Optical scanning apparatus

An optical scanning apparatus in which the luminous flux emitted from a light source is deflected by a deflecting device and formed by an optical device into a spot-shaped image on a plane to be scanned. The optical device has first and second toric lenses. The first toric lens has both lens surfaces formed into aspherical shapes in the main scanning cross section and has a meniscus shape in which a concave surface faces the deflecting device in the vicinity of the center of scanning and which provides a positive refractive power. The second toric lens has both lens surfaces formed into aspherical shapes in the main scanning cross section and has a meniscus shape in which a convex surface faces the deflecting device in the vicinity of the center of scanning and which provides a positive refractive power. Sections of the toric lenses perpendicular to a generating line in a sub-scanning cross section both have meniscus shapes which provide concave surfaces facing the deflecting device and which provide a positive refractive power.

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
The present invention relates to an optical scanning apparatus and, more 
particularly, to an optical scanning apparatus ideally suited to, for 
example, a laser beam printer (LBP) apparatus and a digital copying 
machine having an electrophotographic process wherein a luminous flux 
optically modulated and emitted from a light source means is deflectively 
reflected by an optical deflector composed of a rotary polygon mirror or 
the like, then applied onto a plane to be scanned through an image optical 
system having f.theta. characteristics (an f.theta. lens unit) in order to 
optically scan the plane to record image information. 
2. Description of Related Art 
In a conventional optical scanning apparatus used for a laser beam printer 
or the like, a luminous flux optically modulated and emitted from a light 
source means according to an image signal is periodically deflected by an 
optical deflector constituted by, for example, a rotary polygon mirror, 
then it is converged on a surface of a photosensitive recording medium 
(photosensitive drum) into a spot shape by an image optical system having 
f.theta. characteristics, so that the surface is optically scanned to 
record images. 
FIG. 1 is a schematic cross section in the main scanning direction of an 
essential section of a conventional optical scanning apparatus. In the 
drawing, a divergent luminous flux emitted from a light source means 11 is 
formed into a nearly parallel beam through a collimator lens 12, and it is 
restricted in terms of the quantity of light by a stop 13 before being 
launched into a cylindrical lens 14 having a predetermined refractive 
power only in the sub-scanning direction. The parallel luminous flux 
launched into the cylindrical lens 14 comes out in the main scanning cross 
section as is, whereas it is converged in the sub-scanning cross section 
to form a nearly line-shaped image on a deflection surface, i.e. a 
reflection surface, 15a of an optical deflector 15 composed of a rotary 
polygon mirror. 
The luminous flux deflectively reflected by the deflection surface 15a of 
the optical deflector 15 is guided to the plane of a photosensitive drum 
18 serving as a plane to be scanned through an image optical system (an 
f.theta. lens unit) 16 having f.theta. characteristics. As the optical 
deflector 15 is rotated in the direction indicated by arrow A, the surface 
of the photosensitive drum 18 is optically scanned to record the image 
information. 
For this type of optical scanning apparatus to achieve highly precise 
recording of image information, it is required that the field curvature be 
successfully compensated for, the spot diameter be uniform over the entire 
plane to be scanned, and the distortion aberration exhibit f.theta. 
characteristics wherein the angle of incident light and image height have 
a proportional relationship. Various optical scanning apparatuses having 
such optical characteristics or compensation optical systems (f.theta. 
lens units) therefor have been proposed in the past. 
With an increasing trend toward more compact design and lower cost of laser 
beam printers, digital copying machines, etc., optical scanning 
apparatuses are also being required to be made more compact and less 
expensive. 
Various optical scanning apparatuses having an f.theta. lens unit composed 
of a single lens which satisfy the demands mentioned above have been 
proposed; some of them have been disclosed in, for example, Japanses 
Patent Publication No. 61-48684, Japanses Patent Laid-Open No. 63-157122, 
Japanses Patent Laid-Open No. 4-104213, and Japanses Patent Laid-Open No. 
4-50908 which corresponds to U.S. Pat. No. 5,111,219. 
Among these proposed apparatuses, the ones disclosed in Japanses Patent 
Publication No. 61-48684 and Japanses Patent Laid-Open No. 63-157122 
employ, as the f.theta. lens unit, a single lens having a concave surface 
facing an optical deflector to focus the parallel beam from a collimator 
lens onto the plane of a recording medium. The apparatus disclosed in 
Japanses Patent Laid-Open No. 4-104213 employs a single lens, which has a 
concave surface facing the optical deflector and a toroidal surface facing 
an image surface, as the f.theta. lens into which a luminous flux 
converted to a convergent light beam through the collimator lens is 
launched. The apparatus proposed in Japanses Patent Laid-Open No. 4-50908 
corresponding to U.S. Pat. No. 5,111,219 uses a single lens having 
high-order aspherical surfaces as the f.theta. lens into which a luminous 
flux converted to a convergent light beam through the collimator lens is 
launched. 
The conventional optical scanning apparatus disclosed in Japanses Patent 
Publication No. 61-48684, however, poses a problem in that it cannot 
completely eliminate the field curvature in the sub-scanning direction, 
and that the focal length f from the f.theta. lens to a plane to be 
scanned is long because a parallel beam forms an image on the plane to be 
scanned, thus making it difficult to implement a compact optical scanning 
apparatus. 
The apparatus disclosed in Japanses Patent Laid-Open No. 63-157122 is 
disadvantageous in that the f.theta. lens is thick, making it difficult to 
fabricate it by molding with resultant higher cost. 
The apparatus proposed in Japanses Patent Laid-Open No. 4-104213 has a 
problem in that there is distortion aberration remaining, and that jitters 
are produced at intervals corresponding to the number of the polygon 
facets due to a mounting error of a polygon mirror serving as the optical 
deflector. 
The apparatus disclosed in Japanses Patent Laid-Open No. 4-50908 has a 
high-order aspherical f.theta. lens to successfully compensate for 
aberrations; however, the spot diameter in the sub-scanning direction 
tends to change according to the height of an image because of uneven 
magnification in the sub-scanning direction between the optical deflector 
and the plane to be scanned. 
In addition to those described above, optical scanning apparatuses using 
two lenses to constitute the f.theta. lens unit have been proposed in, for 
example, Japanses Patent Laid-Open No. 56-36622 and Japanses Patent 
Laid-Open No. 61-175607. The sections of the f.theta. lenses of these 
apparatuses are a spherical or slightly aspherical, making it difficult to 
attain more compact design, lower cost, and higher precision. 
SUMMARY OF THE INVENTION 
Accordingly, it is an object of the present invention to provide a compact 
optical scanning apparatus, which is suited for high-resolution printing, 
compensates for field curvature and/or distortion aberration and prevents 
changes in spot diameter in the sub-scanning direction attributable to 
changes in the height of images, by combining two lenses having 
appropriate shapes to constitute an f.theta. lens unit for forming a 
convergent light beam from a collimator lens onto a plane to be scanned 
via an optical deflector. 
To this end, according to one aspect of the present invention, there is 
provided an optical scanning apparatus equipped with: a first optical 
device for converting a luminous flux emitted from light source means into 
a convergent light beam; a second optical device for forming the luminous 
flux into a line-shaped, longitudinal image in a main scanning direction 
on a deflecting surface of a deflecting device; and a third optical device 
for forming the luminous flux which has been deflected by the deflecting 
device into a spot-shaped image on a plane to be scanned; 
wherein the third optical device has a first toric lens and a second toric 
lens disposed in the mentioned order from the deflecting device; the first 
toric lens has both surfaces thereof formed into aspherical shapes in the 
main scanning cross section and has a meniscus shape which provides a 
concave surface facing the deflecting device in the vicinity of the center 
of scanning and which provides a positive refractive power; the second 
toric lens has both surfaces thereof formed into aspherical shapes in the 
main scanning cross section and has a meniscus shape which provides a 
convex surface facing the deflecting device in the vicinity of the center 
of scanning and which provides a positive refractive power; and the 
sections of the first toric lens and the second toric lens perpendicular 
to a generating line in a sub-scanning cross section both have meniscus 
shapes which provide concave surfaces facing the deflecting device and 
which provide a positive refractive power. 
In the second toric lens of the optical scanning apparatus, the curvature 
of the lens surface facing the plane to be scanned continuously changes 
from the center of the lens toward the periphery of the lens in the main 
scanning cross section and the sign of the curvature is reversed in the 
middle part therebetween. 
In the optical scanning apparatus, the first toric lens and the second 
toric lens are fabricated by plastics molding. 
In the optical scanning apparatus, the refractive power of the first toric 
lens in the sub-scanning cross section continuously increases from the 
center of the lens toward the periphery of the lens, whereas the 
refractive power of the second toric lens in the sub-scanning cross 
section continuously decreases from the center of the lens toward the 
periphery of the lens. 
In the optical scanning apparatus, the curvature of a lens surface of the 
second toric lens in the sub-scanning cross section changes laterally 
symmetrically in the main scanning direction from the center of the lens. 
In the optical scanning apparatus, the symmetry axis of the second toric 
lens in the main scanning direction is inclined in the main scanning cross 
section with respect to the normal of the plane to be scanned. 
In the optical scanning apparatus, if the focal lengths of the first toric 
lens and the second toric lens in the main scanning cross section are 
denoted as f6 and f7, respectively, then the following condition is 
satisfied: 
EQU 0.08 &lt;f6/f7&lt;0.17. 
In the first toric lens of the optical scanning apparatus, the curvature of 
its lens surface facing the plane to be scanned in the sub-scanning cross 
section changes laterally asymmetrically from the center of the lens 
toward the periphery of the lens. 
In the optical scanning apparatus, the third optical device satisfies the 
condition given below when the angular magnification at the central part 
of an effective image on the plane to be scanned in the sub-scanning cross 
section between the deflecting device and the plane to be scanned is 
denoted as r.sub.SC : 
EQU 0.25&lt;r.sub.SC &lt;0.67. 
In the optical scanning apparatus, the refractive power of the first toric 
lens in the sub-scanning cross section continuously increases from the 
center of the lens toward the periphery of the lens, whereas the 
refractive power of the second toric lens in the sub-scanning cross 
section continuously decreases from the center of the lens toward the 
periphery of the lens; and the third optical device satisfies the 
condition shown below when the angular magnification at the central part 
of an effective image on the plane to be scanned in the sub-scanning cross 
section between the deflecting device and the plane to be scanned is 
denoted as r.sub.SC and the angular magnification at an arbitrary point in 
the entire image area is denoted as r.sub.So : 
EQU 0.85&lt;r.sub.So /r.sub.SC &lt;1.15. 
According to another aspect of the present invention, there is provided an 
optical scanning apparatus equipped with: a first optical device for 
converting a luminous flux emitted from light source means into a 
convergent light beam; a second optical device for forming the luminous 
flux into a line-shaped, longitudinal image in a main scanning direction 
on a deflecting surface of a deflecting device; and a third optical device 
for forming the luminous flux which has been deflected by the deflecting 
device into a spot-shaped image on a plane to be scanned; 
wherein the third optical device has a first toric lens and a second toric 
lens disposed in the mentioned order from the deflecting device, the first 
toric lens has both surfaces thereof formed into aspherical shapes in the 
main scanning cross section and has a meniscus shape which provides a 
concave surface facing the deflecting device in the vicinity of the center 
of scanning and which provides a positive refractive power, the second 
toric lens has both surfaces thereof formed into aspherical shapes in the 
main scanning cross section and has a meniscus shape which provides a 
convex surface facing the deflecting device in the vicinity of the center 
of scanning and which provides a positive refractive power, and the 
sections of the first toric lens and the second toric lens perpendicular 
to a generating line in a sub-scanning cross section both have meniscus 
shapes which provide concave surfaces facing the deflecting device and 
which provide a positive refractive power; and 
if the focal lengths of the first toric lens and the second toric lens in 
the main scanning cross section are denoted as f6 and f7, respectively, 
then the following condition is satisfied: 
EQU 0.08&lt;f6/f7&lt;2.0. 
In the second toric lens of the optical scanning apparatus, the curvature 
of the lens surface facing the plane to be scanned continuously changes 
from the center of the lens toward the periphery of the lens in the main 
scanning cross section and the sign of the curvature is reversed in the 
middle part therebetween. 
In the optical scanning apparatus, the first toric lens and the second 
toric lens are fabricated by plastics molding. 
In the optical scanning apparatus, the refractive power of the first toric 
lens in the sub-scanning cross section continuously increases from the 
center of the lens toward the periphery of the lens, whereas the 
refractive power of the second toric lens in the sub-scanning cross 
section continuously decreases from the center of the lens toward the 
periphery of the lens. 
In the optical scanning apparatus, the curvature of a lens surface of the 
second toric lens in the sub-scanning cross section changes laterally 
symmetrically in the main scanning direction from the center of the lens. 
In the optical scanning apparatus, the symmetry axis of the second toric 
lens in the main scanning direction is inclined in the main scanning cross 
section with respect to the normal of the plane to be scanned. 
In the first toric lens of the optical scanning apparatus, the curvature of 
its lens surface facing the plane to be scanned in the sub-scanning cross 
section changes laterally asymmetrically from the center of the lens 
toward the periphery of the lens. 
In the optical scanning apparatus, the third optical device satisfies the 
condition given below when the angular magnification at the central part 
of an effective image on the plane to be scanned in the sub-scanning cross 
section between the deflecting device and the plane to be scanned is 
denoted as r.sub.SC : 
EQU 0.25&lt;r.sub.SC &lt;0.67. 
In the optical scanning apparatus, the refractive power of the first toric 
lens in the sub-scanning cross section continuously increases from the 
center of the lens toward the periphery of the lens, whereas the 
refractive power of the second toric lens in the sub-scanning cross 
section continuously decreases from the center of the lens toward the 
periphery of the lens; and the third optical device satisfies the 
condition shown below when the angular magnification at the central part 
of an effective image on the plane to be scanned in the sub-scanning cross 
section between the deflecting device and the plane to be scanned is 
denoted as r.sub.SC and the angular magnification at an arbitrary point in 
the entire image area is denoted as r.sub.So : 
EQU 0.85&lt;r.sub.So /r.sub.SC &lt;1.15.

DESCRIPTION OF THE PREFERRED EMBODIMENTS 
FIG. 2 is a sectional view in the main scanning direction (in the main 
scanning cross section) illustrating an essential part of a first 
embodiment of an optical scanning apparatus in accordance with the present 
invention. The main scanning direction herein refers to the direction in 
which a deflective reflection surface of an optical deflector is 
deflectively scanned with a luminous flux. The main scanning plane means 
the luminous-flux plane formed as time passes by the luminous flux used 
for deflective scanning by the deflective reflection surface of the 
optical deflector. 
In FIG. 2, there are shown a light source means 1 composed of, for example, 
a semiconductor laser, a collimator lens 2 serving as a first optical 
device which converts a luminous flux, i.e. a light beam, emitted from the 
light source means 1 into a convergent light beam on the main scanning 
plane, and an aperture stop 3 for making the diameter of the luminous flux 
passing therethrough uniform. 
There is also shown a cylindrical lens 4 serving as a second optical device 
which has a predetermined refractive power only in the sub-scanning 
direction, i.e. the direction perpendicular to the paper surface of FIG. 
2. A nearly line-shaped image is formed on a deflective reflection plane 
5a of an optical deflector 5 (which will be discussed later) in the 
sub-scanning cross section, which includes an optical axis and which is 
orthogonalized to the main scanning cross section, from the luminous flux 
that has passed through the aperture stop 3. Hence, the luminous flux 
striking the optical deflector 5 forms a longitudinal line-shaped image in 
the main scanning direction. 
The optical deflector 5 is comprised of, for example, a rotary polygon 
mirror having six facets; it is rotated at a constant speed in the 
direction indicated by arrow A in the drawing by a motor or other driving 
means (not shown). 
An f.theta. lens unit 9 is an image forming optical system which serves as 
the third optical device and which has f.theta. characteristics. The third 
optical device 9 has a first toric lens 6 and a second toric lens 7. The 
third optical device 9 forms an image on a photosensitive drum 8, which is 
a recording medium and which provides a plane to be scanned, from a 
luminous flux deflectively reflected by the optical deflector 5 according 
to image information; it also compensates for the inclination of the 
deflection surface of the optical deflector 5. 
In this embodiment, the luminous flux emitted from a semiconductor laser 1 
is converted to a convergent light beam through the collimator lens 2 in 
the main scanning cross section and restricted in the quantity of light 
thereof through the aperture stop 3 before it is launched into a 
cylindrical lens 4. The incident luminous flux is emitted as is in the 
main scanning cross section, whereas it is converged in the sub-scanning 
cross section to form a nearly line-shaped image (a longitudinal 
line-shaped image in the main scanning direction) on the deflection plane 
5a of the optical deflector 5. The luminous flux deflectively reflected by 
the deflection plane 5a of the optical deflector 5 reaches the plane of 
the photosensitive drum 8 through the f.theta. lens unit 9. As the optical 
deflector 5 rotates in the direction indicated by arrow A, the plane of 
the photosensitive drum 8 is optically scanned in the direction indicated 
by arrow B to record images. 
The characteristics of the first toric lens 6 and the second toric lens 7 
making up the third optical device, i.e. the f.theta. lens unit, 9 in this 
embodiment will now be explained. 
The third optical device 9 is constituted by the first toric lens 6 and the 
second toric lens 7 which both have positive refractive powers. The 
refractive powers of the these two lenses are properly adjusted to obtain 
good field curvature characteristics. 
As the distance between the optical deflector 5 and the plane 8 to be 
scanned is decreased, the luminous flux in the main scanning direction 
spreads more widely; hence, the refractive power of the second toric lens 
7 at which the luminous flux grows wider is set so that it is weaker than 
the refractive power of the first toric lens 6. 
The refractive powers are so set that, if the focal lengths of the first 
toric lens 6 and the second toric lens 7 in the main scanning cross 
section are denoted as f6 and f7, respectively, then the condition given 
below is satisfied: 
EQU 0.08&lt;f6/f7&lt;2.0 (1) 
Satisfying conditional formula (1) leads to successful compensation of the 
field curvature and/or distortion aberration. This also permits the 
central thickness of the first toric lens 6 and the second toric lens 7 to 
be nearly equal, so that the cycle time for fabricating the two lenses by 
plastics molding or glass molding is shortened, and the deformation of the 
plane shapes of the first and second toric lenses is reduced when they are 
cooled for fabricating the two lenses by molding. 
In this embodiment, further preferably, the ranges of the values in 
conditional formula (1) are set as shown below so as to obtain even better 
optical performance: 
EQU 0.08&lt;f6/f7 &lt;1.7 (1a) 
EQU 0.08&lt;f6/f7 &lt;0.17 (1b) 
If the f.theta. lens unit 9 were composed of a single toric lens, then it 
would be difficult to maintain a good spot diameter over the entire plane 
to be scanned and also difficult to successfully control the change in the 
field curvature by only two lens surfaces of the single toric lens. 
For this reason, the f.theta. lens unit 9 in this embodiment is composed of 
the two toric lenses having predetermined shapes to satisfactorily 
compensate for the field curvature. 
The first toric lens 6 of the embodiment is aspherical on both surfaces 
thereof in the main scanning cross section primarily to maintain good 
f.theta. characteristics and image curvature characteristics, and it has a 
meniscus so that the concave surface faces the deflector in the vicinity 
of the center of scanning and that positive refractive power is provided. 
Further, the first toric lens 6 is configured such that the curvatures of 
both lens surfaces (the refractive power) continuously increases in the 
sub-scanning cross section. The cross section perpendicular to the 
generating line in the sub-scanning cross section has a meniscus shape of 
a positive refractive power which has its concave surface facing the 
deflector. A lens surface 6b facing the plane to be scanned in the 
sub-scanning cross section is configured so that the curvature (the 
refractive power) thereof greatly changes laterally asymmetrically from 
the center of the lens toward the periphery of the lens. 
This solves the problem of uneven lateral magnification in the sub-scanning 
direction, thus restraining the variation in spot diameter in the 
sub-scanning direction. 
The second toric lens 7 is aspherical on both surfaces 7a and 7b thereof in 
the main scanning cross section, i.e. in the paper surface of FIG. 2, 
primarily to maintain good f.theta. characteristics and image curvature 
characteristics, and it has a meniscus shape so that the convex surface 
thereof faces the deflector 5 in the vicinity of the center of scanning, 
i.e. the center of the lens, and that a positive refractive power is 
provided. 
Further, the second toric lens 7 is configured such that the curvature of a 
lens surface 7b continuously changes from the center of the lens, i.e. the 
center of the main scanning range, toward the periphery of the lens, and 
the positive sign or the negative sign of the curvature is reversed at the 
middle therebetween. 
By configuring the second toric lens 7 as described above, the field 
curvature and distortion aberration are successfully compensate for over 
the whole scanning area. 
Further, the second toric lens 7 is configured such that the curvatures of 
both lens surfaces 7a and 7b in the sub-scanning cross section (the 
section which includes the optical axis and which is orthogonal to the 
main scanning cross section) continuously decrease laterally symmetrically 
away from the center of the lens in the main scanning direction. 
The cross section of the second toric lens 7 perpendicular to the 
generating line in the sub-scanning cross section has a meniscus shape of 
a positive refractive power, the meniscus shape having its concave surface 
facing the deflector 5. This reduces the lateral magnification in the 
sub-scanning direction to control the absolute value of the spot diameter 
to a minimum so as to maintain good image characteristics in the 
sub-scanning direction. 
The f.theta. lens unit 9 is designed so that it satisfies the condition 
shown below when the angular magnification at the center of the effective 
image on the plane 8 to be scanned in the sub-scanning cross section 
between the optical deflector 5 and the plane 8 to be scanned is denotes 
as r.sub.SC : 
EQU 0.25&lt;r.sub.SC &lt;0.67 (2) 
This conditional formula (2) has been established to maintain good image 
characteristics in the sub-scanning cross section while minimizing the 
lens length of the f.theta. lens unit 9 in the main scanning direction. If 
the lower limit value of conditional formula (2) were exceeded, then the 
effective luminous fluxes of the first toric lens 6 and the second toric 
lens 7 would spread, and the lens would grow thicker, making it difficult 
to implement a compact design thereof. If the upper limit value of 
conditional formula (2) were exceeded, then the imaging performance would 
be unstable because the f.theta. lens unit 9 is made of a plastic material 
susceptible to environmental changes including temperature changes. 
Further, the embodiment is designed such that the refractive power of the 
first toric lens 6 in the sub-scanning cross section continuously 
increases from the center to periphery of the lens, while the refractive 
power of the second toric lens 7 continuously decreases from the center to 
periphery of the lens; and the f.theta. lens unit 9 satisfies the 
condition given below when the angular magnification at the center of the 
effective image on the plane 8 to be scanned in the sub-scanning cross 
section between the optical deflector 5 and the plane 8 to be scanned is 
denotes as r.sub.SC and the angular magnification at an arbitrary point in 
the entire image area is denoted as r.sub.SO : 
EQU 0.85&lt;r.sub.SO /r.sub.SC &lt;1.15 (3) 
Conditional formula (3) is provided to make the spot diameter uniform in 
the sub-scanning cross section on the plane 8 to be scanned from the 
center toward the periphery of the plane 8 to be scanned. If the upper 
limit value of conditional formula (3) were exceeded, the spot diameter at 
the edge of an image, i.e. the peripheral area of the plane 8 to be 
scanned, in the main scanning direction would become undesirably small for 
the central part; if the lower limit value of conditional formula (3) were 
exceeded, then the spot diameter at the edge of an image in the main 
scanning direction would become undesirably large for the central part, 
leading to the loss of uniformity of the spot diameter in the sub-scanning 
cross section. 
In this embodiment, the toric lenses are provided with aspheric shapes that 
can be represented by a function having up to a term of the tenth degree 
in the main scanning direction; they have spherical surfaces that 
continuously change in the direction of the height of images in the 
sub-scanning direction. The lenses are shaped such that the direction of 
the generating line corresponding to the main scanning direction is 
expressed by the formula shown below when, for example, the intersection 
point of the toric lenses and the optical axis is defined as the origin, 
the direction of the optical axis is taken on the X-axis, the axis 
orthogonal to the optical axis in the main scanning surface is taken on 
the Y-axis, and the axis orthogonal to the optical axis in the 
sub-scanning surface is taken on the Z-axis: 
##EQU1## 
where R denotes the radius of the curvature; and K, B.sub.4, B.sub.6, 
B.sub.8, and B.sub.10 denote aspherical coefficients. 
The radius of the curvature of the sub-scanning cross section continuously 
changes as the lens surface coordinate in the main scanning direction 
changes. Curvature radius r', in a case where the coordinate on the main 
scanning surface is Y, is expressed by an expression given below: 
EQU r'=r(1+D.sub.2 Y.sup.2 +D.sub.4 Y.sup.4 +D.sub.6 Y.sup.6 +D.sub.8 Y.sup.8 
+D.sub.10 Y.sup.10) 
where r denotes the radius of curvature on the optical axis, and D.sub.2, 
D.sub.4, D.sub.6, D.sub.8, D.sub.10 are coefficients. 
In this case, if the value of Y is positive, then the radius of curvature 
r' is calculated using the coefficients accompanied by subscript "U" as in 
D.sub.2U, D.sub.4U, D.sub.6U, D.sub.8U, D.sub.10U ; if the value of Y is 
negative, then the radius of curvature r' is calculated using the 
coefficients accompanied by subscript "L" as in D.sub.2L, D.sub.4L, 
D.sub.6L, D.sub.8L, D.sub.1OL. 
Table 1 below shows the coefficients representing the surface shapes and 
other characteristics of the lenses used in the first embodiment. FIG. 3 
shows the refractive powers of the first and second toric lenses in the 
sub-scanning direction in the first embodiment. FIG. 4 shows the field 
curvature in relation to distortion aberration in the first embodiment, 
and it also illustrates the changes in angular magnification with the 
center providing the reference; from the diagram, it can be seen that the 
aberrations have been adequately compensated for to the level at which 
there is substantially no problem in practical use. 
TABLE 1 
__________________________________________________________________________ 
DATA 
Working Wavelength 
(nm) 780 Shape of First Lens 
Shape of Second 
__________________________________________________________________________ 
Lens 
Refractive Index of First 
n1 1.5242 First Surface 
Second Surface 
First Surface 
Second Surface 
Lens 
Refractive Index of Second 
n2 1.5242 
R -67.97 -49.663 46.718 45.398 
Lens 
Polygon Incident Angle 
.theta.i 
-60 KU -0.559322 
0.0376036 -9.14521 -9.47556 
Polygon Max. Exiting 
.theta.Max 
41.437 
B4U 0.00000110565 
0.00000120743 
-0.000000563241 
-0.00000103507 
Angle 
Polygon - First Lens 
e1 41.777 
B6U 0.0000000000555 
0.00000000080406 
-0.000000000086 
0.00000000009461 
Central Thickness of First 
d1 9.5 B8U 0 -0.0000000000003 
3.523820000e-14 
-1.308180000e-14 
Lens 
First Lens - Second Lens 
e2 7.133 
B10U 
0 1.6306000000e-16 
-2.32284000e-18 
1.9591800000e-18 
Central Thickness of 
d2 8.6 KL Same as KU 
0.0286653 Same as KU 
Same as KU 
Second Lens 
Second Lens - Plane to be 
Sk 189.74 
B4L Same as B4U 
0.00000115642 
Same as B4U 
Same as B4U 
Scanned 
Focal Length of f.theta. Lens 
ft 251.05 
B6L Same as B6U 
0.00000000084656 
Same as B6U 
Same as B6U 
Convergence of Collimator 
fc 1155 
B8L Same as B8U 
-0.0000000000003 
Same as B8U 
Same as B8U 
Polygon - Natural B10L 
Same as B10U 
1.6307300000e-16 
Same as B10U 
Same as B10U 
Convergent Point 
r -29 -22.884 -68 -25.59 
f6 298.474 D2U 0 -0.000205716 
0.00187374 
0.000996409 
f7 2470.086 D4U 0 0.0000000619717 
0.00000190851 
-0.000000492908 
f6/f7 0.121 D6U 0 0 0 0.00000000012598 
r.sub.sc 0.338 D8U 0 0 0 -1.581770000e-14 
D10U 
0 0 0 7.6398000000e-19 
D2L Same as D2U 
-0.000180986 
Same as D2U 
Same as D2U 
D4L Same as D4U 
0.0000000555638 
Same as D4U 
Same as D4U 
D6L Same as D6U 
0 Same as D6U 
Same as D6U 
D8L Same as D8U 
0 Same as D8U 
Same as D8U 
D10L 
Same as D10U 
0 Same as D10U 
Same as 
__________________________________________________________________________ 
D10U 
In this embodiment, the axis of symmetry of the second toric lens 7 in the 
main scanning direction is inclined by 10 minutes clockwise about the apex 
of the lens surface facing the optical deflector 5 in the main scanning 
cross section with respect to the normal to the plane 8 to be scanned. 
The surface shapes of the second toric lens 7 are symmetric with respect to 
the optical axis of the toric lens itself in the main scanning cross 
section in both directions of the generating line and the generated line. 
FIG. 5 is a sectional view showing an essential part in the main scanning 
direction (the main scanning cross section) of a second embodiment of the 
optical scanning apparatus in accordance with the present invention. In 
the drawing, like elements as those shown in FIG. 2 will be assigned like 
reference numerals. 
As compared with the first embodiment shown in FIG. 2, the second 
embodiment is characterized in that first and second toric lenses 26 and 
27 constituting a third optical device 29 have lens shapes ideally suited 
to the six-facet polygon mirror as shown in Table 2 given below. The rest 
of the configuration and the optical operation of the second embodiment is 
almost identical to those of the first embodiment; hence the second 
embodiment also provides the similar advantages. 
Table 2 below shows the coefficients representing the shapes and other 
characteristics of the lenses used in the second embodiment. FIG. 6 shows 
the refractive powers of the first and second toric lenses in the 
sub-scanning direction in the second embodiment. FIG. 7 shows the field 
curvature in relation to distortion aberration in the second embodiment, 
and it also illustrates the changes in angular magnification with the 
center providing the reference; from the diagram, it can be seen that the 
aberrations have been adequately compensated for to the level at which 
there is substantially no problem in practical use. 
TABLE 2 
__________________________________________________________________________ 
DATA 
Working Wavelength 
(nm) 780 Shape of First Lens 
Shape of Second 
__________________________________________________________________________ 
Lens 
Refractive Index of First 
n1 1.524 First Surface 
Second Surface 
First Surface 
Second Surface 
Lens 
Refractive Index of Second 
n2 1.524 
R -62.49 -55.58 43 49 
Lens 
Polygon Incident Angle 
.theta.i 
-60 KU -3.70908 
0.36798 -5.21726 -6.02921 
Polygon Max. Exiting 
.theta.Max 
41.437 
B4U -0.000000502842 
-0.0000000138321 
-0.000000531432 
-0.000000926641 
Angle 
Polygon - First Lens 
e1 44.1614 
B6U 0.0000000003567 
0.00000000143597 
-0.000000000091 
0.00000000007308 
Central Thickness of 
d1 8.5 B8U 8.144520000e-14 
-0.0000000000007 
3.634640000e-14 
-9.058470000e-15 
First Lens 
First Lens - Second 
e2 1 B10U 
0 3.1129000000e-16 
-2.11543000e-18 
2.1107400000e-18 
Lens 
Central Thickness of 
d2 9.5 KL Same as KU 
0.371585 Same as KU 
Same as KU 
Second Lens 
Second Lens - Plane to be 
Sk 193.578 
B4L Same as B4U 
-0.000000105792 
Same as B4U 
Same as B4U 
Scanned 
Focal Length of f.theta. Lens 
ft 250.45 
B6L Same as B6U 
0.00000000162157 
Same as B6U 
Same as B6U 
Convergence of Collimator 
fc 1157 
B8L Same as B8U 
-0.0000000000008 
Same as B8U 
Same as B8U 
Polygon - Natural B10L 
Same as B10U 
3.4947100000e-16 
Same as B10U 
Same as B10U 
Convergent Point 
r -29.5 -22.99 -67.76 -25.31 
f6 673.942 D2U 0 -0.000167982 
0.00201954 
0.00099252 
f7 433.728 D4U 0 0.0000000618943 
0.00000124093 
-0.000000401812 
f6/f7 1.554 D6U 0 0 0.0000000008935 
0.00000000004964 
r.sub.sc 0.325 D8U 0 0 -4.01765000e-13 
5.3383800000e-15 
D10U 
0 0 5.615490000e-17 
-1.168710000e-18 
D2L Same as D2U 
-0.000148743 
Same as D2U 
Same as D2U 
D4L Same as D4U 
0.0000000634312 
Same as D4U 
Same as D4U 
D6L Same as D6U 
0 Same as D6U 
Same as D6U 
D8L Same as D8U 
0 Same as D8U 
Same as D8U 
D10L 
Same as D10U 
0 Same as D10U 
Same as 
__________________________________________________________________________ 
D10U 
In this embodiment, the axis of symmetry of the second toric lens 27 in the 
main scanning direction is inclined by 10 minutes clockwise about the apex 
of the lens surface facing the optical deflector 5 in the main scanning 
cross section with respect to the normal line of the plane 8 to be 
scanned. 
FIG. 8 is a sectional view showing an essential part in the main scanning 
direction (the main scanning cross section) of a third embodiment of the 
optical scanning apparatus in accordance with the present invention. In 
the drawing, like elements as those shown in FIG. 2 will be assigned the 
like reference numerals. 
As compared with the first embodiment shown in FIG. 2, the third embodiment 
is characterized in that first and second toric lenses 36 and 37 
constituting a third optical device 39 have lens shapes ideally suited to 
the six-facet polygon mirror as shown in Table 3 given below. The rest of 
the configuration and the optical operation of the third embodiment is 
almost identical to those of the first embodiment; hence the third 
embodiment also provides the similar advantages. 
Table 3 below shows the coefficients representing the surface shapes and 
other characteristics of the lenses used in the third embodiment. FIG. 9 
shows the refractive powers of the first and second toric lenses in the 
sub-scanning direction in the third embodiment. FIG. 10 shows the field 
curvature in relation to distortion aberration in the third embodiment, 
and it also illustrates the changes in angular magnification with the 
center providing the reference; from the diagram, it can be seen that the 
aberrations have been adequately compensated for to such a level that 
there is substantially no problem in practical use. 
TABLE 3 
__________________________________________________________________________ 
DATA 
Working Wavelength 
(nm) 780 Shape of First Lens 
Shape of Second 
__________________________________________________________________________ 
Lens 
Refractive Index of First 
n1 1.5242 First Surface 
Second Surface 
First Surface 
Second Surface 
Lens 
Refractive Index of Second 
n2 1.5242 
R -67.97 -49.663 46.718 45.398 
Lens 
Polygon Incident Angle 
.theta.i 
-60 KU -0.386489 
0.0406645 -9.1501 -9.47556 
Polygon Max. Exiting 
.theta.Max 
41.437 
B4U 0.0000011819 
0.00000120604 
-0.000000565087 
-0.00000103507 
Angle 
Polygon - First Lens 
e1 41.73 
B6U 0.0000000000555 
0.00000000080567 
-0.000000000085 
0.00000000009461 
Central Thickness of 
d1 9.5 B8U 0 -0.0000000000003 
3.523820000e-14 
-1.308180000e-14 
First Lens 
First Lens - Second Lens 
e2 7.21 
B10U 
0 1.6306000000e-16 
-2.32284000e-18 
1.9591800000e-18 
Central Thickness of 
d2 8.6 KL Same as KU 
0.0281781 Same as KU 
Same as KU 
Second Lens 
Second Lens - Plane to be 
Sk 189.71 
B4L Same as 134U 
0.0000011631 
Same as B4U 
Same as G4U 
Scanned 
Focal Length of f.theta. Lens 
ft 251.05 
B6L Same as B6U 
0.00000000083822 
Same as B6U 
Same as B6U 
Convergence of Collimator 
fc 1157 
B8L Same as B8U 
-0.0000000000008 
Same as B8U 
Same as B8U 
Polygon - Natural B10L 
Same as B10U 
1.6307300000e-16 
Same as B10U 
Same as B10U 
Convergent Point 
r -29.5 -22.9 -68 -25.59 
f6 298.474 D2U 0 -0.000205789 
0.00187374 
0.000996409 
f7 2470.09 D4U 0 0.0000000614589 
0.00000190851 
-0.000000492908 
f6/f7 0.121 D6U 0 0.0000000000002 
0 0.00000000012598 
r.sub.sc 0.337 D8U 0 0 0 -1.581770000e-14 
D10U 
0 0 0 7.6398000000e-19 
D2L Same as D2U 
-0.000181005 
Same as D2U 
Same as D2U 
D4L Same as D4U 
0.0000000519452 
Same as D4U 
Same as D4U 
D6L Same as D6U 
0.00000000000281 
Same as D6U 
Same as D6U 
D8L Same as D8U 
0 Same as D8U 
Same as D8U 
D10L 
Same as D10U 
0 Same as D10U 
Same as 
__________________________________________________________________________ 
D10U 
In this embodiment, the axis of symmetry of the second toric lens 37 in the 
main scanning direction is inclined by 10 minutes clockwise about the apex 
of the lens surface facing the optical deflector 5 in the main scanning 
cross section with respect to the normal to the plane 8 to be scanned. 
Thus, according to the present invention, it is possible to implement a 
compact optical scanning apparatus which is suited for high-resolution 
printing and which compensates for field curvature and/or distortion 
aberration and prevents changes in spot diameter in the sub-scanning 
direction attributable to changes in the height of images by combining two 
lenses having appropriate shapes to constitute an f.theta. lens unit for 
forming a convergent light beam from a collimator lens onto a plane to be 
scanned via an optical deflector. 
Moreover, configuring the f.theta. lens unit by using the two lenses 
enables the central thickness in the direction of the optical axis of each 
of the lenses to be decreased. This makes it possible to shorten the cycle 
time when plastic-molding the two lenses, resulting in lower cost of the 
optical scanning apparatus.