Abstract:
A scanning optical system that images a laser beam on a surface to be scanned, said laser beam being emitted from a light source and deflected by a deflector in order to scan the surface to be scanned, said scanning optical system, has a first optical functional surface being a reflecting surface having a “twisted” configuration; and a second optical functional surface having a “twisted” configuration different from the “twisted” configuration of the first optical functional surface.

Description:
RELATED APPLICATION 
     This application is based on application No. 11-187194 filed in Japan, the content of which is hereby incorporated by reference. 
     1. Field of the Invention 
     The present invention relates to a scanning optical system and a laser scanning apparatus using the scanning optical system, and more particularly, to a laser scanning apparatus suitable for use as a writing optical apparatus such as a laser beam printer. 
     2. Description of the Prior Art 
     Conventionally, in the field of scanning optical systems, a technology has been proposed to form a laser scanning optical system for scanning a surface at a substantially constant speed by use of a so-called toroidal mirror. An example is described in U.S. Pat. No. 5,353,047. Moreover, a technology has been proposed to form a scanning optical system that realizes high resolution although being small in size and low in cost by use of two rotationally symmetrical mirrors, for example, as described in Japanese Laid-open Patent Application No 08-220440. 
     However, in the above-described structures of the conventional scanning optical systems, only movement and rotation of the rotationally symmetrical surfaces are provided as the degree of freedom for absorbing the asymmetry in the sub scanning direction associated with optical path separation, and it is difficult to simultaneously resolve the problems of so-called bow and out-of-shape beam due to the asymmetry in the sub scanning direction. 
     SUMMARY OF THE INVENTION 
     An object of the present invention is to provide an improved scanning optical system. 
     Another object of the present invention is to provide a scanning optical system having better imaging performance although using a mirror. 
     The above-mentioned objects are achieved by a scanning optical system comprising: a first optical functional surface being a reflecting surface having a “twisted” configuration; and a second optical functional surface having a “twisted” configuration different from the “twisted” configuration of the first optical functional surface. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     These and other objects and features of this invention will become clear from the following description, taken in conjunction with the preferred embodiments with reference to the accompanied drawings in which: 
     FIG. 1 is a perspective view showing a schematic structure of a first embodiment of the present invention 
     FIGS.  2 ( a ) and  2 ( b ) are views showing the structure of a scanning optical system according to the first embodiment of the present invention; 
     FIGS.  3 ( a ) to  3 ( d ) are views showing performances of the scanning optical system according to the first embodiment. 
     FIGS.  4 ( a ) and  4 ( b ) are views showing the structure of a scanning optical system according to a second embodiment of the present invention; 
     FIGS.  5 ( a ) to  5 ( d ) are views showing performances of the scanning optical system according to the second embodiment. 
     FIGS.  6 ( a ) to ( b ) are views showing the structure of a scanning optical system according to a third embodiment of the present invention; 
     FIGS.  7 ( a ) to  7 ( d ) are views showing performances of the scanning optical system according to the third embodiment. 
     FIG. 8 is a perspective view schematically showing a mirror surface or a lens surface configuration the scanning optical system (z is raised to the 0-th power); 
     FIG. 9 is a perspective view schematically showing a mirror surface or a lens surface configuration of the scanning optical system (z is raised to the first power); 
     FIG. 10 is a perspective view schematically showing a mirror surface or a lens surface configuration of the scanning optical system (z is raised to the second power); 
     FIG.  11 . is a perspective view schematically showing a mirror surface or a lens surface configuration of the scanning optical system (z is raised the third power); 
     FIG. 12 is a perspective view schematically showing a mirror surface or a lens surface configuration of the scanning optical system (combination); and 
     FIG. 13 is a view explaining a manner of occurrence of bow when there is no twist. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     Hereinafter, embodiments of the present invention will be described with reference to the drawings. FIG. 1 is a perspective view showing a schematic structure of a first embodiment of a laser scanning apparatus of the present invention. As shown in the figure, a laser beam  2  emanating from a laser light source  1  passes through a collimator lens  3  to be converted into a parallel beam, and then, passes through a cylinder lens  4  and is condensed only in the sub scanning direction in the vicinity of a reflecting surface  5   a  of a polygon mirror  5 . 
     Then, the laser beam  2  is deflected by the polygon mirror  5  rotating about the axis of rotation  5   b  as shown by the arrow, is then reflected at a first mirror  6   a  and a second mirror  6   b  of a scanning mirror  6 , and is condensed on an image surface  7   a  on a cylindrical photosensitive body  7  to form a latent image. The rotation of the polygon mirror  5  rotates the reflecting surfaces  5   a , so that the laser beam  2  scans the image surface  7   a  on the rotating photosensitive body  7  to form a latent image. 
     FIGS.  2 ( a ) and  2 ( b ) are views showing the structure of a scanning optical system according to the first embodiment of the present invention. FIG.  2 ( a ) is a plan view. FIG.  2 ( b ) is a side view. As shown in the figures, the optical axis X (reflected light from the polygon mirror  5  when the angle of deflection is 0; this applies in the following description) is on the x axis, a main scanning direction is the direction of the y axis, and the sub scanning direction is the direction of the z axis. Here, the laser beam  2  is incident on the reflecting surface  5   a  of the polygon mirror  5  from a direction shown by the arrow A of FIG.  2 ( a ). The point where the reflecting surface  5   a  and the optical axis X intersect is the origin. Table 1 shows the surface coordinates of the surfaces of the scanning optical system. In the table, the positions and orientations of the first mirror  6   a , the second mirror  6   b  and the image surface  7   a  (evaluation surface) on the photosensitive body  7  are shown in the form of the origin, the x-axis, vector and the y-axis vector of the local coordinate system of the surfaces. The unit of the values associated with length is mm. The configurations of these surfaces are represented by the following expression:        x   =       ∑     i   =   0     10                       ∑     j   =   0     3                       a     i                 j            y   i            z   j     .                                  
     Here, in the coordinate system, the optical axis is on the x axis, the main scanning direction is the direction of the y axis, and the sub scanning direction is the direction of the z axis as mentioned above. In the expression, the term in which z is raised to the first power represents an inclination in the vicinity of z=0 with respect to a cross-sectional configuration by a plane parallel to the x-z plane, and is used for expressing a configuration being asymmetrical in the sub scanning direction. Moreover, since the term in which z is raised to the first power takes the form of a polynomial of y, an appropriate inclination can be given to an arbitrary y by controlling the coefficients. 
     The configuration representing the inclination in the vicinity of z=0 and being asymmetrical in the sub scanning direction will be described in more detail. Generally, in a case where a surface is expressed as a function x=f(y, z), when the angle between the x axis and a vector of projection of the normal to a tangent plane at a point (y, z)=(y 0 , z 0 ) on the surface onto the x-z plane is θ, the value obtained by substituting y=y 0  and z=z 0  into f partially differentiated with respect to z equals tan θ. Therefore, expressing θ when z=0 as a function of y with respect to the mirror surfaces of the embodiment,        θ   =     arctan          ∑     i   =   0     10                       a   i1            y   i     .                                  
     At this time, if coefficients a i1  are all zero except when i=0, θ is constant irrespective of the value of y. If there is at least one coefficient ail that is not zero when i is not less than 1 and not more than 10, θ changes as y changes. Here, that θ changes as y changes is referred to as that a surface has a twisted configuration. Therefore, in this specification, twisted configurations being used comply with this definition. 
     
       
         
               
               
               
               
             
               
               
               
               
               
             
           
               
                   
                 TABLE 1 
               
               
                   
                   
               
               
                   
                 x-component 
                 y-component 
                 z-component 
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                 1st mirror 
                 Position 
                 105.18 
                 0 
                 0 
               
               
                   
                 x-direction 
                 0.9239 
                 0 
                 −0.3827 
               
               
                   
                 y-direction 
                 0 
                 1 
                 0 
               
               
                 2nd mirror 
                 Position 
                 85.18 
                 0 
                 20 
               
               
                   
                 x-direction 
                 −0.3827 
                 0 
                 0.9239 
               
               
                   
                 y-direction 
                 0 
                 1 
                 0 
               
               
                 Evolution 
                 Position 
                 85.18 
                 0 
                 −167.3 
               
               
                 Surface 
                 x-direction 
                 −0.1736 
                 0 
                 −0.9848 
               
               
                   
                 y-direction 
                 0 
                 1 
                 0 
               
               
                   
               
             
          
         
       
     
     According to the first embodiment of the present invention, the two mirror surfaces each have a finite value where the coefficients a i1  are not zero when i is not more than 1 and not less than 10. Therefore, the two mirror surfaces each have a twisted configuration. 
     In Tables 2 and 3, values of coefficients aij in which y is raised to the i-th power and z is raised to the j-th power in the above surface expression 1 are shown as matrices with i rows and j columns with respect to the mirror surfaces of the first mirror  6   a  and the second mirror  6   b , respectively. In the tables, En (n is an integer) represents ×10 n . For example, E-04 represents ×10 31 4 . 
     
       
         
               
               
               
               
             
               
               
               
               
             
           
               
                 TABLE 2 
               
               
                   
               
               
                 y/z 
                 0 
                 1 
                 2 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 0 
                 0 
                 0 
                 5.7930006E-03 
               
               
                 1 
                 1.7585072E-03 
                 1.9376347E-05 
                 −9.2982281E-07 
               
               
                 2 
                 −7.2025615E-04 
                 7.8637115E-06 
                 −8.6517925E-08 
               
               
                 3 
                 −2.3426228E-07 
                 −1.1444293E-08 
                 4.8174109E-11 
               
               
                 4 
                 1.5112943E-08 
                 2.9953297E-10 
                 1.4449148E-12 
               
               
                 5 
                 1.7069651E-11 
                 1.1955511E-12 
                 −8.4552471E-15 
               
               
                 6 
                 −6.9131564E-13 
                 −4.8237736E-14 
                 2.7566167E-16 
               
               
                 7 
                 −5.2882946E-16 
                 −6.4566469E-17 
                 0 
               
               
                 8 
                 2.0981781E-17 
                 2.9976838E-18 
                 0 
               
               
                 9 
                 7.3088171E-21 
                 1.4014273E-21 
                 0 
               
               
                 10 
                 −4.3978682E-22 
                 −6.5843384E-23 
                 0 
               
               
                   
               
             
          
         
       
     
     
       
         
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE 3 
               
               
                   
               
               
                 y\z 
                 0 
                 1 
                 2 
                 3 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 0 
                 0 
                 0 
                 −5.2597365E-03 
                 1.8435728E-05 
               
               
                 1 
                 0 
                 4.6557208E-06 
                 2.7905216E-07 
                 −1.1401440E-09 
               
               
                 2 
                 −6.6534281E-04 
                 4.0850144E-06 
                 −5.1866305E-08 
                 7.6113064E-10 
               
               
                 3 
                 1.8248434E-07 
                 −3.7431346E-09 
                 4.1586051E-11 
                 −5.5400527E-13 
               
               
                 4 
                 −8.1841951E-09 
                 1.7996978E-10 
                 −2.6309496E-12 
                 8.5466680E-15 
               
               
                 5 
                 −1.0582078E-11 
                 2.3747069E-13 
                 −2.0900791E-15 
                 −5.4837334E-17 
               
               
                 6 
                 3.5070660E-13 
                 −9.0327092E-15 
                 8.4117737E-17 
                 4.2493394E-18 
               
               
                 7 
                 2.2834265E-16 
                 −4.5177540E-18 
                 7.2275337E-20 
                 0 
               
               
                 8 
                 −1.0994187E-17 
                 2.6467607E-19 
                 −5.4917058E-21 
                 0 
               
               
                 9 
                 −2.0922272E-21 
                 −2.7998051E-23 
                 6.9271904E-24 
                 0 
               
               
                 10 
                 2.7269284E-22 
                 −1.4531582E-24 
                 −1.9777168E-25 
                 0 
               
               
                   
               
             
          
         
       
     
     FIGS.  3 ( a ) to  3 ( d ) are views showing performances of the scanning optical system according to this embodiment. FIG.  3 ( a ) shows curvatures of field. FIG.  3 ( b ) shows distortion. FIG.  3 ( c ) shows the height of a ray on the surface to be scanned. FIG.  3 ( d ) shows the shift, in the main scanning direction, of a marginal ray in the sub scanning direction. In FIG.  3 ( a ), the horizontal axis represents the angle of deflection (deg.), the vertical axis represents the defocus amount (mm), and curvatures of field in the sub scanning direction and in the main scanning direction are shown. The curvature of field in the sub scanning direction is represented by ♦ and the solid curved line a. The curvature of field in the main scanning direction is represented by ▪ and the solid curved line b. 
     In FIG.  3 ( b ), the horizontal axis represents the angle of deflection (deg.), the vertical axis represents distortion (%), and distortion is shown. In FIG.  3 ( c ), the horizontal axis represents the angle of deflection (deg.), the vertical axis represents the height (mm) of the ray, and the height, in the sub scanning direction, of the ray on the surface to be scanned is shown. In FIG.  3 ( d ), the horizontal axis represents the angle of deflection (deg.), the vertical axis represents the shift amount (μm), and the shift amount, in the main scanning direction, of the marginal ray in the sub scanning direction is shown. It is apparent that in this embodiment, by the two mirror surfaces each having a twisted configuration, excellent characteristics are obtained in all of these performances. 
     From the fact that a condition in which the height, in the sub scanning direction, of the ray on the surface to be scanned is changed so as to be curved is called bow, bow correction condition is apparent from FIG.  3 ( c ). The shift amount, in the main scanning direction, of the marginal ray in the sub scanning direction in FIG.  3 ( d ) indicates that the ray on the upper end and the ray on the lower end within the pupil reach different positions on the image surface. When the shift amount is large, the imaging condition of the beam deteriorates, so that the problem of so-called out-of-shape beam arises. 
     FIGS.  4 ( a ) and  4 ( b ) are views showing the structure of a scanning optical system according to a second embodiment of the present invention. FIG.  4 ( a ) is a plan view. FIG.  4 ( b ) is a side view. As shown in the figures, the optical axis X is on the x axis, the main scanning direction is the direction of the y axis, and the sub scanning direction is the direction of the z axis. Here, the laser beam  2  is incident on the reflecting surface  5   a  of the polygon mirror  5  from a direction shown by the arrow A of FIG.  4 ( a ). The point where the reflecting surface  5   a  and the optical axis X intersect is the origin. In this embodiment, instead of two mirrors as described above, a scanning mirror  6  and a lens element  8  are used. The lens  8  has a free-form surface  8   a  and a plane  8   b.    
     Table 4 shows the surface coordinates of the surfaces of the scanning optical system. In the table, the positions and orientations of the scanning mirror  6 , the free-form surface  8   a  and the plane  8   b  of the lens  8  (refractive index n), and the image surface  7   a  (evaluation surface) on the photosensitive body  7  are shown in the form of the origin, the x-axis vector and the y-axis vector of the local coordinate system of the surfaces. The unit of the values associated with length is mm. 
     
       
         
               
               
               
               
             
               
               
               
               
               
             
           
               
                   
                 TABLE 4 
               
               
                   
                   
               
               
                   
                 x-component 
                 y-component 
                 z-component 
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                 mirror 
                 Position 
                 50 
                 0 
                 0 
               
               
                   
                 x-direction 
                 0.9848 
                 0 
                 −0.1736 
               
               
                   
                 y-direction 
                 0 
                 1 
                 0 
               
               
                 lens 
                 Position 
                 −25.42 
                 0 
                 27.451 
               
               
                 (free shape) 
                 x-direction 
                 −1 
                 0 
                 0 
               
               
                 n = 1.51882 
                 y-direction 
                 0 
                 1 
                 0 
               
               
                 lens 
                 Position 
                 −30.42 
                 0 
                 27.451 
               
               
                 (plain surface) 
                 x-direction 
                 −1 
                 0 
                 0 
               
               
                   
                 y-direction 
                 0 
                 1 
                 0 
               
               
                 Evolution 
                 Position 
                 −132.42 
                 0 
                 65.731 
               
               
                 Surface 
                 x-direction 
                 −0.8641 
                 0 
                 0.5033 
               
               
                   
                 y-direction 
                 0 
                 1 
                 0 
               
               
                   
               
             
          
         
       
     
     In Tables 5 and 6, values of coefficients aij in which y is raised to the i-th power and z is raised to the j-th power in the above surface expression 1 are shown as matrices with i rows and j columns with respect to the mirror surface of the scanning mirror  6  and the free-form surface  8   a  (lens surface) of the lens  8 , respectively. In the tables, En (n is an integer) represents ×10 n . For example, E-03 represents ×10 −3 . 
     
       
         
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE 5 
               
               
                   
               
               
                 y/z 
                 0 
                 1 
                 2 
                 3 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 0 
                 0 
                 0 
                 −2.7478515E-03 
                 8.7887034E-05 
               
               
                 1 
                 3.2844506E-03 
                 −7.4307105E-06 
                 −1.4494710E-06 
                 −1.6329010E-07 
               
               
                 2 
                 −1.4320348E-03 
                 3.3113815E-06 
                 5.9655542E-07 
                 2.4310991E-08 
               
               
                 3 
                 −6.6003199E-07 
                 1.8942744E-08 
                 2.1031215E-09 
                 5.1497601E-11 
               
               
                 4 
                 1.3835429E-07 
                 −3.1150585E-09 
                 −5.4539675E-10 
                 −7.0404825E-11 
               
               
                 5 
                 3.1917104E-10 
                 −1.2831392E-11 
                 −1.4327293E-12 
                 1.2179974E-15 
               
               
                 6 
                 −2.7224500E-11 
                 1.0908245E-12 
                 2.5479448E-13 
                 2.5045957E-14 
               
               
                 7 
                 −7.9630147E-14 
                 4.1666450E-15 
                 4.5541547E-16 
                 0 
               
               
                 8 
                 5.2868427E-15 
                 −3.3379251E-16 
                 −1.3515453E-16 
                 0 
               
               
                 9 
                 8.9005617E-18 
                 −5.9038004E-19 
                 −7.1799679E-20 
                 0 
               
               
                 10 
                 −8.2046111E-19 
                 6.5674291E-20 
                 3.4602697E-20 
                 0 
               
               
                   
               
             
          
         
       
     
     
       
         
               
               
               
               
             
               
               
               
               
             
           
               
                 TABLE 6 
               
               
                   
               
               
                 y/z 
                 0 
                 1 
                 2 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 0 
                   
                   
                 1.1356484E-02 
               
               
                 1 
                 2.3661243E-03 
                 2.5766269E-05 
                 −8.7447570E-07 
               
               
                 2 
                 −1.2825996E-03 
                 −1.3439291E-05 
                 −2.1671471E-07 
               
               
                 3 
                 −1.0085276E-06 
                 −1.7154363E-08 
                 4.7181139E-12 
               
               
                 4 
                 2.1401860E-07 
                 3.8041639E-09 
                 −1.9538117E-11 
               
               
                 5 
                 1.5136633E-10 
                 3.6573868E-12 
                 1.1717556E-15 
               
               
                 6 
                 −1.5194322E-11 
                 −3.5479908E-13 
                 2.6484209E-15 
               
               
                 7 
                 −1.2158830E-14 
                 −3.6403959E-16 
                 −6.1066027E-19 
               
               
                 8 
                 8.3391294E-16 
                 2.0709827E-17 
                 −2.1863723E-19 
               
               
                 9 
                 4.2991568E-19 
                 1.4677332E-20 
                 5.2030817E-23 
               
               
                 10 
                 −2.6112857E-20 
                 −6.7895371E-22 
                 1.0161679E-23 
               
               
                   
               
             
          
         
       
     
     FIGS.  5 ( a ) to  5 ( d ) are views showing performances of the scanning optical system according to this embodiment. FIG.  5 ( a ) shows curvatures of field. FIG.  5 ( b ) shows distortion. FIG.  5 ( c ) shows the height of a ray on the surface to be scanned. FIG.  5 ( d ) shows the shift, in the main scanning direction, of a marginal ray in the sub scanning direction. In FIG.  5 ( a ), the horizontal axis represents the angle of deflection (deg.), the vertical axis represents the defocus amount (mm), and curvatures of field in the sub scanning direction and in the main scanning direction are shown. The curvature of field in the sub scanning direction is represented by ♦ and the solid curved line a. The curvature of field in the main scanning direction is represented by ▪ and the solid curved line b. 
     In FIG.  5 ( b ), the horizontal axis represents the angle of deflection (deg.), the vertical axis represents distortion (%), and distortion is shown. In FIG.  5 ( c ), the horizontal axis represents the angle of deflection (deg.), the vertical axis represents the height (mm) of the ray, and the height, in the sub scanning direction, of the ray on the surface to be scanned is shown. In FIG.  5 ( d ), the horizontal axis represents the angle of deflection (deg.), the vertical axis represents the shift amount (μm), and the shift amount, in the main scanning direction, of the marginal ray in the sub scanning direction is shown. It is apparent that in this embodiment, by the mirror surface and the free-form surface of the lens each having a twisted configuration, excellent characteristics are obtained in all of these performances. 
     FIGS.  6 ( a ) and  6 ( b ) are views showing the structure of a scanning optical system according to a third embodiment of the present invention. FIG.  6 ( a ) is a plan view. FIG.  6 ( b ) is a side view. As shown in the figures, the optical axis X is on the x axis, the main scanning direction is the direction of the y axis, and the sub scanning direction is the direction of the z axis. Here, the laser beam  2  is incident on the reflecting surface  5   a  of the polygon mirror  5  from a direction shown by the arrow A of FIG.  6 ( a ). The point where the reflecting surface  5   a  and the optical axis X intersect is the origin. Table 7 shows the surface coordinates of the surfaces of the scanning optical system. In the table, the positions and orientations of the first mirror  6   a , the second mirror  6   b  and the image surface  7   a  (evaluation surface) on the photosensitive body  7  are shown in the form of the origin, the x-axis vector and the y-axis vector of the local coordinate system of the surfaces. The unit of the values associated with length is mm. 
     
       
         
               
               
               
               
             
               
               
               
               
               
             
           
               
                   
                 TABLE 7 
               
               
                   
                   
               
               
                   
                 x-component 
                 y-component 
                 z-component 
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                 1st mirror 
                 Position 
                 60 
                 0 
                 0 
               
               
                   
                 x-direction 
                 0.9848 
                 0 
                 −0.1736 
               
               
                   
                 y-direction 
                 0 
                 1 
                 0 
               
               
                 2nd mirror 
                 Position 
                 20 
                 0 
                 14.559 
               
               
                   
                 x-direction 
                 −0.866 
                 0 
                 0.5 
               
               
                   
                 y-direction 
                 0 
                 1 
                 0 
               
               
                 Evolution 
                 Position 
                 154.47 
                 0 
                 −98.276 
               
               
                 Surface 
                 x-direction 
                 0.6428 
                 0 
                 −0.766 
               
               
                   
                 y-direction 
                 0 
                 1 
                 0 
               
               
                   
               
             
          
         
       
     
     In Tables 8 and 9, values of coefficients aij in which y is raised to the i-th power and z is raised to the j-th power in the above surface expression 1 are shown as matrices with i rows and j columns with respect to the mirror surfaces of the first mirror  6   a  and the second mirror  6   b , respectively. In the tables, En (n is an integer) represents ×10 n . For example, E-04 represents ×10 −4 . 
     
       
         
               
               
               
               
             
               
               
               
               
             
           
               
                 TABLE 8 
               
               
                   
               
               
                 y/z 
                 0 
                 1 
                 2 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 0 
                 0 
                 0 
                 1.1749436E-02 
               
               
                 1 
                 1.4551712E-03 
                 -5.5645024E-08 
                 -4.9416272E-06 
               
               
                 2 
                 -8.4647197E-04 
                 4.2545323E-06 
                 -2.7424589E-07 
               
               
                 3 
                 -6.1234167E-07 
                 -1.6663913E-08 
                 1.2846968E-09 
               
               
                 4 
                 1.0282267E-07 
                 2.0442410E-09 
                 -8.6619303E-11 
               
               
                 5 
                 1.6628644E-10 
                 6.5401706E-12 
                 -1.7967436E-13 
               
               
                 6 
                 -1.4682408E-11 
                 -2.9304039E-13 
                 9.5607934E-15 
               
               
                 7 
                 -1.9727172E-14 
                 -1.2044998E-15 
                 0 
               
               
                 8 
                 1.8896042E-15 
                 -2.4199781E-18 
                 0 
               
               
                 9 
                 7.4942283E-19 
                 9.4428804E-20 
                 0 
               
               
                 10 
                 -1.3512133E-19 
                 2.3398063E-21 
                 0 
               
               
                   
               
             
          
         
       
     
     
       
         
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE 9 
               
               
                   
               
               
                 y/z 
                 0 
                 1 
                 2 
                 3 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 0 
                 0 
                 0 
                 −5.6320324E-03 
                 6.6349916E-06 
               
               
                 1 
                 0 
                 −9.8886013E-07 
                 2.9427123E-07 
                 −5.9118946E-09 
               
               
                 2 
                 −4.5843537E-04 
                 1.3791293E-06 
                 −2.6446584E-08 
                 1.5136172E-09 
               
               
                 3 
                 2.8589356E-07 
                 −2.0166937E-09 
                 4.7985474E-12 
                 2.1423237E-12 
               
               
                 4 
                 −4.3075641E-08 
                 2.7774020E-10 
                 −7.2104330E-13 
                 −4.6732429E-13 
               
               
                 5 
                 −3.4874299E-11 
                 4.4720950E-13 
                 −3.5562996E-15 
                 −2.1972985E-16 
               
               
                 6 
                 3.2523975E-12 
                 −2.8093548E-14 
                 3.5025372E-16 
                 4.3162724E-17 
               
               
                 7 
                 1.6592319E-15 
                 −4.0393419E-17 
                 6.3207386E-19 
                 0 
               
               
                 8 
                 −2.1027978E-16 
                 2.3962405E-18 
                 −6.6991125E-20 
                 0 
               
               
                 9 
                 −1.5898246E-20 
                 1.3901528E-21 
                 −3.3440869E-23 
                 0 
               
               
                 10 
                 7.0185824E-21 
                 −1.0394891E-22 
                 4.4504269E-24 
                 0 
               
               
                   
               
             
          
         
       
     
     FIGS.  7 ( a ) to  7 ( d ) are views showing performances of the scanning optical system according to this embodiment. FIG.  7 ( a ) shows curvatures of field. FIG.  7 ( b ) shows distortion. FIG.  7 ( c ) shows the height of a ray on the surface to be scanned. FIG.  7 ( d ) shows the shift, in the main scanning direction, of a marginal ray in the sub scanning direction. In FIG.  7 ( a ), the horizontal axis represents the angle of deflection (deg.), the vertical axis represents the defocus amount (mm), and curvatures of field in the sub scanning direction and in the main scanning direction are shown. The curvature of field in the sub scanning direction is represented by ♦ and the solid curved line a. The curvature of field in the main scanning direction is represented by ▪ and the solid curved line b. 
     In FIG.  7 ( b ), the horizontal axis represents the angle of deflection (deg.), the vertical axis represents distortion (%), and distortion is shown. In FIG.  3 ( c ), the horizontal axis represents the angle of deflection (deg.), the vertical axis represents the height (mm) of the ray, and the height, in the sub scanning direction, of the ray on the surface to be scanned is shown. In FIG.  3 ( d ), the horizontal axis represents the angle of deflection (deg.), the vertical axis represents the shift amount (μm), and the shift amount, in the main scanning direction, of the marginal ray in the sub scanning direction is shown. It is apparent that in this embodiment, by the two mirror surfaces each having a twisted configuration, generally excellent characteristics are obtained in all of these performances. However, in the shift amount shown in FIG.  7 ( d ), fluctuation is comparatively large. 
     FIGS. 8 to  12  are perspective views schematically showing mirror surface and lens surface configurations of the scanning optical systems. In the figures, mirror surfaces and lens surface S having surface configurations represented by the expression 1 are shown in the form of meshes. FIG. 8 shows a surface configuration in which z is raised to the 0-th power. The cross section parallel to the x-z plane is a straight line along the z axis, and the straight line moves in the direction of the x axis along the y axis. 
     FIG. 9 shows a surface configuration in which z is raised to the first power. The cross section parallel to the x-z plane is a straight line, and the straight line rotates along the y axis to be twisted. FIG. 10 shows a surface configuration in which z is raised to the second power. The cross section parallel to the x-z plane is a quadratic curve, and the configuration of the quadratic curve changes along the y axis. 
     FIG. 11 shows a surface configuration in which z is raised to the third power. The cross section parallel to the x-z plane is a cubic curve, and the configuration of the cubic curve changes along the y axis. FIG. 12 shows a combination of the surface configurations in which z is raised to the 0-th to the third powers. As described above, the configurations of the mirror and lens surfaces are decided by selecting or combining various configurations in which z is raised to the powers and which are represented by polynomials of y. 
     Now, a manner in which bow occurs when there is no twist will be described. FIG. 13 shows the position of beam condensation in the sub scanning direction in an example in which a scanning optical system of the present invention is re-designed, for example, so that one of the two mirror surfaces has no twist. In the figure, the horizontal axis represents the angle of deflection (deg.), and the vertical axis represents the sub scanning direction beam condensation position (mm). A case where the surface of the first mirror has no twist is represented by ♦ and the solid curved line a, and a case where the surface of the second mirror has no twist is represented by and the solid curved line b. It is apparent that as shown in the figure, only by one mirror surface having no twist, the sub scanning direction beam condensation position is largely curved to cause bow. 
     The light source as recited in claims corresponds to the laser light source in the embodiments; the deflector, to the polygon mirror; the surface to be scanned, to the image surface on the photosensitive body; the reflecting surface, to the first and the second mirrors; and the lens surface, to the free-form surface of the lens. 
     As described above, according to the laser scanning apparatus of the embodiments, by providing the scanning optical system with two surfaces each having a twist, the problems of bow and out-of-shape beam are resolved, so that a laser scanning apparatus having excellent imaging performance can be provided. 
     Although the present invention has been fully described by way of example with reference to the accompanying drawings, it is to be understood that various changes and modifications will be apparent to those skilled in the art. Therefore, unless otherwise such changes and modification depart from the scope of the present invention, they should be construed as being included therein.