Patent Publication Number: US-9885830-B2

Title: Semiconductor optical waveguide device

Description:
RELATED APPLICATION 
     This application is a division of U.S. patent application Ser. No. 14/810,819, filed Jul. 28, 2015 (now U.S. Pat. No. 9,588,296) which is incorporated herein by reference. 
    
    
     TECHNICAL FIELD 
     The present disclosure relates to optical devices, and in particular to semiconductor optical waveguide devices. 
     BACKGROUND 
     Miniaturization of optical, electro-optical, and optoelectronic components and modules is reaching a stage where complex optical, electro-optical, and opto-electronic functionalities may be realized on a single semiconductor chip termed “photonic integrated circuit”. A photonic integrated circuit may include optical waveguides and other micro-optical structures. 
     Photonic integrated circuits may be used for separation, modulation, demodulation, and detection of optical signals, making them attractive for optical communications systems. Furthermore, photonic integrated circuits may be compatible with electronic circuitry, which enables such functions as transmission, reception, and modulation of light on a single chip. 
     Despite the progress of optical integration of multiple functionalities of photonic integrated circuits, the task of coupling light between different waveguides of a same or a different photonic integrated circuit, and between a photonic integrated circuit and an optical fiber remains challenging. Optical modes guided by planar waveguides of different size and/or different refractive index contrast may differ considerably in size and shape. An optical waveguide mode is usually much smaller in size than an optical mode guided by a singlemode optical fiber or fibers, which are used to optically couple a photonic integrated circuit to an outside environment. A semiconductor-based optical mode converter may be used to provide conversion between optical modes of different sizes, shapes, and different vertical positions relative to the semiconductor substrate. 
     One prior-art solution of a problem of an optical mode conversion and vertical displacement includes using vertical couplers to couple light from a lower optical waveguide to a differently sized upper optical waveguide, or vice versa. Another solution is to use waveguide tapers having physical thickness varying in vertical direction, and/or a width varying in a horizontal direction. These techniques are rather costly and may be difficult to implement in production environment, especially for vertical direction. 
     Waveguide tapers are perhaps most frequently used for conversion between different optical mode sizes of planar waveguides. Waveguide tapers may also be used for coupling light between a waveguide and an external optical fiber. However, waveguide tapers typically have to be made long enough to ensure an adiabatic mode transformation to avoid considerable optical losses. Long waveguide tapers tend to occupy a considerable area on a photonic chip, especially if an array of such tapers is required to optically couple an array of optical fibers to a photonic chip. 
     Therefore, the prior art appears lacking a manufacturable and reproducible semiconductor optical waveguide device capable of optical mode size conversion and/or vertical displacement of optical modes. 
     SUMMARY 
     In accordance with an aspect of the disclosure, there is provided a method of manufacturing a semiconductor optical waveguide device, the method comprising: 
     growing on a substrate a base waveguide comprising one of: 
     i) a gradient index waveguide comprising a local refractive index depending on a growth parameter, wherein the growing comprises varying the growth parameter so as to gradually increase the local refractive index to a maximum value, and then to gradually decrease the local refractive index, whereby upon completion of the growing, the gradient index waveguide comprises a transversal bell-shaped refractive index profile defining an optical axis comprising the maximum value of the transversal bell-shaped refractive index profile; 
     ii) a first step index waveguide comprising a first waveguide core comprising a first core thickness and a first refractive index; and 
     iii) a second step index waveguide comprising a second waveguide core comprising a second core thickness and a second refractive index; 
     forming a first recess in the base waveguide by removing a first portion thereof to a first depth; 
     forming a different one of the gradient index waveguide, the first step index waveguide, and the second step index waveguide in the first recess; 
     forming a second recess in at least one of the waveguides formed heretofore on the substrate, by removing a second portion thereof to a second depth; 
     forming the remaining one of the gradient index waveguide, the first step index waveguide, and the second step index waveguide in the second recess; 
     wherein upon growing the gradient index waveguide and the first and second step index waveguides, an optical path is formed comprising in sequence the first waveguide core, the gradient index waveguide, and the second waveguide core. 
     In one exemplary embodiment, the first and second recesses are formed in the gradient index waveguide, wherein the first step index waveguide is formed in the first recess, and the second step index waveguide is formed in the second recess. Forming at least one of: the gradient index waveguide, the first step index waveguide, and the second step index waveguide may include epitaxial growing. The epitaxial growing may enable the bell-shaped refractive index profile to be varying smoothly and monotonically, substantially without creating micro-steps in the refractive index profile. 
     The method may also include forming a second recess in the gradient index waveguide, and forming the second step index waveguide in the second recess. The second recess may be created by removing a second portion of the gradient index waveguide opposite the first portion to a second depth, thereby defining a length of the gradient index waveguide in between the first and second step index waveguides. 
     In accordance with the disclosure, there is further provided a semiconductor optical waveguide device comprising: 
     a substrate; 
     a first step index waveguide on the substrate, the first step index waveguide comprising a first waveguide core comprising a first core thickness and a first refractive index; 
     a gradient index waveguide on the substrate, the gradient index waveguide abutting the first step index waveguide and comprising a length and a transversal gradually varying bell-shaped refractive index profile defining an optical axis comprising a maximum value of the transversal gradually varying bell-shaped refractive index profile; and 
     a second step index waveguide over the substrate, the second step index waveguide abutting the gradient index waveguide and comprising a second waveguide core comprising a second core thickness and a second refractive index; 
     wherein the semiconductor optical waveguide device comprises an optical path comprising in sequence the first waveguide core, the gradient index waveguide, and the second waveguide core. 
     In one embodiment, the first and second step index waveguides abut the gradient index waveguide on its opposite sides of the gradient index waveguide. 
     The transversal bell-shaped refractive index profile may include a substantially parabolic vertically varying refractive index profile characterized by a repeat length L of an optical field propagating in the gradient index waveguide, 
     wherein the length of the gradient index waveguide between the first and second step index waveguides is substantially equal to LM/4, wherein M is an integer, wherein L=2π/δn eff  k 0 , wherein n eff  is an effective refractive index step between the at least two optical modes, and k 0  is a wavenumber of a zero-order optical mode propagating in the gradient index waveguide. The optical field may include at least two optical modes. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Exemplary embodiments will now be described in conjunction with the drawings, in which: 
         FIG. 1A  illustrates an elevational cross-sectional view of a semiconductor optical waveguide mode converter device of the present disclosure; 
         FIG. 1B  illustrates an elevational cross-sectional view of a semiconductor optical waveguide mode vertical displacer device of the present disclosure; 
         FIG. 1C  illustrates an elevational cross-sectional view of a reflective version of the semiconductor optical waveguide mode vertical displacer device of  FIG. 1B ; 
         FIG. 1D  illustrates an elevational cross-sectional view of a semiconductor optical waveguide device including a gradient index waveguide having two distinct portions; 
         FIGS. 2A to 2F  illustrate elevational cross-sectional views of a semiconductor optical waveguide device of  FIG. 1A  at different progressive stages of manufacturing; 
         FIG. 3  illustrates a side view of a simulated optical field propagating in a gradient index waveguide having a parabolic vertical refractive index profile; 
         FIG. 4  illustrates a side view of a simulated optical field propagating in a semiconductor optical waveguide mode converter of  FIG. 1A , having the gradient index waveguide of  FIG. 3 ; 
         FIG. 5A  illustrates a cross-sectional view of a simulated first optical mode propagating in the gradient index waveguide of  FIG. 3 ; 
         FIG. 5B  illustrates a cross-sectional view of a simulated second optical mode propagating in the gradient index waveguide of  FIG. 3 ; 
         FIG. 5C  illustrates a cross-sectional view of a simulated third optical mode propagating in the gradient index waveguide of  FIG. 3 ; 
         FIG. 5D  illustrates a cross-sectional view of a simulated fourth optical mode propagating in the gradient index waveguide of  FIG. 3 ; 
         FIG. 5E  illustrates a cross-sectional view of a simulated fifth optical mode propagating in the gradient index waveguide of  FIG. 3 ; 
         FIG. 5F  illustrates a cross-sectional view of a simulated sixth optical mode propagating in the gradient index waveguide of  FIG. 3 ; 
         FIG. 6  illustrates a calculated dependence of an effective refractive index n eff  on vertical and horizontal mode numbers; 
         FIG. 7A  illustrates an elevational view of a simulated optical field having a large vertical offset of a narrow input optical field relative to an optical axis of the gradient index waveguide of  FIG. 3 ; 
         FIG. 7B  illustrates an elevational view of a simulated optical field having a medium vertical offset of a narrow input optical field relative to an optical axis of the gradient index waveguide of  FIG. 3 ; 
         FIG. 7C  illustrates an elevational view of a simulated optical field having a zero vertical offset of a narrow input optical field relative to an optical axis of the gradient index waveguide of  FIG. 3 ; 
         FIG. 8A  illustrates an elevational view of a simulated optical field having a non-zero vertical offset of a wide input optical field relative to the optical axis of the gradient index waveguide of  FIG. 3 ; 
         FIG. 8B  illustrates an elevational view of a simulated optical field having a zero vertical offset of a wide input optical field relative to the optical axis of the gradient index waveguide of  FIG. 3 ; 
         FIG. 9A  illustrates an elevational cross-sectional view of a fiber-coupled photodetector including the semiconductor optical waveguide device of  FIG. 1A ; 
         FIG. 9B  illustrates an elevational cross-sectional view of a fiber-coupled optical modulator including the semiconductor optical waveguide device of  FIG. 1A ; and 
         FIG. 10  illustrates an example method for manufacturing a semiconductor optical waveguide device of the present disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     While the present teachings are described in conjunction with various embodiments and examples, it is not intended that the present teachings be limited to such embodiments. On the contrary, the present teachings encompass various alternatives and equivalents, as will be appreciated by those of skill in the art. 
     Referring to  FIGS. 1A to 1C , semiconductor optical waveguide devices  100 A ( FIG. 1A ),  100 B ( FIG. 1B ), and  100 C ( FIG. 1C ) may each include a substrate  102  and a first step index waveguide  114  on the substrate  102 . The first step index waveguide  114  may include a first waveguide core  116  having a first core thickness  117  ( FIG. 1A ) and a first refractive index n 1 . The first waveguide core  116  may be disposed between lower  115 A and upper  115 B cladding layers having refractive indices smaller than the first refractive index n 1 . More than two cladding layers  115 A and  115 B may be provided in the first step index waveguide  114 . 
     A gradient index waveguide  104  abutting the first step index waveguide  114  may be disposed on the substrate  102 . The gradient index waveguide  104  may have a length  155 A ( FIG. 1A );  155 B ( FIG. 1B ); and  155 C ( FIG. 1C ). The gradient index waveguide  104  may have a transversal gradually varying bell-shaped refractive index profile  106  ( FIG. 1A ), which defines a optical axis  108  as including a maximum value of the transversal gradually varying bell-shaped refractive index profile  106 . Herein, the term “transversal” means across to the optical axis  108 , e.g. perpendicular to the optical axis  108 . 
     A second step index waveguide  124  may be disposed over the substrate  102 . The second step index waveguide  124  may abut the gradient index waveguide  104 . The second step index waveguide  124  may have a second waveguide core  126  having a second core thickness  127  ( FIG. 1A ) and a second refractive index n 2 . The second waveguide core  126  may be disposed between lower  125 A and upper  125 B cladding layers having refractive indices smaller than the second refractive index n 2 . More than two cladding layers  125 A and  125 B may be provided in the second step index waveguide  124 . The first step index waveguide  114 , the gradient index waveguide  104 , and the second step index waveguide  124  may form an optical path  130 A ( FIG. 1A ),  130 B ( FIG. 1B ), and  130 C ( FIG. 1C ), shown in thick dashed line. 
     Referring specifically to  FIG. 1A , the first  114  and second  124  step index waveguides of the semiconductor optical waveguide device  100 A may abut the gradient index waveguide  104  on opposite first  131  and second  132  sides of the gradient index waveguide  104 . By way of a non-limiting example, the first  116  and second  126  waveguide cores may be centered on the optical axis  108  as shown. The first core  116  thickness  117  may differ from the second core thickness  127 , and the first core refractive index n 1  may differ from the second core refractive index n 2 . For example, the first core  116  thickness  117  may be larger than the second core  126  thickness  127 , and/or the first core refractive index n 1  may be smaller than the second core refractive index n 2 . The length  155 A of the gradient index waveguide  104  and the gradually varying bell-shaped refractive index profile  106  may be selected so as to cause a mode size transformation by the gradient index waveguide  104  from a mode size of the first step index waveguide  114  to a mode size of the second step index waveguide  124 , as shown by an optical path  130 A. The selection of the length  155 A and the selection of the gradually varying bell-shaped refractive index profile  106  will be considered in detail further below. 
     Referring specifically to  FIG. 1B , the first  114  and second  124  step index waveguides of the semiconductor optical waveguide device  100 B may abut the gradient index waveguide  104  on the opposite sides  131  and  132  of the gradient index waveguide  104 . By way of a non-limiting example, a center of the first waveguide core  116  may be disposed above the optical axis  108 , and a center of the second waveguide core  126  may be disposed below the optical axis  108 , as shown in  FIG. 1B . The length  155 B of the gradient index waveguide  104  and the gradually varying bell-shaped refractive index profile  106  may be selected so as to preserve the mode size. 
     The gradually varying bell-shaped refractive index profile  106  includes continuous refractive indices having a quadratic i.e. parabolic shape through the core of the waveguide  104 . Other bell-shaped refractive index profiles  106  are also contemplated where the index profile  106  includes larger indices of refraction close to the optical axis  108  which indices decrease as the distance from the core or the optical axis increases. The shape of the index profile  106  may be less strictly constrained at distances away from the core (or into the cladding), which are distant from the optical axis. 
     Referring specifically to  FIG. 1C , the first step index waveguide  114  of the semiconductor optical waveguide device  100 C may be disposed under the second step index waveguide  124 , so that the first  114  and second  124  step index waveguides abut the gradient index waveguide  104  on the same first side  131  of the gradient index waveguide  104 . The semiconductor optical waveguide device  100 C may further include a mirror surface  140  optically coupled to the second side  132  of the gradient index waveguide  104 . In operation, light  141  emitted from the first step index waveguide  114  propagates through the gradient index waveguide  104 , is reflected by the mirror surface  140 , propagates back through the gradient index waveguide  104 , and impinges on the second step index waveguide  124 . The length  155 C of the gradient index waveguide  104  and the gradually varying bell-shaped refractive index profile  106  may be selected so as to preserve the mode size. 
     Turning now to  FIG. 1D  with further reference to  FIG. 1A , a semiconductor optical waveguide device  100 D is a variant of the semiconductor optical waveguide device  100 A of  FIG. 1A . The gradient index waveguide  104  of the semiconductor optical waveguide device  100 D of  FIG. 1D  may include a first gradient index waveguide portion  104 A and a second gradient index waveguide portion  104 B abutting the first gradient index waveguide portion  104 A. The first gradient index waveguide portion  104 A may include a transversal gradually varying bell-shaped refractive index profile  106 A having a first width, and the second gradient index waveguide portion  104 B may include a transversal gradually varying bell-shaped refractive index profile  106 B comprising a second width different from the first, for example smaller than the first width, as shown. A length  155 D of the gradient index waveguide  104  is the sum of the first width and the second width. The first step index waveguide  114 , the first gradient index waveguide portion  104 A, the second gradient index waveguide portion  104 B, and the second step index waveguide  124  may form an optical path  130 D shown in  FIG. 1D  in thick dashed line. This enables one to achieve larger magnification or de-magnification factors of the optical mode transformation, to match optical modes of the first  114  and second  124  step index waveguides of different sizes. 
     The semiconductor optical waveguide devices  100 A,  100 B,  100 C,  100 D provide a substantially reduced physical size compared to existing comparable devices, especially for III-V semiconductors and for indium phosphide (InP). In some indium phosphide embodiments, the length  155 A,  155 B, and  155 C may be on the order of 10 micrometers to 50 micrometers. The length  155 D of the gradient index waveguide  104  may be on the order of 20 micrometers to 200 micrometers. 
     A manufacturing method of a semiconductor optical waveguide device will now be considered, using the semiconductor optical waveguide device  100 A of  FIG. 1A  as a non-limiting example. Referring to  FIG. 2A , the gradient index waveguide  104  may be formed on the substrate  102 . For example, the gradient index waveguide  104  may be epitaxially grown on the substrate  102 , so that a local refractive index n depends on a growth parameter. The growing may include varying the growth parameter so as to gradually increase the local refractive index n to a maximum value  109 , and then to gradually decrease the local refractive index n. Upon completion of the growing, the gradient index waveguide  104  may have the transversal bell-shaped refractive index profile  106  n(y), where y is the vertical coordinate. The transversal bell-shaped refractive index profile  106  may define the optical axis  108 , which includes the maximum value  109  of the transversal bell-shaped refractive index profile  106 . 
     Referring to  FIG. 2B , a first recess  110  may be formed in the gradient index waveguide  104  by removing a first portion  112  of the gradient index waveguide  104  to a first depth  113 . To that end, a first mask layer  201  may be formed over a remaining length of the gradient index waveguide, and the first portion  112  of the gradient index waveguide  104  may be etched away using a suitable etchant. 
     Referring to  FIG. 2C , the first step index waveguide  114  may be formed in the first recess  110 , e.g. by epitaxial growth. The lower waveguide cladding layer  115 A, the first waveguide core  116 , and the upper waveguide cladding layer  115 B may be formed in sequence one on top of another, so as to create a first stepped refractive index profile  214 . In some embodiments, more layers may be formed in the first step index waveguide  114  corresponding to multiple steps in the first stepped refractive index profile  214 . Then, the first mask layer  201  may be stripped. 
     The second step index waveguide  124  having a second stepped refractive index profile  224  may be formed, e.g. epitaxially grown, on or over the substrate  102  in a similar manner. By way of a non-limiting example, referring to  FIG. 2D , a second recess  120  may be formed in the gradient index waveguide  104  by removing a second portion  122  of the gradient index waveguide  104  opposite the first portion to a second depth  123 , thereby defining the length  155 A of the gradient index waveguide  104  in between. To form the second recess  120 , a second mask layer  202  may be formed over the length  155 A of the gradient index waveguide  104 , and over the first step index waveguide  114 . Then, the second portion  122  of the gradient index waveguide  104  may be etched away using a suitable etchant. 
     Referring to  FIG. 2E , the second step index waveguide  124  may be formed in the second recess  120 , e.g. by epitaxial growth. The lower waveguide cladding layer  125 A, the second waveguide core  126 , and the upper waveguide cladding layer  125 B may be formed in succession so as to create a second stepped refractive index profile  224 . In some embodiments, more layers may be formed in the second step index waveguide  124  corresponding to multiple steps in the second stepped refractive index profile  224 . Then, the second mask layer  202  may be stripped. 
     Turning now to  FIG. 2F , the manufactured semiconductor optical waveguide device  100 A is shown. Upon growing the gradient index waveguide  104  and the first  114  and second  124  step index waveguides, the optical path  130  is formed. The optical path  130  may include in sequence the first waveguide core  116 , the gradient index waveguide  104 , and the second waveguide core  126 . A similar method may be used to make the semiconductor optical waveguide device  100 B of  FIG. 1B  with offsets to the first  114  and second  124  step index waveguides. The above described method may also be used to manufacture the semiconductor optical waveguide device  100 C of  FIG. 1C . In the latter case, the second recess  120  may be omitted or used to form the vertical mirror surface  140  while the first recess  110  may be of a depth to accommodate forming the second step index waveguide  124  and then the first step index waveguide  114  on top of the second step index waveguide  124 . 
     The above described method may also be used to manufacture the semiconductor optical waveguide device  100 D of  FIG. 1D . Specifically, growing the gradient index waveguide  104  may include growing the first gradient index waveguide portion  104 A and growing the second gradient index waveguide portion  104 B abutting the first gradient index waveguide portion  104 A, for example by etching a recess lithographically and growing the second gradient index waveguide portion  104 B in the recess. Similarly to the semiconductor optical waveguide device  100 A of  FIG. 1A , growing the first gradient index waveguide portion  104 A may include varying the growth parameter so as to gradually increase the local refractive index to a maximum value, and then to gradually decrease the local refractive index, so that upon completion of the growing the first gradient index waveguide portion  104 A, the first gradient index waveguide portion has the transversal bell-shaped refractive index profile  106 A. Similarly, growing the second gradient index waveguide portion  104 B may include varying the growth parameter so as to gradually increase the local refractive index to a maximum value, and then to gradually decrease the local refractive index, so that upon completion of the growing the second gradient index waveguide portion  104 B, the second gradient index waveguide portion has the transversal bell-shaped refractive index profile  106 B. 
     The order of growing the gradient index waveguide  104 , the first step index waveguide  114 , and the second step index waveguide  124  may be varied. As illustrated in  FIG. 10 , a method  1000  of manufacturing the semiconductor optical waveguide devices  100 A to  100 D may include growing  1002  on the substrate  102  “a base waveguide”, which may include one of: the gradient index waveguide  104 , the first step index waveguide  114 , and the second step index waveguide  124 . Then, forming  1004  the first recess  110  in the “base waveguide” by removing the first portion  112  of the base waveguide to the first depth  113 . Then, forming  1006  a different one of the gradient index waveguide  104 , the first step index waveguide  114 , and the second step index waveguide  124  in the first recess  110 . Then, forming  1008  the second recess  120  in at least one of the waveguides formed heretofore on the substrate  102 , for example the gradient index waveguide  104  and the first step index waveguide  114 , or any other two of the three waveguides  104 ,  114 , and  124 , by removing the second portion  122  to the second depth  123 . Then, forming  1010  the remaining one of the gradient index waveguide  104 , the first step index waveguide  114 , and the second step index waveguide  124  in the second recess  120 . Upon growing the gradient index waveguide  104  and the first  114  and second  124  step index waveguides, the optical path  130  may be formed. 
     The semiconductor optical waveguide devices  100 A to  100 D of  FIGS. 1A to 1D , respectively, may be grown epitaxially. Refractive index may be precisely controlled during epitaxial growth, providing a smoothly and gradually varying refractive index n. Thus, the transversal bell-shaped refractive index profile  106  may be precisely defined, which enables the length  155 A,  155 B, and  155 C of the gradient index waveguide  104  to be very small, for example 0.1 mm or less, and even 0.05 mm or less. The growth parameter may include, for example and without limitation, reactive gas pressure, deposition rate, ratios of different metal organic precursor gases (for metal organic chemical vapor deposition), or source element crucible temperatures (for molecular beam epitaxy). The gradient index waveguide  104  may include, for example and without limitation, a III-V semiconductor such as, for example, GaAs/AlGaAs InP/InGaAsP, InGaAlAs, InSb, and GaP. Silicon and germanium may also be used. 
     In one exemplary embodiment, the growth parameter may be varied so that the transversal bell-shaped refractive index profile comprises a substantially parabolic refractive index profile. Referring to  FIG. 3 , a simulated light field  300  emitted by the first step index waveguide  114  and propagating in the gradient index waveguide  104  is shown for a case where the gradient index waveguide  104  has the refractive index profile  106  of a substantially parabolic shape. In  FIG. 3 , the vertical scale is between −8.0 and 8.0 micrometers, and the horizontal scale is between 0 and 85 micrometers. The exemplary light field  300  is repetitive. At first  301 , second  302 , third  303 , and fourth  304  locations, the phase front of the exemplary light field  300  is substantially flat, which makes these locations convenient for placing tips of step-index optical waveguides, because guided light fields propagating in non-tapered step index waveguides have substantially planar wavefront within the waveguide core. 
     Referring now to  FIG. 4 , the second step index waveguide  124  is placed at the second location  302 .  FIG. 4  illustrates a computer simulation of the light field  300  propagating in the semiconductor optical waveguide device  100 A of  FIG. 1A , for the case where the gradually varying bell-shaped refractive index profile  106  has a parabolic shape, or in other words, has a quadratic dependence on vertical coordinate y (thickness) of the gradient index waveguide  104 . In  FIG. 4 , the vertical scale is between −8.0 and 8.0 micrometers, and the horizontal scale between 0 and 40 micrometers. 
     The parabolic or quadratic dependence n(y) of the gradient index waveguide  104  may be expressed as 
     
       
         
           
             
               
                 
                   
                     n 
                     ⁡ 
                     
                       ( 
                       y 
                       ) 
                     
                   
                   = 
                   
                     
                       n 
                       0 
                     
                     + 
                     
                       
                         n 
                         1 
                       
                       ⁢ 
                       
                         y 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     wherein n 0  and n 1  are constants. Optical modes propagating in the gradient index waveguide  104  having the dependence n(y) given by Eq. (1) will have equidistant effective refractive indices n eff , which may be expressed as 
     
       
         
           
             
               
                 
                   
                     n 
                     eff 
                     p 
                   
                   = 
                   
                     
                       n 
                       
                         ⁢ 
                         eff 
                       
                       1 
                     
                     + 
                     
                       
                         ( 
                         
                           p 
                           - 
                           1 
                         
                         ) 
                       
                       ⁢ 
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         n 
                         eff 
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     wherein p is the mode number, e.g. 1, 2, 3, 4, and δn eff  is an effective intermodal refractive index step. In other words, δn eff  is a refractive index difference between neighboring optical modes. Referring to  FIGS. 5A, 5B, 5C, 5D, 5E, and 5F , simulated first  501 , second  502 , third  503 , fourth  504 , fifth  505 , and sixth  506  optical modes are illustrated as an example, with the vertical (y) and horizontal (x) scales shown in micrometers. 
     Turning to  FIG. 6 , a calculated dependence  601  of an effective refractive index n eff  on vertical mode numbers is linear. A calculated dependence  602  of an effective refractive index n eff  on horizontal mode numbers is non-linear, being approximately quadratic. In the calculation of  FIG. 6 , the dependence of the refractive index n on the vertical coordinate y is quadratic as given by Eq. (1), while in the horizontal direction x, the refractive index n is constant. 
     Light propagating in the gradient index waveguide  104  having the refractive index vertical profile  106  represented by Eq. (1) may include a sum of modes, for example the modes  501  to  506  of  FIGS. 5A to 5F  respectively, each mode  501  to  506  having its own exponential propagation term depending on the corresponding n eff  given by Eq. (2): 
     
       
         
           
             
               
                 
                   
                     
                       E 
                       ⁡ 
                       
                         ( 
                         
                           x 
                           , 
                           y 
                           , 
                           z 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           p 
                           = 
                           1 
                         
                         N 
                       
                       ⁢ 
                       
                         
                           
                             E 
                             p 
                           
                           ⁡ 
                           
                             ( 
                             
                               x 
                               , 
                               y 
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           exp 
                           ⁡ 
                           
                             ( 
                             
                               j 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 n 
                                 eff 
                                 p 
                               
                               ⁢ 
                               
                                 k 
                                 0 
                               
                               ⁢ 
                               z 
                             
                             ) 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     wherein N is the total number of modes, k 0  is the wavenumber in free space, j√{square root over (1)} and z is the propagation direction coordinate. The number of modes N may be at least two or at least three. In one embodiment, the number of modes N may be no greater than sixteen. 
     Since the modes  501  to  506  have uniformly spaced effective refractive indices n eff , the optical field may have a period (or repeat length) L, wherein L=2π/δn eff k 0 , because 
     
       
         
           
             
               
                 
                   
                     
                       ∑ 
                       
                         p 
                         = 
                         1 
                       
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                       exp 
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                         ( 
                         
                           j 
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                           ⁢ 
                           
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                           ⁢ 
                           
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                             0 
                           
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                         ) 
                       
                     
                   
                   = 
                   
                     
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                         ( 
                         
                           j 
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                           ⁢ 
                           
                             n 
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                             1 
                           
                           ⁢ 
                           
                             k 
                             0 
                           
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                           L 
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           p 
                           = 
                           1 
                         
                         N 
                       
                       ⁢ 
                       
                         exp 
                         ⁡ 
                         
                           ( 
                           
                             
                               j 
                               ⁡ 
                               
                                 ( 
                                 
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                                   - 
                                   1 
                                 
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                             ⁢ 
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               n 
                               e 
                             
                             ⁢ 
                             
                               ffk 
                               0 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             L 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
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     The periodic character of the light field  300  ( FIGS. 3 and 4 ) may be further illustrated by  FIGS. 7A to 7C . In  FIGS. 7A to 7C , the vertical scale is between −7.0 and 7.0 micrometers, and the horizontal scale is between 0 and 116 micrometers. The optical axis  108 , corresponding to the maximum  109  of the gradually varying parabolic refractive index profile  106 , is disposed at the vertical coordinate y of 0.9 micrometers. The core  116  of the input step-index waveguide  114  is disposed at −1.0 micrometers in  FIG. 7A , at 0.5 micrometers in  FIG. 7B , and at 0.9 micrometers (on-axis) in  FIG. 7C . One can see that light fields  700 A ( FIG. 7A ) and  700 B ( FIG. 7B ) have a repeat period of L≈41 micrometer, whereas a light field  700 C ( FIG. 7C ) has a repeat period of L/2≈20.5 micrometers. This is because in case of  FIG. 7C , the excited light field  700 C may only include even modes, which effectively doubles the effective intermodal refractive index step δn eff  between neighboring optical modes. 
     Referring to  FIGS. 8A and 8B  with further reference to  FIGS. 7A and 7C ,  FIGS. 8A and 8B  illustrate results of similar computations as those represented by  FIGS. 7A and 7C , respectively, and have the same geometrical scale. In the case of  FIGS. 8A and 8B , larger input optical fields are used than in the case of  FIGS. 7A and 7C . Similarly to  FIGS. 7A and 7C , an asymmetrically launched light field  800 A has a repeat period of L≈41 micrometers, whereas a symmetrically launched light field  800 B has a repeat period of L/2≈20.5 micrometers. 
     The above simulation results indicate that, for the substantially parabolic transversal bell-shaped refractive index profile  106  characterized by the repeat length L of an optical field (e.g.  300  of  FIG. 3 ) propagating in the gradient index waveguide  104 , the length  155 A ( FIG. 1A ) of the gradient index waveguide  104  between the first  114  and second  124  step index waveguides may be substantially equal to LM/4, wherein M is an integer. This is because for an on-axis first  114  and second  124  step index waveguides, the repeat period is L/2 and one needs one half of that value, that is L/4, to obtain a mode size transformation. More generally, to obtain a mode size transformation, M may need to be an odd number, e.g. 1, 3, 5, . . . , with the length  155 A substantially equal to LM/4. To merely obtain a vertical translation, such as in the semiconductor optical waveguide device  100 B of  FIG. 1B , M may need to be an even number, e.g. 2, 4, 6, . . . . Furthermore, for the reflective semiconductor optical waveguide device  100 C of  FIG. 1C , the length  155 C of the gradient index waveguide  104  between the first  114  and second  124  step index waveguides may be substantially equal to LM/4, wherein M is an odd integer. 
     Referring now to  FIG. 9A , a fiber-coupled photodetector  900 A may include the semiconductor optical waveguide device  100 A of  FIG. 1A , an optical fiber  914  butt-coupled to the first step index waveguide  114 , and a photodetector  910  optically coupled to the second step index waveguide  124 . In operation, an optical signal  905  propagates in the optical fiber  914 , and is coupled to the first step index waveguide  114 . The optical mode sizes of the optical fiber  914  and the first step index waveguide  114  are similar, so that optical coupling loss may not be significant e.g. less than 1 dB. The gradient index waveguide  104  may effectively convert the optical mode size from the size of the first step index waveguide  114  to the size of the second step index waveguide  124 , which is optically coupled to the photodetector  910  for detecting the optical signal  900 A. The fiber-coupled photodetector  900 A may be manufactured by following the steps described above with reference to  FIGS. 2A to 2F , followed by a step of butt-coupling the optical fiber  914  to the first step index waveguide  114 . 
     Turning to  FIG. 9B , a fiber-coupled optical modulator  900 B may include the semiconductor optical waveguide device  100 A of  FIG. 1A , the optical fiber  914  butt-coupled to the first step index waveguide  114 , and an optical modulator  920 , for example electro-absorption or Mach-Zehnder optical modulator, optically coupled to the second step index waveguide  124 . In operation, a modulated optical signal  906  propagates in the second step index waveguide  124 , and is coupled to the first step index waveguide  114  by the gradient index waveguide  104 , with a corresponding mode size transformation. The larger mode size of the first step index waveguide  114  may enable a low-loss, e.g. less than 1 dB, optical coupling to the optical fiber  914 . 
     The present disclosure is not to be limited in scope by the specific embodiments described herein. Indeed, other various embodiments and modifications, in addition to those described herein, will be apparent to those of ordinary skill in the art from the foregoing description and accompanying drawings. For example, in  FIGS. 2C-2F , the index profiles  214 ,  224  are illustrated as single steps; however, the step index profiles may have multiple steps, for example, when the step index waveguides  114 ,  124  comprise more than three layers. Thus, such other embodiments and modifications are intended to fall within the scope of the present disclosure. Further, although the present disclosure has been described herein in the context of a particular implementation in a particular environment for a particular purpose, those of ordinary skill in the art will recognize that its usefulness is not limited thereto and that the present disclosure may be beneficially implemented in any number of environments for any number of purposes. Accordingly, the claims set forth below should be construed in view of the full breadth and spirit of the present disclosure as described herein.