Laser beam optical scanning device

A laser beam optical scanning device which has a laser diode, a collimator lens, a cylindrical lens, a polygonal scanner and an f .theta. lens. The polygonal scanner is made of resin, and when the polygonal scanner is driven to rotate, the reflective facets of the polygonal scanner are distorted to be concave or convex because of a centrifugal force. If the reflective facets are distorted to be concave with rotation of the polygonal scanner, the image surface shifts along the optical axis toward the polygonal scanner. Therefore, in this case, the elements of the optical scanning device are positioned such that the image surface is located behind a light receiving surface while the polygonal scanner is stationary and is located nearer to the light receiving surface while the polygonal scanner is rotating.

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
The present invention relates to a laser beam optical scanning device, and 
more particularly to a laser beam optical scanning device which has a 
polygonal scanner and which is to be installed in an image forming 
apparatus such as a laser printer and a facsimile, or in an image reading 
apparatus. 
2. Description of Related Art 
Recently, it is tried in various ways to manufacture polygonal scanners 
from resin by an injection molding method. Using resin as the material has 
an advantage that by use of accurate molds, products with high accuracy 
can be manufactured with low cost. 
However, a resin polygonal scanner has a disadvantage that when the 
polygonal scanner is rotated, its reflective facets are slightly distorted 
to be concave or convex because of the centrifugal force. This lowers the 
performance of the optical scanning system. More specifically, because of 
the distortion of the reflective facets of the polygonal scanner, the 
image surface is displaced from a light receiving surface of the 
photosensitive drum, which causes jitter and degrades the picture quality. 
In order to solve this problem, the polygonal scanner shall be processed 
into a special configuration, but this is practically impossible because 
it requires complicated processes and special skill. 
SUMMARY OF THE INVENTION 
It is an object of the present invention to provide a laser beam optical 
scanning device which performs excellently, that is, can form an image of 
good quality with no jitter, not requiring a polygonal scanner subjected 
to special processing. 
In order to attain the object, a laser beam optical scanning device 
according to the present invention is so made that the beam waist of the 
laser beam in the main scanning direction shifts along the optical axis 
closer to a light receiving surface when the polygonal scanner is driven 
to rotate at a specified speed. 
While a polygonal scanner is rotating at a high speed, its reflective 
facets are distorted to be slightly concave or convex because of the 
centrifugal force. If the reflective facets are distorted to be concave, 
the laser beam reflected by the reflective facets is slightly converged. 
Accordingly, the image surface at that time is located nearer to the 
polygonal scanner than the image surface while the polygonal scanner is 
stationary. In the light of this fact, the members of the optical scanning 
device are positioned such that the image surface will be located behind 
the light receiving surface while the polygonal scanner is stationary and 
will shift toward the polygonal scanner and come closer to the light 
receiving surface when the polygonal scanner is driven to rotate at the 
specified speed. Consequently, in spite of the distortion of the polygonal 
scanner, the performance of the optical scanning device is not degraded, 
that is, an image of good quality with no jitter can be formed. 
If the reflective facets of the polygonal scanner are distorted to be 
convex with rotation of the polygonal scanner, the laser beam reflected by 
the reflective facets is slightly diverged. Accordingly, the image surface 
while the polygonal scanner is rotating is farther from the polygonal 
scanner than the image surface while the polygonal scanner is stationary. 
In the light of this fact, the members of the optical scanning device are 
positioned such that the image surface will be located before the light 
receiving surface while the polygonal scanner is stationary and will shift 
farther from the polygonal scanner and come closer to the light receiving 
surface when the polygonal scanner is driven to rotate at the specified 
speed. Consequently, in spite of the distortion of the polygonal scanner, 
the performance of the optical scanning device is not degraded, that is, 
an image of good quality with no jitter can be formed. 
The laser beam optical scanning device further has, in an object side of 
the polygonal scanner, at least a lens which has a power in a main 
scanning direction and a power in a sub scanning direction, and a lens 
which has only a power in the sub scanning direction. Preferably, the 
lenses are movable along the optical axis. By adjusting the positions of 
the lenses, the position of the beam waist can be adjusted, and the 
adjustment does not interfere with correction of an error caused by 
misalignment of the reflective facets of the polygonal scanner. 
Additionally, in an optical scanning device employing a polygonal scanner 
whose reflective facets are distorted to be concave with rotation of the 
polygonal scanner, preferably, the optical scanning device acts with a 
weaker positive power on a laser beam which is incident to a reflective 
facet and reflected therefrom at a larger angle. On the other hand, in an 
optical scanning device employing a polygonal scanner whose reflective 
facets are distorted to be convex with rotation of the polygonal scanner, 
preferably, the optical scanning device acts with a stronger positive 
power on a laser beam which is incident to a reflective facet and 
reflected therefrom at a larger angle. With this arrangement, beam waists 
of laser beams which are reflected from the reflective facets of the 
polygonal scanner at large angles will not be displaced from the light 
receiving surface so largely.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
Some embodiments of the present invention are hereinafter described with 
reference to the accompanying drawings. 
First Embodiment: FIGS. 1-5 
FIG. 1 shows the general structure of a laser beam optical scanning system 
which is a first embodiment of the present invention. The laser beam 
optical scanning system comprises a laser diode 1, a collimator lens 2, a 
cylindrical lens 3, a resin polygonal scanner 4, a toric lens 20, a 
spherical mirror 21 and a reflecting mirror 22. 
The laser diode 1 is modulated (turned on and off) in accordance with image 
data stored in a control section (not shown). When the laser diode 1 is 
turned on, a laser beam is emitted through a cover glass of the laser 
diode 1. The laser beam passes through the collimator lens 2 and the 
cylindrical lens 3. The cylindrical lens 3 changes the spot shape of the 
laser beam such that the laser beam will be imaged on a reflective facet 
4a of the polygonal scanner 4 in a linear form (long in a main scanning 
direction). 
The polygonal scanner 4 is rotated in a direction of arrow a at a constant 
speed, and with the rotation, the laser beam is deflected in a plane 
perpendicular to the axis of the rotation at a constant angular velocity. 
Then, the deflected laser beam enters the toric lens 20. The toric lens 20 
has a power in a direction in parallel with the deflection plane (main 
scanning direction) and a power in a direction perpendicular to the 
deflection plane (sub scanning direction). The toric lens 20 cooperates 
with the cylindrical lens 3 to correct an error in the sub scanning 
direction caused by misalignment of the reflective facets 4a of the 
polygonal scanner 4. 
Then, the laser beam is reflected by the spherical mirror 21 and is imaged 
on a light receiving surface of a photosensitive drum 30 via the 
reflecting mirror 22. Image formation on the photosensitive drum 30 is 
carried out by main scanning resulting from the rotation of the polygonal 
scanner 4 in the direction of arrow a and sub scanning resulting from 
rotation of the photosensitive drum 30 in a direction of arrow b. The 
spherical mirror 21 functions to correct distortion by adjusting the main 
scanning speed of the laser beam and functions to correct curvature of 
field on the photosensitive drum 30. 
Now, distortion of the resin polygonal scanner 4 during its rotation is 
described. 
As shown in FIG. 2, the polygonal scanner 4 has four reflective facets 4a 
and is a square which has sides of 30 mm. The dashed line 5 indicates the 
shape of the polygonal scanner 4 during its rotation. Distortion of the 
polygonal scanner 4 was simulated in a finite element method, and the 
result is shown in Table 1. Table 2 shows the characteristics of the 
material used for the simulation. 
TABLE 1 
______________________________________ 
Rotating Speed 
6000 rpm 15000 rpm 
______________________________________ 
Amount of Concavity 
1.5 .mu.m 9.5 .mu.m 
(d1-d2) 
Radius of Curvature 
-45000 mm -7600 mm 
of Reflective Facets 
Radius of Inscribed 
30 mm 30 mm 
Circle 
______________________________________ 
TABLE 2 
______________________________________ 
Material polycarbonate 
______________________________________ 
Young's Modulus 210 kg/mm.sup.2 
Specific Gravity 
1.2 
Poisson's Ratio 0.35 
______________________________________ 
Each of the reflective facets 4a is distorted to be concave, and the amount 
of concavity in Table 1 means the difference between the maximum value d1 
and the minimum value d2 of the distortion (see FIG. 2). 
The concave distortion of the reflective facets 4a influences the location 
of the image surface in the main scanning direction most. With the concave 
distortion, each of the reflective facets 4a obtains a positive power in 
the main scanning direction. Accordingly, the laser beam reflected by the 
reflective facets 4a becomes convergent, and consequently, the image 
surface in the main scanning direction shifts in a minus direction (toward 
the polygonal scanner 4). Thus, the image surface in the main scanning 
direction and that in the sub scanning direction come to disagree with 
each other. If the amount of the concavity becomes larger, the image 
surface will be inclined in the main scanning direction, and the scanning 
performance will be lowered, that is, curvature of field and distortion 
will be larger. Since the distortion becomes larger in a plus direction, 
the magnification (scanning width) in the main scanning direction will be 
larger. 
The distortion of the image surface caused by the concave distortion of the 
reflective facets 4a of the polygonal scanner 4 can be calculated 
beforehand, and it is possible to design an f .theta. system which causes 
opposite distortion which will offset the distortion. By increasing the 
power in the sub scanning direction, the disagreement between the image 
surface in the sub scanning direction and that in the main scanning 
direction can be prevented. The inclination of the image surface can be 
corrected by decentering the f .theta. system in the main scanning 
direction. 
It is not always necessary to carry out all these corrections. Corrections 
are necessary at least to things which are largely influenced by the 
distortion of the reflective facets 4a. For example, only the correction 
to the distortion is considered on the designing stage, and the shift of 
the image surface in the main scanning direction is corrected after 
assembly of the scanning system. 
Preferred positions of the elements of the optical scanning system which 
has the polygonal scanner 4, whose characteristics are shown in Table 1 
and Table 2, and the photosensitive drum 30 were studied. Table 3 shows a 
preferred positional relation among these members such that the image 
surface of the optical scanning system will shift closer to the light 
receiving surface of the photosensitive drum 30 when the polygonal scanner 
4 is driven to rotate at a speed of 6000 rpm. 
The data shown in Table 3 are values which were measured while the 
polygonal scanner 4 is stationary. While the polygonal scanner 4 is 
rotating at a speed of 6000 rpm, the radius of curvature of the reflective 
facets 4a of the polygonal scanner 4 is -45000 mm (see Table 1). 
TABLE 3 
______________________________________ 
Radius of Curvature 
(mm) 
main scanning 
sub scanning 
Distance 
Refractive 
Elements 
direction direction (mm) Index 
______________________________________ 
laser diode 
0.48 1.0 (air) 
infinite 
cover glass 0.25 1.51118 
of laser (glass) 
diode 
infinite 
5.86045 1.0 (air) 
infinite 
collimator 2.8 1.78571 
lens (SF6) 
-6.286 
224.432 
1.0 (air) 
infinite 14.537 
cylindrical 1.5 1.48457 
lens (AC) 
infinite 
28.25 1.0 (air) 
polygonal 
infinite 
scanner (reflective facets) 
40 1.0 (air) 
-50.962 29.25 
toric lens 8 1.48457 
(AC) 
-54.561 
95 1.0 (air) 
spherical 
-480 
mirror (reflective surface) 
165 1.0 (air) 
photo- 
sensitive 
drum 
______________________________________ 
(for rotation of 6000 rpm, in a stationary state) 
The light emergent side of the collimator lens 2 is aspherical and defined 
as follows: 
##EQU1## 
x: position in a direction of the optical axis .PHI.: height from the 
optical axis 
.epsilon.: quadratic curve parameter 
C.sub.o : paraxial curvature 
A.sub.1 : high-order parameter (aspherical coefficients from second order 
to sixteenth order A.sub.2 through A.sub.16) 
By substituting the following aspherical data (2) and -6.286 mm, which is 
indicated in Table 3 as the radius of curvature (1/C.sub.o) of the 
emergent side of the collimator lens 2, into the expression (1), the 
emergent side of the collimator lens 2 can be embodied. 
##EQU2## 
In the optical scanning system shown by Table 3, the toric lens 20 is set 
such that the vertex of its light incident side is -0.21 mm out of the 
optical axis in the main scanning direction and such that the vertex of 
its light emergent side is -0.5 mm out of the optical axis in the main 
scanning direction and -9 mm out of the optical axis in the sub scanning 
direction. Additionally, the spherical mirror 21 is set such that the 
vertex of its reflective surface is -10.3 mm out of the optical axis in 
the sub scanning direction. 
Next, the operation and effect of the laser beam optical scanning system 
whose elements are positioned as shown in Table 3 are described. 
In the optical scanning system, the laser beam incident to the polygonal 
scanner 4 is a slightly divergent pencil of rays. While the polygonal 
scanner 4 is stationary, the image surface is behind the light receiving 
surface of the photosensitive drum 30. While the polygonal scanner 4 is 
rotating at a speed of 6000 rpm, the reflective facets 4a of the polygonal 
scanner 4 are distorted to be concave, and thereby the laser beam 
reflected from the polygonal scanner 4 is collimated and is incident to 
the toric lens 20 as substantially a parallel pencil of rays. 
Consequently, the image surface shifts toward the polygonal scanner 4 in 
the direction of the optical axis and comes closer to the light receiving 
surface of the photosensitive drum 30. 
Table 4 and Table 5 show curvature of field which occurs in the optical 
scanning system of Table 3 while the polygonal scanner 4 is stationary and 
while the polygonal scanner 4 is rotating at a speed of 6000 rpm 
respectively. In Table 4 and Table 5, a field angle means the angle of the 
laser beam to the optical axis B in the main scanning direction, and a 
scanning start side and a scanning end side with the optical axis B in the 
center are indicated with positive values and negative values 
respectively. The curvature of field is shown by listing the distances 
between the image points of laser beams at the respective field angles and 
the light receiving surface of the photosensitive drum 30. When the image 
point is behind the light receiving surface of the photosensitive drum 30 
in the direction of the optical axis, the distance is indicated as a 
positive value. When the image point is before the light receiving surface 
of the photosensitive drum 30, the distance is indicated as a negative 
value. 
TABLE 4 
______________________________________ 
Curvature of Field 
Curvature of Field 
in the Sub in the Main 
Field Angle 
Scanning Direction 
Scanning Direction 
(degrees) (mm) (mm) 
______________________________________ 
-30.00 -1.3099 1.8602 
-25.00 -0.0257 1.9577 
-20.00 0.7426 2.2174 
-15.00 1.0880 2.5332 
-10.00 1.0983 2.8248 
-5.00 0.8570 3.0347 
0.00 0.4448 3.1268 
5.00 -0.0603 3.0857 
10.00 -0.5816 2.9165 
15.00 -1.0434 2.6466 
20.00 -1.3693 2.3259 
25.00 -1.4811 2.0303 
30.00 -1.2973 1.8645 
______________________________________ 
(for rotation of 6000 rpm, in a stationary state) 
TABLE 5 
______________________________________ 
Curvature of Field 
Curvature of Field 
in the Sub in the Main 
Field Angle 
Scanning Direction 
Scanning Direction 
(degrees) (mm) (mm) 
______________________________________ 
-30.00 -1.3221 -0.8221 
-25.00 -0.0338 -0.6309 
-20.00 0.7374 -0.2855 
-15.00 1.0848 0.1100 
-10.00 1.0963 0.4768 
-5.00 0.8556 0.7586 
0.00 0.4434 0.9198 
5.00 -0.0619 0.9457 
10.00 -0.5839 0.8420 
15.00 -1.0465 0.6356 
20.00 -1.3734 0.3770 
25.00 -1.4864 0.1419 
30.00 -1.3044 0.0344 
______________________________________ 
(in a state of 6000 rpm rotation) 
FIG. 3 is a graph drawn from the data about the curvature of field in the 
main scanning direction provided in Table 4 and Table 5. In FIG. 3, the 
solid line 32 shows the curvature of field while the polygonal scanner 4 
is stationary, and the dashed line 33 shows the curvature of field while 
the polygonal scanner 4 is rotating at a speed of 6000 rpm. 
As is apparent from FIG. 3, while the polygonal scanner 4 is stationary, 
the curvature of field in the main scanning direction is large, but the 
curvature of field becomes small when the polygonal scanner 4 is driven to 
rotate. The difference is approximately 2 mm. On the other hand, the 
curvature of field in the sub scanning direction hardly changes and is 
always small whether the polygonal scanner 4 is stationary or rotates (see 
Table 4 and Table 5). Thus, in the laser beam optical scanning system of 
the first embodiment, when the polygonal scanner 4 is driven for image 
formation, the curvature of field becomes small enough such that an image 
of high quality with no jitter can be obtained. 
For comparison, an optical scanning system whose elements are positioned 
such that the image surface is close to the light receiving surface of the 
photosensitive drum 30 while the polygonal scanner 4 is stationary is 
described. Table 6 shows positions of the elements of this comparative 
optical scanning system. Values indicated in Table 6 were measured while 
the polygonal scanner 4 is stationary. While the polygonal scanner 4 is 
rotating at a speed of 6000 rpm, the radius of curvature of the polygonal 
scanner 4 is -45000 mm (see Table 1). 
TABLE 6 
______________________________________ 
Radius of Curvature 
(mm) 
main scanning 
sub scanning 
Distance 
Refractive 
Elements 
direction direction (mm) Index 
______________________________________ 
laser diode 
0.48 1.0 (air) 
infinite 
cover glass 0.25 1.51118 
of laser (glass) 
diode 
infinite 
5.86245 1.0 (air) 
infinite 
collimator 2.8 1.78571 
lens (SF6) 
-6.286 
224.48 1.0 (air) 
infinite 14.537 
cylindrical 1.5 1.48457 
lens (AC) 
infinite 
28.2 1.0 (air) 
polygonal 
infinite 
scanner (reflective facets) 
40 1.0 (air) 
-50.962 29.25 
toric lens 8 1.48457 
(AC) 
-54.561 
95 1.0 (air) 
spherical 
-480 
mirror (reflective surface) 
165 1.0 (air) 
photo- 
sensitive 
drum 
______________________________________ 
(comparative example, in a stationary state) 
As is apparent from Table 3 and Table 6, in the comparative example, the 
collimator lens 2 and the cylindrical lens 3 are positioned closer to the 
polygonal scanner 4 than those in the first embodiment by 0.002 mm and 
0.05 mm respectively. In the optical scanning system of the comparative 
example, while the polygonal scanner 4 is stationary, the laser beam 
reflected from the polygonal scanner 4 and incident to the toric lens 20 
is substantially a parallel pencil of rays. Accordingly, the image surface 
is substantially on the light receiving surface of the photosensitive drum 
30. When the polygonal scanner 4 is driven to rotate, the laser beam 
incident to the toric lens 20 becomes a slightly convergent pencil of rays 
because of distortion of the polygonal scanner 4. Accordingly, the image 
surface shifts toward the polygonal scanner 4 along the optical axis, that 
is, comes before the light receiving surface of the photosensitive drum 
30. 
Table 7 and Table 8 show curvature of field which occurs in the optical 
scanning system of the comparative example while the polygonal scanner 4 
is stationary and while the polygonal scanner 4 is rotating at a speed of 
6000 rpm respectively. 
TABLE 7 
______________________________________ 
Curvature of Field 
Curvature of Field 
in the Sub in the Main 
Field Angle 
Scanning Direction 
Scanning Direction 
(degrees) (mm) (mm) 
______________________________________ 
-30.00 -1.2193 -0.8602 
-25.00 0.0645 -0.4488 
-20.00 0.8323 -0.2191 
-15.00 1.1772 0.0729 
-10.00 1.1869 0.3474 
-5.00 0.9451 0.5473 
0.00 0.5324 0.6367 
5.00 0.0270 0.6003 
10.00 -0.4945 0.4431 
15.00 -0.9562 0.1919 
20.00 -1.2816 -0.1036 
25.00 -1.3927 -0.3685 
30.00 -1.2082 -0.4991 
______________________________________ 
(comparative example, in a stationary state) 
TABLE 8 
______________________________________ 
Curvature of Field 
Curvature of Field 
in the Sub in the Main 
Field Angle 
Scanning Direction 
Scanning Direction 
(degrees) (mm) (mm) 
______________________________________ 
-30.00 -1.2303 -3.1717 
-25.00 0.0576 -3.0144 
-20.00 0.8283 -2.6981 
-15.00 1.1751 -2.3259 
-10.00 1.1860 -1.9761 
-5.00 0.9447 -1.7047 
0.00 0.5321 -1.5466 
5.00 0.0265 -1.5168 
10.00 -0.4956 -1.6097 
15.00 -0.9581 -1.7985 
20.00 -1.2846 -2.0334 
25.00 -1.3968 -2.2395 
30.00 -1.2136 -2.3136 
______________________________________ 
(comparative example, in a state of 6000 rpm rotation) 
FIG. 4 is a graph drawn from the data about the curvature of field in the 
main scanning direction provided in Table 7 and Table 8. In FIG. 4, the 
solid line 35 shows the curvature of field while the polygonal scanner 4 
is stationary, and the dashed line 36 shows the curvature of field while 
the polygonal scanner 4 is rotating at a speed of 6000 rpm. 
As is apparent from FIG. 4, the curvature of field in the main scanning 
direction is small while the polygonal scanner 4 is stationary, and when 
the polygonal scanner 4 is driven to rotate, the curvature of field 
becomes large. Thus, in the optical scanning system of the comparative 
example, when the polygonal scanner 4 is driven for image formation, the 
curvature of field becomes larger. In other words, the performance of this 
optical scanning system is not sufficient to form an image of good quality 
with no jitter. By changing the positions of the collimator lens 2 and the 
cylindrical lens 3 so as to have the positional relations shown in Table 
3, the optical scanning system of the first embodiment which has 
sufficient performance can be obtained. 
Table 9 shows a preferred positional relation among the elements of an 
optical scanning system, including the polygonal scanner 4 with the 
characteristics indicated in Table 1 and Table 2, and the photosensitive 
drum 30 such that the image surface will be very near the light receiving 
surface of the photosensitive drum 30 while the polygonal scanner 4 is 
rotating at a speed of 15000 rpm. 
Values provided in Table 9 were measured while the polygonal scanner 4 is 
stationary. While the polygonal scanner 4 is rotating at a speed of 15000 
rpm, the radius of curvature of the polygonal scanner 4 is -7600 mm (see 
Table 1). 
TABLE 9 
______________________________________ 
Radius of Curvature 
(mm) 
main scanning 
sub scanning 
Distance 
Refractive 
Elements 
direction direction (mm) Index 
______________________________________ 
laser diode 
0.48 1.0 (air) 
infinite 
cover glass 0.25 1.51118 
of laser (glass) 
diode 
infinite 
5.85045 1.0 (air) 
infinite 
collimator 2.8 1.78571 
lens (SF6) 
-6.286 
224.492 1.0 (air) 
infinite 14.537 
cylindrical 1.5 1.48457 
lens (AC) 
infinite 
28.25 1.0 (air) 
polygonal 
infinite 
scanner (reflective facets) 
40 1.0 (air) 
-50.962 29.00 
toric lens 8 1.48457 
(AC) 
-54.561 
95 1.0 (air) 
spherical 
-480 
mirror (reflective surface) 
165 1.0 (air) 
photo- 
sensitive 
drum 
______________________________________ 
(for rotation of 15000 rpm, in a stationary state) 
The light emergent side of the collimator lens 2 has the same configuration 
of that of the optical scanning system shown in Table 3. The toric lens 20 
is set such that the vertex of its light incident side is -0.21 mm out of 
the optical axis in the main scanning direction and such that the vertex 
of its light emergent side is -1.5 mm out of the optical axis in the main 
scanning direction and -9 mm out of the optical axis in the sub scanning 
direction. Additionally, the spherical mirror 21 is set such that the 
vertex of its reflective surface is -10.3 mm out of the optical axis in 
the sub scanning direction. 
The operation and effect of the laser beam optical scanning system whose 
elements are positioned as shown in Table 9 are the same as those of the 
optical scanning system shown in Table 3. Table 10 and Table 11 show 
curvature of field which occurs in the optical scanning system of Table 9 
while the polygonal scanner 4 is stationary and while the polygonal 
scanner 4 is rotating at a speed of 15000 rpm respectively. 
TABLE 10 
______________________________________ 
Curvature of Field 
Curvature of Field 
in the Sub in the Main 
Field Angle 
Scanning Direction 
Scanning Direction 
(degrees) (mm) (mm) 
______________________________________ 
-30.00 0.1119 14.7971 
-25.00 1.2986 14.7935 
-20.00 1.9494 14.8913 
-15.00 2.1625 14.9907 
-10.00 2.0308 15.0188 
-5.00 1.6427 14.9266 
0.00 1.0826 14.6870 
5.00 0.4321 14.2936 
10.00 -0.2297 13.7606 
15.00 -0.8250 13.1236 
20.00 -1.2762 12.4412 
25.00 -1.5045 11.7970 
30.00 -1.4290 11.3030 
______________________________________ 
(for rotation of 15000 rpm, in a stationary state) 
TABLE 11 
______________________________________ 
Curvature of Field 
Curvature of Field 
in the Sub in the Main 
Field Angle 
Scanning Direction 
Scanning Direction 
(degrees) (mm) (mm) 
______________________________________ 
-30.00 0.2348 -1.2147 
-25.00 1.4153 -0.6989 
-20.00 2.0519 -0.1248 
-15.00 2.2444 0.4193 
-10.00 2.0870 0.8686 
-5.00 1.6691 1.1800 
0.00 1.0761 1.3307 
5.00 0.3904 1.3171 
10.00 -0.3084 1.1556 
15.00 -0.9419 0.8830 
20.00 -1.4319 0.5580 
25.00 -1.6995 0.2641 
30.00 -1.6632 0.1121 
______________________________________ 
(in a state of 15000 rpm rotation) 
FIG. 5 is a graph drawn from the data about the curvature of field in the 
main scanning direction provided in Table 10 and Table 11. In FIG. 5, the 
solid line 37 shows the curvature of field while the polygonal scanner 4 
is stationary, and the dashed line 38 shows the curvature of field while 
the polygonal scanner 4 is rotating at a speed of 15000 rpm. 
Further, the polygonal scanner 4 of the optical scanning system of the 
comparative example shown in Table 6 is rotated at a speed of 15000 rpm, 
and curvature of field which occurs at that time is shown in Table 12 and 
by the dashed line 39 in FIG. 5. 
TABLE 12 
______________________________________ 
Curvature of Field 
Curvature of Field 
in the Sub in the Main 
Field Angle 
Scanning Direction 
Scanning Direction 
(degrees) (mm) (mm) 
______________________________________ 
-30.00 -1.2846 -15.9727 
-25.00 0.0236 -15.3409 
-20.00 0.8087 -14.5986 
-15.00 1.1650 -13.8370 
-10.00 1.1816 -13.1255 
-5.00 0.9431 -12.5138 
0.00 0.5309 -12.0322 
5.00 0.0238 -11.6923 
10.00 -0.5012 -11.4865 
15.00 -0.9677 -11.3868 
20.00 -1.2991 -11.3430 
25.00 -1.4171 -11.2809 
30.00 -1.2403 -11.0938 
______________________________________ 
(comparative example, in a state of 15000 rpm rotation) 
The following describes how to adjust the laser beam optical scanning 
system so as to perform sufficiently to form an image of good quality with 
no jitter in spite of the distortion of the polygonal scanner 4 with its 
rotation. 
In a first way, while the polygonal scanner 4 is rotating, the diameter of 
a beam spot is measured, and adjustment is carried out. The polygonal 
scanner 4 is driven to rotate at an usual speed in use (6000 rpm or 15000 
rpm). A driving signal is sent to the laser diode 1 in synchronization 
with an image writing start signal, and accordingly, the laser diode 1 
emits a laser beam for one dot. The beam spot of the laser beam is 
monitored by a monitor camera which is disposed in a place corresponding 
to the light receiving surface of the photosensitive drum 30. The diameter 
of the beam spot is measured by counting picture elements of the monitor 
camera which receive luminous intensities not less than 1/e.sup.2 of the 
peak intensity and multiplying the number by the length of a side of one 
picture element. While the beam spot is monitored in this way, first, the 
position of the collimator lens 2 is adjusted such that the diameter of 
the beam spot in the main scanning direction will be within a specified 
value. Since the collimator lens 2 has not only a power in the main 
scanning direction but also a power in the sub scanning direction, the 
diameter of the beam spot in the sub scanning direction may get over a 
specified value with this adjustment. In this case, the position of the 
cylindrical lens 3, which has a power only in the sub scanning direction, 
is adjusted such that the diameter of the beam spot in the sub scanning 
direction will be within the specified value. In this way, the optical 
scanning system is adjusted, and the diameter of the beam spot in the main 
scanning direction and that in the sub scanning direction are within the 
respective specified values. 
In the above-described way, the monitor camera is disposed in a place 
corresponding to the light receiving surface of the photosensitive drum 
30. For more accurate monitoring, however, the following camera setting is 
possible. The monitor camera is disposed behind the light receiving 
surface, and a lens is disposed between the light receiving surface and 
the monitor camera such that the beam spot on the light receiving surface 
will be magnified and projected on the monitor camera. 
In a second way, while the polygonal scanner 4 is stationary, the diameter 
of a beam spot is measured, and adjustment is carried out. As described 
above, the amount of a shift of the image surface caused by rotation of 
the polygonal scanner 4 can be calculated by simulation. The monitor 
camera is disposed behind a place corresponding to the light receiving 
surface of the photosensitive drum 30 by the calculated amount. While a 
beam spot is monitored with the monitor camera, the positions of the 
collimator lens 2 and the cylindrical lens 3 are adjusted such that the 
diameter of the beam spot in the main scanning direction and that in the 
sub scanning direction will be within the respective specified values. 
Thus, the optical scanning system is adjusted. In mass production, only 
several samples of optical scanning systems are subjected to the 
adjustment in these two ways. Others are assembled in accordance with 
collected data. 
In the above two adjusting ways, the position of the beam waist is adjusted 
by moving the lenses 2 and 3 which are in the object side of the polygonal 
scanner 4 along the optical axis, and the adjustment does not weaken the 
effect of the lenses 2 and 3 of correcting an error caused by misalignment 
of the reflective facets 4a of the polygonal scanner 4. More specifically, 
the concave distortion of the reflective facets 4a of the polygonal 
scanner 4 with rotation of the polygonal scanner 4 moves the beam waist in 
respect to the main scanning direction in a minus direction. As measures 
to prevent this, the collimator lens 2 is moved in the minus direction 
along the optical axis such that the laser beam will be incident to the 
reflective facets 4a as a slightly divergent pencil of rays. Accordingly, 
the beam waist in the main scanning direction while the polygonal scanner 
4 is stationary shifts in the plus direction (behind the light receiving 
surface). In this condition, when the polygonal scanner 4 is driven to 
rotate, the beam waist in the main scanning direction comes closer to the 
light receiving surface. However, since the collimator lens 2 has a power 
in the sub scanning direction as well as a power in the main scanning 
direction, this adjustment also shifts the beam waist in the sub scanning 
direction. In order to offset the shift of the beam waist in the sub 
scanning direction, the cylindrical lens 3 which has a power only in the 
sub scanning direction is moved along the optical axis. Thus, the beam 
waist in the sub scanning direction can be kept close to the light 
receiving surface. 
As described, if the position of the beam waist in the main scanning 
direction is adjusted by moving a lens which is disposed in the object 
side of the polygonal scanner 4 and has powers both in the main scanning 
direction and in the sub scanning direction, it is preferred that the 
position of the beam waist in the sub scanning direction is adjusted by 
moving a lens which is disposed in the object side of the polygonal 
scanner 4 and has a power only in the sub scanning direction. More 
specifically, the position of the beam waist in the sub scanning direction 
is adjusted preferably by moving the cylindrical lens 3. When the 
cylindrical lens 3 is set in such a position that the beam waist in the 
sub scanning direction comes closer to the light receiving surface, the 
image point of the laser beam emergent from the cylindrical lens 3 comes 
closer to the reflective facet 4a of the polygonal scanner 4, and an error 
caused by misalignment of the reflective facets 4a can be certainly 
corrected. 
Although adjustment of the laser beam optical scanning system have been 
described in connection with the above two ways, it is to be noted that 
other ways are possible. Any method may be adopted as long as the method 
can correct the shift of the image surface caused by the distortion of the 
polygonal scanner 4 with its rotation. 
When the reflective facets 4a of the polygonal scanner 4 are distorted to 
be concave with rotation of the scanner 4, the reflective facets 4a have a 
larger power (converge the laser beam more strongly) toward a laser beam 
at a field angle of -30 degrees than toward a laser beam at a field angle 
of 30 degrees. Accordingly, the beam waist of the laser beam at a field 
angle of -30 degrees is located in the minus side of that of the laser 
beam at a field angle of 30 degrees along the optical axis. This is 
apparent from the dashed line 39 in the graph of FIG. 5 and Table 12. In 
order to correct the inclination of the image surface, in the optical 
scanning system shown by Table 9, the vertex of the light emergent side of 
the toric lens 20 is decentered by -1.5 mm toward the laser beam incident 
to the polygonal scanner 4 as shown in FIG. 6. 
The solid line 37 in the graph of FIG. 5 and Table 10 show the positions of 
beam waists in the above-described structure while the polygonal scanner 4 
is stationary. The beam waist of a laser beam at a field angle of -30 
degrees is displaced from the light receiving surface in the plus 
direction more largely than that of a laser beam at a field angle of 30 
degrees. Thereby, while the polygonal scanner 4 is rotating, the beam 
waist of the laser beam at a field angle of -30 degrees is not displaced 
in the minus direction so largely (see the dashed line 38 in FIG. 5). In 
other words, the laser beam at a field angle of -30 degrees is converged 
by the toric lens 20 less strongly than the laser beam of a field angle of 
30 degrees such that displacement of the beam waist of the laser beam at a 
field angle of 30 degrees and that of the beam waist of the laser beam at 
a field angle of -30 degrees will be almost in the same degree. The larger 
the angle between the laser beam incident to a reflective facet 4a and the 
laser beam reflected therefrom is, the smaller the positive power of the 
toric lens 20 acting on the laser beam in the main scanning direction is. 
Second Embodiment: FIGS. 7 and 8 
FIG. 7 shows the general structure of a laser beam optical scanning system 
which is a second embodiment. The laser beam optical scanning system 
comprises a laser diode 41, a collimator lens 42, a slit board 43, a 
cylindrical lens 44, a resin polygonal scanner 45, an aspherical lens 46, 
an aspherical toric lens 47 and a reflecting mirror 48. A laser beam 
emitted from the laser diode 41 passes through the collimator lens 42, a 
slit 43a made in the slit board 43 and the cylindrical lens 44, and is 
incident to the polygonal scanner 45. The polygonal scanner 45 is driven 
to rotate at a constant angular velocity, and the laser beam is deflected 
with the rotation of the polygonal scanner 45. The deflected laser beam 
passes through the aspherical lens 46, the aspherical toric lens 47 and is 
reflected by the reflecting mirror 48. Then, the laser beam passes through 
a cover glass 49 and is imaged on a light receiving surface of a 
photosensitive drum 50. The optical members 41 through 48 are well-known 
type, and the description thereof is omitted. 
The polygonal scanner 45 has four reflective facets 45a and has the 
characteristics shown in Table 1 and Table 2 provided in the description 
of the first embodiment. Table 13 shows a preferred positional relation 
among the optical members 41 through 48 and the photosensitive drum 50 
which makes the image surface during rotation of the polygonal scanner 45 
come closer to the light receiving surface of the photosensitive drum 50. 
The data shown in Table 13 are values which were measured while the 
polygonal scanner 45 is stationary. While the polygonal scanner 45 is 
rotating at a speed of 6000 rpm, the radius of curvature of the reflective 
facets 45a of the polygonal scanner 45 is -45000 mm (see Table 1). 
TABLE 13 
______________________________________ 
Radius of Curvature 
(mm) 
main scanning 
sub scanning 
Distance 
Refractive 
Elements 
direction direction (mm) Index 
______________________________________ 
infinite 197 
cylindrical 10 1.51072 
lens infinite 
389 1.0 (air) 
polygonal 
infinite 
scanner (reflective facets) 
58 1.0 (air) 
aspherical 
-408 
lens 24 1.48495 
-105.2 
210 1.0 (air) 
aspherical 
-1900 32.15 
toric lens 8 1.48495 
1300 
78.4 1.0 (air) 
photo- 
sensitive 
drum 
______________________________________ 
(for rotation of 6000 rpm, in a stationary state) 
The light emergent side of the aspherical lens 46 and the light emergent 
side of the aspherical toric lens 47 are defined by the expression (1) 
provided in the description of the first embodiment. By substituting the 
following aspherical data (3) and (4), and -105.2 mm and 1300 mm, which 
are indicated in Table 13 as the radius of curvature (1/C.sub.o) of the 
light emergent side of the lens 46 and that of the light emergent side of 
the lens 47 respectively, into the expression (1), the light emergent side 
of the aspherical lens 46 and the light emergent side of the aspherical 
toric lens 47 can be embodied. 
##EQU3## 
The light incident side of the aspherical toric lens 47 is defined by the 
following expressions (5), (6), (7), (8), (9) and (10). 
##EQU4## 
x: position in a direction of the optical axis y: position in the main 
scanning direction 
z: position in the sub scanning direction 
K: curvature in the main scanning direction 
C: curvature in the sub scanning direction 
.mu.: quadratic curve parameter in the main scanning direction 
.epsilon.: quadratic curve parameter in the sub scanning direction 
A.sub.ij : aspherical additional term 
By substituting the following aspherical data (11), and -1900 mm and 32.15 
mm, which are indicated in Table 13 as the radius of curvature of the 
light incident side in the main scanning direction (1/K) and that in the 
sub scanning direction (1/C) of the aspherical toric lens 47, into the 
expressions (5) through (10), the light incident side of the aspherical 
toric lens 47 can be embodied. 
##EQU5## 
Further, the position of the collimator lens 42 is adjusted such that the 
object distance between the reflective facets 45a of the polygonal scanner 
45 and an object point S1 in the main scanning direction will be -17196 
mm. If the laser beam incident to the polygonal scanner 45 is a parallel 
pencil of rays, the object distance is infinite. If the laser beam 
incident to the polygonal scanner 45 is a divergent pencil of rays, the 
object distance is indicated with a negative value, and if the laser beam 
is a convergent pencil of rays, the object distance is indicated with a 
positive value. Therefore, in the optical scanning system wherein the 
object distance is -17196 mm, the laser beam emitted from the laser diode 
41 is incident to the polygonal scanner 45 as a slightly divergent pencil 
of rays. The laser beam is reflected by the reflective facets 45a of the 
polygonal scanner 45. While the polygonal scanner 45 is stationary, the 
reflected laser beam is incident to the aspherical lens 46 as a slightly 
divergent pencil of rays. Accordingly, in this state, the image surface is 
located behind the light receiving surface of the photosensitive drum 50 
in the direction of the optical axis. While the polygonal scanner 45 is 
rotating at a speed of 6000 rpm, the reflective facets 45a of the 
polygonal scanner 45 are distorted to be concave, and the laser beam 
reflected by the reflective facets 45a is incident to the aspherical lens 
46 as a parallel pencil of rays. Accordingly, the image surface shifts 
along the optical axis and comes closer to the light receiving surface of 
the photosensitive drum 50. 
Table 14 and Table 15 show curvature of field which occurs in the optical 
scanning system of the second embodiment while the polygonal scanner 45 is 
stationary and while the polygonal scanner 45 is rotating at a speed of 
6000 rpm respectively. 
TABLE 14 
______________________________________ 
Curvature of Field 
Curvature of Field 
in the Sub in the Main 
Field Angle 
Scanning Direction 
Scanning Direction 
(degrees) (mm) (mm) 
______________________________________ 
-45.00 0.3310 3.6325 
-36.00 -1.7150 3.1909 
-27.00 -1.2481 3.0839 
-18.00 0.2791 4.1585 
-9.00 1.8013 5.2469 
0.00 2.4228 5.5937 
9.00 1.8067 5.2779 
18.00 0.3258 4.3358 
27.00 -1.0952 3.3194 
36.00 -1.3655 3.0615 
45.00 1.0269 2.7721 
______________________________________ 
(for rotation of 6000 rpm, in a stationary state) 
TABLE 15 
______________________________________ 
Curvature of Field 
Curvature of Field 
in the Sub in the Main 
Field Angle 
Scanning Direction 
Scanning Direction 
(degrees) (mm) (mm) 
______________________________________ 
-45.00 0.3396 -0.8345 
-36.00 -1.7143 -1.1957 
-27.00 -1.2493 -1.2732 
-18.00 0.2779 -0.3556 
-9.00 1.8009 0.6091 
0.00 2.4227 1.0286 
9.00 1.8063 0.9999 
18.00 0.3248 0.4965 
27.00 -1.0962 -0.0882 
36.00 -1.3647 -0.0790 
45.00 1.0343 -0.1386 
______________________________________ 
(in a state of 6000 rpm rotation) 
FIG. 8 is a graph drawn from the data about the curvature of field in the 
main scanning direction provided in Table 14 and Table 15. In FIG. 8, the 
solid line 61 shows the curvature of field while the polygonal scanner 45 
is stationary, and the dashed line 62 shows the curvature of field while 
the polygonal scanner 45 is rotating at a speed of 6000 rpm. 
As is apparent from FIG. 8, the curvature of field in the main scanning 
direction is large while the polygonal scanner 45 is stationary, and when 
the polygonal scanner 45 is driven to rotate, the curvature of field 
becomes small. The difference is approximately 4 mm. On the other hand, 
the curvature of field in the sub scanning direction is always small 
whether the polygonal scanner 45 is stationary or is rotating (see Table 
14 and Table 15). Thus, in the laser beam optical scanning system of the 
second embodiment, when the polygonal scanner 45 is driven for image 
formation, the curvature of field becomes small enough such that an image 
of high quality with no jitter can be obtained. 
For comparison, the following describes an optical scanning system wherein 
its optical elements are positioned such that the image surface is very 
near the light receiving surface of the photosensitive drum 50 while the 
polygonal scanner 45 is stationary. Table 16 shows the positional relation 
among the optical members of this optical scanning system. Data provided 
in Table 16 are values which were measured while the polygonal scanner 45 
is stationary. While the polygonal scanner 45 is rotating at a speed of 
6000 rpm, the radius of curvature of the reflective facets 45a of the 
polygonal scanner 45 becomes -45000 mm (see Table 1). 
TABLE 16 
______________________________________ 
Radius of Curvature 
(mm) 
main scanning 
sub scanning 
Distance 
Refractive 
Elements 
direction direction (mm) Index 
______________________________________ 
infinite 197 
cylindrical 10 1.51072 
lens infinite 
380 1.0 (air) 
polygonal 
infinite 
scanner (reflective facets) 
58 1.0 (air) 
aspherical 
-408 
lens 24 1.48495 
-105.2 
210 1.0 (air) 
aspherical 
-1900 32.15 
toric lens 8 1.48495 
1300 
78.4 1.0 (air) 
photo- 
sensitive 
drum 
______________________________________ 
(comparative example, in a stationary state) 
As is apparent from Table 13 and Table 16, in the comparative example, the 
cylindrical lens 44 is positioned mm closer to the polygonal scanner 45 
than that in the second embodiment. Further, in the comparative example, 
the collimator lens 42 is positioned such that the object distance between 
the reflective facets 45a of the polygonal scanner 45 and an object point 
S1 will be infinite. In the structure, the laser beam is incident to the 
polygonal scanner 45 as a parallel pencil of rays. While the polygonal 
scanner 45 is stationary, the laser beam is reflected by the reflective 
facet 45a and is incident to the aspherical lens 46 as a parallel pencil 
of rays. Accordingly, the image surface in this state is located 
substantially on the light receiving surface of the photosensitive drum 
50. While the polygonal scanner 45 is rotating at a speed of 6000 rpm, the 
reflective facets 45a of the polygonal scanner 45 are distorted to be 
concave, and the laser beam reflected by the reflective facets 45a is 
incident to the aspherical lens 46 as a slightly convergent pencil of 
rays. Accordingly, the image surface shifts toward the polygonal scanner 
45 and comes before the light receiving surface of the photosensitive drum 
50. 
Table 17 and Table 18 show curvature of field which occurs in the optical 
scanning system of the comparative example shown in Table 16 while the 
polygonal scanner 45 is stationary and while the polygonal scanner 45 is 
rotating at a speed of 6000 rpm respectively. 
TABLE 17 
______________________________________ 
Curvature of Field 
Curvature of Field 
in the Sub in the Main 
Field Angle 
Scanning Direction 
Scanning Direction 
(degrees) (mm) (mm) 
______________________________________ 
-45.00 0.3817 0.0532 
-36.00 -1.6674 -0.6472 
-27.00 -1.1995 -1.0191 
-18.00 0.3303 -0.3666 
-9.00 1.8551 0.3452 
0.00 2.4775 0.5502 
9.00 1.8605 0.3749 
18.00 0.3771 -0.1970 
27.00 -1.0465 -0.7987 
36.00 -1.3176 -0.7995 
45.00 1.0781 -0.8405 
______________________________________ 
(comparative example, in a stationary state) 
TABLE 18 
______________________________________ 
Curvature of Field 
Curvature of Field 
in the Sub in the Main 
Field Angle 
Scanning Direction 
Scanning Direction 
(degrees) (mm) (mm) 
______________________________________ 
-45.00 0.3903 -4.4097 
-36.00 -1.6666 -4.9979 
-27.00 -1.2008 -5.3103 
-18.00 0.3291 -4.7811 
-9.00 1.8547 -4.1684 
0.00 2.4775 -3.8854 
9.00 1.8601 -3.7885 
18.00 0.3760 -3.9514 
27.00 -1.0476 -4.1545 
36.00 -1.3167 -3.9139 
45.00 1.0855 -3.7481 
______________________________________ 
(comparative example, in a state of 6000 rpm rotation) 
As is apparent from Table 17 and Table 18, the curvature of field in the 
main scanning direction is small while the polygonal scanner 45 is 
stationary, and when the polygonal scanner 45 rotates, that is, when the 
polygonal scanner 4 is driven for image formation, the curvature of field 
becomes larger. In other words, the performance of this optical scanning 
system is not sufficient to form an image of good quality with no jitter. 
Third Embodiment: FIGS. 9-12 
It has been known that a resin polygonal scanner which has five or more 
reflective facets, during its rotation, is distorted such that the 
reflective facets become convex. A third embodiment is a laser beam 
optical scanning system provided with such a polygonal scanner. 
FIG. 9 shows the general structure of the optical scanning system. The 
optical scanning system has a polygonal scanner 14 which has six 
reflective facets 14a. The other members are the same as those of the 
optical scanning system shown in FIG. 1, and these same members are 
provided with the same reference numbers and marks. 
As shown in FIG. 10, the polygonal scanner 14 has a shape indicated with 
the solid line while it is stationary, and when the polygonal scanner 14 
is driven to rotate, it is distorted as indicated with the dashed line. 
The distortion was simulated by the finite element method, and the result 
is indicated in Table 19 as the radius of curvature of the reflective 
facets 14a. Table 19 also shows the characteristics of the material used 
for the simulation. 
TABLE 19 
______________________________________ 
Material polycarbonate 
______________________________________ 
Young's Modulus 210 kg/mm.sup.2 
Specific Gravity 1.2 
Poisson's Ratio 0.35 
Radius of Inscribed 30 mm 
Circle 
Rotating Speed 15000 rpm 
Radius of Curvature 22500 mm 
of Reflective Facets 
______________________________________ 
The convex distortion of the reflective facets 14a influences the location 
of the image surface in the main scanning direction most as the cases of 
the first and the second embodiments. More specifically, each of the 
reflective facets 14a, when it is distorted to be convex, obtains a 
negative power in the main scanning direction. Accordingly, the laser beam 
reflected by the convex reflective facets 14a becomes a slightly divergent 
pencil of rays, and consequently, the image surface in the main scanning 
direction shifts in the plus direction. This phenomenon is opposite to 
those in the first embodiment and in the second embodiment. Distortion of 
the image surface caused by the convex distortion of the reflective facets 
14a can be calculated beforehand, and an f .theta. system shall be 
designed to have a power in the opposite direction to offset the 
distortion. 
Table 20 shows a preferred positional relation among the elements of the 
optical scanning system of the third embodiment and the photosensitive 
drum 30 which makes the image surface during rotation of the polygonal 
scanner 14 (15000 rpm) come closer to the light receiving surface of the 
photosensitive drum 30. 
The data shown in Table 20 are values which were measured while the 
polygonal scanner 14 is stationary. While the polygonal scanner 14 is 
rotating at a speed of 15000 rpm, the radius of curvature of the 
reflective facets 14a of the polygonal scanner 14 is 22500 mm (see Table 
19). 
TABLE 20 
______________________________________ 
Radius of Curvature 
(mm) 
main scanning 
sub scanning 
Distance 
Refractive 
Elements 
direction direction (mm) Index 
______________________________________ 
laser diode 
0.48 1.0 (air) 
infinite 
cover glass 0.25 1.5112 
of laser (glass) 
diode 
infinite 
5.866 1.0 (air) 
infinite 
collimator 2.8 1.7857 
lens (SF6) 
-6.286 
219.58 1.0 (air) 
infinite 14.54 
cylindrical 1.5 1.4846 
lens (AC) 
infinite 
28.1 1.0 (air) 
polygonal 
infinite 
scanner (reflective facets) 
40 1.0 (air) 
-50.96 29.25 
toric lens 8 1.4846 
(AC) 
-54.56 
95 1.0 (air) 
spherical 
-480 
mirror (reflective surface) 
165 1.0 (air) 
photo- 
sensitive 
drum 
______________________________________ 
(for rotation of 15000 rpm, in a stationary state) 
The light emergent side of the collimator lens 2 is defined by the 
expression (1). By substituting the following aspherical data (12) and 
-6.286 mm, which is indicated in Table 20, into the expression (1), the 
emergent side of the collimator lens 2 can be embodied. 
##EQU6## 
In the optical scanning system shown in Table 20, the toric lens 20 is set 
such that the vertex of its light incident side is -0.2 mm out of the 
optical axis in the main scanning direction and that the vertex of its 
light emergent side is -0.5 mm out of the optical axis in the main 
scanning direction and -9 mm out of the optical axis in the sub scanning 
direction. Further, the spherical mirror 21 is set such that the vertex of 
its reflective surface is -10.3 mm out of the optical axis in the main 
scanning direction. 
Next, the operation and effect of the laser beam optical scanning system 
which has optical positioning shown by Table 20 is described. 
The laser beam is incident to the polygonal scanner 14 as a slightly 
divergent pencil of rays. While the polygonal scanner 14 is stationary, 
the image surface is located before the light receiving surface of the 
photosensitive drum 30 in the direction of the optical axis. When the 
polygonal scanner 14 rotates at a speed of 15000 rpm, the reflective 
facets 14a of the polygonal scanner 14 are distorted to be convex, and the 
laser beam reflected by the reflective facets 14a further diverges. 
Accordingly, the image surface shifts toward the photosensitive drum 30 
along the optical axis and comes closer to the light receiving surface of 
the photosensitive drum 30. 
FIG. 11 shows curvature of field which occurs in the optical scanning 
system of this third embodiment. In FIG. 11, the solid line 65 shows the 
curvature of field while the polygonal scanner 14 is stationary, and the 
dashed line 66 shows the curvature of filed while the polygonal scanner 14 
is rotating at a speed of 15000 rpm. As is apparent from FIG. 11, while 
the polygonal scanner 14 is stationary, the image surface in the main 
scanning direction is distant from the light receiving surface in the 
minus direction, and when the polygonal scanner 14 is driven to rotate, 
the image surface comes closer to the light receiving surface. 
For comparison, the following describes an optical scanning system wherein 
the optical elements are positioned such that the image surface will be 
substantially on the light receiving surface of the photosensitive drum 30 
while the polygonal scanner 14 is stationary. Table 21 shows the 
positional relation among the optical elements of this optical scanning 
system. Data provided in Table 21 are values which were measured while the 
polygonal scanner 14 is stationary. While the polygonal scanner 14 is 
rotating at a speed of 15000 rpm, the radius of curvature of the 
reflective facets 14a is 22500 mm (see Table 19). 
TABLE 21 
______________________________________ 
Radius of Curvature 
(mm) 
main scanning 
sub scanning 
Distance 
Refractive 
Elements 
direction direction (mm) Index 
______________________________________ 
laser diode 
0.48 1.0 (air) 
infinite 
cover glass 0.25 1.5112 
of laser (glass) 
diode 
infinite 
5.862 1.0 (air) 
infinite 
collimator 2.8 1.7857 
lens (SF6) 
-6.286 
219.48 1.0 (air) 
infinite 14.54 
cylindrical 1.5 1.4846 
lens (AC) 
infinite 
28.2 1.0 (air) 
polygonal 
infinite 
scanner (reflective facets) 
40 1.0 (air) 
-50.96 29.25 
toric lens 8 1.4846 
(AC) 
-54.56 
95 1.0 (air) 
spherical 
-480 
mirror (reflective surface) 
165 1.0 (air) 
photo- 
sensitive 
drum 
______________________________________ 
(comparative example, in a stationary state) 
In the optical scanning system of the third embodiment shown by Table 20, 
the collimator lens 2 is disposed a little farther from the laser diode 1 
compared with that in the comparative example such that the divergence of 
the laser beam will be smaller. Further, the cylindrical lens 3 is 
disposed closer to the polygonal scanner 14 such that a shift of the image 
surface in the sub scanning direction caused by the change in the position 
of the collimator lens 2 can be corrected. 
FIG. 12 shows curvature of field in the main scanning direction which 
occurs in the optical scanning system (comparative example) shown by Table 
21. In FIG. 12, the solid line 67 shows curvature of field while the 
polygonal scanner 14 is stationary, and the dashed line 68 shows curvature 
of field while the polygonal scanner 14 is rotating at a speed of 15000 
rpm. 
In the optical scanning system of the third embodiment and in the optical 
scanning system of the comparative example, there occur curvature of field 
and distortion in the sub scanning direction in almost the same degree. 
Fourth Embodiment: FIGS. 13-15 
FIG. 13 shows a laser beam optical scanning system of a fourth embodiment. 
The optical scanning system has a polygonal scanner 55 whose reflective 
facets are distorted to be convex with rotation of the scanner 55. The 
general structure of the optical scanning system is almost the same as 
that of the second embodiment shown by FIG. 7, and the same members are 
provided with the same reference numbers and marks. 
The polygonal scanner 55 has characteristics shown by Table 19. Table 22 
shows a positional relation among the members 41 through 48 of the optical 
scanning system and a photosensitive drum 50 which makes the image surface 
be located near a light receiving surface of the photosensitive drum 50 
during rotation of the polygonal scanner 55. Data provided in Table 22 are 
values which were measured while the polygonal scanner 55 is stationary. 
While the polygonal scanner 55 is rotating at a speed of 15000 rpm, the 
radius of curvature of its reflective facets is 22500 mm (see Table 19). 
TABLE 22 
______________________________________ 
Radius of Curvature 
(mm) 
main scanning 
sub scanning 
Distance 
Refractive 
Elements 
direction direction (mm) Index 
______________________________________ 
infinite 197 
cylindrical 10 1.51072 
lens infinite 
380 1.0 (air) 
polygonal 
infinite 
scanner (reflective facets) 
58 1.0 (air) 
aspherical 
-408 
lens 24 1.48495 
-103.5 
210 1.0 (air) 
aspherical 
-1900 32.15 
toric lens 8 1.48495 
1300 
78.4 1.0 (air) 
photo- 
sensitive 
drum 
______________________________________ 
(for rotation of 15000 rpm, in a stationary state) 
The light emergent sides of the aspherical lens 46 and the aspherical toric 
lens 47 are defined by the expression (1) provided in the first 
embodiment. By substituting the following aspherical data (13) and (14), 
and -103.5 mm and 1300 mm, which are indicated in Table 22 as the radius 
of curvatures (1/C.sub.o) of the light emergent sides of the lenses 46 and 
47, into the expression (1), the light emergent sides of the aspherical 
lens 46 and the aspherical toric lens 47 can be embodied. 
##EQU7## 
The light incident side of the aspherical toric lens 47 is defined by the 
expressions (6), (7), (8), (9) and (10) provided in the third embodiment. 
By substituting the following aspherical data (15), and -1900 mm and 32.15 
mm, which are indicated in Table 22 as the radius of curvature in the main 
scanning direction (1/K) and the radius of curvature in the sub scanning 
direction (1/C) of the light incident side of the aspherical toric lens 
47, into the expressions (5) through (10), the light incident side of the 
lens 47 can be embodied. 
##EQU8## 
Further, the collimator lens 42 is positioned such that an object distance 
between a reflective facet 55a of the polygonal scanner 55 and an object 
point S1 in the main scanning direction will be -17196 mm. If the laser 
beam is incident to the polygonal scanner 55 as a parallel pencil of rays, 
the object distance is infinite. If the laser beam is incident to the 
polygonal scanner 55 as a divergent pencil of rays, the object distance is 
indicated as a negative value. If the laser beam is incident to the 
polygonal scanner 55 as a convergent pencil of rays, the object distance 
is indicated as a positive value. Accordingly, in this optical scanning 
system, since the object distance is -17196 mm, the laser beam emitted 
from the laser diode 41 is incident to the polygonal scanner 55 as a 
slightly divergent pencil of rays. While the polygonal scanner 55 is 
stationary, the laser beam is reflected by a reflective facet 55a of the 
scanner 55, and the reflected laser beam is incident to the aspherical 
lens 46 as an almost parallel pencil of rays. Then, the image surface is 
located before the light receiving surface of the photosensitive drum 50 
in a direction of the optical axis. While the polygonal scanner 55 is 
rotating at a speed of 15000 rpm, the reflective facets 55a of the 
polygonal scanner 55 are distorted to be convex, and accordingly, the 
laser beam reflected from the reflective facets 55a is incident to the 
aspherical lens 46 as a slightly divergent pencil of rays. Thereby, the 
image surface shifts along the optical axis and comes closer to the light 
receiving surface of the photosensitive drum 50. 
FIG. 14 shows curvature of field which occurs in the optical scanning 
system of the fourth embodiment. In FIG. 14, the solid line 71 shows 
curvature of field while polygonal scanner 55 is stationary, and the 
dashed line 72 shows curvature of field while the polygonal scanner 55 is 
rotating at a speed of 15000 rpm. As is apparent from FIG. 14, the image 
surface in the main scanning direction, while the polygonal scanner 55 is 
stationary, is displaced from the light receiving surface in the minus 
direction. However, when the polygonal scanner 55 is driven to rotate, the 
image surface comes closer to the light receiving surface of the 
photosensitive drum 50. 
For comparison, the following describes an optical scanning system whose 
optical elements are positioned such that the image surface is located 
near the light receiving surface of the photosensitive drum 50 while the 
polygonal scanner 55 is stationary. In this optical scanning system, the 
optical elements has a positional relation shown by Table 23. Data 
provided in Table 23 are values which were measured while the polygonal 
scanner 55 is stationary. While the polygonal scanner 55 is rotating at a 
speed of 15000 rpm, the radius of curvature of its reflective facets 55a 
is 22500 mm (see Table 19). 
TABLE 23 
______________________________________ 
Radius of Curvature 
(mm) 
main scanning 
sub scanning 
Distance 
Refractive 
Elements 
direction direction (mm) Index 
______________________________________ 
infinite 197 
cylindrical 10 1.51072 
lens infinite 
380 1.0 (air) 
polygonal 
infinite 
scanner (reflective facets) 
58 1.0 (air) 
aspherical 
-408 
lens 24 1.48495 
-105.2 
210 1.0 (air) 
aspherical 
-1900 32.15 
toric lens 8 1.48495 
1300 
78.4 1.0 (air) 
photo- 
sensitive 
drum 
______________________________________ 
(comparative example, in a stationary state) 
In the comparative example, the collimator lens 42 is positioned such that 
an object distance between the reflective facets 55a and an object point 
S1 will be infinite. In other words, the laser beam is incident to the 
polygonal scanner 55 as a parallel pencil of rays. While the polygonal 
scanner 55 is stationary, the laser beam is reflected by a reflective 
facet 55a and is incident to the aspherical lens 46 as a parallel pencil 
of rays. Then, the image surface is located substantially on the light 
receiving surface of the photosensitive drum 50. While the polygonal 
scanner 55 is rotating at a speed of 15000 rpm, its reflective facets 55a 
are distorted to be convex, and the laser beam reflected therefrom is 
incident to the aspherical lens 46 as a slightly divergent pencil of rays. 
Thereby, the image surface shifts in the plus direction along the optical 
axis and comes behind the light receiving surface of the photosensitive 
drum 50. 
FIG. 15 shows curvature of field in the main scanning direction which 
occurs in the optical scanning system shown by Table 23. In FIG. 15, the 
solid line 73 shows curvature of field while the polygonal scanner 55 is 
stationary, and the dashed line 74 shows curvature of field while the 
polygonal scanner 55 is rotating at a speed of 15000 rpm. 
In the optical scanning system of the comparative example shown by Table 23 
and in the optical scanning system of the fourth embodiment shown by Table 
22, there occur curvature of field and distortion in the sub scanning 
direction in almost the same degree. 
Incidentally, in the third embodiment and in the fourth embodiment, the 
positions of the beam waists are adjusted such that an image of high 
quality with no jitter can be obtained in spite of the distortion of the 
polygonal scanner 14 or 55. The adjustment, as in the first and the second 
embodiments, is carried out by moving the collimator lens 2 or 42 and the 
cylindrical lens 3 or 44, which are in the object side of the polygonal 
scanner 14 or 55, along the optical axis. This adjustment does not 
interfere with corrections to an error caused by misalignment of the 
reflective facets of the polygonal scanner 14 or 55. The adjustment is 
described in connection with the third embodiment. The reflective facets 
14a of the polygonal scanner 14 are distorted to be convex with rotation 
of the polygonal scanner 14, and thereby, the beam waist in the main 
scanning direction shifts in the plus direction. In order to prevent this, 
the collimator lens 2 is moved in the plus direction, and thereby, the 
divergence of the laser beam incident to the reflective facets 14a of the 
polygonal scanner 14 becomes smaller. Accordingly, while the polygonal 
scanner 14 is stationary, the beam waist is located in the minus side of 
the light receiving surface, and when the polygonal scanner 14 is driven 
to rotate, the beam waist comes closer to the light receiving surface. 
Since the collimator lens 2 has not only a power in the main scanning 
direction but also a power in the sub scanning direction, the movement of 
the collimator lens 2 results in shifting the beam waist in the sub 
scanning direction. In order to offset the shift of the beam waist in the 
sub scanning direction, the cylindrical lens 3, which has only a power in 
the sub scanning direction, is moved along the optical axis. 
Thus, if the position of the beam waist in the main scanning direction is 
adjusted by moving a lens which is located in the object side of the 
polygonal scanner 14 and has a power in the sub scanning direction as well 
as a power in the main scanning direction, the position of the beam waist 
in the sub scanning direction is adjusted preferably by moving a lens 
which is located in the object side of the polygonal scanner 14 and has 
only a power in the sub scanning direction. If the cylindrical lens 3 is 
positioned such that the beam waist in the sub scanning direction will be 
located substantially on the light receiving surface, the laser beam 
emergent from the cylindrical lens 3 will be imaged substantially on the 
reflective facets 14a of the polygonal scanner 14, and thus, an error 
caused by misalignment of the reflective facets 14a will be sufficiently 
corrected. 
When the reflective facets of the polygonal scanner are distorted to be 
convex with rotation of the polygonal scanner, each of the reflective 
facets has a larger power toward a laser beam at a field angle of 30 
degrees than toward a laser beam at a field angle of -30 degrees, that is, 
each of the reflective facets converges the laser beam at a field angle of 
30 degrees more strongly than the laser beam at a field angle of -30 
degrees. Accordingly, the beam waist of the laser beam at a field angle of 
30 degrees is located in the minus side of the beam waist of the laser 
beam at a field angle of -30 degrees. This is apparent from the dashed 
line 74 in FIG. 15. In order to correct the inclination of the image 
surface, in the optical scanning system of the fourth embodiment shown by 
Table 22, the aspherical lens 46 and the aspherical toric lens 47 are 
decentered toward the side from which the laser beam is incident to the 
polygonal scanner 55. More specifically, the aspherical lens 46 is 
disposed such that the vertex of its light incident side is -5 mm distant 
from the optical axis, and the aspherical toric lens 47 is disposed such 
that the vertex of its light emergent side is -5 mm distant from the 
optical axis. 
With this arrangement, while the polygonal scanner 55 is stationary, the 
beam waist of the laser beam at a field angle of 30 degrees is located in 
the plus side of the beam waist of the laser beam at a field angle of -30 
degrees (see the solid line 71 in FIG. 14), and this offsets the shift of 
the beam waist of the laser beam at a field angle of 30 degrees in the 
minus direction during rotation of the polygonal scanner 55 (see the 
dashed line 72 in FIG. 14). In other words, the laser beam at a field 
angle of 30 degrees is converged in the main scanning direction less 
strongly than the laser beam at a field angle of -30 degrees such that the 
beam waists of the laser beams will not be displaced from the image 
surface so largely. The larger the angle between a laser beam incident to 
a reflective facet and the laser beam reflected therefrom is, the smaller 
the positive power acting on the laser beam in the main scanning direction 
is. 
Other Embodiments 
Although the present invention has been described in connection with the 
preferred embodiments above, it is to be noted that various changes and 
modifications are possible to those who are skilled in the art. Such 
changes and modifications are to be understood as being within the scope 
of the invention. 
The number of reflective facets of a polygonal scanner is at least one, and 
may be two or eight as well as four and six. As the material of the 
polygonal scanner, not only the material described in connection with the 
embodiments but also polystyrene, acrylonitrile styrene copolymer, 
tetrapolymethylene pentene and such resin processed by aroylation can be 
used. Further, even a metal polygonal scanner is distorted like a resin 
polygonal scanner when it rotates at extremely a high speed, and the 
arrangements described in the embodiments may be necessary in an optical 
scanning system which has a metal polygonal scanner. 
Another way of adjusting the laser beam optical scanning system so as to 
form an image of high quality with no jitter in spite of distortion of the 
polygonal scanner is providing a concave lens between the cylindrical lens 
and the polygonal scanner in a right position. 
According to the embodiments above, the optical scanning system is designed 
in consideration of distortion of the reflective facets of the polygonal 
scanner, and when the reflective facets of the polygonal scanner are 
distorted with rotation of the polygonal scanner, the beam waists of laser 
beams in the main scanning direction come closer to the light receiving 
surface, and the inclination of the image surface becomes smaller. 
However, the following structures are also possible. The collimator lens 
is automatically moved along the optical axis in accordance with the field 
angle so as to correct the inclination of the image surface, and/or the 
collimator lens is automatically moved in accordance with the degree of 
distortion of the reflective facets of the polygonal scanner so as to 
adjust the positions of the beam waists in the main scanning direction. 
Further, the present invention is applicable to an optical scanning system 
which has an f .theta. system comprising other types of lenses.