Patent Publication Number: US-8531770-B2

Title: Diffractive optical element and optical device

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application claims priority to Japanese Patent Application No. 2010-074039 filed on Mar. 29, 2010, the disclosure of which including the specification, the drawings, and the claims is hereby incorporated by reference in its entirety. 
     BACKGROUND 
     The present disclosure relates to a diffractive optical element in which two optical members are stacked, and a diffraction grating is formed at an interface between the two optical members, and to an optical device including the diffractive optical element. 
     A diffractive optical element has been known, in which a plurality of optical members are stacked in close contact with each other, and a relief pattern is formed at an interface between the optical members (see Japanese Patent Publication No. H09-127321). 
     In a diffractive optical element described in, e.g., Japanese Patent Publication No. H09-127321, a plurality of optical members are stacked, and a diffraction grating having a sawtooth cross section is formed at an interface between the optical members. 
     SUMMARY 
     At the interface where the diffraction grating is formed, an absolute value of a refractive index difference between mediums sandwiching the interface is smaller as compared to that at an interface between air and the optical member. Typically, a larger absolute value of a refractive index difference results in larger scattering at an interface. Thus, the scattering becomes smaller at the interface where the diffraction grating is formed. Since the scattering is smaller at the interface where the diffraction grating is formed, precise surface roughness as required for the interface between air and the optical member is not required for the interface where the diffraction grating is formed. Because a required surface roughness level is low, a manufacturing of the optical member is facilitated. 
     However, there is no findings regarding allowable surface roughness at the interface. 
     A technique disclosed herein has been made in view of the foregoing, and it is an objective of the present disclosure to facilitate a manufacturing of a close-contact type multilayer diffractive optical element while keeping scattering small. 
     A diffractive optical element disclosed herein includes a first optical member including a first diffraction grating having a sawtooth cross section; and a second optical member which includes a second diffraction grating having an inverted shape of the first diffraction grating and which has a refractive index different from that of the first optical member. The first and second optical members are stacked in a state in which the first and second diffraction gratings closely contact each other, and a diffraction surface of the first diffraction grating satisfies the following expression (1): 
     
       
         
           
             
               
                 
                   
                     4 
                     × 
                     Ra_s 
                   
                   ≤ 
                   Ra_a 
                   ≤ 
                   
                     
                       Ra_s 
                       · 
                       
                          
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             n 
                             d 
                           
                           ⁢ 
                           _s 
                         
                          
                       
                     
                     
                        
                       
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           n 
                           d 
                         
                       
                        
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     where “Ra_a” represents arithmetical mean roughness (μm) of the diffraction surface, “Δn d ” represents a refractive index difference between the first and second optical members, “Ra_s” represents arithmetical mean roughness (μm) of a surface causing the largest scattering among light entering surfaces other than the diffraction surface in the diffractive optical element, and “Δn d     —    s” represents a refractive index difference between mediums sandwiching the surface at which the scattering is largest. 
     A diffractive optical element disclosed herein includes a first optical member including a first diffraction grating having a sawtooth cross section; and a second optical member which includes a second diffraction grating having an inverted shape of the first diffraction grating and which has a refractive index different from that of the first optical member. The first and second optical members are stacked in a state in which the first and second diffraction gratings closely contact each other. The first diffraction grating is formed by a mold. An inverted structure of tool marks of the mold is formed in a diffraction surface of the first diffraction grating. The inverted structure is raised so as to have a segmental section. The diffraction surface of the first diffraction grating satisfies the following expression (2): 
     
       
         
           
             
               
                 
                   
                     4 
                     × 
                     Ra_s 
                   
                   ≤ 
                   
                     R 
                     ⁢ 
                     
                       { 
                       
                         1 
                         - 
                         
                           
                             1 
                             - 
                             
                               
                                 ( 
                                 
                                   P 
                                   
                                     2 
                                     ⁢ 
                                     R 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                       } 
                     
                   
                   ≤ 
                   
                     
                       Ra_s 
                       · 
                       
                          
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             n 
                             d 
                           
                           ⁢ 
                           _s 
                         
                          
                       
                     
                     
                        
                       
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           n 
                           d 
                         
                       
                        
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     where “R” represents a curvature radius (μm) of the cross section of the inverted structure, “P” represents a pitch (μm) of the inverted structure, Δn d  represents a refractive index difference between the first and second optical members, “Ra_s” represents arithmetical mean roughness (μm) of a surface causing the largest scattering among light entering surfaces other than the diffraction surface in the diffractive optical element, and “Δn d     —   s” represents a refractive index difference between mediums sandwiching the surface at which the scattering is largest. 
     Further, a diffractive optical element disclosed herein includes a first optical member including a first diffraction grating having a sawtooth cross section; and a second optical member which includes a second diffraction grating having an inverted shape of the first diffraction grating and which has a refractive index different from that of the first optical member. The first and second optical members are stacked in a state in which the first and second diffraction gratings closely contact each other, and a diffraction surface of the first diffraction grating satisfies the following expression (3): 
     
       
         
           
             
               
                 
                   
                     λ 
                     50 
                   
                   ≤ 
                   Ra_a 
                   ≤ 
                   
                     
                       Ra_s 
                       · 
                       
                          
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             n 
                             d 
                           
                           ⁢ 
                           _s 
                         
                          
                       
                     
                     
                        
                       
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           n 
                           d 
                         
                       
                        
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     where “λ” represents the shortest wavelength (μm) in a wavelength band to be used, “Ra_a” represents arithmetical mean roughness (μm) of the diffraction surface, “Δn d ” represents a refractive index difference between the first and second optical members, “Ra_s” represents arithmetical mean roughness (μm) of a surface causing the largest scattering among light entering surfaces other than the diffraction surface in the diffractive optical element, and “Δn d     —   s” represents a refractive index difference between mediums sandwiching the surface at which the scattering is largest. 
     An optical device disclosed herein includes an optical imaging system configured to focus light bundles on a predetermined surface. The optical imaging system includes the diffractive optical element. 
     According to the present disclosure, a manufacturing of the diffractive optical element can be facilitated while keeping the scattering small. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic view of a camera to which an interchangeable lens of an embodiment of the present disclosure is attached. 
         FIG. 2  is a schematic cross-sectional view of a diffractive optical element. 
         FIG. 3  is an enlarged cross-sectional view illustrating a detail of a structure at an interface between a first optical member and a second optical member. 
         FIG. 4  is a cross-sectional view illustrating a model in a simulation of a calculation of scatter components. 
         FIG. 5  is a graph showing a relationship of arithmetical mean roughness and a refractive index difference with scattering. 
     
    
    
     DETAILED DESCRIPTION 
     Example embodiments of the present disclosure will be described below in detail with reference to the drawings. 
     Embodiments 
       FIG. 1  is a schematic view of an interchangeable lens  200  including a diffractive optical element  1  of the example embodiment, and a camera  100  to which the interchangeable lens  200  is attached.  FIG. 2  is a schematic cross-sectional view of the diffractive optical element  1 . 
     The interchangeable lens  200  is detachable from the camera  100 . The interchangeable lens  200  is, e.g., a telephoto zoom lens. In the interchangeable lens  200 , the diffractive optical element  1  functions as a lens element in addition to refractive lenses  210 ,  220 . The refractive lenses  210 ,  220  and the diffractive optical element  1  form an optical imaging system  230  configured to focus light bundles on an imaging device  110  of the camera  100 . The interchangeable lens  200  forms an optical device. 
     The diffractive optical element  1  is a close-contact type multilayer diffractive optical element in which a first optical member  10  and a second optical member  11  having light transmission properties are stacked in close contact with each other. In the present embodiment, the first optical member  10  is made of glass, and the second optical member  11  is made of resin. Note that the materials of the first and second optical members  10 ,  11  are not limited to the foregoing materials. The first optical member  10  and the second optical member  11  are bonded together. A diffraction grating  13  having a sawtooth cross section is formed at an interface  12  defined by a bonding surface  10   a  of the first optical member  10  and a bonding surface  11   a  of the second optical member  11 . Optical power of the diffraction grating  13  has wavelength dependency. Thus, the diffraction grating  13  provides the substantially same phase difference to light having different wavelengths, and diffracts the light having different wavelengths at diffraction angles which are different from each other. The diffraction grating  13  is formed of a first diffraction grating  14  formed in the bonding surface  10   a  of the first optical member  10  and having a sawtooth cross section, and a second diffraction grating  15  formed in the bonding surface  11   a  of the second optical member  11  and having a sawtooth cross section. 
     Specifically, the first diffraction grating  14  includes a plurality of ridge-like raised portions  14   a  which extend in a circumferential direction around an optical axis X of the diffractive optical element  1 , and which are concentrically and regularly arranged around the optical axis X. Each of the raised portions  14   a  has a vertical surface  14   b  which is substantially parallel to the optical axis X (i.e., extends along the optical axis X), and a diffraction surface  14   c  which is inclined to the optical axis X (i.e., inclined to the vertical surface  14   b ). Each of the raised portions  14   a  has a substantially triangular cross section. 
     The second diffraction grating  15  includes a plurality of valley-like recessed portions  15   a  which extend in the circumferential direction around the optical axis X of the diffractive optical element  1 , and which are concentrically and regularly arranged around the optical axis X. Each of the recessed portions  15   a  has a vertical surface  15   b  which is substantially parallel to the optical axis X, and a diffraction surface  15   c  which is inclined to the optical axis X. Each of the recessed portions  15   a  has a substantially triangular cross section. 
     The first diffraction grating  14  and the second diffraction grating  15  have the same grating height and the same grating pitch. That is, the second diffraction grating  15  has an inverted shape of the first diffraction grating  14 , and the first diffraction grating  14  and the second diffraction grating  15  are engaged with each other in a state in which the raised portions  14   a  of the first diffraction grating  14  closely contact the recessed portions  15   a  of the second diffraction grating  15 . Thus, the first diffraction grating  14  and the second diffraction grating  15  monolithically form the diffraction grating  13 . 
     Note that the diffraction surface  14   c  of the first diffraction grating  14  and the diffraction surface  15   c  of the second diffraction grating  15  may be curved so as to define an aspherical or spherical surface. 
     A surface  10   b  of the first optical member  10  on an opposite side of the bonding surface  10   a , and a surface  11   b  of the second optical member  11  on an opposite side of the bonding surface  11   a  are formed into flat surfaces parallel to each other. As illustrated in  FIG. 1 , e.g., light entering the diffractive optical element  1  from the first optical member  10  side is diffracted at the diffraction grating  13  (specifically the diffraction surfaces  14   c ,  15   c ) to exit to the second optical member  11  side. Note that the surface  10   b  of the first optical member  10  and the surface  11   b  of the second optical member  11  may not be parallel to each other. 
     Next, an example of a manufacturing method of the diffractive optical element  1  configured as described above will be briefly described. First, a mold having the inverted shape of the first diffraction grating  14  is prepared. The mold is filled with softened glass (e.g., glass heated to a temperature equal to or higher than a glass transition point or molten glass). Then, the first optical member  10  is molded. Subsequently, the first optical member  10  is arranged in another mold so that the first diffraction grating  14  faces an inside of the mold. The mold in which the first optical member  10  is arranged is filled with molten resin. Then, the second optical member  11  is stacked on the first diffraction grating  14  of the first optical member  10 . In such a state, the molten resin flows into the valley portions of the first diffraction grating  14 , thereby molding the second optical member  11  which has the second diffraction grating  15  contacting the first diffraction grating  14 . Note that the manufacturing method is one example, and any manufacturing methods may be applied as long as the diffractive optical element  1  can be manufactured. For example, if the second optical member  11  is made of glass, the diffractive optical element  1  in which the second optical member  11  is stacked on the first optical member  10  may be manufactured as follows: a mold in which the first optical member  10  is arranged is filled with soften glass (e.g., glass heated to a temperature equal to or higher than a glass transition point or molten glass), and the soften glass is pressed against the first diffraction grating  14  of the first optical member  10 . In either case of forming the second optical member  11  with resin or glass, soften optical material is deformed into a shape patterned after the first diffraction grating  14  of the first optical member  10 , and the second diffraction grating  15  is formed in close contact with the first diffraction grating  14 . 
     Even a microscopic recessed-raised structure of the diffraction surface  14   c  of the first diffraction grating  14  is transferred to the diffraction surface  15   c  of the second diffraction grating  15 . Consequently, the recessed-raised structure of the diffraction surface  14   c  of the first diffraction grating  14  exactly fits the recessed-raised structure of the diffraction surface  15   c  of the second diffraction grating  15 . In other words, even the microscopic recessed-raised structure of the diffraction surface  15   c  of the second diffraction grating  15  is formed in shape patterned after the diffraction surface  14   c  of the first diffraction grating  14 . That is, a surface shape of the diffraction surface  15   c  of the second diffraction grating  15  is an inverted surface shape of the diffraction surface  14   c  of the first diffraction grating  14 . Thus, surface roughness of the diffraction surface  14   c  of the first diffraction grating  14  is surface roughness at the interface  12  between the first optical member  10  and the second optical member  11 . As will be described later, the surface roughness of the diffraction surfaces  14   c ,  15   c  has an influence on scattering at the diffraction surfaces  14   c ,  15   c.    
     The surface roughness of the diffraction surface  14   c  of the first diffraction grating  14  is determined depending on surface roughness of the mold. That is, a surface shape of a portion corresponding to the diffraction surface  14   c  in the mold of the first optical member  10  is transferred to the diffraction surface  14   c  of the first diffraction grating  14 . In other words, the surface roughness of the diffraction surface  14   c  of the first diffraction grating  14  is the substantially same as the surface roughness of the mold in the portion corresponding to the diffraction surface  14   c.    
     The mold surface (particularly a surface having a complex shape such as the inverted shape of the first diffraction grating  14 ) is typically finished by cutting with a turning tool. Specifically, the turning tool having a curvature radius R at a tip end cuts the mold surface while moving with a predetermined fine pitch P. In such a manner, a plurality of grooves having an arc-shaped cross section with the radius R are formed as tool marks in the mold surface with the pitch P. 
     As illustrated in  FIG. 3 , the tool marks of the mold surface are transferred to the diffraction surface  14   c  of the first diffraction grating  14  formed by the mold.  FIG. 3  is a cross-sectional view of the first optical member  10  and the second optical member  11 . For convenience of description, the diffraction surfaces  14   c ,  15   c  are drawn in the horizontal direction. Specifically, a plurality of raised striations  14   d , each of which has the segmental cross section with the radius R, are formed in the diffraction surface  14   c  of the first optical member  10  with the pitch P. The striations  14   d  form an inverted structure of the tool marks. The surface shape of the diffraction surface  14   c  is transferred to the diffraction surface  15   c  of the second optical member  11 , and therefore a plurality of grooves  15   d , each of which has the arc-shaped cross section with the radius R, are formed in the diffraction surface  15   c  of the second optical member  11  with the pitch P. The striations  14   d  exactly fit the grooves  15   d.    
     The curvature radius R at the tip end of the turning tool and the pitch P are determined depending on surface roughness required for the mold surface (i.e., surface roughness required for the diffraction surface  14   c  of the first diffraction grating  14 ) and a shape required for the ridge portion (a crest defined by the vertical surface  14   b  and the diffraction surface  14   c ) of the first diffraction grating  14 . In the cutting, the curvature radius R at the tip end of the turning tool is increased, resulting in smaller surface roughness. On the other hand, the degree of sharpening at the valley bottom portion having the inverted shape of the ridge portion of the first diffraction grating  14  in the mold is determined depending on the curvature radius R at the tip end of the turning tool. That is, a larger curvature radius R at the tip end of the turning tool results in a rounder valley bottom portion of the mold. Consequently, the ridge portion of the first diffraction grating  14  is also rounded. The rounded ridge portion does not contribute to diffraction. Thus, the first diffraction grating  14  cannot properly diffract light, thereby degrading diffraction efficiency. In the view of the foregoing, it is necessary to decrease the curvature radius R at the tip end of the turning tool. However, if the curvature radius R at the tip end of the turning tool is decreased, the surface roughness becomes larger as described above. In order to avoid the large surface roughness, if the curvature radius R at the tip end of the turning tool is small, it is necessary to narrow the moving pitch of the turning tool. However, in such a case, a cutting distance by the turning tool (moving distance of the turning tool during the mold processing) is increased. The longer cutting distance may cause abrasion or breakage of the turning tool. In addition, due to the longer cutting distance, environmental conditions such as a temperature and a humidity during the mold processing are changed, thereby providing adverse effects on finished quality of the mold. As in such a case, it is not easy to manufacture the mold having the smaller surface roughness. 
     Considering the scattering at the diffraction surface  14   c , the surface roughness of the diffraction surface  14   c  may not be so small. Specifically, if surface roughness at an interface becomes larger, scattering at such an interface is typically increased. However, a factor determining the degree of scattering is not only surface roughness at an interface but also a refractive index difference between mediums sandwiching the interface. That is, larger surface roughness at an interface and a larger absolute value of a refractive index difference between mediums sandwiching the interface result in larger scattering at the interface. The mediums sandwiching the diffraction surface  14   c  are the first optical member  10  and the second optical member  11 . An absolute value of a refractive index difference between the mediums sandwiching the diffraction surface  14   c  is smaller than that between the surface  10   b  of the first optical member  10  or the surface  11   b  of the second optical member  11  and air. In the case where the absolute value of the refractive index difference between the mediums sandwiching the interface is small, even if the surface roughness at the interface becomes somewhat larger, the scattering at the interface is not significantly increased. 
     The surface roughness of the diffraction surface  14   c  of the first diffraction grating  14  satisfies the following expression (1): 
     
       
         
           
             
               
                 
                   
                     4 
                     × 
                     Ra_s 
                   
                   ≤ 
                   Ra_a 
                   ≤ 
                   
                     
                       Ra_s 
                       · 
                       
                          
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             n 
                             d 
                           
                           ⁢ 
                           _s 
                         
                          
                       
                     
                     
                        
                       
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           n 
                           d 
                         
                       
                        
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     where “Ra_a” represents arithmetical mean roughness (μm) of the diffraction surface  14   c , “Δn d ” represents a refractive index difference between the first optical member  10  and the second optical member  11 , “Ra_s” represents arithmetical mean roughness (μm) of a surface causing the largest scattering among light entering surfaces other than the diffraction surface  14   c  in the diffractive optical element  1 , and “Δn d     —   s” represents a refractive index difference between mediums sandwiching the surface at which the scattering is largest. 
     That is, in the present embodiment, the arithmetical mean roughness Ra_a of the diffraction surface  14   c  is set to equal to or greater than 4×Ra_s. In other words, the arithmetical mean roughness Ra_a of the diffraction surface  14   c  may be more than four times larger than the arithmetical mean roughness Ra_s of the surface causing the largest scattering among light entering surfaces other than the diffraction surface  14   c  in the diffractive optical element  1 . Thus, if the arithmetical mean roughness Ra_a of the diffraction surface  14   c  is equal to or greater than 4×Ra_s, arithmetical mean roughness Ra of the mold surface may be also equal to or greater than 4×Ra_s. In such a case, the cutting distance during the mold processing is substantially half as compared to a case where the arithmetical mean roughness Ra of the mold surface is Ra_s (i.e., a case where the mold is processed so as to have surface roughness equal to that of the surface causing the largest scattering among light entering surfaces other than the diffraction surface  14   c  in the diffractive optical element  1 ). Consequently, the mold can be easily manufactured, thereby facilitating the manufacturing (including the manufacturing of the mold) of the diffractive optical element  1 . 
     Specifically, in the diffractive optical element  1  in which the first and second optical members  10 ,  11  are stacked, the surface causing the largest scattering among light entering surfaces other than the diffraction surface  14   c  in the diffractive optical element  1  is the surface  10   b  of the first optical member  10  or the surface  11   b  of the second optical member  11 . The surfaces  10   b ,  11   b  are interfaces with air, and arithmetical mean roughness Ra of the surfaces  10   b ,  11   b  is typically set to about 0.002 μm. In a method for measuring surface roughness, such as an optical interference method, an asperity of about 0.0025-0.005 μm is a typically limit of the measurement. In other words, light cannot recognize an asperity smaller than about 0.0025-0.005 μm. Thus, an interface with air, at which light enters is typically set to arithmetical mean roughness Ra of about 0.002 μM in order to reduce scattering. 
     As a result, the expression (1) provides the following expression (4): 
     
       
         
           
             
               
                 
                   
                     0.008 
                     ⁢ 
                     
                         
                     
                     [ 
                     µm 
                     ] 
                   
                   ≤ 
                   Ra_a 
                   ≤ 
                   
                     
                       Ra_s 
                       · 
                       
                          
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             n 
                             d 
                           
                           ⁢ 
                           _s 
                         
                          
                       
                     
                     
                        
                       
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           n 
                           d 
                         
                       
                        
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     That is, the arithmetical mean roughness Ra_a of the diffraction surface  14   c  is set to equal to or greater than 0.008 μm. The surface roughness of the diffraction surface  14   c  may be significantly larger as compared to the arithmetical mean roughness Ra of the surfaces  10   b ,  11   b  which are the interfaces with air. Thus, if the arithmetical mean roughness Ra_a of the diffraction surface  14   c  is equal to or greater than 0.008 μm, the arithmetical mean roughness Ra of the mold surface may be also equal to or greater than 0.008 μm. In such a case, the cutting distance during the mold processing is substantially half as compared to a case where the arithmetical mean roughness Ra of the mold surface is about 0.002 μm. Consequently, the mold can be easily manufactured, thereby facilitating the manufacturing (including the manufacturing of the mold) of the diffractive optical element  1 . 
     Meanwhile, an upper limit of the arithmetical mean roughness Ra_a of the diffraction surface  14   c  is set as in the expression (1), thereby keeping the scattering small at the diffraction surface  14   c . That is, as described above, a larger surface roughness at an interface and a larger absolute value of a refractive index difference between mediums sandwiching the interface result in larger scattering at the interface. As will be described in detail later, a product of the arithmetical mean roughness Ra and the absolute value of the refractive index difference Δn d  can be used as a parameter representing the degree of scattering (hereinafter referred to as a “scattering parameter”). That is, a greater scattering parameter results in larger scattering at an interface. The diffraction surface  14   c  is configured so that a scattering parameter at the diffraction surface  14   c  is equal to or less than a scattering parameter at a surface causing the largest scattering among light entering surfaces other than the diffraction surface  14   c  in the diffractive optical element  1  (hereinafter referred to as a “maximum scattering surface”). That is, a relationship between the scattering parameter at the diffraction surface  14   c  and the scattering parameter at the maximum scattering surface (e.g., the surface  10   b  of the first optical member  10 ) satisfies the following expression (5):
 
 Ra   —   a·Δn   d   ≦Ra   —   s·Δn   d     —     s   (5)
 
     The expression (5) is rearranged for the arithmetical mean roughness Ra_a of the diffraction surface  14   c , thereby specifying the upper limit obtained by the expression (1). That is, if the arithmetical mean roughness Ra_a of the diffraction surface  14   c  is set to equal to or less than the upper limit obtained by the expression (1), the scattering at the first diffraction grating  14  can be kept smaller at least as compared to the scattering at the maximum scattering surface. 
     Thus, the first diffraction grating  14  is configured so that the arithmetical mean roughness Ra_a of the diffraction surface  14   c  satisfies the expression (1), thereby keeping the scattering small and facilitating the manufacturing of the diffractive optical element  1 . 
     As described above, the tool marks of the mold are transferred to the diffraction surface  14   c  of the first optical member  10 , and the striations  14   d  are formed in the diffraction surface  14   c  of the first optical member  10  as the inverted structure of the tool marks. When a curvature radius in a cross section of the striation  14   d  of the diffraction surface  14   c  is “R (μm),” and a pitch between the striations  14   d  is “P (μm),” the first diffraction grating  14  satisfies the following expression (2): 
     
       
         
           
             
               
                 
                   
                     4 
                     × 
                     Ra_s 
                   
                   ≤ 
                   
                     R 
                     ⁢ 
                     
                       { 
                       
                         1 
                         - 
                         
                           
                             1 
                             - 
                             
                               
                                 ( 
                                 
                                   P 
                                   
                                     2 
                                     ⁢ 
                                     R 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                       } 
                     
                   
                   ≤ 
                   
                     
                       Ra_s 
                       · 
                       
                          
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             n 
                             d 
                           
                           ⁢ 
                           _s 
                         
                          
                       
                     
                     
                        
                       
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           n 
                           d 
                         
                       
                        
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     If the plurality of raised striations  14   d , each of which has the segmental cross section with the radius R, are formed in the diffraction surface  14   c  with the pitch P, the arithmetical mean roughness Ra_a of the diffraction surface  14   c  is represented by the following expression (6): 
     
       
         
           
             
               
                 
                   Ra_a 
                   = 
                   
                     R 
                     ⁢ 
                     
                       { 
                       
                         1 
                         - 
                         
                           
                             1 
                             - 
                             
                               
                                 ( 
                                 
                                   P 
                                   
                                     2 
                                     ⁢ 
                                     R 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     That is, by substituting the expression (6) into the expression (1), the expression (2) is obtained. If the first diffraction grating  14  is configured so that not the arithmetical mean roughness Ra_a of the diffraction surface  14   c  but the curvature radius R and the pitch P in the inverted structure of the tool marks transferred to the diffraction surface  14   c  satisfy the relationship of the expression (2), as described above, the manufacturing of the diffractive optical element  1  can be facilitated while keeping the scattering small. 
     Note that, if arithmetical mean roughness Ra of the surface causing the largest scattering among light entering surfaces other than the diffraction surface  14   c  in the diffractive optical element  1  is 0.002 μm, the expression (2) provides the following expression (7): 
     
       
         
           
             
               
                 
                   
                     0.008 
                     ⁢ 
                     
                         
                     
                     [ 
                     µm 
                     ] 
                   
                   ≤ 
                   
                     R 
                     ⁢ 
                     
                       { 
                       
                         1 
                         - 
                         
                           
                             1 
                             - 
                             
                               
                                 ( 
                                 
                                   P 
                                   
                                     2 
                                     ⁢ 
                                     R 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                       } 
                     
                   
                   ≤ 
                   
                     
                       Ra_s 
                       · 
                       
                          
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             n 
                             d 
                           
                           ⁢ 
                           _s 
                         
                          
                       
                     
                     
                        
                       
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           n 
                           d 
                         
                       
                        
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     Further, the arithmetical mean roughness Ra_a of the diffraction surface  14   c  of the first diffraction grating  14  may be set so as to satisfy the following expression (3) when the shortest wavelength in the wavelength band to be used is λ (μm): 
     
       
         
           
             
               
                 
                   
                     λ 
                     50 
                   
                   ≤ 
                   Ra_a 
                   ≤ 
                   
                     
                       Ra_s 
                       · 
                       
                          
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             n 
                             d 
                           
                           ⁢ 
                           _s 
                         
                          
                       
                     
                     
                        
                       
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           n 
                           d 
                         
                       
                        
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     According to the expression (3), the arithmetical mean roughness Ra_a of the diffraction surface  14   c  is set to equal to or greater than λ/50. If a target for the diffractive optical element  1  is visible light, the wavelength band to be used is about 400-800 nm. In such a case, a lower limit of the arithmetical mean roughness Ra_a of the diffraction surface  14   c  is about 0.008 μm. As described above, such a limit is four times as large as the typical value of the arithmetical mean roughness Ra at the interface with air (0.002 μm), and is a value at which the cutting distance during the mold processing is substantially half as compared to the cutting distance at such an interface. Thus, the manufacturing (including the manufacturing of the mold) of the diffractive optical element  1  can be facilitated. 
     If the inverted structure of the tool marks, which is transferred to the diffraction surface  14   c  is the striation  14   d  having the curvature radius R and the pitch P, the expression (3) provides the following expression (8): 
     
       
         
           
             
               
                 
                   
                     λ 
                     50 
                   
                   ≤ 
                   
                     R 
                     ⁢ 
                     
                       { 
                       
                         1 
                         - 
                         
                           
                             1 
                             - 
                             
                               
                                 ( 
                                 
                                   P 
                                   
                                     2 
                                     ⁢ 
                                     R 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                       } 
                     
                   
                   ≤ 
                   
                     
                       Ra_s 
                       · 
                       
                          
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             n 
                             d 
                           
                           ⁢ 
                           _s 
                         
                          
                       
                     
                     
                        
                       
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           n 
                           d 
                         
                       
                        
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     That is, the first diffraction grating  14  is configured so that the curvature radius R and the pitch P of the inverted structure of the tool marks, which is transferred to the diffraction surface  14   c  satisfy a relationship of the expression (8). This facilitates the manufacturing of the diffractive optical element  1  while keeping the scattering small as described above. 
     A relationship of surface roughness at an interface and a refractive index difference between mediums sandwiching the interface with scattering will be described with reference to  FIGS. 4 and 5 .  FIG. 4  illustrates a model of a multilayer optical element used for a simulation, and  FIG. 5  illustrate results of the scattering simulation. 
     First, suppose that a model  3  illustrated in  FIG. 4  was used. In the model  3 , a first optical member  31  and a second optical member  32  are stacked, and a recessed-raised structure is formed at an interface  33  between the first optical member  31  and the second optical member  32 . Specifically, a plurality of raised striations  31   c , each of which has a segmental cross section with a radius R, are formed at an interface  31   a  of the first optical member  31  with a pitch P. On the other hand, a plurality of grooves  32   c , each of which has the arc-shaped cross section with the radius R, are formed at an interface  32   a  of the second optical member  32  with the pitch P. The first optical member  31  and the second optical member  32  are stacked in a state in which the striations  31   c  at the interface  31   a  and the grooves  32   c  at the interface  32   a  closely contact each other. A refractive index at a d line (0.587562 μm) of the first optical member  31  is “n d1 ,” and a refractive index at a d line of the second optical member  32  is “n d2  (&lt;n d1 ).” Suppose that light at the d line enters the first optical member  31  through a surface  31   b , and exits from the second optical member  32  through a surface  32   b . At this point, a part of the light is scattered at the interface  31   a  ( 32   a ). A proportion of scatter components was calculated by a rigorous coupled-wave analysis (RCWA) method while changing arithmetical mean roughness Ra of the interface  31   a  ( 32   a ) and a refractive index difference Δn d  (=n d1 −n d2 ) at the interface  31   a  ( 32   a ). A proportion of a scattered light amount to a total transmitted light amount (=a zero-order light amount+a total scattered light amount) was considered as the proportion of scatter components. Specifically, the pitch P was set to about 0.001 μm, and the radius R was changed within a range of about 1.1 μm-about 62.5 μm, thereby changing the arithmetical mean roughness Ra of the interface  31   a  ( 32   a ). Note that Ra=about 0.120 μm when R=about 1.1 μm, or Ra=about 0.002 μm when R=about 62.5 μm. The arithmetical mean roughness Ra was calculated based on the radius R in the cross section and the pitch P by using the expression (3). The refractive index n d1  of the first optical member  31  was fixed to about 1.7 (equivalent to a refractive index of glass K—LaFn3 manufactured by Sumita Optical Glass Inc.), and the refractive index n d2  of the second optical member  32  was changed. Consequently, the refractive index difference Δn d  was changed. 
     As can be appreciated from the simulation results of  FIG. 5 , greater arithmetical mean roughness Ra results in an increase in proportion of scatter components. In addition, a greater absolute value of a refractive index difference Δn d  results in a greater proportion of scatter components. For example, when an absolute value of the refractive index difference Δn d  is reduced by half, arithmetical mean roughness Ra is substantially doubled. That is, when an absolute value of the refractive index difference Δn d  is in 1/n, arithmetical mean roughness Ra is multiplied by n times in order to realize the substantially same degree of scattering as that before changing the absolute value of the refractive index difference Δn d . Thus, the substantially same scattering parameters (surface roughness Ra at the center line×an absolute value of the refractive index difference Δn d ) produce the substantially same degree of scattering. That is, the upper limit of the arithmetical mean roughness Ra_a of the diffraction surface  14   c  is set as in the expression (1), and therefore scattering at the diffraction surface  14   c  can be kept to substantially equal to or less than scattering at the maximum scattering surface of the diffractive optical element  1 . 
     EXAMPLES 
     Examples of the diffractive optical element will be described below. 
     First Example 
     In a first example, a first optical member  10  was made of hypothetical glass having a refractive index n 1 (λ d ) of about 1.57 and an Abbe number ν d1  of about 71.2 at a d line, and a second optical member  11  was made of hypothetical resin having a refractive index n 2 (λ d ) of about 1.54 and an Abbe number ν d2  of about 42.1 at the d line. Consequently, a refractive index difference Δn d  of about 0.03 was obtained. A blaze wavelength λ b  was set to the d line. In such a state, a grating height was about 19.59 (μm). In addition, a pitch P of a diffraction grating was about 0.1 (mm). In order to realize higher diffraction efficiency, a curvature radius R of a ridge portion of the diffraction grating was set to about 10 (μm). That is, a condition was assumed, in which a turning tool having a curvature radius of about 10 (μm) at a tip end cuts a mold for the diffraction grating. 
     Suppose that, e.g., a mold corresponding to a the first optical member  10  having a diffraction surface diameter of about 60 (mm) is processed. If the mold is processed by the turning tool having the curvature radius of about 10 (μm) at the tip end so that arithmetical mean roughness Ra of a mold surface is equal to or less than about 0.002 (μm) required for a typical optical element surface, a moving pitch of the turning tool is about 0.40 (μm), and a cutting distance is equal to or longer than about 7000 (m). However, in the case where the refractive index difference Δn d  between mediums sandwiching the diffraction surface is about 0.03 as in the present example, if a surface  10   b  of the first optical member  10  is a surface at which scattering is largest, and arithmetical mean roughness Ra of the surface  10   b  is about 0.002 (μm) (the value required for the typical optical element surface), the arithmetical mean roughness Ra at of the mold surface may be equal to or less than about 0.038 (μm) based on the expression (1). If the mold surface is processed by the turning tool having the curvature radius of about 10 (μm) at the tip end so that the arithmetical mean roughness Ra is about 0.038 (μm), the moving pitch of the turning tool is about 1.74 (μm). Consequently, a cutting distance of about 1600 (m) is obtained. The cutting distance is significantly shortened as compared to the case where the arithmetical mean roughness is about 0.002 (μm). 
     By forming the first optical member  10  with the mold formed in this manner, a diffraction surface  14   c  of the first optical member  10  has surface roughness similar to that of the mold. That is, arithmetical mean roughness Ra of the diffraction surface  14   c  is about 0.038 (μm). If the diffraction surface  14   c  contacts air, and a proportion of scatter components is calculated by a simulation using the RCWA method, about 0.50(%) of light at the d line, which vertically enters the diffraction surface  14   c  is changed into scattered light. On the other hand, if the second optical member  11  having a refractive index difference Δn d  of about 0.03 from the first optical member  10  is stacked on the first optical member  10 , about 0.01(%) or less of the light at the d line, which vertically enters the diffraction surface  14   c  is changed into scattered light. As compared to the case where the diffraction surface  14   c  contacts airspace, the scattering is significantly reduced. 
     Second Example 
     In a second example, a first optical member  10  was made of glass (glass K-VC78 manufactured by Sumita Optical Glass Inc.) having a refractive index n 1 (λ d ) of about 1.66955 and an Abbe number ν d1  of about 55.4 at a d line, and a second optical member  11  was made of hypothetical resin having a refractive index n 2 (λ d ) of about 1.606 and an Abbe number ν d2  of about 27.3 at the d line. Consequently, a refractive index difference Δn d  of about 0.06355 was obtained. A blaze wavelength λ b  was set to the d line. In such a state, a grating height was about 9.25 (μm). In addition, a pitch P of a diffraction grating was about 0.4 (mm). In order to realize higher diffraction efficiency, a curvature radius R of a ridge portion of the diffraction grating was set to about 15 (μm). That is, a condition was assumed, in which a turning tool having a curvature radius of about 15 (μm) at a tip end cuts a mold for the diffraction grating. 
     Suppose that, e.g., a mold corresponding to a the first optical member  10  having a diffraction surface diameter of about 80 (mm) is processed. If the mold is processed by the turning tool having the curvature radius of about 15 (μm) at the tip end so that arithmetical mean roughness Ra of a mold surface is equal to or less than about 0.002 (μm), a moving pitch of the turning tool is about 0.49 (μm), and a cutting distance is equal to or longer than about 10000 (m). However, in the case where the refractive index difference Δn d  between mediums sandwiching the diffraction surface is about 0.06355 as in the present example, if a surface  10   b  of the first optical member  10  is a surface at which scattering is largest, and arithmetical mean roughness Ra of the surface  10   b  is about 0.002 (μm) (a value required for a typical optical element surface), the arithmetical mean roughness Ra of the mold surface may be equal to or less than about 0.021 (μm) based on the expression (1). If the mold surface is processed by the turning tool having the curvature radius of about 15 (μm) at the tip end so that the arithmetical mean roughness Ra is about 0.021 (μm), the moving pitch of the turning tool is about 1.59 (μm). Consequently, a cutting distance of about 3200 (m) is obtained. The cutting distance is significantly shortened as compared to the case where the arithmetical mean roughness is about 0.002 (μm). 
     By forming the first optical member  10  with the mold formed in this manner, a diffraction surface  14   c  of the first optical member  10  has surface roughness similar to that of the mold. That is, arithmetical mean roughness Ra of the diffraction surface  14   c  is about 0.021 (μm). If the diffraction surface  14   c  contacts air, and a proportion of scatter components is calculated by a simulation using the RCWA method, about 0.22(%) of light at the d line, which vertically enters the diffraction surface  14   c  is changed into scattered light. On the other hand, if the second optical member  11  having a refractive index difference Δn d  of about 0.06355 from the first optical member  10  is stacked on the first optical member  10 , about 0.01(%) or less of the light at the d line, which vertically enters the diffraction surface  14   c  is changed into scattered light. As compared to the case where the diffraction surface  14   c  contacts airspace, the scattering is significantly reduced. 
     Third Example 
     Suppose that a mold surface for a diffraction grating is cut by a turning tool having a curvature radius of about 20 (μm) at its tip end, with a pitch with which arithmetical mean roughness Ra is about 0.008 (μm); and that a first optical member  10  is formed with such a mold. In addition, suppose that a refractive index n 1 (λ d ) of the first optical member  10  at a d line is about 1.85, and an Abbe number ν d1  is about 42. An amount of scattered light when light having a wavelength of about 400 nm vertically enters a diffraction surface of the first optical member  10  configured in this manner was simulated by the RCWA method. The wavelength of about 400 nm is the shortest wavelength when light to be used is visible light. 
     If the diffraction surface of the first optical member  10  contacts air, about 0.11(%) of the light vertically entering the diffraction surface and having the wavelength of about 400 nm becomes scattered light. On the other hand, if the diffraction surface of the first optical member  10  contacts an optical member having a refractive index n 1 (λ d ) of about 1.70 and an Abbe number ν d1  of about 17 at the d line, less than about 0.01(%) of the light vertically entering the diffraction surface and having the wavelength of about 400 nm becomes scattered light. This shows that the scattering is significantly reduced as compared to the case where the diffraction surface contacts air. 
     Other Embodiments 
     The present disclosure may have the following configurations in the foregoing embodiment. 
     That is, in the foregoing embodiment, the diffractive optical element  1  is employed in the interchangeable lens  200 , but the present disclosure is not limited to such a configuration. The diffractive optical element  1  may be applied as a lens element inside the camera  100 . In addition, the present disclosure is not limited to the diffractive optical element  1  serving as a lens, and the diffractive optical element  1  may be applied for purposes other than the foregoing purpose. 
     The shape of the first and second diffraction gratings  14 ,  15  is not limited to that in the foregoing embodiment. That is, the first and second diffraction gratings  14 ,  15  may be formed in any shapes as long as the first and second diffraction gratings  14 ,  15  has the sawtooth cross section. 
     Further, the shape of the first and second diffraction gratings  14 ,  15  may be rectangular in cross section, or trapezoidal in cross section. 
     In the foregoing embodiment, the first optical member  10  is made of resin material, and the second optical member  11  is made of glass material. However, the present disclosure is not limited to such a configuration. The first optical member  10  may be made of glass material, and the second optical member  11  may be made of resin material, or both of the first and second optical members  10 ,  11  may be made of glass material or resin material. Alternatively, transparent ceramic may be used instead of glass material and resin material. In short, the first and second optical members  10 ,  11  may be made of material having transparency in a wavelength band to be used. 
     In a case where the inverted structure of the tool marks formed in the mold by the turning tool is also formed in the maximum scattering surface of the diffractive optical element  1 , if a curvature radius of a cross section of the inverted structure of the tool marks in the maximum scattering surface is “R_s,” and a pitch in such an inverted structure is “P_s,” the first diffraction grating  14  may satisfy the following expression (9): 
     
       
         
           
             
               
                 
                   
                     4 
                     × 
                     A 
                   
                   ≤ 
                   
                     R 
                     ⁢ 
                     
                       { 
                       
                         1 
                         - 
                         
                           
                             1 
                             - 
                             
                               
                                 ( 
                                 
                                   P 
                                   
                                     2 
                                     ⁢ 
                                     R 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                       } 
                     
                   
                   ≤ 
                   
                     A 
                     · 
                     
                       
                          
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             n 
                             d 
                           
                           ⁢ 
                           _s 
                         
                          
                       
                       
                          
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             n 
                             d 
                           
                         
                          
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     where “A” is a value satisfying the following expression (10): 
     
       
         
           
             
               
                 
                   A 
                   = 
                   
                     R_s 
                     ⁢ 
                     
                       { 
                       
                         1 
                         - 
                         
                           
                             1 
                             - 
                             
                               
                                 ( 
                                 
                                   P_s 
                                   
                                     2 
                                     · 
                                     R_s 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     Note that a lower limit of the expression (9) may be about 0.008 (μm), or may be λ/50 when the shortest wavelength in the wavelength band to be used is λ. 
     If the diffractive optical element  1  and other optical elements such as lenses together form an optical system, the maximum scattering surface may be a surface causing the largest scattering among light entering surfaces other than the diffraction surface  14   c  of the diffractive optical element  1  in the optical system. 
     The foregoing embodiments have been set forth merely for purposes of preferred examples in nature, and are not intended to limit the scope, applications, and use of the present disclosure. 
     As described above, the present disclosure is useful for the diffractive optical element in which the two optical members are stacked, and the diffraction grating is formed at the interface between the two optical members, and for the optical device including the diffractive optical element.