Patent Publication Number: US-6222662-B1

Title: Optical scanning device and a scanning lens therefor

Description:
This application is a DIV of application Ser. No. 09/352,872 filed Jul. 13, 1999, U.S. Pat. No. 6,141,133, which is a continuation of application Ser. No. 09/177,658, filed Oct. 23, 1998, U.S. Pat. No. 5,986,791, which is a DIV of application Ser. No. 08/738,979, filed Oct. 24, 1996, U.S. Pat. No. 5,875,051. 
    
    
     BACKGROUND OF THE INVENTION 
     The present invention relates to an optical scanning device for use in an image forming apparatus, and a scanning lens therefor. 
     In an optical scanning device for an optical printer, digital copier or similar image forming apparatus, a deflector deflects, at a uniform angular velocity, a beam issuing from a light source and forming a line image elongate in a direction corresponding to a main scanning direction. A scanning lens converges the deflected beam to a desired surface to be scanned in the form of a beam spot. The scanning device scans the desired surface at a uniform velocity with the beam spot. The direction corresponding to the main scanning direction refers to a direction parallel to the main scanning direction at any position on a virtual optical path which is formed by developing linearly an optical path between the light source and the desired surface along the optical axis. Likewise, a direction corresponding to a subscanning direction refers to a direction parallel to a subscanning direction at any position on the above virtual path. 
     It is a common practice with the scanning device to use a single scanning lens in order to reduce the size and cost of the device. A single scanning lens is taught in, e.g., Japanese Patent Laid-Open Publication No. 7-174998. An image forming lens for scanning is disclosed in, e.g., Japanese Patent Laid-Open Publication No. 4-50908. 
     To prevent the scanning lens from increasing in size, it is preferable that the lens be positioned as close to the deflector as possible. This, however, brings about a problem that the magnification of the lens in the subscanning direction increases. For example, if the position where the line image is formed is even slightly deviated on the virtual optical path due to the slight positional error of optical system for forming the line image in the optical axis direction, the focusing position of the beam spot in the subscanning direction is noticeably shifted. As a result, field curvatures are noticeably deteriorated, compared to designed field curvatures. This is particularly true with the field curvature in the subscanning direction. 
     While the scanning lens functions to cause the deflected beam to converge to the desired surface in the directions corresponding to the main and subscanning directions, it must implement uniform velocity scanning in the direction corresponding to the main scanning direction. To realize uniform velocity scanning and desirable field curvatures with a single lens, it is necessary that at least one surface of the lens has a special shape different from an arc in the direction corresponding to the main scanning direction. Generally, the scanning lens is implemented as a convex lens in order to converge the deflected beam to the desired surface in the direction corresponding to the main scanning direction. For this reason, the lens is often formed by molding plastics. However, the problem is that when the lens is provided with a biconvex configuration, the thickness is apt to increase in the paraxial portion and noticeably differs from the paraxial portion to the peripheral portion. Consequently, the lens implemented as a molding of plastics often suffers from sinking, waving and other errors in configuration. The lens with such errors has critically low performance in practice although the designed performance may be optimal. This problem is particularly serious when a wide-angle lens is desired to render the scanning device compact or to extend the scanning angle range, or when a lens is located in the vicinity of the deflector in order to reduce the size of the lens. 
     When use is made of a single lens, uniform velocity scanning capability will be effectively promoted if the surface of the lens facing the deflector is provided with a convex shape. In the case, however, the thickness of the lens decreases with a decrease in the radius of curvature, aggravating the difference in thickness between the paraxial portion and the peripheral portion. Moreover, if the radius of curvature of the surface facing the deflector is small, the angle at which the deflected beam is incident to the above surface, i.e., the angle between the outward normal of the surface and the principal ray of the incident beam increases in the direction corresponding to the main scanning direction. This is likely to deteriorate the field curvatures and uniform velocity scanning due to the errors in the configuration of the above surface. 
     In addition, the scanning device should preferably have a compact and broad effective main scanning width. The deflector therefore must have a broad effective deflection angle. 
     SUMMARY OF THE INVENTION 
     It is therefore an object of the present invention to provide an optical scanning device having desirable performance including field curvatures in main and subscanning directions and uniform velocity scanning capability, and implementing a broad effective main scanning width, and a scanning lens therefor. 
     It is another object of the present invention to provide an optical scanning device which frees a single scanning lens implemented as a molding of plastics from errors in configuration, and the scanning lens. 
     It is yet another object of the present invention to provide an optical scanning device reducing the limitation on the tolerance of optical elements in the optical axis elements as to assembly, and a scanning lens therefor. 
     It is a further object of the present invention to provide an optical scanning device capable of reducing the deviation of a single scanning lens as to a beam spot diameter ascribable to the image height of the beam spot, and the scanning lens. 
     It is an additional object of the present invention to provide a compact and high performance optical scanning device using a single scanning lens. 
     In accordance with the present invention, in an optical scanning device for causing a deflector to deflect at a uniform angular velocity a beam forming a line image elongate in a direction corresponding to a main scanning direction with a reflection surface adjoining a position where the line image is formed, and for causing a scanning lens to converge a deflected beam output from the deflector to a desired surface to be scanned in the form of a beam spot, thereby optically scanning the desired surface at a uniform velocity, the lens is a single lens whose opposite surfaces are convex in a deflection plane. At least one of opposite surfaces of the lens has, in the deflection plane, a nonarcuate shape determined by R, K, A, B, C, D . . . included in: 
      X=Y 2 /[R+R{square root over ({1+L −(1+L +K)(Y/R) 2 +L })}]+A·Y 4 +B·Y 6 +C·Y 8 +D·Y 10 + 
     where X is the coordinate in the optical axis direction, Y is the coordinate in a direction perpendicular to the optical axis, R is the paraxial radius of curvature, K is a cone constant, and A, B, C, D . . . are high degree coefficients. At least one of the opposite surfaces is a special toric surface such that a line connecting the centers of curvatures of the at least one surface in planes perpendicular to the deflection plane forms a curve different in shape from the at least one surface. An effective main scanning width W, a thickness d 1  of the lens on the optical axis and a distance L between the deflection start point of the deflector and the desired surface satisfy the following relations (1) and (2): 
     
       
         W/L&gt;0.9  (1) 
       
     
     
       
         10&lt;(W/L) 2 ·(L/d 1 )&lt;30  (2) 
       
     
     In addition, the lens is implemented as a moulding of plastics. 
     Also, in accordance with the present invention, in an optical scanning device for causing a deflector to deflect at a uniform angular velocity a beam forming a line image elongate in a direction corresponding to a main scanning direction with a reflection surface adjoining a position where the line image is formed, and for causing an imaging forming lens to converge a deflected beam output from the deflector to a desired surface to be scanned in the form of a beam spot, thereby optically scanning the desired surface at a uniform velocity, the lens is a single lens. At least one of opposite surfaces of the lens has, in the deflection plane, a nonarcuate shape determined by R, K, A, B, C, D . . . included in: 
     
       
         X=Y 2 /[R+R{square root over ({1+L −(1+L +K)(Y 2 +L /R 2 +L })}]+A·Y 4 +B·Y 6 +C·Y 8 +D·Y 10 + 
       
     
     where X is the coordinate in an optical axis direciton, Y is the coordinate in a direction perpendicular to the optical axis, R is the paraxial radius of curvature, K is a cone constant, and A, B, C, D . . . are high degree coefficients. At least one of the opposite surfaces is a special toric surface such that a line connecting the centers of curvature of the at least one surface in planes perpendicular to the deflection plane forms a curve different in shape from the at least one surface. A shape in a plane perpendicular to the deflection plane containing the optical axis is a meniscus shape with the concave surface facing the deflector. 
     Further, in accordance with the present invention, a scanning lens converges a beam deflected by a deflector at a uniform angular velocity to a desired surface to be scanned with respect to a direction corresponding to a main scanning direction, and causes the desired surface to be scanned at a uniform velocity in the main scanning direction. The lens is a biconvex single lens and implemented as a molding of plastics. A first and a second surface of the lens, as counted from the deflector side, each has, in a deflection plane, a nonarcuate shape determined by R, K, A, B, C, D . . . included in: 
     
       
         X=Y 2 /[R+R{square root over ({1+L −(1+L +K)(Y/R) 2 +L })}]+A·Y 4 +B·Y 6 +C·Y 8 +D·Y 10 + 
       
     
     where X is the coordinate in the optical axis direciton, Y is the coordinate in a direction perpendicular to the optical axis, R is the paraxial radius of curvature, K is a cone constant, and A, B, C, D . . . are high degree coefficients. An effective main scanning width W, a thickness d 1  of the scanning lens on the optical axis and a distance L between the deflection start point of the deflector and the desired surface satisfy following relations (1) and (2): 
      W/L&gt;0.9  (1) 
     
       
         10&lt;(W/L) 2 ·(L/d 1 )&lt;30  (2) 
       
     
     Moreover, in accordance with the present invention, a scanning lens converges a beam deflected by a deflector at a uniform angular velocity to a desired surface to be scanned with respect to a direction corresponding to a main scanning direction, and causes the desired surface to be scanned at a uniform velocity. The lens is a single lens in which at least a surface facing the deflector has a convex shape in a paraxial portion in a deflection plane. The lens is implemented as a molding of plastics. A first and a second surface of the lens, as counted from the deflector side, each has, in the deflection plane, a nonarcuate shape determined by R, K, A, B, C, D . . . included in: 
     
       
         X=Y 2 /[R+R{square root over ({1+L −(1+L +K)(Y/R) 2 +L })}]+A·Y 4 +B·Y 6 +C·Y 8 +D·Y 10 + 
       
     
     where X is the coordinate in the optical axis direciton, Y is the coordinate in a direction perpendicular to the optical axis, R is the paraxial radius of curvature, K is a cone constant, and A, B, C, D . . . are high degree coefficients. An effective main scanning width W, a thickness d 1  of the scanning lens in the optical axis direction and a distance L between the deflection start point of the deflector and the desired surface satisfy the following relations (1) and (2): 
     
       
         W/L&gt;0.9  (1) 
       
     
     
       
         10&lt;(W/L) 2 ·(L/d 1 )&lt;30  (2) 
       
     
     In addition, a focal distance fm in the deflection plane and the paraxial radius of curvature R 1  of the first surface in the deflection plane satisfy a condition (3): 
      1.0&lt;R1/fm&lt;3.0  (3). 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The above and other objects, features and advantages of the present invention will become apparent from the following detailed description taken with the accompanying drawings in which: 
     FIGS. 1A and 1B show a first embodiment of the optical scanning device in accordance with the present invention; 
     FIGS. 2A-2C are views showing a nonarcuate toric surface and special toric surfaces; 
     FIG. 3 are views representative of conditions (3) and (4) to be described; 
     FIGS. 4-16 plot field curvatures, uniform velocity scanning characteristic and linearity relating to Examples 1-13 of the first embodiment; 
     FIGS. 17A-17C show a second embodiment of the optical scanning device in accordance with the present invention; and 
     FIGS. 18-32 plot field curvatures and distortional aberrations relating to Examples 1-15 of the second embodiment. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     Preferred embodiments of the optical scanning device and a scanning lens therefore in accordance with the present invention will be described hereinafter. 
     1st Embodiment 
     Referring to FIGS. 1A and 1B of the drawings, an optical scanning device embodying the present invention is shown. As shown, the device includes a laser diode or emission source (LD)  10  and a coupling lens  12 . The coupling lens  12  constitutes a light source together with the LD  10 . A divergent beam issuing from the LD  10  is coupled by the coupling lens  12  and then converged by a cylinder lens  14  in the direction corresponding to the subscanning direction (perpendicular to the sheet surface of FIG.  1 A). As a result, the beam forms an line image elongate in the direction corresponding to the main scanning direction on a reflection surface  16  included in a deflector. In the illustrative embodiment, the deflector is assumed to be a rotary single-surface mirror. The reflection surface  16  has an axis of rotation  16 A lying in the surface  16 . The line image is formed such that its principal ray runs through the position of the axis  16 A. This embodiment is free from the deviation between the line image and the reflection surface  16 , i.e., so-called sag ascribable to the rotation of the surface  16 . The cylinder lens  14  may be replaced with a concave cylinder mirror, if desired. 
     When the reflection surface  16  rotates at a uniform angular velocity, the reflected beam is also deflected at a uniform angular velocity and turns out a deflected beam. The deflected beam is incident to a single scanning lens  18 . The lens  18  focuses the incident beam in the form of a spot onto a desired surface  20  to be scanned. As a result, the surface  20  is scanned at a uniform velocity. Usually, a photoconductive element is located at the surface  20  and scanned by the above beam spot to form an image thereon. 
     As shown in FIG. 1A, the lens  18  is a biconvex lens, as seen in a deflection plane. The lens  18  may have any shape in the plane perpendicular to the deflection plane and including an optical axis AX; it may also be biconvex. Assume that the lens  18  has a meniscus shape with the concave surface facing the deflector, as seen in the plane perpendicular to the deflection plane and containing the optical axis. Then, as shown in FIG. 1B, both the front principal plane H and the rear principal plane H′ can be set at the surface  20  side. Even when the lens  18  is positioned in the vicinity of the deflector, the principal planes H and H′ are positioned closer to the surface  20  than the substance of the lens  18 . This successfully reduces the increases in the lateral magnification in the direction corresponding to the subscanning direction. As a result, the limitation on the tolerance as to the assembly of the cylinder lens  14  in the optical axis direction is effectively relaxed. 
     Assume that the effective main scanning width of the lens  18  is W, and that the distance between the deflection start point of the deflector and the surface  20  is L. Then, the width W and distance L satisfy a condition: 
      W/L&gt;0.9  condition (1) 
     The above condition (1) holds when the image height is zero; the distance L is equal to S(0)+d(0)+l(0) shown in FIG.  1 A. As the ratio W/L increases, the angle of view of the lens  18  increases and renders the scanning device more compact. When the ratio W/L decreases below the lower limit of 0.9, a broad effective main scanning width and therefore a compact arrangement is not achievable. 
     Assuming that the lens  18  has a thickness d 1 , then the width W, length L and thickness d 1  satisfy a condition: 
     
       
         10&lt;(W/L) 2 ·(L/d 1 )&lt;30  condition (2) 
       
     
     To implement a compact scanning device with a scanning lens whose thickness differs little from its paraxial portion to its peripheral portion, it is preferable that the ratio L/d 1  (d 1 =d(0) shown in FIG. 1A) and the product (W/L) 2 ·(L/d 1 ) satisfy the above condition (2). 
     As the parameter (W/L) 2 ·(L/d 1 ) of the condition (2) decreases with the condition (1) satisfied, the thickness d 1  on the optical axis and therefore the difference in thickness between the paraxial portion and the peripheral portion increases. This is undesirable when it comes to the molding of plastics. As the parameter (W/L) 2 ·(L/d 1 ) decreases below the lower limit of 10 included in the condition (2), the above disadvantage is aggravated. This brings about sinking or waving in the event of molding. While values above the upper limit of 30 included in the condition (2) reduce the difference in thickness and promotes accurate molding, the resulting lenses will be too thin to implement field curvatures and uniform velocity scanning. 
     On the other hand, when the beam incident to the lens  18  is a parallel beam, i.e., when the lens  18  is used as an f-θ lens, the condition (2) poses a heavy limitation on design freedom. Specifically, to satisfy the condition (2) when the beam incident to the lens  18  is a parallel beam, the ratio d(0)/l(0) shown in FIG. 1A must be limited to about 0.2. As a result, the position of the lens  18  is extremely limited. This limitation on design freedom can be obviated if the beam incident to the lens  18  is divergent or convergent with respect to the direction corresponding to the main scanning direction. 
     A nonarcuate toric surface and a special toric surface will be described. A toric surface is defined as a curved surface which an arc forms when rotated about a line or axis lying in the same plane as the arc, but not passing through the center of the arc. A nonarcuate toric surface is a plan which a nonarcuate shape forms when rotated about a line or axis lying in the same plane as the nonarcuate shape. 
     FIG. 2A shows a specific nonarcuate toric surface. In FIG. 2A, X(Y) is representative of a nonarcuate shape produced by: 
     
       
         X=Y 2 /[R+R{square root over ({1+L −(1+L +K)(Y/R) 2 +L })}]+A·Y 4 +B·Y 6 +C·Y 8 +D·Y 10 +  Eq. (1) 
       
     
     where X is the coordinate of the lens  18  in the optical axis direction, Y is the coordinate in the deflection plane in the direction perpendicular to the optical axis direction, R is the paraxial radius of curvature, K is the cone constant, and A, B, C, D and so forth are higher degree coefficients. 
     The nonarcuate toric surface shown in FIG. 2A is formed when the nonarcuate shape (X(Y)) is rotated about an axis ax parallel to the Y axis. Assume that the Y coordinate of a section in a plane parallel to the XZ plane is Y=η. Then, the radius of curvature of the nonarcuate toric surface in the above section is r(η), as shown in FIG.  2 A. In the nonarcuate toric surface, the centers of curvatures in the planes parallel to the XZ plane coincide with the axis ax without exception. That is, a line connecting the above centers of curvatures is the axis ax itself. 
     FIGS. 2B and 2C each shows a particular special toric surface. In FIGS. 2B and 2C, X(Y) is representative of a curve lying in the XY plane. While X(Y) is shown as being nonarcuate, it may be a simple arc or a line parallel to the Y axis. In the special toric surfaces, at each point of the curve X(Y), the radius of curvature in a plane parallel to the XZ plane is defined as a function of the coordinate Y. Further, the center of curvature at each Y coordinate position lies in the XY plane. Moreover, a curve L(Y) connecting the centers of curvatures also lies in the XY plane. In addition, the curve L(Y) is different in shape from the curve X(Y). In the special toric surface of FIG. 2B, the curve L(Y) is located at the opposite side to the paraxial center of curvature (paraxial radius of curvature R) of the curve X(Y). On the other hand, in the special toric surface of FIG. 2C, the curve L(Y) is located at the same side as the paraxial center of curvature of the curve X(Y). 
     In the following description, the surfaces of the lens  18  lying in the deflection plane and facing the deflector and the surface  20 , respectively, will be represented by X 1  (Y) and X 2 (Y). Y is the coordinate in the direction corresponding to the main scanning direction and having the optical axis AX as an origin. Let the surfaces of the lens  18  facing the deflector and the surface  20 , respectively, and in the plane in the direction corresponding to the subscanning direction be represented by r s1 (Y) and r s2 (Y); r s1 (Y) is the radius of curvature, at the coordinate Y, of the surface facing the deflector and lying in the plane perpendicular to the deflection plane while r s2 (Y) is the radius of curvature, at the coordinate Y, of the surface facing the surface  20  and lying in the plane perpendicular to the deflection plane. 
     In the illustrative embodiment, at least one of the lens surfaces X 1  (Y) and X 2  (Y) has a nonarcuate shape while at least one of them is a special toric surface. Therefore, by selecting the nonarcuate surface adequately, it is possible to correct field curvature and uniform velocity scanning characteristic in the main scanning direction optimally. In addition, by selecting the special toric surface adequately, it is possible to correct field curvature and uniform velocity scanning characteristic in the subscanning direction optimally. For example, both surfaces of the lens  18  may be nonarcuate surfaces, as seen in the deflection plane, or may be special toric surfaces. This enhances the free correction of field curvature and uniform velocity scanning characteristic in the main and subscanning directions, and thereby make the correction easier. 
     FIG. 3A shows the chief principal ray of the beam deflected by an angle θ at the position of the axis of rotation  16 A of the reflection surface  16 . The principal ray, labeled ln 1 , is incident to the surface of the lens  18  facing the deflector at a position Y in the direction corresponding to the main scanning direction. The principal ray ln 1  is refracted by the lens  18  to turn out a ray ln 2 . The ray ln 2  is propagated through the lens  18  until it leaves the lens  18  as a ray ln 3 . The ray ln 3  is output from the surface of the lens  18  facing the surface  20  at a position Y′ in the direction corresponding to the main scanning direction. Then, the ray ln 3  is incident to the surface  20 . 
     As shown in FIG. 3A, the distance between the position where the line image is formed (position of the axis of rotation  16 A) and the point of incidence, i.e., the length of the ray ln 1  is S(θ). The distance between the incident point and the exit point, i.e., the length of the ray ln 2  in the lens  18  is d(θ). Further, the distance between the exit point and the image forming point in the direction corresponding to the subscanning direction, i.e., the length of the ray ln 3  is l(θ). The plane containing the parts ln 1 , ln 2  and ln 3  of the principal ray and perpendicular to the sheet surface of FIG. 3A is the plane perpendicular to the deflection plane. In such a perpendicular plane containing the ray ln 1 , the radius of curvature of the lens surface in the proximity of the ray at a position Y is r s1 ′(Y). Likewise, in the perpendicular plane containing the ray ln 2 , the radius of curvature of the lens surface in the proximity of the ray at a position Y′ is r s2 ′(Y). Therefore, Y is equal to Y′ when the deflection angle θ is zero degree. In this condition, S(0), d(0) and l(0) shown in FIG. 1A hold, and the sum thereof is the previously stated length L of the optical path. 
     FIG. 3B is a section of the lens  18  in the plane perpendicular to the deflection plane containing the optical axis. In FIG. 3B, A indicates the deflection start point of the principal ray by the reflection surface  16  while B indicates the image forming position (in the subscanning direction) of the beam spot. H and H′ respectively indicate the front and rear principal planes on the optical axis in the direction corresponding to the subscanning direction. Assume that the deviation between the surface of the lens  18  facing the deflector and the principal plane H is ΔH(0), and that the deviation between the other surface of the lens  18  facing the surface  20  and the rear principal plane H′ is ΔH′(0). Then, the deviations ΔH(0) and ΔH′(0) are expressed as:                Δ                   H        (   0   )         =       -       r   S1   ′          (   0   )         ×       d        (   0   )       /     [       n        {         r   s2   ′          (   0   )       -       r   s1   ′          (   0   )         }       +       (     n   -   1     )     ·     d        (   0   )           ]                     =       -       r   s1   ′          (   0   )         ×     P        (   0   )                         Δ                     H   ′          (   0   )         =       -       r   s2   ′          (   0   )         ×     P        (   0   )                         
     In this condition, the lateral magnification β(0) in the direction corresponding to the subscanning direction has an absolute value produced by:                     β        (   0   )            =       {       1        (   0   )       -     Δ                     H   ′          (   0   )           }     /     {       S        (   0   )       +     Δ                   H        (   0   )           }                   =       {       1        (   0   )       +         r   s2   ′          (     Y   ′     )       ·     P        (   0   )           }     /     {       S        (   0   )       -         r   s1   ′          (   Y   )            P        (   0   )           }                             
     where r S1 ′(0)=r S1 (0) and r S2 ′(0)=r S2 (0). 
     Therefore, by providing the lens  18  with the above meniscus shape (r s2 (0)&lt;0 and r s1 (0)&lt;0), it is possible to bring both the front and rear principal points closer to the image, i.e., to the surface  20  and reduce the magnification |β(0)| in the direction corresponding to the subscanning direction. In this case, the longitudinal magnification is β 2 (0). 
     In FIG. 1A, when the position of the cylinder lens  14 , for example, is deviated by Δ in the optical axis direction, the imaging forming position of the beam spot in the subscanning direction is shifted by Δ×β 2 (0) in the optical axis direction. Therefore, to accommodate a greater positional error Δ, the absolute value |β(0)| of the magnification should preferably be as small as possible. While the lens  18  may be brought closer to the surface  20  in order to reduce |β(0)| such an approach is apt to increase the thickness and effective diameter (in the main scanning direction) of the lens  18 . 
     The illustrative embodiment accommodates a greater tolerance as to the assembly of the optical elements in the optical axis direction while implementing to make the lens  18  be compact by positioning it as close to the deflector as possible. This can be done if the following condition is satisfied: 
      1.45≦|β(0)|≦4.75  condition (3) 
     If |β(0)| decreases below the above lower limit of the condition (3), the thickness of the lens  18  increases, and moreover it is difficult to correct the field curvatures and uniform velocity scanning characteristic desirably. If |β(0)| increases above the upper limit, the tolerance in the main scanning direction is severely restricted. 
     FIG. 3C shows an image forming condition with respect to the principal ray of the beam deflected by the angle θ. As shown, ΔH(θ) and ΔH′(θ) are expressed as:                Δ                   H        (   θ   )         =       -       r   s1   ′          (   Y   )         ×       d        (   θ   )       /     [       n        {         r   s2   ′          (     Y   ′     )       -       r   s1   ′          (   Y   )         }       +       (     n   -   1     )     ·     d        (   θ   )           ]                     =       -       r   s1   ′          (   Y   )         ×     P        (   θ   )                         Δ                     H   ′          (   θ   )         =       -       r   s2   ′          (     Y   ′     )         ×     P        (   θ   )                         
     The lateral magnification in the direction corresponding to the subscanning direction with respect to the beam with the deflection angle θ has the following absolute value: 
     
       
         |B(θ)|={I(θ)+r s2 ′(Y′)·P(θ)}/{S(θ)−r s1 ′(Y)P(θ)} 
       
     
     Then, when the beam with the deflection angle θ forms a spot at a point B, the spot has a diameter: 
     
       
         ω 1 =|β(θ)|·ω 0   
       
     
     where ω0 is the beam width, in the direction corresponding to the subscanning direction, of the line image formed at the deflection start point. 
     If |β(θ)| is noticeably deviated depending on the deflection angle θ, the diameter of the spot in the subscanning direction also noticeably changes. The allowable deviation of the spot diameter between image heights is usually less than ±5%. 
     In the illustrative embodiment, |β(θ)| satisfies the following condition in the effective scanning range: 
     
       
         0.95|β(0)|≦|β(θ)|≧1.05|β(0)|  condition (4) 
       
     
     The above condition (4) must be satisfied in order to stabilize the spot diameter in the subscanning direction. The values r s1 ′(Y) and r s2 ′(Y′) are so selected as to satisfy the condition (4). 
     In the foregoing description, the deflector is assumed to be a rotary single-surface or two-surface mirror or a polygonal mirror free from sag. Although a polygonal mirror (used as the deflector) is apt to bring about sag and thereby slightly deteriorate the field curvatures and uniform velocity scanning characteristic, this can be improved on the basis of, e.g., the shift or the tilt of the lens  18 . 
     Specific examples of the illustrative embodiment will be described hereinafter. Referring again to FIG. 1A, the beam coupled by the coupling lens  12  may turn out a parallel beam, a weak divergent beam or a weak convergent beam. Assume that the beam coupled by the coupling lens  12  turns out a weak convergent beam, that the deflection angle θ of the reflection surface  16  is zero degree, and that the cylinder lens  14  and lens  18  are absent. Then, the beam converges to a point Q rearward of the rear of the surface  20 . Let this point Q be referred to as a natural convergence point. The distance between the deflection start point of the reflection surface  16  and the natural convergence point Q is labeled as S. Assume that the distance S is positive when the point Q is positioned at the surface  20  side or negative when it is positioned closer to the light source than the reflection surface  16 . When the distance S is negative, the coupled beam is a weak divergent beam. When the coupled beam is a parallel beam, there holds S=∞. 
     Examples to be described assume a deflector free from sag. η is the refractive index associated with the wavelength (780 nm) of the material of the lens  18  used. The nonarcuate shape is determined on the basis of the Eq. (1) using R, K, A, B, C and D. As for a special toric surface, the radius of curvature r si (Y) (i=1, 2) at the coordinate in the plane perpendicular to the deflection plane is determined by giving coefficients a, b, c, d, e, f . . . in the following equation: 
     
       
         r s1 (y)=a+bY 2 +cY 4 +dY 6 +eY 8 +fY 10 + . . .   Eq. (2) 
       
     
     In the notation of the higher degree coefficients A, B, C, D, b, c, d, e, f, . . . , E and a numerical value following it indicate power. For example, “E-9” means 10 −9 . The unit of an amount having a dimension of distance is mm. 
     EXAMPLE 1 
     
       
         
           
               
             
               
                   
               
             
            
               
                 W = 216, L = 213.2, S = 3245.2 (beam coupled by the coupling lens 12 is 
               
               
                 a weak convergent beam) 
               
            
           
           
               
               
               
            
               
                   
                 S(0) = 49.0 
                   
               
               
                 nonarcuate toric surface 
                 d(0) = 21.754 
                 n = 1.53664 
               
               
                 special toric surface 
                 l(0) = 142.446 
               
            
           
           
               
            
               
                 surface facing deflector: nonarcuate toric surface 
               
            
           
           
               
               
            
               
                   
                 X 1 (Y) (arcuate shape) 
               
               
                   
                 R = 461.302, K = 35.937, 
               
               
                   
                 A = −2.205E−7, B = 2.732E−11, 
               
               
                   
                 C = −4.9587E−15, D = 1.6248E−19 
               
               
                   
                 r s1 (Y) 
               
               
                   
                 r s1 (0) = −73.845 
               
            
           
           
               
            
               
                 surface facing surface 20: special toric surface 
               
            
           
           
               
               
            
               
                   
                 X 2 (Y) (arcuate shape) 
               
               
                   
                 R = −97.244 
               
            
           
           
               
               
               
            
               
                   
                 in r s2 (Y) = 
                 a + bY 2  + cY 4  + dY 6  + eY 8  + fY 10   
               
               
                   
                 a = 
                 −20.63952, b = 8.2661E−4 
               
               
                   
                 c = 
                 1.3218E−7, d = 2.09082E−11 
               
               
                   
                 e = 
                 −5.3157E−15, f = −4.71E−19 
               
            
           
           
               
            
               
                 in conditon (1): W/L = 1.01 
               
               
                 in condition (2) (W/L) 2  · (L/d 1 ) = 10.1 
               
               
                   
               
            
           
         
       
     
     FIG. 4 shows field curvatures, uniform scanning velocity characteristic and linearity relating to Example 1. The uniform velocity scanning characteristic is defined as [{h′(θ)−h(θ)}×100(%) where h(θ)(=constant×θ) is the ideal image height for the deflection angle θ while h′(θ) is the actual image height. In FIG. 4, a field curvature in the main scanning direction and a field curvature in the subscanning direction are indicated by a dashed curve and a solid curve, respectively. 
     EXAMPLE 2 
     
       
         
           
               
             
               
                   
               
             
            
               
                 W = 216, L = 174.9, S = ∞ (beam coupled by the coupling lens 12 is a 
               
               
                 parallel beam) 
               
            
           
           
               
               
               
            
               
                   
                 S(0) = 29.887 
                   
               
               
                 coaxial nonspherical surface 
                 d(0) = 12.364 
                 n = 1.53664 
               
               
                 special toric surface 
                 l(0) = 132.649 
               
            
           
           
               
            
               
                 surface facing deflector: coaxial nonspherical surface shape appearing 
               
               
                 when a nonarcuate shape is rotated about the optical axis (X axis) and 
               
               
                 determined by 
               
            
           
           
               
               
            
               
                   
                 R = 137.503 K = −92.438, 
               
               
                   
                 A = −1.11822E−6, B = 7.28745E−10 
               
               
                   
                 C = −3.20311E−13, D = 9.55204E−17 
               
            
           
           
               
            
               
                 surface facing surface 20: special toric surface 
               
            
           
           
               
               
            
               
                   
                 X2(Y) (nonarcuate shape) 
               
               
                   
                 R = −154.248, K = 5.36873, 
               
               
                   
                 A = −2.51300E−6, B = 1.95625E−9, 
               
               
                   
                 C = −1.18490E−12, D = 3.38272E−16, 
               
            
           
           
               
               
               
            
               
                   
                 in r s2 (Y) = 
                 a + bY 2  + cY 4  + dY 6  + eY 8  + fY 10   
               
               
                   
                 a = 
                 −17.0750, b = −4.7190E−3 
               
               
                   
                 c = 
                 1.1918E−5, d = −2.3684E−8 
               
               
                   
                 e = 
                 2.2979E−11, f = −8.0723E−15 
               
            
           
           
               
            
               
                 in condition (1): W/L = 1.23 
               
               
                 in condition (2): (W/L) 2  · (L/d 1 ) = 21.6 
               
               
                   
               
            
           
         
       
     
     FIG. 5 shows field curvatures, uniform velocity scanning characteristic and linearity relating to Example 2. 
     EXAMPLE 3 
     
       
         
           
               
             
               
                   
               
             
            
               
                 W = 216, L = 174.9, S = −387.690 (beam coupled by the coupling lens 12 
               
               
                 is a weak divergent beam) 
               
            
           
           
               
               
               
            
               
                   
                 S(0) = 22.381 
                   
               
               
                 coaxial nonspherical surface 
                 d(0) = 10.000 
                 n = 1.53664 
               
               
                 special toric surface 
                 l(0) = 142.519 
               
            
           
           
               
            
               
                 surface facing deflector: coaxial nonspherical surface shape appearing 
               
               
                 when a nonarcuate shape is rotated about the optical axis (X) and 
               
               
                 determined by 
               
            
           
           
               
               
            
               
                   
                 R = 108.503, K = −56.32541, 
               
               
                   
                 A = −3.46610E−6, B = 1.98195E−9 
               
               
                   
                 C = −132.194E−13, D = 5.00528E−17 
               
            
           
           
               
            
               
                 surface facing surface 20: special toric surface 
               
            
           
           
               
               
            
               
                   
                 S 2 (Y) (nonarcuate shape) 
               
               
                   
                 R = −121.259, K = 4.91312, 
               
               
                   
                 A = −3.24924E−6, B = 1.44308E−9, 
               
               
                   
                 C = −1.89357E−12, D = 1.43613E−15, 
               
            
           
           
               
               
               
            
               
                   
                 in r s2 (Y) = 
                 a + bY 2  + cY 4  + dY 6  + eY 8  + fY 10   
               
               
                   
                 a = 
                 −13.9150, b = −6.6641E−3, 
               
               
                   
                 c = 
                 1.0956E−5, d = −6.2958E−9, 
               
               
                   
                 e = 
                 3.4232E−12, f = −3.6141E−15 
               
            
           
           
               
            
               
                 in condition (1): W/L = 1.23 
               
               
                 in condition (2): (W/L) 2  · (L/d 1 ) = 26.7 
               
               
                   
               
            
           
         
       
     
     FIG. 6 shows field curvatures, uniform velocity scanning characteristic and linearity relating to Example 3. 
     EXAMPLE 4 
     
       
         
           
               
             
               
                   
               
             
            
               
                 W = 216, L = 224.939, S = 774.830 (beam coupled by the coupling lens 
               
               
                 12 is a weak convergent beam) 
               
            
           
           
               
               
               
            
               
                   
                 S(0) = 51.669 
                   
               
               
                 coaxial nonspherical surface 
                 d(0) = 10.390 
                 n = 1.53664 
               
               
                 special toric surface 
                 l(0) = 162.880 
               
            
           
           
               
            
               
                 surface facing deflector: coaxial nonspherical surface shape appearing 
               
               
                 when a nonarcuate shape is rotated about the optical axis (X axis) and 
               
               
                 determined by 
               
            
           
           
               
               
            
               
                   
                 R = 148.864, K = 5.4534, 
               
               
                   
                 A = −7.09267E−7, B = −2.21975E−10, 
               
               
                   
                 C = 6.07139E−14, D = −8.33979E−18 
               
            
           
           
               
            
               
                 surface facing surface 20: special toric surface 
               
            
           
           
               
               
            
               
                   
                 X 2 (Y) (nonarcuate shape) 
               
               
                   
                 R = −562.372, K = −462.3035, 
               
               
                   
                 A = −4.60398E−7, B = −2.89720E−11, 
               
               
                   
                 C = −5.93656E−14, D = 1.75926E−17 
               
            
           
           
               
               
               
            
               
                   
                 in r s2 (Y) = 
                 a + bY 2  + cY 4  + dY 6  + eY 8  + fY 10   
               
               
                   
                 a = 
                 −26.3310, b = −5.2874E−3 
               
               
                   
                 c = 
                 1.9092E−6, d = −3.4661E−10, 
               
               
                   
                 e = 
                 8.6372E−13, f = −3.6498E−16 
               
            
           
           
               
            
               
                 in condition (1): W/L = 0.96 
               
               
                 in condition (2): (W/L) 2  · (L/d 1 ) = 20.0 
               
               
                   
               
            
           
         
       
     
     FIG. 7 shows field curvatures, uniform velocity scanning characteristic and linearity relating to Example 4. 
     EXAMPLE 5 
     
       
         
           
               
             
               
                   
               
             
            
               
                 W = 216, L = 175.201, S = −375.922 (beam coupled by the coupling lens 
               
               
                 12 is a weak divergent beam) 
               
            
           
           
               
               
               
            
               
                   
                 S(0) = 22.274 
                   
               
               
                 nonarcuate toric surface 
                 d(0) = 10.0 
                 n = 1.53664 
               
               
                 special toric surface 
                 l(0) = 142.927 
               
            
           
           
               
            
               
                 surface facing deflector: nonarcuate toric surface 
               
            
           
           
               
               
            
               
                   
                 X 1 (Y) (nonarcuate shape) 
               
               
                   
                 R = 107.1, K = −60.196, 
               
               
                   
                 A = −3.5107E−6, B = 2.06849E−9 
               
               
                   
                 C = −8.9722E−14, D = 1.1672E−17 
               
               
                   
                 r si (Y) 
               
               
                   
                 r s1 (0) = −60.0 
               
            
           
           
               
            
               
                 surface facing surface 20: special toric surface 
               
            
           
           
               
               
            
               
                   
                 X 2 (Y) (nonarcuate shape) 
               
               
                   
                 R = −119.99, K = 4.78039, 
               
               
                   
                 A = −3.4156E−6, B = 1.5437E−9, 
               
               
                   
                 C = −18765E−12, D = 1.4637E−15, 
               
            
           
           
               
               
               
            
               
                   
                 in r s2 (Y) = 
                 a + bY 2  + cY 4  + dY 6  + eY 8  + fY 10   
               
               
                   
                 a = 
                 −11.4579, b = −4.3509E−3, 
               
               
                   
                 c = 
                 8.8457E−6, d = −6.9161E−9 
               
               
                   
                 e = 
                 5.3831E−12, f = 3.2985E−15 
               
            
           
           
               
            
               
                 in condition (1): W/L = 1.23 
               
               
                 in condition (2): (W/L) 2  · (L/d 1 ) = 26.6 
               
               
                   
               
            
           
         
       
     
     FIG. 8 shows field curvatures, uniform velocity scanning characteristic and linearity relating to Example 5. 
     EXAMPLE 6 
     
       
         
           
               
             
               
                   
               
             
            
               
                 W = 216, L = 175.201, S = −375.922 (beam coupled by the coupling lens 12 is a 
               
               
                 weak divergent beam) 
               
            
           
           
               
               
               
            
               
                   
                 S(0) = 22.274 
                   
               
               
                 special toric surface 
                 d(0) = 10.0 
                 n = 1.53664 
               
               
                 special toric surface 
                 l(0) = 142.927 
               
            
           
           
               
            
               
                 surface facing deflector: special toric surface 
               
            
           
           
               
               
            
               
                   
                 X 1 (Y) (nonarcuate shape) 
               
               
                   
                 R = 107.1, K = −60.197 
               
               
                   
                 A = −3.5107E−6, B = 2.0684E−9, 
               
               
                   
                 C = −8.9722E−14, D = 1.1672E−17 
               
            
           
           
               
               
               
            
               
                   
                 in r s1 (Y) = 
                 a + bY 2  + cY 4  + dY 6  + eY 8  + fY 10  + gY 12   
               
               
                   
                 a = 
                 −22, b = −2.5751E−1, 
               
               
                   
                 c = 
                 4.49657E−3, d = −4.466449E−5, 
               
               
                   
                 e = 
                 1.956944E−7, f = −3.87E−10, 
               
               
                   
                 g = 
                 2.792783E−13 
               
            
           
           
               
            
               
                 surface facing surface 20: special toric surface 
               
            
           
           
               
               
            
               
                   
                 X 2 (Y) (nonarcuate shape) 
               
               
                   
                 R = −119.11, K = 4.78039, 
               
               
                   
                 A = −3.4156E−6, B = 1.5437E−9 
               
               
                   
                 C = −1.8756E−12, D = 1.4637E−15 
               
            
           
           
               
               
               
            
               
                   
                 in r s2 (Y) = 
                 a + bY 2  + cY 4  + dY 6  + eY 8  + fY 10  + gY 12   
               
               
                   
                 a = 
                 −9.807, b = −1.662605E−2 
               
               
                   
                 c = 
                 1.870374E−4, d = −12.7425E−6, 
               
               
                   
                 e = 
                 4.079722E−9, f = −6.029437E−12, 
               
               
                   
                 g = 
                 3.315003E−15 
               
            
           
           
               
            
               
                 in condition (1): W/L = 1.23 
               
               
                 in condition (2): (W/L) 2  · (L/d 1 ) = 26.6 
               
               
                 S(θ), d(θ), l(θ), r s1 ′(Y′) are listed below. 
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 θ 
                 0 
                 10 
                 20 
                 30 
                 36 
                 40 
                 45 
               
               
                   
               
               
                 S(θ) 
                 22.274 
                 22.689 
                 23.997 
                 26.399 
                 28.509 
                 30.261 
                 32.924 
               
               
                 d(θ) 
                 10 
                 9.876 
                 9.452 
                 8.551 
                 7.635 
                 6.793 
                 5.386 
               
               
                 l(θ) 
                 142.887 
                 144.327 
                 148.338 
                 154.765 
                 159.674 
                 163.608 
                 168.608 
               
               
                 r s1 ′(Y) 
                 −22 
                 −24.732 
                 −27.589 
                 −36.35 
                 −43.735 
                 −55.259 
                 −69.233 
               
               
                 r s2 ′(Y′) 
                 −9.807 
                 −10.101 
                 −10.495 
                 −11.382 
                 −12.042 
                 −12.761 
                 −13.572 
               
               
                   
               
            
           
         
       
     
     B(θ) are calculated by substituting the above data for the right side of P(θ). Then, |β(θ)| and |β(θ)|/|β(0)| with respect to the deflection angle θ are produced, as listed below. 
     
       
         
           
               
               
               
               
               
               
               
               
             
               
                   
               
             
            
               
                 θ (°) 
                 0 
                 10 
                 20 
                 30 
                 36 
                 40 
                 45 
               
               
                   
               
               
                 Y 
                 0 
                 3.94 
                 8.207 
                 13.2 
                 16.757 
                 19.451 
                 23.281 
               
               
                 |β(θ)| 
                 4.42 
                 4.47 
                 4.49 
                 4.53 
                 4.53 
                 4.55 
                 4.52 
               
               
                 |β(θ)|/|β(0)| 
                 1.0 
                 1.01 
                 1.02 
                 1.03 
                 1.03 
                 1.03 
                 1.02 
               
            
           
           
               
            
               
                 in condition (3): |β(θ)| = 4.42 
               
               
                 in condition (4): |β(0)| ≦ |β(θ)| ≦ 1.03 |β(0)| 
               
               
                   
               
            
           
         
       
     
     FIG. 9 shows field curvatures, uniform velocity scanning characteristic and linearity relating to Example 6. 
     EXAMPLE 7 
     
       
         
           
               
             
               
                   
               
             
            
               
                 W = 216, L = 174.9, S = 554.6 (beam coupled by the coupling lens 12 
               
               
                 is a weak convergent beam) 
               
            
           
           
               
               
               
            
               
                   
                 S(0) = 38.916 
                   
               
               
                 special toric surface 
                 d(0) = 15.0 
                 n = 1.53664 
               
               
                 special toric surface 
                 l(0) = 120.984 
               
            
           
           
               
            
               
                 surface facing deflector: special toric surface 
               
            
           
           
               
               
            
               
                   
                 X 1 (Y) (nonarcuate shape) 
               
               
                   
                 R = 168.191, K = −39.924, 
               
               
                   
                 A = −3.4224E−7, B = 2.2729E−13, 
               
               
                   
                 C = 8.0109E−15, D = −8.7059E−19 
               
            
           
           
               
               
               
            
               
                   
                 in r s1 (Y) = 
                 a + bY 2  + cY 4  + dY 6  + dY 8  + fY 10  + gY 12   
               
               
                   
                 a = 
                 −31, b = −3.26507E−2, 
               
               
                   
                 c = 
                 −2.90654E−4, d = 5.99426E−7, 
               
               
                   
                 e = 
                 −3.979759E−10, f = 4.109081E−14, 
               
               
                   
                 g = 
                 2.443981E−17 
               
            
           
           
               
            
               
                 surface facing surface 20: special toric surface 
               
            
           
           
               
               
            
               
                   
                 X 2 (Y) (nonarcuate shape) 
               
               
                   
                 R = −188.994, K = −4.83792, 
               
               
                   
                 A = −4.9817E−7, B = 7.7236E−12, 
               
               
                   
                 C = −1.6386E−14, D = 1.7827E−18 
               
            
           
           
               
               
               
            
               
                   
                 in r s2 (Y) = 
                 a + bY 2  + cY 4  + dY 6  + dY 8  + fY 10  + gY 12   
               
               
                   
                 a = 
                 −13.91, b = −2.866184E−3, 
               
               
                   
                 c = 
                 −1.21239E−5, d = 2.690094E−8 
               
               
                   
                 e = 
                 2.090761E−11, f = 6.668513E−15, 
               
               
                   
                 g = 
                 −6.734702E−19 
               
            
           
           
               
            
               
                 in condition (1): W/L = 1.23 
               
               
                 in condition (2): (W/L) 2  · (L/d 1 ) = 17.8 
               
               
                 S(θ), d(θ), l(θ), r s1 ′(Y) and r s2 ′(Y′) are listed below. 
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 θ 
                 0 
                 10 
                 20 
                 30 
                 36 
                 40 
                 45 
               
               
                   
               
               
                 S(θ) 
                 38.916 
                 39.656 
                 42.01 
                 46.417 
                 50.376 
                 53.81 
                 58.794 
               
               
                 d(θ) 
                 15 
                 14.747 
                 13.884 
                 12.038 
                 10.135 
                 8.354 
                 5.324 
               
               
                 l(θ) 
                 121.012 
                 122.213 
                 125.713 
                 131.319 
                 135.629 
                 138.898 
                 143.746 
               
               
                 r s1 ′(Y) 
                 −31 
                 −32.676 
                 −43.121 
                 −62.437 
                 −69.544 
                 −84.248 
                 −101.250 
               
               
                 r s2 ′(Y′) 
                 −13.91 
                 −14.145 
                 −15.21 
                 −16.769 
                 −17.478 
                 −18.376 
                 −19.395 
               
               
                   
               
            
           
         
       
     
     B(θ) are calculated by substituting the above data for the right side (?) of P(θ). 
     Then, |β(θ)| and |β(θ)|/|β(0)| with respect to the deflection angle θ are produced, as listed below. 
     
       
         
           
               
               
               
               
               
               
               
               
             
               
                   
               
             
            
               
                 θ (°) 
                 0 
                 10 
                 20 
                 30 
                 36 
                 40 
                 45 
               
               
                   
               
               
                 y 
                 0 
                 6.886 
                 14.368 
                 23.208 
                 29.610 
                 34.524 
                 41.573 
               
               
                 |β(θ)| 
                 2.19 
                 2.20 
                 2.25 
                 2.29 
                 2.28 
                 2.27 
                 2.27 
               
               
                 |β(θ)|/ 
                 1.0 
                 1.01 
                 1.03 
                 1.04 
                 1.04 
                 1.04 
                 1.04 
               
               
                 |β(0)| 
               
            
           
           
               
            
               
                 in condition (3): |β(θ)| = 2.19 
               
               
                 in condition (4): |β(0)| ≦ |β(θ)| ≦ 1.04 |β(0)| 
               
               
                   
               
            
           
         
       
     
     FIG. 10 shows field curvatures, uniform velocity scanning characteristic and linearity relating to Example 7. 
     EXAMPLE 8 
     
       
         
           
               
             
               
                   
               
             
            
               
                 W = 216, L = 174.9, S = 316.7 (beam coupled by the coupling lens 12 is a weak 
               
               
                 convergent beam) 
               
            
           
           
               
               
               
            
               
                   
                 S(0) = 48.618 
                   
               
               
                 special toric surface 
                 d(0) = 20.0 
                 n = 1.53664 
               
               
                 special toric surface 
                 l(0) = 106.282 
               
            
           
           
               
            
               
                 surface facing deflector: special toric surface 
               
            
           
           
               
               
            
               
                   
                 X 1 (Y) (nonarcuate shape) 
               
               
                   
                 R = 200.224, K = −30.5827, 
               
               
                   
                 A = −1.5926E−7, B = 2.2729E−13, 
               
               
                   
                 C = 8.0109E−15, D = −3.98579E−19 
               
            
           
           
               
               
               
            
               
                   
                 in r s1 (Y) = 
                 a + bY 2  + cY 4  + dY 6  + dY 8  + fY 10  + gY 12   
               
               
                   
                 a = 
                 30, b = 1.521146E−2, 
               
               
                   
                 c = 
                 −6.399798E−5, d = 1.166608E−7, 
               
               
                   
                 e = 
                 −9.651438E−11, f = 3.652588E−14, 
               
               
                   
                 g = 
                 −5.073881E−18 
               
            
           
           
               
            
               
                 surface facing surface 20: special toric surface 
               
            
           
           
               
               
            
               
                   
                 X 2 (Y) (nonarcuate shape) 
               
               
                   
                 R = −224.092, K = 1.19901, 
               
               
                   
                 A = −3.2013E−7, B 7.0624E−12, 
               
               
                   
                 C = −9.2418E−15, D 1.9593E−18 
               
            
           
           
               
               
               
            
               
                   
                 in r s2 (Y) = 
                 a + bY 2  + cY 4  + dY 6  + eY 8  + fY 10  + gY 12   
               
               
                   
                 a = 
                 −15.32, b = 1.12584E−3, 
               
               
                   
                 c = 
                 −3.490667E−6, d = 5.91427E−9 
               
               
                   
                 e = 
                 −4.057836E−12, f = 1.284796E−15, 
               
               
                   
                 g = 
                 −1.500234E−19 
               
               
                   
                   
               
            
           
         
       
     
     S(θ), d(θ), l(θ), r S1 ′(Y) and r S2 ′(Y′) are listed below. 
     
       
         
           
               
               
               
               
               
               
               
               
             
               
                   
               
               
                 θ 
                 0 
                 10 
                 20 
                 30 
                 36 
                 40 
                 45 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 S(θ) 
                 48.618 
                 49.552 
                 52.533 
                 58.152 
                 63.255 
                 67.611 
                 74.444 
               
               
                 d(θ) 
                 20 
                 19.662 
                 18.494 
                 15.956 
                 13.296 
                 10.774 
                 6.411 
               
               
                 l(θ) 
                 106.282 
                 1073.76 
                 110.608 
                 115.846 
                 119.932 
                 123.124 
                 127.823 
               
               
                 r s1 ′(Y) 
                 −30.0 
                 −28.769 
                 −27.175 
                 −24.561 
                 −22.426 
                 −19.745 
                 −13.291 
               
               
                 r s2 ′(Y) 
                 −15.316 
                 −15.220 
                 −15.123 
                 −14.736 
                 −14.155 
                 −13.233 
                 −10.366 
               
               
                   
               
            
           
         
       
     
     B(θ) are calculated by substituting the above data for the right side of P(θ). Then, |β(θ)| and |β(θ)|/|β(0)| with respect to the deflection angle θ are produced, as listed below. 
     
       
         
           
               
               
               
               
               
               
               
               
             
               
                   
               
             
            
               
                 θ (°) 
                 0 
                 10 
                 20 
                 30 
                 36 
                 40 
                 45 
               
               
                   
               
               
                 Y 
                 0 
                 8.605 
                 17.967 
                 29.076 
                 37.180 
                 43.459 
                 52.640 
               
               
                 |β(θ)| 
                 1.45 
                 1.45 
                 1.44 
                 1.42 
                 1.41 
                 1.41 
                 1.40 
               
               
                 |β(θ)|/ 
                 1.0 
                 0.99 
                 0.99 
                 0.97 
                 0.97 
                 0.97 
                 0.96 
               
               
                 |β(0)| 
               
            
           
           
               
            
               
                 in condition (3): |β(0)| = 1.46 
               
               
                 in condition (4): |β(0)| ≦ |β(θ)| ≦ 0.96 |β(0)| 
               
               
                   
               
            
           
         
       
     
     FIG. 11 shows field curvatures, uniform velocity scanning characteristic and linearity relating to Example 8. 
     EXAMPLE 9 
     
       
         
           
               
             
               
                   
               
             
            
               
                 S = −375.922 (beam coupled by the coupling lens 12 is a weak divergent 
               
               
                 beam). 
               
            
           
           
               
               
               
            
               
                   
                 S(0) = 22.274 
                   
               
               
                 X 1 (Y), r s1 (Y) 
                 d(0) = 10 
                 n = 1.53664 
               
               
                 X 2 (&amp;), r s2 (Y) 
                 l(0) = 142.927 
               
            
           
           
               
            
               
                 surface facing deflector: nonarcuate toric surface 
               
            
           
           
               
               
            
               
                   
                 X 1 (Y) (nonarcuate shape) 
               
               
                   
                 R = 107.1, K = −60.197, 
               
               
                   
                 A = −3.5107E−6, B = 2.0684E−9, 
               
               
                   
                 C = −8.9722E−14, D = 1.1672E−17 
               
               
                   
                 r s1 (Y) 
               
               
                   
                 r s1 (0) = −60.0 
               
            
           
           
               
            
               
                 surface facing surface 20: special toric surface 
               
            
           
           
               
               
            
               
                   
                 X 2 (Y) (nonarcuate shape) 
               
               
                   
                 R = −119.99, K = 4.78039 
               
               
                   
                 A = −3.4156E−6, B = 1.5437E−9, 
               
               
                   
                 C = −1.8765E−12, D = 1.4637E−15 
               
               
                   
                   
               
            
           
         
       
     
     As for r S2 (Y), r S2 (Y′) with respect to the deflection angle 0 are listed below. 
     
       
         
           
               
               
               
               
               
               
               
               
             
               
                   
               
               
                 θ (°) 
                 0 
                 10 
                 20 
                 30 
                 36 
                 40 
                 45 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 Y′ 
                 0 
                 4.935 
                 10.107 
                 15.775 
                 19.523 
                 22.190 
                 25.721 
               
               
                 r s2 (Y′) 
                 −11.46 
                 −11.56 
                 −11.82 
                 −12.08 
                 −12.13 
                 −12.06 
                 −11.86 
               
               
                   
               
            
           
         
       
     
     By using a polynomial, r S2 (Y′) may be approximated as: 
     
       
         r S2 (Y′)=a+b·Y′ 2 +c·Y′ 4 +d·Y′ 6 +e·Y′ 8 +f·Y′ 10   
       
     
     Then, by use of Y′ and corresponding r S2 (Y′) shown above, the coefficients a, b, c, d, e and f are determined by a minimum square method as follows: 
     a=−11.4579, b=−4.3509E-3, c=8.8457E-6, 
     d=−6.9161E-9, e=5.3831E-12, 
     f=−3.2985E-15 
     The above approximation was used for the calculation of aberration. 
     |β(0)|=4.74 holds and satisfies the above-mentioned condition (3). 
     In this connection, when the power in the direction corresponding to the subscanning direction is assigned to only one surface of the lens  18 , |β(0)| is 4.97 and does not satisfy the condition (3). 
     FIG. 12 shows field curvatures, uniform velocity scanning characteristic and linearity relating to Example 9. 
     The uniform velocity scanning characteristic is defined as: 
     
       
         [{h′(θ)−h(θ)}/h(θ)]×100(%) 
       
     
     where h(θ) is the ideal image height for the deflection angle θ (=constant×θ) while h′(θ) is the actual image height. 
     In FIG. 12, a field curvature indicated by a dashed line and a field curvature indicated by a solid line correspond to the main scanning direction and subscanning direction, respectively. 
     EXAMPLE 10 
     
       
         
           
               
             
               
                   
               
             
            
               
                 S = −∞ (beam coupled by the coupling lens is a parallel beam) 
               
            
           
           
               
               
               
            
               
                   
                 S(0) = 51.871 
                   
               
               
                 X 1 (Y), r s1 (Y) 
                 d(0) = 21.754 
                 n = 1.52664 
               
               
                 X 2 (Y), r s2 (Y) 
                 l(0) = 149.07 
               
            
           
           
               
            
               
                 surface facing deflector: nonarcuate toric surface 
               
            
           
           
               
               
            
               
                   
                 X 1 (Y) (nonarcuate shape) 
               
               
                   
                 R = 461.302, K = −35.937, 
               
               
                   
                 A = −2.205E−7, B = 2.732E−11, 
               
               
                   
                 C = −4.9587E−15, D = 1.6048E−19 
               
               
                   
                 r s1 (Y) 
               
               
                   
                 r s1 (0) = −73.845 
               
            
           
           
               
            
               
                 surface facing surface: special toric surface 
               
            
           
           
               
               
            
               
                   
                 X 2 (Y): arcuate shape with R = −97.244 
               
               
                   
                   
               
            
           
         
       
     
     As for r s2 (Y), r s2 (Y′) with respect to the deflection angle θ are listed below. 
     
       
         
           
               
               
               
               
               
               
               
               
             
               
                   
               
               
                 θ (°) 
                 0 
                 6 
                 14 
                 26 
                 32 
                 36 
                 41 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 Y′ 
                 0 
                 6.831 
                 16.005 
                 30.152 
                 37.565 
                 42.713 
                 49.470 
               
               
                 r s2 (Y′) 
                 −20.64 
                 −20.6 
                 −20.42 
                 19.77 
                 19.18 
                 18.64 
                 −17.75 
               
               
                   
               
            
           
         
       
     
     By using a polynomial, r s2 (Y′) may be approximated as: 
     
       
         r s2 (Y′)=a+b·Y′ 2 +c·Y′ 4 +d·Y′ 6 +e·P′ 8 +f·Y′ 10   
       
     
     Then, by use of Y′ and corresponding r s2 (Y′) shown above, the coefficients a, b, c, d, e and f are determined by a minimum square method as follows: 
     a=−20.63952, b=0.82661E-3, c=0.13218E-6, 
     d=0.209082E-10, e=−0.53157E-14, 
     f=4.71E-19 
     |β(0)|=2.09 holds and satisfies the condition (3). 
     FIG. 13 shows field curvatures, uniform velocity scanning characteristic and linearity relating to Example 10. In this example, because the coupled beam is a parallel beam, the constant mentioned previously in relation to the uniform velocity scanning characteristic is the focal distance fin in the direction corresponding to the main scanning direction. The uniform velocity scanning characteristic is the usual f-θ characteristic. 
     EXAMPLE 11 
     
       
         
           
               
             
               
                   
               
             
            
               
                 S = −375.922 (beam coupled by the coupling lens 12 is a weak divergent 
               
               
                 beam). 
               
            
           
           
               
               
               
            
               
                   
                 S(0) = 22.274 
                   
               
               
                 X 1 (Y), r s1 (Y) 
                 d(0) = 10 
                 n = 1.53664 
               
               
                 X 2 (&amp;), r s2 (Y) 
                 l(0) = 142.927 
               
            
           
           
               
            
               
                 surface facing deflector: nonarcuate toric surface 
               
            
           
           
               
               
            
               
                   
                 X 1 (Y) (nonarcuate shape) 
               
               
                   
                 R = 107.1, K = −60.197, 
               
               
                   
                 A = −3.5107E−6, B = 2.0684E−9, 
               
               
                   
                 C = −8.9722E−14, D = 1.1672E−17 
               
            
           
           
               
            
               
                 The values of r s1 (Y) with respect to the deflection angle θ are listed 
               
               
                 below. 
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 θ (°) 
                 0 
                 10 
                 20 
                 30 
                 36 
                 40 
                 45 
               
               
                   
               
               
                 Y 
                 0 
                 3.940 
                 8.207 
                 13.200 
                 16.757 
                 19.451 
                 23.281 
               
               
                 r s1 (Y) 
                 −22.0 
                 −25.07 
                 −29.076 
                 −40.588 
                 −50.791 
                 −65.907 
                 −85.507 
               
               
                   
               
            
           
           
               
            
               
                 surface facing surface 20: special toric surface 
               
            
           
           
               
               
            
               
                   
                 X 2 (Y) (nonarcuate shape) 
               
               
                   
                 R = −119.99, K = 4.78039 
               
               
                   
                 A = −3.4156E−6, B = 1.5437E−9, 
               
               
                   
                 C = −1.8765E−12, D = 1.4637E−15 
               
               
                   
                   
               
            
           
         
       
     
     As for r s2 (Y), r s2 (Y′) with respect to the deflection angle θ are listed below. 
     
       
         
           
               
               
               
               
               
               
               
               
             
               
                   
               
               
                 θ (°) 
                 0 
                 10 
                 20 
                 30 
                 36 
                 40 
                 45 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 Y′ 
                 0 
                 4.935 
                 10.107 
                 15.775 
                 19.523 
                 22.190 
                 25.721 
               
               
                 r s2 (Y′) 
                 −9.807 
                 −10.12 
                 −10.53 
                 −11.32 
                 −11.77 
                 −12.22 
                 −12.58 
               
               
                   
               
            
           
         
       
     
     Let r s1 (Y′) be approximated as: 
     
       
         r s1 (Y)=a+b·Y 2 +c·Y 4 +d·Y 6 +e·Y 8 +f·Y 10 +g·Y 12   
       
     
     Then, the coefficients a, b, c, d, e and f are determined by a minimum square method as follows: 
     a=−22, b=−2.5751E-1, c=0.449657, 
     d=4.466449E-5, e=1.956944E-7, 
     f=−3.87E-10, g=2.79283E-13. 
     Let r S2 (Y′) be approximated as: 
     
       
         r S2 (Y′)=a′+b′·Y 2 +c′·Y′ 4 +d′·Y′ 6 , +e′·Y′ 8 +f·Y′ 10 +g′·Y 12   
       
     
     Then, the coefficients a′, b′, c′, d′, e′, f′ and g′ are determined by a minimum square method as follows: 
     a′=−9.807, b′=−1.662605E-2, c′=1.870374E-4, 
     d′=−1.27425E-6, e′=4,079722E-9, 
     f′=−6.029437E-12, g′=3.315003E-15 
     S(θ), d(θ), l(θ), r S1 ′(Y) and r S2 ′(Y) are listed below. 
     
       
         
           
               
               
               
               
               
               
               
               
             
               
                   
               
               
                 θ 
                 0 
                 10 
                 20 
                 30 
                 36 
                 40 
                 45 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 S (θ) 
                 22.274 
                 22.689 
                 23.997 
                 26.399 
                 28.509 
                 30.261 
                 32.924 
               
               
                 d (θ) 
                 10 
                 9.876 
                 9.452 
                 8.551 
                 7.635 
                 6.793 
                 5.386 
               
               
                 l (θ) 
                 142.887 
                 144.327 
                 148.338 
                 154.765 
                 159.674 
                 163.608 
                 168.608 
               
               
                 r s1 ′ (Y) 
                 −22 
                 −24.732 
                 −27.589 
                 −36.35 
                 −43.735 
                 −55.259 
                 −69.233 
               
               
                 r s2 ′ (Y′) 
                 −9.807 
                 −10.101 
                 −10.495 
                 −11.382 
                 −12.042 
                 −12.761 
                 −13.572 
               
               
                   
               
            
           
         
       
     
     β(θ) are calculated by substituting the above data for the above-mentioned equation including P(θ). then, |β(θ)| and |β(θ)/β(0)| with respect to the deflection angle θ are produced, as listed below. 
     
       
         
           
               
               
               
               
               
               
               
               
             
               
                   
               
               
                 θ (°) 
                 0 
                 10 
                 20 
                 30 
                 36 
                 40 
                 45 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 Y 
                 0 
                 3.94 
                 8.207 
                 13.2 
                 16.757 
                 19.451 
                 23.281 
               
               
                 |β(θ)| 
                 4.42 
                 4.47 
                 4.49 
                 4.53 
                 4.53 
                 4.55 
                 4.52 
               
               
                 |β(θ)/β(0)| 
                 1.0 
                 1.01 
                 1.02 
                 1.03 
                 1.03 
                 1.03 
                 1.02 
               
               
                   
               
            
           
         
       
     
     |β(0)|=4.24 holds and satisfies the above-mentioned condition (3). |β(θ)| lies between |β(0)| and 1.03 |β(0)| and satisfies the above-mentioned condition (4). 
     FIG. 14 shows field curvatures, uniform velocity scanning characteristic and linearity relating to Example 11. Because the surface of the lens  18  facing the deflector in Example 11 is identical in shape as the same surface of the lens  18  in Example 9, the field curvature in the main scanning direction, uniform velocity scanning characteristic and linearity of Example 9 also hold. 
     EXAMPLE 12 
     
       
         
           
               
             
               
                   
               
             
            
               
                 S = 554.6 (beam coupled by the coupling lens 12 is a weak convergent beam). 
               
            
           
           
               
               
               
            
               
                   
                 S(0) = 38.916 
                   
               
               
                 X 1 (Y), r s1 (Y) 
                 d(0) = 15 
                 n = 1.53664 
               
               
                 X 2 (Y), r s2 (Y) 
                 l(0) = 120.984 
               
            
           
           
               
            
               
                 surface facing deflector: special toric surface 
               
            
           
           
               
               
            
               
                   
                 X 1 (Y) (nonarcuate shape) 
               
               
                   
                 R = 168.191, K = −39.924, 
               
               
                   
                 A = −3.4224E−7, B = 2.2792E−13, 
               
               
                   
                 C = 8.0209E−15, D = −8.7059E−19 
               
               
                   
                   
               
            
           
         
       
     
     The values of r S1 (Y) with respect to the deflection angle θ are listed below. 
     
       
         
           
               
               
               
               
               
               
               
               
             
               
                   
               
             
            
               
                 θ (°) 
                 0 
                 10 
                 20 
                 30 
                 36 
                 40 
                 45 
               
               
                   
               
               
                 Y 
                 0 
                 6.886 
                 14.368 
                 23.208 
                 29.610 
                 34.524 
                 41.574 
               
               
                 r s1 (Y) 
                 −31.0 
                 −33.14 
                 −45.56 
                 −70.28 
                 −81.84 
                 −102.13 
                 −126.96 
               
            
           
           
               
               
            
               
                   
                 surface facing surface 20: special toric surface 
               
            
           
           
               
               
            
               
                   
                 X 2 (Y) (nonarcuate shape) 
               
               
                   
                 R = −188.994, K = −4.83793 
               
               
                   
                 A = −4.9817E−7, B = 7.7236E−12, 
               
               
                   
                 C = −1.6386E−14, D = 1.7827E−18 
               
               
                   
                   
               
            
           
         
       
     
     As for r S2 (Y), r S2 (Y′) with respect to the deflection angle θ are listed below. 
     
       
         
           
               
               
               
               
               
               
               
               
             
               
                   
               
               
                 θ (°) 
                 0 
                 10 
                 20 
                 30 
                 36 
                 40 
                 45 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 Y′ 
                 0 
                 8.348 
                 17.093 
                 26.710 
                 33.136 
                 37.760 
                 43.915 
               
               
                 r s2 (Y′) 
                 −13.91 
                 −14.16 
                 −15.25 
                 −16.63 
                 −16.99 
                 −17.44 
                 −17.54 
               
               
                   
               
            
           
         
       
     
     Let r S2 (Y′) be approximated as: 
     
       
         r S2 (Y′)=a′+b′·Y′ 2 +c′·Y′ 4 +d′·Y′ 6 +e′·Y′ 8 +f′·Y′ 10 +g′·Y′ 12   
       
     
     Then the coefficients a′, b′, c′, d′, e′, f′ and g′ are determined by a minimum square method as follows: 
     a′=−13.91, b′=−2.866184E-3, c′=−1.21239E-5, 
     d′=2.690094E-8, e′=−2.090761E-11, 
     f′=6.668513E-15, g′=−6.73470E-19 
     S(θ), d(θ), l(θ), r S1 ′(Y) and r S2 ′(Y′) are listed below. 
     
       
         
           
               
               
               
               
               
               
               
               
             
               
                   
               
               
                 θ 
                 0 
                 10 
                 20 
                 30 
                 36 
                 40 
                 45 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 S (θ) 
                 38.916 
                 39.656 
                 42.01 
                 46.417 
                 50.376 
                 53.81 
                 58.794 
               
               
                 d (θ) 
                 15 
                 14.747 
                 13.884 
                 12.038 
                 10.135 
                 8.354 
                 5.324 
               
               
                 l (θ) 
                 121.012 
                 122.213 
                 125.713 
                 131.319 
                 135.629 
                 138.898 
                 143.746 
               
               
                 r s1 ′ (Y) 
                 −31 
                 −32.676 
                 −43.121 
                 −62.437 
                 −69.544 
                 −84.248 
                 −101.250 
               
               
                 r s2 ′ (Y′) 
                 −13.91 
                 −14.145 
                 −15.21 
                 −16.769 
                 −17.478 
                 −18.376 
                 −19.395 
               
               
                   
               
            
           
         
       
     
     β(θ) are calculated by substituting the above data for the above-mentioned equation including P(θ). Then, |β(θ)| and |β(θ)/β(0)| with respect to the polarization angle θ are produced, as listed below. 
     
       
         
           
               
               
               
               
               
               
               
               
             
               
                   
               
               
                 θ (°) 
                 0 
                 10 
                 20 
                 30 
                 36 
                 40 
                 45 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 Y 
                 0 
                 6.886 
                 14.368 
                 23.208 
                 29.610 
                 34.524 
                 41.573 
               
               
                 |β(θ)| 
                 2.19 
                 2.20 
                 2.25 
                 2.29 
                 2.28 
                 2.27 
                 2.27 
               
               
                 |β(θ)/β(0)| 
                 1.0 
                 1.01 
                 1.03 
                 1.04 
                 1.04 
                 1.04 
                 1.04 
               
               
                   
               
            
           
         
       
     
     |β(0)|=4.24 holds and satisfies the condition (3). |β(θ)| lies between |β(0)| and 1.04 |β(0)| and satisfies the condition (4). 
     FIG. 15 shows field curvatures, uniform velocity scanning characteristic and linearity relating to Example 12. 
     EXAMPLE 13 
     
       
         
           
               
             
               
                   
               
             
            
               
                 S = 316.7 (beam coupled by the coupling lens 12 is a weak convergent beam). 
               
            
           
           
               
               
               
            
               
                   
                 S(0) = 48.618 
                   
               
               
                 X 1 (Y), r s1 (Y) 
                 d(0) = 20.0 
                 n = 1.53664 
               
               
                 X 2 (Y), r s2 (Y) 
                 l(0) = 106.282 
               
            
           
           
               
            
               
                 surface facing deflector: special toric surface 
               
            
           
           
               
               
            
               
                   
                 X 1 (Y) (nonarcuate shape) 
               
               
                   
                 R = 200.224, K = −30.5827, 
               
               
                   
                 A = −1.5926E−7, B = −4.0453E−12, 
               
               
                   
                 C = 6.0017E−15, D = −3.9857E−19 
               
               
                   
                   
               
            
           
         
       
     
     The values of r S1 (Y) with respect to the deflection angle θ are listed below. 
     
       
         
           
               
               
               
               
               
               
               
               
             
               
                   
               
             
            
               
                 θ (°) 
                 0 
                 10 
                 20 
                 30 
                 36 
                 40 
                 45 
               
               
                   
               
               
                 Y 
                 0 
                 8.605 
                 17.967 
                 29.208 
                 37.180 
                 43.459 
                 52.640 
               
               
                 r s1 (Y) 
                 −30.0 
                 −29.18 
                 −28.76 
                 −27.74 
                 −26.56 
                 −24.18 
                 −17.02 
               
               
                   
               
            
           
           
               
               
            
               
                   
                 surface facing surface 20: special toric surface 
               
            
           
           
               
               
            
               
                   
                 X 2 (Y) (nonarcuate shape) 
               
               
                   
                 R = −224.092, K = 1.19901 
               
               
                   
                 A = −3.2013E−7, B = 7.0634E−12, 
               
               
                   
                 C = −9.2418E−15, D = 1.9593E−18 
               
               
                   
                   
               
            
           
         
       
     
     As for r S2 (Y), r S2 (Y′) with respect to the deflection angle θ are listed below. 
     
       
         
           
               
               
               
               
               
               
               
               
             
               
                   
               
               
                 θ (°) 
                 0 
                 10 
                 20 
                 30 
                 36 
                 40 
                 45 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 Y′ 
                 0 
                 10.583 
                 21.558 
                 33.647 
                 41.714 
                 47.523 
                 55.334 
               
               
                 r s2 (Y′) 
                 −15.32 
                 −15.23 
                 −15.12 
                 −14.53 
                 −13.64 
                 −12.44 
                 −9.34 
               
               
                   
               
            
           
         
       
     
     Let r S1 (Y) be approximated as: 
     
       
         r S1 (Y)=a+b·Y 2 +c·Y 4 +d·Y 6 ,+e·Y 8 +f·Y 10 +g·Y 12   
       
     
     Then, the coefficients a, b, c, d, e, f, and g are determined by a minimum square method as follows: 
     a=−30, b=1.521146E-2, c=−6.399798E-5, 
     d=1.166608E-7, e=−9.651438E-11, 
     f=3.652588E-14, g=−5.073881E-18 
     Let r S1 (Y′) be approximated as: 
     
       
         r S1 (Y′)=a′+b′·Y′ 2 +c′·Y′ 4 +d′·Y′ 6 ,+e′·Y′ 8 +f·Y′ 10 +g′·Y′ 12   
       
     
     Then coefficients a′, b′, c′, d′, e′, f′ and g′ are determined by a minimum square method as follows: 
     a′=−15.32, b′=1.12584E-3, c′=−3.490667E-6, 
     d′=5.91427E-9, e′=−4.057836E-12, 
     f′=1.284796E-15, g′=−1.500234E-19 
     S(θ), d(θ), l(θ), r S1 ′(Y) and r S2 ′(Y) are listed below. 
     
       
         
           
               
               
               
               
               
               
               
               
             
               
                   
               
               
                 θ 
                 0 
                 10 
                 20 
                 30 
                 36 
                 40 
                 45 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 S (θ) 
                 48.618 
                 49.552 
                 52.533 
                 58.152 
                 63.255 
                 67.611 
                 74.444 
               
               
                 d (θ) 
                 20 
                 19.662 
                 18.494 
                 15.956 
                 13.296 
                 10.774 
                 6.411 
               
               
                 l (θ) 
                 106.282 
                 107.376 
                 110.608 
                 115.846 
                 119.932 
                 123.124 
                 127.823 
               
               
                 r 1 ′ (Y) 
                 −30.0 
                 −28.769 
                 −27.175 
                 −24.561 
                 −22.426 
                 −19.745 
                 −13.291 
               
               
                 r s2 ′ (Y′) 
                 −15.316 
                 −15.220 
                 −15.123 
                 −14.736 
                 −14.155 
                 −13.233 
                 −10.366 
               
               
                   
               
            
           
         
       
     
     β(θ) are calculated by substituting the above data for the equation including P(θ). Then, |β(θ)| and |β(θ)/β(0)| with respect to the deflection angle θ are produced, as listed below. 
     
       
         
           
               
               
               
               
               
               
               
               
             
               
                   
               
               
                 θ (°) 
                 0 
                 10 
                 20 
                 30 
                 36 
                 40 
                 45 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 Y 
                 0 
                 8.605 
                 17.967 
                 29.076 
                 37.180 
                 43.459 
                 52.640 
               
               
                 |β(θ)| 
                 1.46 
                 1.45 
                 1.44 
                 1.42 
                 1.41 
                 1.41 
                 1.40 
               
               
                 |β(θ)/β(0)| 
                 1.0 
                 0.99 
                 0.99 
                 0.97 
                 0.97 
                 0.97 
                 0.96 
               
               
                   
               
            
           
         
       
     
     |β(θ)|=1.46 holds and satisfies the condition (3). |β(θ)| lies between |β(θ)| and 0.96 |β(θ)| and satisfies the condition (4). 
     FIG. 16 shows field curvatures, uniform scanning characteristic and linearity relating to Example 13. 
     As stated above, the illustrative embodiment has the following advantages. 
     (1) In an optical scanning device using a single scanning lens, a broad effective main scanning range and a compact configuration are achieved. 
     (2) When the above lens is molded out of plastics, sinking, waving and other errors in configuration are effectively obviated so as to enhance accurate configuration. 
     (3) Limitation on the tolerance of assembly of an optical element in the optical axis direction is reduced. 
     (4) In the above scanning device, field curvatures and uniform velocity scanning, characteristic in the main and subscanning directions are improved. 
     (5) In the above scanning device, the fluctuation of the diameter of a beam spot in the subscanning direction is effectively reduced. 
     2nd Embodiment 
     Referring to FIG. 17A, a second embodiment of the present invention is shown. In FIG. 17A, the same structural elements as the elements shown in FIG. 1A are designated by the same reference numerals. 
     As shown, a divergent beam issuing from the LD  10  is coupled by the coupling lens  12  which constitute a light source together with the LD  10 . The beam from the light source is converged by the convex cylinder lens or line image forming optical system  14  in the direction corresponding to the subscanning direction (perpendicular to the sheet surface of FIG.  17 A). As a result, the beam forms a line image elongate in the direction corresponding to the main scanning direction on the deflector or polygonal mirror  16 . When the polygonal mirror  16  rotates, it deflects the incident beam at a uniform angular velocity. The deflected beam is incident to the scanning lens  18 . The lens  18  focuses the incident beam in the form of a spot onto the surface  20  to be scanned. As a result, the surface  20  is scanned in the main scanning direction (up-and-down direction in the figure). Usually, a photoconductive element is located at the surface  20  and scanned by the above beam spot to form an image thereon. The effective main scanning width is indicated by letter W. The cylinder lens  14  may be replaced with a concave cylinder mirror. 
     In the illustrative embodiment, the coupling lens  12  functions to transform the bean issuing from the LD  10  to a weak convergent beam. Therefore, in the direction corresponding to the main scanning direction, the deflected beam with a deflection angle of zero degree is focused onto the previously stated natural convergence point Q if the lens  18  is absent. Specifically, assuming a virtual optical path formed by linearly developing an optical path between a light source and a surface to be scanned along an optical axis, the beam coupled by the coupling lens  12  converges to the point Q if the lens  18  is absent on the above virtual optical path. 
     The distance between the reflection surface  16  and the point Q is labeled S. When the point Q is located to the side of the surface  20  with respect to the deflector, the distance S is greater than zero; the coupled beam is a weak convergent beam. When the point Q is located to the side of the light source with respect to the polarizer, the distance S is smaller than zero; the coupled beam is a weak divergent beam. If the coupled beam is a parallel beam, the distance is infinite. 
     In FIG. 17A, X 1 (Y) is representative of the shape of the first surface, as counted from the deflector side, as seen in the deflection plane (i.e.,shape shown in FIG. 17A) while X 2 (Y) is representative of the shape of the second surface. Both of these shapes are nonarcuate shapes determined on the basis of R, K, A, B, C, D, . . . of the Eq. (1). 
     Further, in this embodiment, the lens  18  constitutes an image forming lens itself. Therefore, the lens  18  maintains the image forming position of the line image and the surface  20  in a geometrically conjugate relation in the direction corresponding to the subscanning direction. In addition, the lens  18  is so configured as to desirably correct the field curvature in the subscanning direction. It follows that the first and/or the second surface of the lens  18 , as counted from the deflector side, has a special toric surface, as shown in FIG. 17B or  17 C. Let the shapes of the opposite surfaces of the lens  18  with respect to the direction corresponding to the subscanning direction (direction perpendicular to the sheet surface of FIG. 17A) be labeled x 1 (Y) and x 2 (Y), respectively. 
     In FIGS. 17B and 17C, a curve X(Y) shows the nonarcuate shape (R: paraxial radius of curvature in the Eq. (1)). A special toric surface is a surface in which the radius of curvature r(Y) of a circle of curvature in the direction corresponding to the subscanning direction is changed in accordance with the individual Y coordinate position of a nonarcuate shape in such a manner as to desirably correct the conjugate function and the field curvature in the subscanning direction. Generally, a line connecting the centers of curvatures for the radii of curvatures r(Y) is generally a curve, as shown in FIGS. 17A and 17C. 
     In the illustrative embodiment, the nonarcuate shape, as seen in the deflection plane, may be convex toward the deflector in the paraxial portion of the first lens surface and may be convex toward the surface  20  in the paraxial portion of the second lens surface. 
     Specific examples of the second embodiment will be described hereinafter. In the examples, as shown in FIG. 17A, the distance between the deflection start point of the deflector and the incidence surface of the lens  18  is assumed to be d 0 . The lens  18  is assumed to have a thickness d 1  in the optical axis direction. The surface of the lens  18  facing the surface  20  and the surface  20  are assumed to be spaced a distance d 2  from each other. Therefore, L is equal to d 0 +d 1 +d 2 . In the polarization plane, the lens  18  is assumed to have a paraxial radius of curvature R 1  on the side facing the polarizer and a paraxial radius of curvature R 2  on the side facing the surface  20 . The refractive index (waveform: 780 mm) of the material of the lens  18  is labeled N. Further, the focal distance in the polarization plane is labeled fm. 
     The nonarcuate shape X 1 (Y) of the first lens surface, as counted from the deflector side, is determined by giving R 1 , K 1 , A 1 , B 1 , C 1  and D 1 . The nonarcuate shape X 2 (Y) of the second lens surface is determined by giving R 2 , K 2 , A 2 , B 2 , C 2  and D 2 . 
     When the first and/or the second lens surface is a special toric surface, a radius of curvature r 1 (θ) and/or r 2 (θ) is given to x 1 (Y)=r 2 (Y) and/or x 2 Y=r 2 (Y) for each of the deflection angles θ=0, 10, 20, 30, 36, 40 and 45 degrees (or=10, 20, 30 and 34 degrees in Example 6). It is to be noted that Y and θ have a relation of Y=d 0 ·θ. 
     In Examples 1-5 to be described, the effective main scanning width W is 216 mm while the effective half field angle θmax is 45 degrees. In Example 6, W and θmax are 216 mm and 34 degrees, respectively. 
     EXAMPLE 1 
     In this example, the beam coupled by the coupling lens is a parallel beam. 
     
       
         
           
               
               
            
               
                   
                   
               
               
                   
                 S = ∞, fm = 137.503 
               
            
           
           
               
               
               
               
               
            
               
                   
                 i 
                 Ri 
                 di 
                 N 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
            
               
                   
                 0 
                   
                 29.887 
                   
               
               
                   
                 1 
                 137.503 
                 12.364 
                 1.53664 
               
               
                   
                 2 
                 −154.248 
                 132.649 
               
               
                   
                   
               
            
           
         
       
     
     The lens surface facing the deflector is a coaxial nonspherical surface and symmetrical with respect to the optical axis. Let this be denoted by X 1 (Y)=x 1 (Y). The lens surface facing the surface  20  is a special toric surface. 
     
       
         
           
               
             
               
                   
               
             
            
               
                 X 1 (Y): 
               
               
                 R 1  = 237.503, K 1  = −92.438, 
               
               
                 A 1  = 1.11822 × 10 −6 , B 1  = 7.28745 × 10 −10 , 
               
               
                 C 1  = −3.20311 × 10 −13 , D 1  = 9.55204 × 10 −17   
               
               
                 X2(Y): 
               
               
                 R 2  = −154.248, K 1  = 5.36873, 
               
               
                 A 2  = −2.51300 × 10 −6 , B 2  = 1.95625 × 10 −9 , 
               
               
                 C 2  = −1.18490 × 10 −12 , D 2  = 3.38372 × 10 −16   
               
               
                 r 2 (θ): 
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 θ (°) 
                 0 
                 10 
                 20 
                 30 
                 36 
                 40 
                 45 
               
               
                   
               
               
                 r2 
                 −17.083 
                 −17.243 
                 −17.658 
                 −18.116 
                 −18.463 
                 −18.607 
                 −18.803 
               
               
                 fm = 
                 136.503 
               
            
           
           
               
            
               
                 (W/L = 1.23, (W/L) 2  · (L/d 1 ) = 21.6, 
               
               
                 (d 0 /d 2 ) = 0.23 
               
               
                   
               
            
           
         
       
     
     In Examples 2 and 3 to be described, the coupled beam is a weak divergent beam, so that S is smaller than zero. Again, the lens surface facing the deflector and the surface facing the surface  20  are a coaxial nonspherical surface and a special toric surface, respectively. 
     EXAMPLE 2 
     
       
         
           
               
             
               
                   
               
             
            
               
                 S = −391.925, fm = 108.193 
               
            
           
           
               
               
               
               
            
               
                 i 
                 R i   
                 d i   
                 N 
               
               
                   
               
               
                 0 
                   
                 22.381 
               
               
                 1 
                 108.193 
                 10.000 
                 1.53664 
               
               
                 2 
                 −121.259 
                 142.519 
               
               
                   
               
            
           
           
               
            
               
                 X 1 (Y) (= x 1 (Y)): 
               
               
                 R 1  = 108.503, K 1  = −56.32541, 
               
               
                 A 1  = −3.46610 × 10 −6 , B 1  = 1.98195 × 10 −9 , 
               
               
                 C 1  = −1.32194 × 10 −13 , D 1  = 5.00528 × 10 −17   
               
               
                 X 2 (Y): 
               
               
                 R 2  = 121.259, K 2  = 4.91312, 
               
               
                 A 2  = −3.24924 × 10 −6 , B 2  = 1.44308 × 10 −9 , 
               
               
                 C 2  = −1.89357 × 10 −12 , D 2  = 1.43613 × 10 −15   
               
               
                 r 2  (θ): 
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 θ (°) 
                 0 
                 10 
                 20 
                 30 
                 36 
                 40 
                 45 
               
               
                   
               
               
                 r 2   
                 −13.913 
                 −14.070 
                 −14.487 
                 −14.979 
                 −15,160 
                 −15.181 
                 −15.136 
               
               
                   
               
            
           
           
               
            
               
                 (W/L) = 1.23, (W/L) 2  · (L/d 1 ) = 26.7, 
               
               
                 (d 0 /d 2 ) = 0.16 
               
               
                   
               
            
           
         
       
     
     EXAMPLE 3 
     
       
         
           
               
             
               
                   
               
             
            
               
                 S = −796.535, fm = 125.223 
               
            
           
           
               
               
               
               
            
               
                 i 
                 R i   
                 d i   
                 N 
               
               
                   
               
               
                 0 
                   
                 25.505 
               
               
                 1 
                 125.223 
                 12.700 
                 1.48578 
               
               
                 2 
                 −114.374 
                 136.695 
               
               
                   
               
            
           
           
               
            
               
                 X 1 (Y) (= x 1 (Y)): 
               
               
                 R 1  = 125.223, K 1  = −12.16377, 
               
               
                 A 1  = −3.41094 × 10 −6 , B 1  = 1.79586 × 10 −9 , 
               
               
                 C 1  = −2.13309 × 10 −13 , D 1  = 1.22926 × 10 −16   
               
               
                 X 2 (Y): 
               
               
                 R 2  = −114.374, K 2  = 4.94342, 
               
               
                 A 2  = −1.42312 × 10 −6 , B 2  = 1.07576 × 10 −9 , 
               
               
                 C 2  = −1.88925 × 10 −12 , D 2  = 1.20601 × 10 −15   
               
               
                 r 2  (θ): 
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 θ (°) 
                 0 
                 10 
                 20 
                 30 
                 36 
                 40 
                 45 
               
               
                   
               
               
                 r 2   
                 −13.246 
                 −13.396 
                 −13.815 
                 −14.395 
                 −14.717 
                 −14.871 
                 −15.096 
               
               
                   
               
            
           
           
               
            
               
                 (W/L) = 1.23, (W/L) 2  · (L/d 1 ) = 21.0, 
               
               
                 (d 0 /d 2 ) = 0.19 
               
               
                   
               
            
           
         
       
     
     In Examples 4-6 to be described, the coupled beam is a weak convergent flux, so that S is greater than zero. In Examples 4 and 5, both the surface facing the deflector and the surface facing the surface  20  are special toric surfaces. 
     EXAMPLE 4 
     
       
         
           
               
             
               
                   
               
             
            
               
                 S = 551.935, fm = 168.191 
               
            
           
           
               
               
               
               
            
               
                 i 
                 R i   
                 d i   
                 N 
               
               
                   
               
               
                 0 
                   
                 38.916 
               
               
                 1 
                 168.191 
                 15.000 
                 1.5370 
               
               
                 2 
                 −188.994 
                 120.984 
               
               
                   
               
            
           
           
               
            
               
                 X 1 (Y) (= x 1 (Y)): 
               
               
                 R 1  = 168.191, K 1  = −−39.925, 
               
               
                 A 1  = −3.42235 × 10 −7 , B 1  = 2.27288 × 10 −13 , 
               
               
                 C 1  = 8.02089 × 10 −15 , D 1  = −8.70591 × 10 −19   
               
               
                 r 1  (θ): 
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 θ (°) 
                 0 
                 10 
                 20 
                 30 
                 36 
                 40 
                 45 
               
               
                   
               
               
                 r 1   
                 −31.000 
                 −33.138 
                 −45.561 
                 −70.282 
                 −81.839 
                 −102.128 
                 −126.958 
               
               
                   
               
            
           
           
               
            
               
                 X 2 (Y): 
               
               
                 R 2  = −188.994, K 2  = 4.83792, 
               
               
                 A 2  = −4.98172 × 10 −7 , B 2  = 7.72365 × 10 −12 , 
               
               
                 C 2  = −1.63863 × 10 −14 , D 2  = 1.7827 × 10 −18   
               
               
                 r 2  (θ): 
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 θ (°) 
                 0 
                 10 
                 20 
                 30 
                 36 
                 40 
                 45 
               
               
                   
               
               
                 r 2   
                 −13.910 
                 −14.163 
                 −15.245 
                 −16.629 
                 −16.990 
                 −17.441 
                 −17.540 
               
               
                   
               
            
           
           
               
            
               
                 (W/L) = 1.23, (W/L) 2  · (L/d 1 ) = 17.8, 
               
               
                 (d 0 /d 2 ) = 0.46 
               
               
                   
               
            
           
         
       
     
     EXAMPLE 5 
     
       
         
           
               
             
               
                   
               
             
            
               
                 S = 316.745, fm = 200.243 
               
            
           
           
               
               
               
               
            
               
                 i 
                 R i   
                 d i   
                 N 
               
               
                   
               
               
                 0 
                   
                 48.618 
               
               
                 1 
                 200.244 
                 20.000 
                 1.537664 
               
               
                 2 
                 −224.142 
                 106.382 
               
               
                   
               
            
           
           
               
            
               
                 X 1 (Y): 
               
               
                 R 1  = 200.244, K 1  = −30.283, 
               
               
                 A 1  = −1.59263 × 10 −7 , B 1  = −4.04532 × 10 −12 , 
               
               
                 C 1  = 6.02170 × 10 −15 , D 1  = −3.98571 × 10 −19   
               
               
                 r 1  (θ): 
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 θ (°) 
                 0 
                 10 
                 20 
                 30 
                 36 
                 40 
                 45 
               
               
                   
               
               
                 r 1   
                 −30.000 
                 −29.182 
                 −28.764 
                 −27.743 
                 −26.556 
                 −24.175 
                 −17.022 
               
               
                   
               
            
           
           
               
            
               
                 X 2 (Y): 
               
               
                 R 2  = −224.142, K 2  = 1.1990 
               
               
                 A 2  = −3.20126 × 10 −7 , B 2  = 7.06344 × 10 −12 , 
               
               
                 C 2  = −9.24177 × 10 −15 , D 2  = 1.95829 × 10 −18   
               
               
                 r 2  (θ): 
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 θ (°) 
                 0 
                 10 
                 20 
                 30 
                 36 
                 40 
                 45 
               
               
                   
               
               
                 r 2   
                 −15.316 
                 −15.232 
                 −15.119 
                 −14.531 
                 −13.641 
                 −12.436 
                 −9.338 
               
               
                   
               
            
           
           
               
            
               
                 (W/L) = 1.23, (W/L) 2  · (L/d 1 ) = 13.3, 
               
               
                 (d 0 /d 2 ) = 0.37 
               
               
                   
               
            
           
         
       
     
     In Example 6 to be described, the lens surface facing the deflector is a coaxial nonspherical surface, i.e., X 1 (Y)=x 1 (Y) while the lens surface facing the surface  20  is a special toric surface. 
     EXAMPLE 6 
     
       
         
           
               
             
               
                   
               
             
            
               
                 S = +728.528, fm = 225.700 
               
            
           
           
               
               
               
               
            
               
                 i 
                 R i   
                 d i   
                 N 
               
               
                   
               
               
                 0 
                   
                 51.669 
               
               
                 1 
                 148.846 
                 10.390 
                 1.5241 
               
               
                 2 
                 −562.372 
                 162.880 
               
               
                   
               
            
           
           
               
            
               
                 X 1 (Y) (= x 1 (Y)): 
               
               
                 R 1  = 148.846, K 1  = 5.4534, 
               
               
                 A 1  = −7.09267 × 10 −7 , B 1  = −2.21975 × 10 −10   
               
               
                 C 1  = 6.07139 × 10 −14 , D 1  = −8.33979 × 10 −18   
               
               
                 X 2 (Y): 
               
               
                 R 2  = −562.372, K 2  = −462.3035, 
               
               
                 A 2  = −4.60398 × 10 −7 , B 2  = −2.89720 × 10 −11   
               
               
                 C 2  = −5.93656 × 10 −14 , D 2  = 1.73926 × 10 −17   
               
               
                 r 2  (θ): 
               
            
           
           
               
               
               
               
               
               
            
               
                 θ (°) 
                 0 
                 10 
                 20 
                 30 
                 34 
               
               
                   
               
               
                 r 2   
                 −25.651 
                 −26.163 
                 −27.558 
                 −28.895 
                 −28.836 
               
               
                   
               
            
           
           
               
            
               
                 (W/L) = 0.96, (W/L) 2  · (L/d 1 ) = 19.9, 
               
               
                 (d 0 /d 2 ) = 0.32 
               
               
                   
               
            
           
         
       
     
     As for the above special toric surface r(Y), an analytical representation is attainable by substituting the values Y(=d 0 ×θ) corresponding to the values θ for the following equation: 
      r(Y)=a+b·Y 2 +c·Y 4 +d·Y 6 +e·Y 8 +f·Y 10 +g·Y 12   
     and determining the coefficients a-g by use of simultaneous equations. 
     FIGS. 18-23 respectively show field curvatures and distortional aberrations relating to Examples 1-6. In each figure, a field curvature represented by a solid and a field curvature represented by a dashed line relate to the main scanning direction and the subscanning direction, respectively. The field curvature in the main scanning direction and the distortional aberration are effectively corrected by the nonarcuate shapes X 1 (Y) and X 2 (Y) determining the configuration of the lens in the deflection plane. Particularly, in Examples 4 and 5, the field curvature in the main scanning direction and the distortional aberration are both desirable. 
     The distortional aberration relates to the uniform velocity scanning function. The scanning velocity of the beam spot tends to increase with an increase in image height due to the beam polarized at a uniform angular velocity. In light of this, uniform velocity scanning is desirably realized by causing the aberration to increase at the portion where the image height increases, as illustrated. Further, the field curvature in the subscanning direction is desirably corrected by the special toric surface. 
     In Examples 7-15 to be described, both the first and second lens surfaces, as counted from the deflector side, are convex in the deflection plane, and each is a special toric surface. FIG. 17B shows the first lens surface while FIG. 17C shows the second lens surface. Only the data relating to the performance (field curvatures and uniform velocity scanning) in the direction corresponding to the main scanning direction, i. e., shape data in the deflection plane are given. The data relating to the direction corresponding to the subscanning direction, i.e., the radii of curvatures r(Y) in the above direction at the coordinate Y relating to the main scanning direction are optimized in each Example in a particular manner. 
     EXAMPLE 7 
     
       
         
           
               
             
               
                   
               
             
            
               
                 S = ∞, fm = 139.256 
               
            
           
           
               
               
               
               
            
               
                 i 
                 R i   
                 d i   
                 N 
               
               
                   
               
               
                 0 
                   
                 33.047 
               
               
                 1 
                 153.181 
                 11.223 
                 1.57210 
               
               
                 2 
                 −161.581 
                 134.898 
               
               
                   
               
            
           
           
               
            
               
                 X 1 (Y): 
               
               
                 R 1  = 153.181, K 1  = −15.522, 
               
               
                 A 1  = −4.90025 × 10 −7 , B 1  = −3.620007 × 10 −11   
               
               
                 C 1  = 1.57778 × 10 −14 , D 1  = 4.52347 × 10 −18   
               
               
                 X 2 (Y): 
               
               
                 R 2  = −161.581, K 2  = 4.90839, 
               
               
                 A 2  = 2.30522 × 10 −8 , B 2  = −1.60484 × 10 −10 , 
               
               
                 C 2  = −5.10123 × 10 −14 , D 2  = 4.52347 × 10 −17   
               
               
                 (W/L) = 1.17, (W/L) 2  · (L/d 1 ) = 21.9, 
               
               
                 (d 0 /d 2 ) = 0.24, R 1 /fm = 1.1 
               
               
                   
               
            
           
         
       
     
     EXAMPLE 8 
     
       
         
           
               
             
               
                   
               
             
            
               
                 S = ∞, fm = 137.047 
               
            
           
           
               
               
               
               
            
               
                 i 
                 R i   
                 d i   
                 N 
               
               
                   
               
               
                 0 
                   
                 34.511 
               
               
                 1 
                 274.094 
                 12.045 
                 1.57210 
               
               
                 2 
                 −108.062 
                 133.443 
               
               
                   
               
            
           
           
               
            
               
                 X 1 (Y): 
               
               
                 R 1  = 274.094, K 1  = −−0.488153 
               
               
                 A 1  = −3.90047 × 10 −7 , B 1  = −3.64736 × 10 −11   
               
               
                 C 1  = 1.52644 × 10 −14 , D 1  = −4.46187 × 10 −18   
               
               
                 X 2 (Y): 
               
               
                 R 2  = −108.062, K 2  = 2.98552, 
               
               
                 A 2  = 3.11749 × 10 −7 , B 2  = 2.04686 × 10 −11 , 
               
               
                 C 2  = −4.65135 × 10 −14 , D 2  = 1.16507 × 10 −17   
               
               
                 (W/L) = 1.17, (W/L) 2  · (L/d 1 ) = 20.3, 
               
               
                 (d 0 /d 2 ) = 0.26, R 1 /fm = 2.0 
               
               
                   
               
            
           
         
       
     
     EXAMPLE 9 
     
       
         
           
               
             
               
                   
               
             
            
               
                 S = ∞, fm = 137.041 
               
            
           
           
               
               
               
               
            
               
                 i 
                 R i   
                 d i   
                 N 
               
               
                   
               
               
                 0 
                   
                 34.305 
               
               
                 1 
                 205.561 
                 11.945 
                 1.57210 
               
               
                 2 
                 −124.060 
                 133.750 
               
               
                   
               
            
           
           
               
            
               
                 X 1 (Y): 
               
               
                 R 1  = 205.561, K 1  = −−4.42977, 
               
               
                 A 1  = −4.58583 × 10 −7 , B 1  = −3.92890 × 10 −11 , 
               
               
                 C 1  = 1.47898 × 10 −14 , D 1  = −8.64090 × 10 −19   
               
               
                 X 2 (Y): 
               
               
                 R 2  = −124.060, K 2  = 3.52004, 
               
               
                 A 2  = 3.24804 × 10 −7 , B 2  = −9.05801 × 10 −11 , 
               
               
                 C 2  = −4.80823 × 10 −14 , D 2  = 1.45322 × 10 −17   
               
               
                 (W/L) = 1.17, (W/L) 2  · (L/d 1 ) = 20.5, 
               
               
                 (d 0 /d 2 ) = 0.26, R 1 /fm = 1.5 
               
               
                   
               
            
           
         
       
     
     EXAMPLE 10 
     
       
         
           
               
             
               
                   
               
             
            
               
                 S = 413.282, fm = 179.979 
               
            
           
           
               
               
               
               
            
               
                 i 
                 R i   
                 d i   
                 N 
               
               
                   
               
               
                 0 
                   
                 42.725 
               
               
                 1 
                 197.976 
                 17.879 
                 1.51933 
               
               
                 2 
                 −171.592 
                 114.304 
               
               
                   
               
            
           
           
               
            
               
                 X 1 (Y): 
               
               
                 R 1  = 197.976, K 1  = 5.03422, 
               
               
                 A 1  = −4.91745 × 10 −7 , B 1  = −1.83806 × 10 −11 , 
               
               
                 C 1  = 1.63101 × 10 −14 , D 1  = −8.38595 × 10 −19   
               
               
                 X 2 (Y): 
               
               
                 R 2  = −171.592, K 2  = 4.60861, 
               
               
                 A 2  = −3.38197 × 10 −8 , B 2  = 3.65873 1.60484 × 10 −11 , 
               
               
                 C 2  = −5.14531 × 10 −14 , D 2  = 1.28251 × 10 −17   
               
               
                 (W/L) = 1.20, (W/L) 2  · (L/d 1 ) = 14.1 
               
               
                 (d 0 /d 2 ) = 0.37, R 1 /fm = 1.1 
               
               
                   
               
            
           
         
       
     
     EXAMPLE 11 
     
       
         
           
               
             
               
                   
               
             
            
               
                 S = 492.845, fm = 168.271 
               
            
           
           
               
               
               
               
            
               
                 i 
                 R i   
                 d i   
                 N 
               
               
                   
               
               
                 0 
                   
                 45.430 
               
               
                 1 
                 252.407 
                 17.300 
                 1.51933 
               
               
                 2 
                 −130.535 
                 117.270 
               
               
                   
               
            
           
           
               
            
               
                 X 1 (Y): 
               
               
                 R 1  = 252.407, K 1  = −5.43257, 
               
               
                 A 1  = −3.90428 × 10 −7 , B 1  = −6.84195 × 10 −12   
               
               
                 C 1  = 1.88037 × 10 −14 , D 1  = −2.13891 × 10 −18   
               
               
                 X 2 (Y): 
               
               
                 R 2  = −130.535, K 2  = −0.03984, 
               
               
                 A 2  = 2.85512 × 10 −8 , B 2  = 2.12059 × 10 −11 , 
               
               
                 C 2  = −2.76749 × 10 −14 , D 2  = 1.01019 × 10 −17   
               
               
                 (W/L) = 1.17, (W/L) 2  · (L/d 1 ) = 14.2, 
               
               
                 (d 0 /d 2 ) = 0.39, R 1 /fm = 1.5 
               
               
                   
               
            
           
         
       
     
     EXAMPLE 12 
     
       
         
           
               
             
               
                   
               
             
            
               
                 S = 746.803, fm = 156.939 
               
            
           
           
               
               
               
               
               
            
               
                   
                 i 
                 R i   
                 d i   
                 N 
               
               
                   
                   
               
               
                   
                 0 
                   
                 41.457 
               
               
                   
                 1 
                 313.879 
                 14.327 
                 1.57210 
               
               
                   
                 2 
                 −123.669 
                 124.216 
               
               
                   
                   
               
            
           
           
               
            
               
                 X 1 (Y): 
               
               
                 R 1  = 313.879, K 1  = 11.63499, 
               
               
                 A 1  = −2.71261 × 10 −7 , B 1  = −2.54808 × 10 −11 , 
               
               
                 C 1  = 1.96321 × 10 −14 , D 1  = −2.73780 × 10 −18   
               
               
                 X 2 (Y): 
               
               
                 R 2  = −123.669, K 2  = −0.03984, 
               
               
                 A 2  = 1.30110 × 10 −7 , B 2  = 5.58410 × 10 −11 , 
               
               
                 C 2  = −3.31374 × 10 −14 , D 2  = 1.13522 × 10 −17   
               
               
                 (W/L) = 1.17, (W/L) 2  · (L/d 1 ) = 17.1 
               
               
                 (d 0 /d 2 ) = 0.33, R 1 /fm = 2.0 
               
               
                   
               
            
           
         
       
     
     EXAMPLE 13 
     
       
         
           
               
             
               
                   
               
             
            
               
                 S = −1066.595, fm = 124.231 
               
            
           
           
               
               
               
               
               
            
               
                   
                 i 
                 R i   
                 d i   
                 N 
               
               
                   
                   
               
               
                   
                 0 
                   
                 27.989 
               
               
                   
                 1 
                 136.654 
                 10.110 
                 1.57210 
               
               
                   
                 2 
                 −144.109 
                 136.660 
               
               
                   
                   
               
            
           
           
               
            
               
                 X 1 (Y): 
               
               
                 R 1  = 136.654, K 1  = 89.8649, 
               
               
                 A 1  = 3.58249 × 10 −6 , B 1  = −1.82151 × 10 −9 , 
               
               
                 C 1  = −1.84033 × 10 −13 , D 1  = 5.22868 × 10 −17   
               
               
                 X 2 (Y): 
               
               
                 R 2  = −144.109, K 2 = 7.53274,   
               
               
                 A 2  = 4.10849 × 10 −6 , B 2  = 2.30006 × 10 −9 ; 
               
               
                 C 2 = −1.92530 × 10 −12 , D 2  = 9.95898× 10 −16   
               
               
                 (W/L) = 1.20, (W/L) 2  · (L/d 1 ) = 25.0, 
               
               
                 (d 0 /d 2  = 0.20, R 1 /fm = 1.1 
               
               
                   
               
            
           
         
       
     
     EXAMPLE 14 
     
       
         
           
               
             
               
                   
               
             
            
               
                 S = −646.979, fm = 118.255 
               
            
           
           
               
               
               
               
               
            
               
                   
                 i 
                 R i   
                 d i   
                 N 
               
               
                   
                   
               
               
                   
                 0 
                   
                 28 
               
               
                   
                 1 
                 177.382 
                 10.725 
                 1.57210 
               
               
                   
                 2 
                 −106.959 
                 140.375 
               
               
                   
                   
               
            
           
           
               
            
               
                 X 1 (Y): 
               
               
                 R 1  = 177.382, K 1  = −187.605; 
               
               
                 A 1  = −3.29591 × 10 −6 , B 1  = 1.90010 × 10 −9 , 
               
               
                 C 1  = −1.89600 × 10 −13 , D 1  = 2.05754 × 10 −17   
               
               
                 X 2 (Y): 
               
               
                 R 2  = −106.959, K 2  = 5.99435, 
               
               
                 A 2  = −3.44944 × 10 −6, B   2  = 2.49373 × 10 −9,   
               
               
                 C 2  = −1.83255 × 10 −12, D   2  = 1.03695 × 10 −15   
               
               
                 (W/L) = 1.17(W/L)2 · (L/d 1 )) = 24.5, 
               
               
                 (d 0 d 2 ) = 0.21, R 1 fm = 1.5 
               
               
                   
               
            
           
         
       
     
     EXAMPLE 15 
     
       
         
           
               
             
               
                   
               
             
            
               
                 S = −607.344, fm= 117.840 
               
            
           
           
               
               
               
               
               
            
               
                   
                 i 
                 R i   
                 d i   
                 N 
               
               
                   
                   
               
               
                   
                 0 
                   
                 26.108 
               
               
                   
                 1 
                 235.681 
                 13.292 
                 1.57210 
               
               
                   
                 2 
                 −92.489 
                 140.500 
               
               
                   
                   
               
            
           
           
               
            
               
                 X 1 (Y) (=x 1 (Y)): 
               
               
                 R 1  = 235.681, K 1  = 426.354, 
               
               
                 A 1  = −3.07964 × 10 −6 , B 1  = 2.02800 × 10 19 , 
               
               
                 C 1  = 1.70546 × 10 −13, D   1  = −9.47523 × 10 −17   
               
               
                 X 2 (Y): 
               
               
                 R 2  = −92.489, K 2 = 4.42119, 
               
               
                 A 2  = −2.74485 × 10 −6 , B 2  = 2.00025 × 10 −9   
               
               
                 C 2  = −1.83661 × 10 −12, D   2  = 1.05160 × 10 −15   
               
               
                 (W/L) = 1.17, (W/L) 2  · (L/d 1 ) = 18.44, 
               
               
                 (d 0 d 2 = 0.19, Rb/fm = 2.0 
               
               
                   
               
            
           
         
       
     
     FIGS. 24-32 respectively show field curvatures and distortional aberrations relating to Examples 7-15. Both the field curvatures and distortional aberrations relate to the main scanning direction and effectively corrected by the nonarcuate shapes X 1 (Y) and X 2 (Y). Adjustment is so made as to enhance the uniform velocity scanning characteristic because the distortional aberration relates to the uniform velocity scanning capability, as stated earlier. 
     As stated above, the second embodiment achieves the following advantages. 
     (1) Despite that a scanning lens has a single structure, it implements both the uniform velocity scanning function and the field curvature correcting function in the main scanning direction which are indispensable for a focusing lens as to the main scanning direction. In addition, the lens is free from errors in configuration ascribable to the molding of plastics. 
     (2) Because an image forming lens includes a scanning lens therein, a compact, yet broad, effective main scanning range is achievable and reduces the cost. 
     Various modifications will become possible for those skilled in the art after receiving the teachings of the present disclosure without departing from the scope thereof.