  2  which are disposed in parallel with each other, a slab portion  16  (intermediate region) which is formed to connect the core portions  1  and  2 , and slab portions  17  and  18  (outer extension regions) that are formed to extend outward in the width directions from the respective core portions  1  and  2 . Shading is given to the slab portions  16  to  18  in  FIG. 53A . 
     The slab portions  17  and  18  are formed to be lower in height than the core portions  1  and  2 , similar to the slab portion  16 . 
     Lower surfaces of the slab portions  16  to  18  are continuous with lower surfaces of the core portions  1  and  2 . Thus, the core portions  1  and  2  protrude upward from upper surfaces of the slab portions  16  to  18 . 
     The core  175  forms a so-called rib waveguide since the slab portions  17  and  18  are also provided, similar to one side (inner side) of each of the core portions  1  and  2  in the width direction, on the other side (outer side) thereof. 
     The core  175  includes a preceding-stage mode conversion section  178  and a subsequent-stage mode conversion section  179 . The slab portions  17  and  18  are not formed in an output portion  3  of the subsequent-stage mode conversion section  179 . 
     In the planar optical waveguide device  170 , since the core  175  includes the slab portions  16  to  18 , it is possible to strengthen light coupling between waveguides and to shorten a device length, compared with a case where there is no slab portion. 
     Since the planar optical waveguide device  170  includes the slab portions  17  and  18  that extend outward from the core portions  1  and  2 , leakage of light outward from the core portions  1  and  2  becomes larger. Thus, a coupling coefficient χ becomes smaller compared with the planar optical waveguide device  120  shown in  FIGS. 44A and 44B . On the other hand, there is an advantage in that it is possible to reduce a detrimental influence (increase in loss due to light scattering) due to roughness on side walls of core portions (surface roughening), generated in a process such as dry etching. 
     The reason why the influence of roughness on the side walls of the core portions can be reduced is that since the planar optical waveguide device  170  is provided with the slab portions  16  to  18  on both sides of the core portions  1  and  2 , the areas of the side surfaces of the core portions  1  and  2  are small. 
     Accordingly, from the viewpoint of reducing loss due to roughness of side walls, this example (rib waveguide) is preferable. 
     Example 10 
     &lt;Planar Optical Waveguide Device&gt; 
       FIGS. 54A to 54C  illustrate a planar optical waveguide device  190  (mode conversion element) according to Example 10 of the invention. Example 10 corresponds to a second example of the planar optical waveguide device that employs the rib waveguide structure.  FIG. 54A  is a plan view thereof,  FIG. 54B  is a sectional view of an output portion  3 , and  FIG. 54C  is a sectional view of core portions  1  and  2 . 
     The planar optical waveguide device  190  has the same configuration as that of the planar optical waveguide device  170  illustrated in  FIGS. 53A to 53C  except that a core  195  instead of the core  175  is provided. 
     The core  195  includes a preceding-stage mode conversion section  178  and a subsequent-stage mode conversion section  199 . 
     The subsequent-stage mode conversion section  199  is different from the subsequent-stage mode conversion section  179  in  FIG. 53A  in that the slab portions  17  and  18  are also formed in the output portion  3 . 
     In the planar optical waveguide device  190  can achieve the same effects as in the planar optical waveguide device  170  in  FIGS. 53A to 53C . 
     Since the planar optical waveguide device  190  includes the subsequent-stage mode conversion section  199  that employs the rib waveguide structure, it is possible to reduce loss when being connected to a high-order polarization conversion element (see  FIGS. 64A to 64D ) having an asymmetric structure in a height direction. 
     Example 11 
     &lt;Planar Optical Waveguide Device&gt; 
       FIGS. 55A to 55C  illustrate a planar optical waveguide device  200  (mode conversion element) according to Example 11 of the invention. Example 11 corresponds to a third example of the planar optical waveguide device that employs the rib waveguide structure.  FIG. 55A  is a plan view thereof,  FIG. 55B  is a sectional view of an output portion  3 , and  FIG. 55C  is a sectional view of core portions  1  and  2 . 
     The planar optical waveguide device  200  has the same configuration as that of the planar optical waveguide device  190  illustrated in  FIGS. 54A to 54C  except that a core  205  instead of the core  195  is provided. 
     The core  205  includes a preceding-stage mode conversion section  208  and a subsequent-stage mode conversion section  209 . 
     The core  205  is formed so that slab portions  17  and  18  (outer extension regions) are formed in the output portion  3 , but the slab portions  17  and  18  are not formed in the core portions  1  and  2 . 
     The planar optical waveguide device  200  shows the same effects as in the planar optical waveguide device  170  in  FIGS. 53A to 53C . 
     Since the planar optical waveguide device  200  includes the subsequent-stage mode conversion section  209  that employs the rib waveguide structure, it is possible to reduce loss when being connected to a high-order polarization conversion element (see  FIGS. 64A to 64D ) having an asymmetric structure in a height direction. 
     Example 12 
     &lt;Planar Optical Waveguide Device&gt; 
       FIG. 56  is a plan view illustrating a planar optical waveguide device  180  according to Example 12 of the invention. 
     The planar optical waveguide device  180  (mode conversion element) has the same configuration as that of the planar optical waveguide device  120  illustrated in  FIGS. 44A and 44B  except that a first core portion  1  includes a one-side tapered waveguide  25  and a second core portion  2  includes a one-side tapered waveguide  26 . 
     The one-side tapered waveguide  25  is provided on an input side of a linear waveguide  23 , and the one-side tapered waveguide  26  is provided on an input side of a bent waveguide  24 . 
       FIGS. 57A to 57C  are diagrams illustrating an example of the one-side tapered waveguide  26 , in which  FIG. 57A  is a sectional view of an input end,  FIG. 57B  is a plan view thereof, and  FIG. 57C  is a sectional view of an output end. 
     The one-side tapered waveguide  26  includes a rib portion  181 , and a slab portion  182  that is formed to extend from the rib portion  181  on one side surface of the rib portion  181 . 
     The rib portion  181  has a uniform width in a length direction. 
     The slab portion  182  is formed in a tapered shape in which the width gradually increases from one end  181   a  to the other end  181   b  in the length direction of the rib portion  181 . 
     Since the slab portion  182  is formed using the end  181   a  of the rib portion  181  as a starting point, the one-side tapered waveguide  26  is a rectangular waveguide at an input end  26   a  and is a half-rib waveguide at an output end  26   b.    
     The one-side tapered waveguide  25  may also employ the same structure. 
     Thus, by using the one-side tapered waveguides  25  and  26 , it is possible to smoothly change the waveguide structure in the length direction at connecting portions between the rectangular waveguides  183  and  184  and the half-rib waveguides (linear waveguide  23  and bent waveguide  24 ), to thereby realize connection with low loss. 
     In the planar optical waveguide device  180  illustrated in  FIG. 56 , connection between a half-rib waveguide in which a slab portion is present only one side and a rectangular waveguide is assumed. However, this structure is not particularly limiting, and in the case of connection between a rib waveguide in which a slab portion is present on both sides and a rectangular waveguide, a double-side tapered waveguide  27  illustrated in  FIGS. 58A to 58C  may be used. 
     The double-sided tapered waveguide  27  includes a rib portion  181  and a slab portion  182  formed on each of both side surfaces of the rib portion  181 . 
     Since the slab portion  182  is formed using one end  181   a  as a starting point, the double-side tapered waveguide  27  is a rectangular waveguide at an input end and is a rib waveguide at an output end. 
     Thus, by using the double-side tapered waveguide  27 , it is possible to realize connection with low loss at a connecting portion between an external rectangular waveguide and a rib waveguide in which a slab portion is present on both sides thereof. 
     Example 13 
     &lt;Planar Optical Waveguide Device&gt; 
       FIG. 59  is a plan view illustrating a planar optical waveguide device  130  (mode conversion element) according to Example 13. Example 13 corresponds to a fourth example having a structure in which an intermediate core portion is provided between a preceding-stage mode conversion section and a subsequent-stage mode conversion section. 
     The planar optical waveguide device  130  has the same configuration as that of the planar optical waveguide device  30  illustrated in  FIG. 23  except that a slab portion  16  is formed between core portions  31  and  32 . 
     In the planar optical waveguide device  130 , similar to the planar optical waveguide device  30 , since an intermediate core portion  34  having a tapered structure is provided, it is possible to connect a preceding-stage mode conversion section  38  and a subsequent-stage mode conversion section  39  with low loss. 
     Example 14 
     &lt;Planar Optical Waveguide Device&gt; 
       FIG. 60  is a plan view illustrating a planar optical waveguide device  140  (mode conversion element) according to Example 14. Example 14 corresponds to a fifth example having a structure in which an intermediate core portion is provided between a preceding-stage mode conversion section and a subsequent-stage mode conversion section. 
     The planar optical waveguide device  140  has the same configuration as that of the planar optical waveguide device  40  illustrated in  FIG. 24  except that a slab portion  16  is formed between core portions  41  and  42 . 
     In the planar optical waveguide device  140 , similar to the planar optical waveguide device  40 , since an intermediate core portion  44  of a linear shape is provided, it is possible to enhance the degree of freedom in disposition of a subsequent-stage mode conversion section  49 . 
     Example 15 
     &lt;Planar Optical Waveguide Device&gt; 
       FIG. 61  is a plan view illustrating a planar optical waveguide device  150  (mode conversion element) according to Example 15. Example 15 corresponds to a sixth example having a structure in which an intermediate core portion is provided between a preceding-stage mode conversion section and a subsequent-stage mode conversion section. 
     The planar optical waveguide device  150  has the same configuration as that of the planar optical waveguide device  50  illustrated in  FIG. 25  except that a slab portion  16  is formed between core portions  51  and  52 . 
     In the planar optical waveguide device  150 , similar to the planar optical waveguide device  50 , since an intermediate core portion  54  of a curved shape is provided, it is possible to enhance the degree of freedom in disposition of a subsequent-stage mode conversion section  59 . 
     Example 16 
     &lt;Planar Optical Waveguide Device&gt; 
       FIG. 62  is a plan view illustrating a planar optical waveguide device  260  (mode conversion element) according to Example 16. 
     The planar optical waveguide device  260  is a planar optical waveguide device that employs a rib waveguide structure, and has the same configuration as that of the planar optical waveguide device  80  illustrated in  FIG. 26  except that a core  185  instead of the core  85  is provided. 
     The core  185  includes a preceding-stage mode conversion section  178  and a subsequent-stage mode conversion section  189 , and is different from the planar optical waveguide device  80  illustrated in  FIG. 26  in that slab portions  16  to  18  are formed in the core portions  1  and  2 . 
     The subsequent-stage mode conversion section  189  may be used as a 1×2MMI (multi-mode interferometer) since two core portions  13  and  14  are connected to an input side of the output portion  3  and one output-side core portion  86  is connected to an output-side (rear end of the output portion  3 ). 
     Example 17 
     &lt;Planar Optical Waveguide Device (Polarization Conversion Element)&gt; 
       FIG. 63  is schematic view illustrating a planar optical waveguide device  160  (polarization conversion element) according to Example 17. 
     The planar optical waveguide device  160  has a configuration in which a high-order polarization conversion element  101  (high-order polarization-converting section) is provided on the output-side of the planar optical waveguide device  110  (mode conversion element) illustrated in  FIG. 30A . The high-order polarization conversion refers to conversion between TE 1  and TM 0 . 
     As the high-order polarization conversion element  101 , the high-order polarization conversion element shown in  FIG. 28B  may be used. Instead of the high-order polarization conversion element  101 , the high-order polarization conversion element  61  shown in  FIG. 29A  may be used. 
     The planar optical waveguide device  160  in  FIG. 63  is capable of converting TE 0  into TE 1  by the planar optical waveguide device  110 , and is capable of converting TE 1  into TM 0  by the high-order polarization conversion element  101 . 
     Since TE 0  is not converted into a separate mode in the high-order polarization conversion element  101 , TE 0  (written as TE 0 ′ for distinction) inputted to a core portion  2  and output from an output portion  3  is not converted. 
     Thus, an output obtained by combining TM 0  and TE 0 ′ is output from an output-side of the high-order polarization conversion element  101 . Accordingly, the planar optical waveguide device  160  can be used as an element for performing polarization multiplexing. 
     Example 18 
     &lt;Planar Optical Waveguide Device (Polarization Conversion Element)&gt; 
       FIGS. 64A and 64B  illustrate a planar optical waveguide device  210  (polarization conversion element) according to Example 18, in which  FIG. 64A  is an overall plan view thereof,  FIG. 64B  is a plan view of the high-order polarization conversion element,  FIG. 64C  is a sectional view of an ending portion of the high-order polarization conversion element, and  FIG. 64D  is a sectional view of a starting portion of the high-order polarization conversion element. 
     A core  215  of the planar optical waveguide device  210  includes a preceding-stage mode conversion section  178  and a subsequent-stage mode conversion section  199 . 
     The planar optical waveguide device  210  includes a high-order polarization conversion element  111  (high-order polarization-converting section) on an output-side of the subsequent-stage mode conversion section  199 . 
     As shown in  FIG. 64A , the preceding-stage mode conversion section  178  and the subsequent-stage mode conversion section  199  may be formed to be the same as the preceding-stage mode conversion section and the subsequent-stage mode conversion section used in the planar optical waveguide device  190  illustrated in  FIGS. 54A to 54C . 
     As shown in  FIGS. 64B to 64D , in a high-order polarization conversion element  111 , a core  112  is formed by a lower core  114  and an upper core  113 , and the upper core  113  and the lower core  114  are formed in a tapered shape in which the widths continuously decrease in a light waveguide direction. 
     The core  112  has an asymmetric structure in a height direction, in which the width of the upper core  113  and the width of the lower core  114  are different from each other. 
     The lower core  114  may be integrally formed with slab portions  17  and  18  of the subsequent-stage mode conversion section  199 . 
     The high-order polarization conversion element  111  may perform polarization conversion between TE 1  of a starting portion  118  and TM 0  of an ending portion  119 . 
     The planar optical waveguide device  210  is capable of converting TE 0  into TE 1  by the preceding-stage mode conversion section  178  and the subsequent-stage mode conversion section  199 , and is capable of converting TE 1  into TM 0  by the high-order polarization conversion element  111 . 
     TE 0  (written as TE 0 ′ for distinction) which is input to the core portion  2  and is input to the high-order polarization conversion element  111  is not converted. 
     Thus, an output obtained by combining TM 0  and TE 0 ′ is input from an output-side of the high-order polarization conversion element  111  to an output portion  213  which is a rectangular waveguide. 
     Accordingly, the planar optical waveguide device  210  can be used as an element for performing polarization multiplexing. 
     Since the planar optical waveguide device  210  includes the subsequent-stage mode conversion section  199  that employs a rib waveguide structure, it is possible to perform connection to the high-order polarization conversion element  111  having an asymmetric structure with low loss. 
     Example 19 
     &lt;Planar Optical Waveguide Device (Polarization Conversion Element)&gt; 
       FIG. 65  is a plan view illustrating a planar optical waveguide device  250  (polarization conversion element) according to Example 19. 
     The planar optical waveguide device  250  has the same configuration as that of the planar optical waveguide device  210  illustrated in  FIGS. 64A to 64D  except that a core  255  instead of the core  195  is provided. 
     A subsequent-stage mode conversion section  209  of the core  255  is different from the subsequent-stage mode conversion section  199  in  FIGS. 64A to 64D  in that slab portions  17  and  18  (outer extension regions) are formed in an output portion  3  but the slab portions  17  and  18  are not present in core portions  1  and  2 . 
     A preceding-stage mode conversion section  208  of the core  205  is different from the preceding-stage mode conversion section  178  in  FIGS. 64A to 64D  in that the slab portions  17  and  18  are not present in core portions  11  and  12 . 
     The preceding-stage mode conversion section  208  and the subsequent-stage mode conversion section  209  may have the same configuration as in the preceding-stage mode conversion section and the subsequent-stage mode conversion section used in the planar optical waveguide device  200  illustrated in  FIGS. 55A to 55C . 
     The planar optical waveguide device  250  is capable of converting TE 0  into TE 1  by the preceding-stage mode conversion section  208  and the subsequent-stage mode conversion section  209 , and is capable of converting TE 1  into TM 0  by the high-order polarization conversion element  111 . 
     TE 0  (written as TE 0 ′ for distinction) which is input to the core portion  2  and is input to the high-order polarization conversion element  111  is not converted. 
     Thus, an output obtained by combining TM 0  and TE 0 ′ is input from an output-side of the high-order polarization conversion element  111  to an output portion  213  which is a rectangular waveguide. 
     Accordingly, the planar optical waveguide device  250  can be used as an element for performing polarization multiplexing. 
     Example 20 
     &lt;Polarization Multiplexing 4-Value Phase (Dual Polarization-Quadrature Phase Shift Keying (DP-QPSK)) Modulator&gt; 
     The planar waveguide device of the invention may be used, for example, for a DP-QPSK modulator disclosed in P. Dong, C. Xie, L. Chen, L. L. Buhl, and Y.-K. Chen, “112-Gb/s Monolithic PDM-QPSK Modulator in Silicon,” in European Conference and Exhibition on Optical Communication (2012), Vol. 1, p. Th. 3. B. 1. 
       FIG. 66  is a diagram schematically illustrating an example of a DP-QPSK modulator. 
     A DP-QPSK modulator  220  performs DP-QPSK modulation having a QPSK signal independent of both modes of TE 0  and TM 0  using the fact that two modes of TE 0  and TM 0  can be present in a normal optical waveguide. 
     Specifically, the DP-QPSK modulator  220  splits light input in the TE 0  mode from an input section  221  into two optical waveguides  222  and  222 , and modulates the split light components into QPSK signals using QPSK modulators  223  and  223 , respectively. Then, the DP-QPSK modulator  220  converts TE 0  of one of optical waveguides  224  and  224  into TM 0  by a polarization conversion element  225 , combines the two modes on the same optical waveguide using a polarization beam combiner, and then, outputs signals independent of TE 0  and TM 0  to an output portion  226 . 
     Example 21 
     &lt;Coherent Receiver&gt; 
     A planar optical waveguide device according to this example may be used, for example, for a polarization diversity coherent receiver on an Si optical waveguide of a polarization multiplexing signal through which TE 0  and TM 0  are simultaneously transmitted, as disclosed in C. Doerr, et al., “Packaged Monolithic Silicon 112-Gb/s Coherent Receiver,” IEEE Photonics Technology Letters. Vol. 23, p.p. 762, 2011. 
       FIG. 67  is a diagram schematically illustrating an example of a polarization diversity coherent receiver. 
     In a coherent receiver  230 , an optical waveguide  231  of a polarization multiplexing signal through which TE 0  and TM 0  are simultaneously transmitted is connected to a polarization conversion element  232  capable of simultaneously performing polarization conversion and polarization beam splitter, and a signal in TE 0  is input to one of optical waveguides  233  and  233  and a signal in TE 0  converted from TM 0  is input to the other one of the optical waveguides  233  and  233 . As a local light-emitting section  234 , a semiconductor laser light source, which is generally used, uses only a single polarized wave, for example, an output in TE 0  (local). In a case where such a light source is used, in the related art, it is necessary to perform polarization conversion of local light emission. 
     However, in the coherent receiver  230 , since any signal light after polarization separation becomes a TE 0  signal (signal), it is not necessary to perform polarization conversion of local light emission. Signal light and local light emission unit are output from a coupling section  236  through an optical multiplexing section  235 . 
     In a case where an optical waveguide type structure is used in the polarization conversion element  232 , light coupling with the outside of an element in the coupling section  236  may be performed using a coupler that does not have a polarization split function, such as a reverse taper type mode field converter in which coupling is performed from a lateral side of a substrate. The coupler may employ a reverse taper type structure disclosed in Qing Fang, et al., “Suspended optical fiber-to-waveguide mode size converter for Silicon photonics”, OPTICS EXPRESS, Vol. 18, No. 8, 7763 (2010), for example. 
     Example 22 
     &lt;Polarization Diversity&gt; 
     A planar optical waveguide device according to this example may be used for executing a polarization diversity method in a case where an element for assigning the same operation is to be used with respect to both modes in polarization multiplexing transmission in which TE 0  and TM 0  are simultaneously transmitted or in random transmission of one polarization, as disclosed in Hiroshi Fukuda, et al., “Silicon photonic circuit with polarization diversity,” Optics Express, Vol. 16, No. 7, 2008, for example. 
     In a polarization diversity  240  shown in  FIG. 68 , an optical waveguide  241  of a polarization multiple signal through which TE 0  and TM 0  are simultaneously transmitted is connected to a polarization conversion element  242  capable of simultaneously performing polarization conversion and polarization beam splitter. Further, a signal in TE 0  is input to one of the optical waveguides  243  and  243  and a signal in TE 0  converted from TM 0  is input to the other one of the optical waveguides  243  and  243 . Signal light components in TE 0  operated by elements  244  and  244  are combined by a polarization conversion element  246  from optical waveguides  245  and  245 , and are output to an optical waveguide  247  of a polarization multiple signal through which TE 0  and TM 0  are simultaneously transmitted. 
     Similar to the polarization diversity coherent receiver, the polarization conversion element according to the present embodiment capable of simultaneously performing polarization conversion and polarization beam splitter may be used as the polarization conversion element  242 . 
     Further, similar to the DP-QPSK modulator, the polarization conversion element according to the present embodiment capable of simultaneously performing polarization conversion and polarization beam combiner may be used as the polarization conversion element  246 . 
     The invention is not limited to the above-described embodiments, and various modifications may be made in a range without departing from the concept of the invention.