Imaging optical system for light beam scanning system

An imaging optical system for a light beam scanning system for causing a light bundle deflected by a deflector to form an image on a predetermined surface to be scanned and causing the image of the light bundle to scan the surface at a constant speed is provided with an anamorphic lens for compensating for tilt of the deflecting surface of the deflector. The imaging optical system satisfies formula EQU 0.ltoreq..phi.1/.phi..ltoreq.0.1.vertline.M.vertline.+0.45 wherein .phi.1 represents the power of the anamorphic lens in the main scanning direction, .phi. represents the power of the entire imaging optical system in the main scanning direction and M represents the lateral magnification when the deflecting angle by the deflector is 0.

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
This invention relates to an imaging optical system for a light beam 
scanning system which deflects a light bundle such as a laser beam to 
cause the light bundle to scan a predetermined surface. 
2. Description of the Related Art 
There has been known a light beam scanning system in which a light beam is 
deflected by a deflector such as a rotary polygonal mirror and is caused 
to scan a surface. Such a light beam scanning system is used, for 
instance, in a laser recording system or a laser read-out system. The 
light beam scanning system generally comprises a laser for emitting a 
laser beam, a rotary polygonal mirror which deflects the light bundle 
emitted from the laser and an imaging optical system including a f.theta. 
lens which focuses the deflected light bundle on a predetermined surface 
to be scanned to form an image in a height proportional to the deflecting 
angle at the rotary polygonal mirror. 
In such a light beam scanning system, there has been known a surface tilt 
compensation optical system which corrects shift of the scanning lines due 
to tilt of the deflecting surface of the deflector. FIG. 10 shows an 
example of an imaging optical system in which the surface tilt 
compensation is applied. The imaging optical system comprises a deflector 
50, a positive meniscus lens 52 made of plastic and an anamorphic lens 54 
made of glass arranged in this order. By arranging the optical system by 
use of a combination of a plastic lens and a glass lens, the plastic lens 
may be weak in power (refracting power). Accordingly increase in the ratio 
of the thickness of the central portion to that of the peripheral portion 
can be suppressed, thereby suppressing change in optical properties of the 
plastic lens due to change of environment, production of the plastic lens 
is facilitated, the size of the optical system can be small, and the 
optical system can be better in its aberration and f.theta. 
characteristics. 
In the surface tilt compensation optical system employing an anamorphic 
lens, the lateral magnification (magnification in the sub-scanning 
direction) varies depending on the scanning angle due to the anamorphic 
lens having power in the main scanning direction. This is mainly because 
the anamorphic lens must be large in the ratio of the thickness of the 
central portion to that of the peripheral portion in order to compensate 
for the surface tilt of the deflector. 
FIG. 11 shows the relation between the beam diameter and the lateral 
magnification of an imaging optical system in which the surface tilt 
compensation is applied. In FIG. 11, the diameter Wo of the beam in the 
sub-scanning direction on the image plane (the surface to be scanned) can 
be expressed by the following formula. 
EQU Wo=M(.theta.).times.Wi 
wherein Wi represents the diameter in the sub-scanning direction of a beam 
impinging upon the deflecting surface and M(.theta.) represents the 
lateral magnification as a function of the scanning angle .theta.. 
As can be seen from the formula above, when the lateral magnification 
M(.theta.) fluctuates with the scanning angle .theta., the beam diameter 
Wo in the sub-scanning direction fluctuates according to the scanning 
angle .theta. and the scanning spot cannot be uniform in diameter. 
SUMMARY OF THE INVENTION 
In view of the foregoing observations and description, the primary object 
of the present invention is to provide an imaging optical system for a 
light beam scanning system which is provided with an anamorphic lens for 
surface tilt compensation and in which fluctuation in lateral 
magnification can be suppressed. 
In accordance with the present invention, there is provided an imaging 
optical system for a light beam scanning system for causing a light bundle 
deflected by a deflector to form an image on a predetermined surface to be 
scanned and causing the image of the light bundle to scan the surface at a 
constant speed, wherein the improvement comprises that 
the imaging optical system is provided with an anamorphic lens for 
compensating for tilt of the deflecting surface of the deflector, and 
the imaging optical system satisfies formula 
EQU 0.ltoreq..phi.1/.phi..ltoreq.0.1.vertline.M.vertline.+0.45 
wherein .phi.1 represents the power of the anamorphic lens in the main 
scanning direction, .phi. represents the power of the entire imaging 
optical system in the main scanning direction and M represents the lateral 
magnification when the deflecting angle by the deflector is 0. 
In the imaging optical system of the present invention, fluctuation in 
lateral magnification with change in the scanning angle can be suppressed 
to not larger than about 23%. 
When the imaging optical system satisfies formula 
0.ltoreq..phi.1/.phi..ltoreq.0.1.vertline.M.vertline.+0.35, fluctuation in 
lateral magnification with change in the scanning angle can be suppressed 
to not larger than about 20%. 
Preferably the imaging optical system of the present invention comprises a 
positive meniscus lens, an anamorphic lens having positive power in the 
main scanning direction, and a cylindrical mirror having power only in the 
sub-scanning direction arranged in this order from the deflector. 
More preferably the meniscus lens is made of plastic and/or the meniscus 
lens is aspheric in at least one side. 
Further it is preferred that the surface of the anamorphic lens facing the 
deflector be a concave cylindrical surface having power only in the 
sub-scanning direction and the surface of the anamorphic lens facing the 
surface to be scanned be a convex spherical surface. 
As will be described in more detail later, when the ratio .phi.1/.phi. of 
the power of the anamorphic lens in the main scanning direction to the 
power of the entire imaging optical system in the main scanning direction 
is in the range described above, fluctuation in lateral magnification with 
change in the scanning angle can be suppressed to not larger than about 
23% or to not larger than about 20%. 
Further when the imaging optical system of the present invention is formed 
by a combination of a plastic lens (a positive meniscus lens) and a glass 
lens (an anamorphic lens), the size of the optical system can be small, 
and the optical system can be better in its aberration and f.theta. 
characteristics. 
The reason why fluctuation in lateral magnification with change in the 
scanning angle can be suppressed to not larger than about 23% or to not 
larger than about 20% when the power ratio .phi.1/.phi. of the power of 
the anamorphic lens in the main scanning direction to the power of the 
entire imaging optical system in the main scanning direction is in the 
range described above will be described with reference to FIG. 12, 
hereinbelow. FIG. 12 shows a result of simulation in which the relation 
between fluctuation in lateral magnification with change in the scanning 
angle and the power ratio .phi.1/.phi. is calculated for cases where the 
lateral magnification M when the deflecting angle by the deflector is 0 
(will be referred to as "axial lateral magnification" hereinbelow) is 
-0.5, -1.0 and -1.5. In FIG. 12, the result for the axial lateral 
magnification M of -0.5 is shown by the fine dashed line, that for the 
axial lateral magnification M of -1.0 is shown by the solid line and that 
for the axial lateral magnification M of -1.5 is shown by the rough dashed 
line. The triangles, squares and diamonds indicated at a to h on the 
respective lines show the values for imaging optical systems to be 
described later in the description of preferred embodiments. 
As can be understood from FIG. 12, fluctuation in the lateral magnification 
is increased as the power of the anamorphic lens in the main scanning 
direction becomes stronger (as the power ratio .phi.1/.phi. increases) 
though the values slightly vary depending on the axial lateral 
magnification M. Thus it can be understood that in order to make 
fluctuation in the lateral magnification, for instance, not larger than 
23%, the power ratio .phi.1/.phi. should be not larger than about 0.55 
when the axial lateral magnification is -0.5 and should be not larger than 
about 0.6 when the axial lateral magnification is -1.5. Since fluctuation 
in lateral magnification changes with the power ratio .phi.1/.phi. in 
substantially the same manner for different axial lateral magnifications, 
the aforesaid formula 
0.ltoreq..phi.1/.phi..ltoreq.0.1.vertline.M.vertline.+0.45 is derived as 
the condition for making fluctuation in the lateral magnification not 
larger than 23%. 
Similarly in order to make fluctuation in the lateral magnification not 
larger than 20%, the power ratio .phi.1/.phi. should be not larger than 
about 0.4 when the axial lateral magnification is -0.5 and should be not 
larger than about 0.5 when the axial lateral magnification is -1.5. Thus 
the aforesaid formula 
0.ltoreq..phi.1/.phi..ltoreq.0.1.vertline.M.vertline.+0.35 is derived as 
the condition for making fluctuation in the lateral magnification not 
larger than 20%.

DESCRIPTION OF THE PREFERRED EMBODIMENTS 
In FIG. 1, a light beam scanning system provided with an imaging optical 
system in accordance with a first embodiment of the present invention 
comprises a laser 11, a collimator lens 12 which collimates a laser beam 
emitted from the laser 11, and an incident light imaging optical system 14 
such as a cylindrical lens which causes the parallel light bundle passing 
through the collimator lens 12 to form a line image near a deflecting 
surface of a polygonal mirror 13, which is rotated by a motor (not shown) 
to deflect the incident light bundle toward the imaging optical system of 
the first embodiment. The imaging optical system of the first embodiment 
causes the light bundle deflected by the polygonal mirror 13 to form an 
image on a predetermined surface to be scanned and causes the image of the 
light bundle to scan the surface at a constant speed. 
The imaging optical system of the first embodiment comprises a single 
positive meniscus lens L1 which is aspheric in both the faces and is 
concave toward the polygonal mirror 13 and is made of plastic, an 
anamorphic lens L2 having positive power in the main scanning direction, 
and a cylindrical mirror M1 having power only in the sub-scanning 
direction. The face of the anamorphic lens L2 facing toward the polygonal 
mirror 13 is a concave cylindrical face having power only in the 
sub-scanning direction and the face of the anamorphic lens L2 facing 
toward the surface to be scanned is a convex spherical face. 
The radii of curvature R1 to R4 (mm) of the refracting surfaces of the 
positive meniscus lens L1 and the anamorphic lens L2, the radius of 
curvature R5 (mm) of the cylindrical mirror M1, the axial surface 
separation d.sub.0 (mm) between the deflecting point at the polygonal 
mirror 13 and the meniscus lens L1, the central thickness d.sub.1 (mm) of 
the meniscus lens L1, the axial surface separation d.sub.2 (mm) between 
the meniscus lens L1 and the anamorphic lens L2, the central thickness 
d.sub.3 (mm) of the anamorphic lens L2, the axial surface separation 
d.sub.4 (mm) between the anamorphic mirror L2 and the cylindrical lens M1, 
and the refractive indexes n.sub.1 and n.sub.3 for the sodium d-line of 
the meniscus lens L1 and the anamorphic lens L2 of the imaging optical 
system of the first embodiment are as shown in the following table 1. 
In table 1 and the tables to be described later, the radius of curvature R 
is positive when the surface is convex toward the deflecting point and is 
negative when the surface is convex toward the surface to be scanned. 
The value designated by * represents the radius of curvature on the optical 
axis (at the pole of an aspheric surface) and means that the aspheric 
surface has a shape defined by the following formula (1). 
EQU z=ch.sup.2 /[1+{1-(1+K)c.sup.2 h.sup.2 }.sup.1/2 ]+a.sub.1 h.sup.4 +a.sub.2 
h.sup.6 +a.sub.3 h.sup.8 +a.sub.4 h.sup.10 (1) 
wherein z represents the sag of the surface parallel to the z-axis, h 
represents the height above the optical axis, c represents the curvature 
at the pole of the surface, K represents a conic constant, and a.sub.1 to 
a.sub.4 respectively represent fourth-order, sixth-order, eighth-order and 
tenth-order aspheric coefficients. 
The imaging optical system of this embodiment is -1.0 in axial lateral 
magnification (the lateral magnification when the deflecting angle (the 
scanning angle) by the deflector is 0), and 0.54 in power ratio 
.phi.1/.phi. and satisfies the following formula 
EQU 0.ltoreq..phi.1/.phi..ltoreq.0.1.vertline.M.vertline.+0.45 (2) 
wherein .phi.1 represents the power of the anamorphic lens in the main 
scanning direction, and .phi. represents the power of the entire imaging 
optical system in the main scanning direction. In the imaging optical 
system of this embodiment, fluctuation in lateral magnification is not 
larger than 23% (square mark a in FIG. 12). 
TABLE 1 
______________________________________ 
M = -1.0 .phi.1/.phi. = 0.54 
R d n note 
______________________________________ 
R1 = -148.564 * 
d.sub.0 = 34.18 
K = -14.644 
a.sub.1 = -0.10281 E-05 
a.sub.2 = 0.63246 E-09 
a.sub.3 = -0.31298 E-12 
a.sub.4 = 0.39528 E-16 
d.sub.1 = 20.94 
n.sub.1 = 1.526750 
R2 = -93.193 * 
K = 1.3152 
a.sub.1 = -0.37030 E-07 
a.sub.2 = 0.816683 E-10 
a.sub.3 = 0.134247 E-13 
d.sub.2 = 31.55 
a.sub.4 = -0.836380 E-17 
R3 = -43.493 : sub 
= .infin. : main 
d.sub.3 = 32.00 
n.sub.3 = 1.614977 
R4 = -238.697 
d.sub.4 = 77.22 
R5 = -172.020 CYM plane 
d5 = 161.53 
image plane 
______________________________________ 
The operation of the imaging optical system of this embodiment will be 
described, hereinbelow. A laser beam emitted from the laser 11 is 
collimated by the collimator lens 12 and a line image of the collimated 
laser beam is formed on the deflecting surface of the polygonal mirror 13 
by the incident light imaging optical system 14. 
After being reflected by the deflecting surface of the polygonal mirror 13, 
the laser beam travels through the positive meniscus lens L1 and the 
anamorphic lens L2 and is reflected by the cylindrical mirror M1 and is 
focused on the surface to be scanned to form a scanning spot. Since the 
polygonal mirror 13 is rotated in the direction of arrow R at a high 
speed, the scanning spot repeatedly scans the surface of the cylindrical 
mirror M1 in the direction of arrow X (main scanning) and accordingly 
scans the surface to be scanned after being reflected by the cylindrical 
mirror M1. 
FIGS. 2A and 2B respectively show curvature of field and f.theta. 
characteristics of the imaging optical system of this embodiment. In FIG. 
2A, the solid line shows the curvature of field in the main scanning 
direction and the dashed line shows that in the sub-scanning direction 
(the same for the following drawings). As can be understood from FIGS. 2A 
and 2B, in the imaging optical system of this embodiment, aberrations are 
well corrected and also the f.theta. characteristics are excellent. 
Thus in the imaging optical system of this embodiment, fluctuation in 
lateral magnification by the scanning angle can be suppressed by limiting 
the power in the main scanning direction of the anamorphic lens L2 within 
the range defined by formula (2). Further since the imaging optical system 
of this embodiment is formed by a combination of a plastic lens (a 
positive meniscus lens) and a glass lens (a anamorphic lens), the ratio of 
the thickness of the central portion to that of the peripheral portion of 
the plastic lens may be relatively small, whereby production of the 
plastic lens is facilitated and fluctuation in the position on which the 
laser beam is focused due to change in the environmental condition such as 
the environmental temperature can be suppressed. 
An imaging optical system in accordance with a second embodiment will be 
described with reference to table 2 and FIGS. 3A to 3D. 
The imaging optical system of this embodiment comprises a single positive 
meniscus lens L1 which is aspheric in both the faces and is concave toward 
the polygonal mirror 13 and is made of plastic, an anamorphic lens L2 
having positive power in the main scanning direction, and a cylindrical 
mirror M1 having power only in the sub-scanning direction. The face of the 
anamorphic lens L2 facing toward the polygonal mirror 13 is a concave 
cylindrical face having power only in the sub-scanning direction and the 
face of the anamorphic lens L2 facing toward the surface to be scanned is 
a convex spherical face. 
The specification of the imaging optical system of this embodiment is shown 
in the following table 2. In this embodiment, the radius of curvature of 
the concave cylindrical face of the anamorphic lens L2 and the radius of 
curvature of the cylindrical mirror M1 are selected so that the axial 
lateral magnification M of the imaging optical system becomes -1.5. 
Further the power ratio .phi.1/.phi. is 0.54. Accordingly the imaging 
optical system of this embodiment satisfies formula (2) and accordingly is 
not larger than 23% in fluctuation in lateral magnification (triangular 
mark b in FIG. 12). 
TABLE 2 
______________________________________ 
M = -1.5 .phi.1/.phi. = 0.54 
R d n note 
______________________________________ 
R1 = -148.564 * 
d.sub.0 = 34.18 
K = -14.644 
a.sub.1 = -0.10281 E-05 
a.sub.2 = 0.63246 E-09 
a.sub.3 = -0.31298 E-12 
a.sub.4 = 0.39528 E-16 
d.sub.1 = 20.94 
n.sub.1 = 1.526750 
R2 = -93.193 * 
K = 1.3152 
a.sub.1 = -0.37030 E-07 
a.sub.2 = 0.816683 E-10 
a.sub.3 = 0.134247 E-13 
d.sub.2 = 31.55 
a.sub.4 = -0.836380 E-17 
R3 = -43.915 : sub 
= .infin. : main 
d.sub.3 = 32.00 
n.sub.3 = 1.614977 
R4 = -238.697 
d.sub.4 = 9.53 
R5 = -152.818 CYM plane 
d5 = 184.74 
image plane 
______________________________________ 
Also in this embodiment, fluctuation in lateral magnification is not larger 
than 23%. Further as can be understood from FIGS. 3C and 3D, in the 
imaging optical system of this embodiment, aberrations are well corrected 
and also the f.theta. characteristics are excellent. 
An imaging optical system in accordance with a third embodiment where 
fluctuation in lateral magnification is suppressed to not larger than 20% 
will be described with reference to table 3 and FIGS. 4A to 4D. 
The imaging optical system of this embodiment comprises a single positive 
meniscus lens L1 which is aspheric in both the faces and is concave toward 
the polygonal mirror 13 and is made of plastic, an anamorphic lens L2 
having positive power in the main scanning direction, and a cylindrical 
mirror M1 having power only in the sub-scanning direction. The face of the 
anamorphic lens L2 facing toward the polygonal mirror 13 is a concave 
cylindrical face having power only in the sub-scanning direction and the 
face of the anamorphic lens L2 facing toward the surface to be scanned is 
a convex spherical face. 
The specification of the imaging optical system of this embodiment is shown 
in the following table 3. The imaging optical system of this embodiment is 
designed so that the axial lateral magnification M becomes -1.0 and the 
power of the anamorphic lens in the main scanning direction becomes 
weaker. The power ratio .phi.1/.phi. is 0.34. Accordingly the imaging 
optical system of this embodiment satisfies the following formula (3) and 
accordingly is not larger than 20% in fluctuation in lateral magnification 
(square mark c in FIG. 12). 
EQU 0.ltoreq..phi.1/.phi..ltoreq.0.1.vertline.M.vertline.+0.35 (3) 
TABLE 3 
______________________________________ 
M = -1.0 .phi.1/.phi. = 0.34 
R d n note 
______________________________________ 
R1 = -176.108 * 
d.sub.0 = 35.00 
K = -23.840 
a.sub.1 = -0.10387 E-05 
a.sub.2 = 0.40783 E-09 
a.sub.3 = -0.22064 E-12 
a.sub.4 = 0.12564 E-16 
d.sub.1 = 33.97 
n.sub.1 = 1.526750 
R2 = -88.828 * 
K = 1.1038 
a.sub.1 = -0.27541 E-07 
a.sub.2 = 0.19677 E-11 
a.sub.3 = 0.21825 E-13 
d.sub.2 = 32.45 
a.sub.4 = -0.51223 E-17 
R3 = -52.590 : sub 
= .infin. : main 
d.sub.3 = 30.00 
n.sub.3 = 1.614977 
R4 = -384.044 
d.sub.4 = 30.19 
R5 = -172.664 CYM plane 
d5 = 157.78 
image plane 
______________________________________ 
In this embodiment, fluctuation in lateral magnification is not larger than 
20%. Further as can be understood from FIGS. 4C and 4D, in the imaging 
optical system of this embodiment, aberrations are well corrected and also 
the f.theta. characteristics are excellent. 
Examples of imaging optical systems where fluctuation in lateral 
magnification is larger than 23% will be described with reference to 
tables 4 to 8 and FIGS. 5 (5A to 5D) to 9 (9A to 9D). 
As shown in FIGS. 5A and 5B, 6A and 6B, 7A and 7B, 8A and 8B, and 9A and 
9B, each of the imaging optical systems basically comprises, as the 
imaging optical systems of the first to third embodiments, a single 
positive meniscus lens L1 which is aspheric in both the faces and is 
concave toward the polygonal mirror 13 and is made of plastic, an 
anamorphic lens L2 having positive power in the main scanning direction, 
and a cylindrical mirror M1 having power only in the sub-scanning 
direction. The face of the anamorphic lens L2 facing toward the polygonal 
mirror 13 is a concave cylindrical face having power only in the 
sub-scanning direction and the face of the anamorphic lens L2 facing 
toward the surface to be scanned is a convex spherical face. 
The specifications of the imaging optical systems are shown in the 
following tables 4 to 8. 
The imaging optical system shown in table 4 and FIGS. 5A and 5B is designed 
so that the axial lateral magnification M becomes -0.5 and the power of 
the anamorphic lens in the main scanning direction becomes weaker. The 
power ratio .phi.1/.phi. is 0.54. Accordingly the imaging optical system 
of this example cannot satisfy formula (2) and accordingly is larger than 
23% in fluctuation in lateral magnification (diamond mark d in FIG. 12). 
TABLE 4 
______________________________________ 
M = -0.5 .phi.1/.phi. = 0.54 
R d n note 
______________________________________ 
R1 = -148.564 * 
d.sub.0 = 34.18 
K = -14.644 
a.sub.1 = -0.10281 E-05 
a.sub.2 = 0.63246 E-09 
a.sub.3 = -0.31298 E-12 
a.sub.4 = 0.39528 E-16 
d.sub.1 = 20.94 
n.sub.1 = 1.526750 
R2 = -93.193 * 
K = 1.3152 
a.sub.1 = -0.37030 E-07 
a.sub.2 = 0.816683 E-10 
a.sub.3 = 0.134247 E-13 
d.sub.2 = 32.00 
a.sub.4 = -0.836380 E-17 
R3 = -44.002 : sub 
= .infin. : main 
d.sub.3 = 32.00 
n.sub.3 = 1.614977 
R4 = -238.697 
d.sub.4 = 77.22 
R5 = -178.662 CYM plane 
d5 = 117.05 
image plane 
______________________________________ 
In the imaging optical system shown in table 5 and FIGS. 6A and 6B, the 
radius of curvature in the sub-scanning direction of the anamorphic lens 
L2 and the radius of curvature of the cylindrical lens M1 are changed so 
that the axial lateral magnification M becomes -1.0 and the power ratio 
.phi.1/.phi. is 0.71. Accordingly the imaging optical system of this 
example cannot satisfy formula (2) and accordingly is larger than 23% in 
fluctuation in lateral magnification (square mark e in FIG. 12). 
TABLE 5 
______________________________________ 
M = -1.0 .phi.1/.phi. = 0.71 
R d n note 
______________________________________ 
R1 = -109.024 * 
d.sub.0 = 36.50 
K = -1.4243 
a.sub.1 = -0.10257 E-05 
a.sub.2 = 0.78737 E-09 
a.sub.3 = -0.28244 E-12 
a.sub.4 = 0.68860 E-17 
d.sub.1 = 14.40 
n.sub.1 = 1.526750 
R2 = -87.901 * 
K = 1.3381 
a.sub.1 = -0.25258 E-06 
a.sub.2 = 0.21942 E-09 
a.sub.3 = 0.17566 E-13 
d.sub.2 = 26.60 
a.sub.4 = -0.118285 E-16 
R3 = -43.376 : sub 
= .infin. : main 
d.sub.3 = 32.00 
n.sub.3 = 1.614977 
R4 = -180.668 
d.sub.4 = 40.15 
R5 = -181.903 CYM plane 
d5 = 165.25 
image plane 
______________________________________ 
In the imaging optical systems shown in tables 6 to 8 and FIGS. 7 (7A and 
7B) to 9 (9A and 9B), the radius of curvature of the positive meniscus 
lens L1, the radius of curvature in the sub-scanning direction of the 
anamorphic lens L2 and the radius of curvature of the cylindrical lens M1 
are changed. 
The imaging optical system shown in table 6 and FIGS. 7A and 7B is -0.5 in 
the axial lateral magnification M and 0.92 in the power ratio .phi.1/.phi. 
(diamond mark f in FIG. 12). Accordingly it cannot satisfy formula (2) and 
is larger than 23% in fluctuation in lateral magnification. The imaging 
optical system shown in table 7 and FIGS. 8A and 8B is -1.0 in the axial 
lateral magnification M and 0.92 in the power ratio .phi.1/.phi. (square 
mark g in FIG. 12). Accordingly it cannot satisfy formula (2) and is 
larger than 23% in fluctuation in lateral magnification. The imaging 
optical system shown in table 8 and FIGS. 9A and 9B is -1.5 in the axial 
lateral magnification M and 0.92 in the power ratio .phi.1/.phi. 
(triangular mark h in FIG. 12). Accordingly it cannot satisfy formula (2) 
and is larger than 23% in fluctuation in lateral magnification. 
TABLE 6 
______________________________________ 
M = -0.5 .phi.1/.phi. = 0.92 
R d n note 
______________________________________ 
R1 = -105.513 * 
d.sub.0 = 38.00 
K = -1.0715 
a.sub.1 = -0.11059 E-05 
a.sub.2 = 0.80000 E-09 
a.sub.3 = -0.19827 E-12 
a.sub.4 = 0.10760 E-16 
d.sub.1 = 15.00 
n.sub.1 = 1.526750 
R2 = -106.424 * 
K = 1.2965 
a.sub.1 = -0.34827 E-06 
a.sub.2 = 0.23039 E-09 
a.sub.3 = -0.45024 E-14 
d.sub.2 = 5.00 
a.sub.4 = -0.24601 E-17 
R3 = -40.946 : sub 
= .infin. : main 
d.sub.3 = 32.00 
n.sub.3 = 1.614977 
R4 = -140.591 
d.sub.4 = 104.32 
R5 = -187.083 CYM plane 
d5 = 114.72 
image plane 
______________________________________ 
TABLE 7 
______________________________________ 
M = -1.0 .phi.1/.phi. = 0.92 
R d n note 
______________________________________ 
R1 = -105.513 * 
d.sub.0 = 38.00 
K = -1.0715 
a.sub.1 = -0.11059 E-05 
a.sub.2 = 0.80000 E-09 
a.sub.3 = -0.19827 E-12 
a.sub.4 = 0.10760 E-16 
d.sub.1 = 15.00 
n.sub.1 = 1.526750 
R2 = -106.424 * 
K = 1.2965 
a.sub.1 = -0.34827 E-06 
a.sub.2 = 0.23039 E-09 
a.sub.3 = -0.45024 E-14 
d.sub.2 = 5.00 
a.sub.4 = -0.24601 E-17 
R3 = -40.532 : sub 
= .infin. : main 
d.sub.3 = 32.00 
n.sub.3 = 1.614977 
R4 = -140.591 
d.sub.4 = 56.24 
R5 = -189.965 CYM plane 
d5 = 162.79 
image plane 
______________________________________ 
TABLE 8 
______________________________________ 
M = -1.5 .phi.1/.phi. = 0.92 
R d n note 
______________________________________ 
R1 = -105.513 * 
d.sub.0 = 38.00 
K = -1.0715 
a.sub.1 = -0.11059 E-05 
a.sub.2 = 0.80000 E-09 
a.sub.3 = -0.19827 E-12 
a.sub.4 = 0.10760 E-16 
d.sub.1 = 15.00 
n.sub.1 = 1.526750 
R2 = -106.424 * 
K = 1.2965 
a.sub.1 = -0.34827 E-06 
a.sub.2 = 0.23039 E-09 
a.sub.3 = -0.45024 E-14 
d.sub.2 = 5.00 
a.sub.4 = -0.24601 E-17 
R3 = -41.036 : sub 
= .infin. : main 
d.sub.3 = 32.00 
n.sub.3 = 1.614977 
R4 = -140.591 
d.sub.4 = 30.12 
R5 = -173.276 CYM plane 
d5 = 188.91 
image plane 
______________________________________ 
As can be understood from the description above, in the imaging optical 
system in accordance with the present invention, when the ratio 
.phi.1/.phi. of the power of the anamorphic lens in the main scanning 
direction to the power of the entire imaging optical system in the main 
scanning direction satisfies formula 
"0.ltoreq..phi.1/.phi..ltoreq.0.1.vertline.M.vertline.+0.45", fluctuation 
in lateral magnification by the scanning angle can be suppressed to not 
larger than 23%, and when the ratio .phi.1/.phi. satisfies formula 
"0.ltoreq..phi.1/.phi..ltoreq.0.1.vertline.M.vertline.+0.35", fluctuation 
in lateral magnification by the scanning angle can be suppressed to not 
larger than 20%. 
Further in such an imaging optical system, aberrations are well corrected 
and also the f.theta. characteristics are excellent.