Optical scanning system including two hologram elements

An optical scanning system includes a laser source, a deflector deflecting a laser beam emitted from the laser source, and at least two hologram elements located between the deflector and an image forming member.

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
The present invention generally relates to optical scanning systems, and 
more particularly to an optical scanning system having a laser beam 
source, a deflector and hologram elements and performing highly-precise 
laser scan. 
Optical scanning systems are applied to various devices such as a laser 
beam printer and an optical instrument, and scan an object by deflecting 
the laser beam. It is required that these optical scanning systems can 
realize a highly-precise laser scan. 
2. Description of the Prior Art 
An optical scanning system having a polygon mirror and an f-.zeta. lens 
system is well known as an optical scanning system in which the laser beam 
scans an object. 
However, such an optical scanning system has a disadvantage in that it is 
very expensive because the f-.THETA. lens system is made up of a plurality 
of lens groups. With the above in mind, recently, there has been 
considerable activity in the development of an optical scanning system 
that uses a hologram disk. As shown in FIG. 1, an optical scanning system 
using a hologram disk utilizes a characteristic in which the diffracting 
direction of the incident beam depends on the position of the incident 
beam on a lens. Recently, it has been proposed that a plurality of 
hologram elements are used instead of a lens. 
FIG. 2 shows such an application of hologram elements, in which a hologram 
disk is illustrated. The hologram disk includes a plurality of hologram 
elements disposed on a disk in a ring formation. The hologram disk is 
rotated to move the incident laser beam that scans an object. 
Optical scanning systems using hologram disks as described above have a 
disadvantage in that the image plane of the laser beam is moved due to a 
variation in the wavelength of the laser beam. In order to eliminate the 
above disadvantage, other hologram elements are additionally used to 
compensate for a variation in the wavelength of the laser beam to make the 
laser beam project onto an identical point even when the wavelength of the 
laser beam varies. 
The optical scanning system using the above-mentioned hologram disk 
realizes highly-precise laser scan. However, the hologram disk does not 
have strong mechanical strength, and cannot be rotated at high speeds. 
Hence, the optical scanning system using the hologram disk cannot realize 
high-speed laser scanning. 
SUMMARY OF THE INVENTION 
It is a general object of the present invention to provide an optical 
scanning system in which the above disadvantages are eliminated. 
A more specific object of the present invention is to provide an optical 
scanning system capable of performing highly-precise, high-speed laser 
scan. 
These objects of the present invention are achieved by an optical scanning 
system comprising: 
a laser source; 
a deflector deflecting a laser beam emitted from the laser source; and 
at least two hologram elements located between the deflector and an image 
forming member. 
Preferably, each of said at least two hologram elements has the following 
phase transfer function .phi.(x, y): 
EQU .phi.(x, y)=(2.pi./.lambda.).multidot..SIGMA.c.sub.i r.sup.2i ! 
where x denotes a position in a main scanning direction of a hologram, y 
denotes a position in a direction perpendicular to the main scanning 
direction, r denotes a distance from a center of the hologram, namely 
(x.sup.2 +y.sup.2).sup.0.5, .lambda. is a wavelength of the laser beam, 
c.sub.i denotes coefficients, and i is an arbitrary integer. 
Preferably, each of said at least two hologram elements has the following 
phase transfer function .phi.(x, y): 
EQU .phi.(x, y)=(2.pi./.lambda.).multidot..SIGMA.c.sub.i r.sup.2i 
+sin(.alpha.)y! 
where x denotes a position in a main scanning direction of a hologram, y 
denotes a position in a direction perpendicular to the main scanning 
direction, r denotes a distance from a center of the hologram, namely 
(x.sup.2 +y.sup.2).sup.0.5, .lambda. is a wavelength of the laser beam, 
c.sub.i denotes coefficients, i is an arbitrary integer, and a denotes an 
arbitrary angle; and 
wherein signs of terms sin(.alpha.) in the phase transfer functions of said 
at least two hologram elements are different from each other.

DESCRIPTION OF THE PREFERRED EMBODIMENTS 
FIG. 3A is a perspective view of the basic structure of an optical scanning 
system according to the present invention. The optical scanning system 
shown in FIG. 3A includes a laser source 1, a collimating lens 2, a 
deflector 3, a front-stage (first stage) hologram element 4, and a 
rear-stage (second stage) hologram element 5. The laser source 1 is formed 
with, for example, a semiconductor laser, and emits a laser beam. The 
collimating lens 2 collimates the laser beam emitted from the laser source 
1. The deflector 3 is formed with, for example, a polygon mirror, and 
deflects the laser beam from the collimating lens 2. The first-stage 
hologram element 4 diffracts the laser beam deflected by the deflector 3. 
The second-stage hologram element 5 diffracts the laser beam diffracted by 
the first-stage hologram element 4, and irradiates the diffracted laser 
beam onto an image forming plane 6. 
As shown in FIG. 3B, the first-stage hologram element 4 and the 
second-stage hologram element 5 respectively have a phase transfer 
function .phi.(x, y) defined from interference fringes formed by the 
object light and the reference light. More particularly, each of the 
first-stage hologram element 4 and the second-stage hologram element 5 can 
be mathematically expressed by the phase transfer function .phi.(x, y) 
written as follows: 
EQU .phi.(x, y)=(2.pi./.lambda.).multidot..SIGMA.c.sub.i r.sup.2i ! 
where: 
x denotes the position in the main scanning direction of the hologram; 
y denotes the position in the direction perpendicular to the main scanning 
direction; 
r denotes the distance from the center of the hologram and is equal to 
(x.sup.2 +y.sup.2).sup.0.5 ; 
.lambda. denotes the wavelength of the laser beam; 
c.sub.i is a constant; 
i denotes positive integers; and 
.SIGMA. denotes the sum of items with respect to i. 
Each of the first-stage hologram element 4 and the second-stage hologram 
element 5 can also be mathematically expressed by the phase transfer 
function .phi.(x, y) written as follows: 
EQU .phi.(x, y)=(2.pi./.lambda.).multidot.C.SIGMA.c.sub.i 
r.sup.2i)+sin(.alpha.)y! 
where: 
.alpha. denotes an arbitrary angle. 
According to the present invention, the phase transfer functions .phi.(x, 
y) of the first-stage hologram element 4 and the second-stage hologram 
element 5 are expanded with even powers of r. That is, the phase transfer 
functions .phi.(x, y) are rotationally symmetrical about the optical axis. 
Coma aberration does not take place in the waveform plane reproduced by the 
holograms having the above-mentioned phase transfer functions .phi.(x, y), 
so that the beam spot size can be reduced. Further, the phase transfer 
functions .phi.(x, y) of the hologram elements 4 and 5 are symmetrical 
with respect to the main scanning direction x, so that a uniform linear 
scan becomes possible. The uniform linear scan can be achieved by 
determining the values of the coefficients c.sub.i using an algorithm of 
the attenuation least square method. 
It is preferable that the first-stage hologram element 4 and the 
second-stage hologram element 5 have the mutually different signs of the 
term sin(.alpha.). When the phase transfer function .phi.(x, y) contains 
the term "sin(.alpha.)y", an off-axis hologram is formed. In this case, 
the scanning line is curved. In order to cancel curvature of the scanning 
line, the signs of the term sin(.alpha.) of the first-stage hologram 
element 4 and the second-stage hologram element 5 are selected so as to be 
opposite to each other. 
Further, the first-stage hologram element 4 and the second-stage hologram 
element 5 can be formed so that the expressions .SIGMA. of the phase 
transfer functions .phi.(x, y) contain one or a plurality of products of 
even powers of x multiplied by the coefficient values and odd powers of y 
multiplied by the above coefficient values, or one or a plurality of 
products of even powers of x multiplied by the coefficient values and even 
powers of y multiplied by the above coefficient values. The beam 
aberration can be reduced by determining the above coefficient values by 
the attenuation least square algorithm. 
Furthermore, there is a case where the phase characteristic of the phase 
transfer function .phi.(x, y) of either the first-stage hologram element 4 
or the second-stage hologram element 5 indicates convergence (convex lens 
function), and the phase characteristic of the other hologram element 
indicates divergence (concave lens function). In this case, it is 
preferable that the phase transfer function .phi.(x, y) of the first-stage 
hologram element 4 indicates a convergent phase characteristic and the 
phase transfer function .phi.(x, y) of the second-stage hologram element 5 
indicates a divergent phase characteristic. With the combination of the 
two different lens functions, it becomes possible to further reduce the 
beam aberration. 
Moreover, according to the present invention, it becomes possible to cancel 
a jitter due to a variation in the wavelength of the laser beam by the 
combination of two holograms, that is, the front-stage hologram element 4 
and the second-stage hologram element 5. Hence, the deflector 3 can be 
formed with a highly rotatable deflector such as a polygon mirror, whereby 
a high-speed optical scanning system can be achieved. 
A description will now be given, with reference to FIG. 4, of a first 
embodiment of the present invention. In FIG. 4, parts that are the same as 
those shown in FIG. 3A are given the same reference numbers as previously. 
An optical scanning system shown in FIG. 4 is made up of the laser diode 
1, the collimating lens 2, the polygon mirror 3, the first-stage hologram 
element 4, the second-stage hologram element 5, a flat plane/convex 
spherical lens 7, and a transparent glass plate 8. The flat plane/convex 
spherical lens 7 has a flat plane receiving the laser beam emitted from 
the laser diode 1 and deflected by the polygon mirror 3, and a convex 
spherical surface opposite to the flat plane. The first-stage hologram 
element 4 is attached to a first surface of the glass plate 8 and the 
second-stage hologram element 5 is attached to a second surface thereof 
opposite to the first surface. The laser beam outgoing from the 
second-stage hologram element 5 is irradiated on a photosensitive member 
6a of, for example, a drum shape. 
The flat plane/convex spherical lens 7 is used to reduce the distance 
between the polygon mirror 3 and the photosensitive member 6a. The glass 
plate 8 is used to support the first-stage hologram element 4 and the 
second-stage hologram element 5 by a single base and to achieve compact 
mounting of these two hologram elements 4 and 5. It is possible to 
assemble a less-expensive unit of the first-stage hologram element 4, the 
glass plate 8 and the second-stage hologram element 5 by injection using a 
plastic material. 
It will now be assumed that the phase transfer function .phi.(x, y) of the 
first-stage hologram element 4 shown in FIG. 4 is mathematically expressed 
as follows: 
##EQU1## 
The parameters x, y and .lambda. are the same as defined previously. The 
above mathematical expression can be rewritten as follows: 
The following will now be assumed: 
EQU c.sub.1 =c.sub.2 =c.sub.4 =c.sub.6 =c.sub.8 =c.sub.11 =c.sub.13 =c.sub.15 
=c.sub.17 =c.sub.19 =0. 
In this case, the above mathematical expression can be rewritten as 
follows: 
##EQU2## 
Similarly, the phase transfer function .phi.(x, y) will be assumed as 
follows: 
##EQU3## 
The phase transfer functions .phi.(x, y) of the first-stage hologram 
element 4 and the second-stage hologram element 5 thus assumed have the 
following basic formulation: 
EQU .phi.(x, y)=(2.pi./-80 ).multidot..alpha..sub.1 (x.sup.2 
+y.sup.2)+.alpha..sub.2 (x.sup.2 +y.sup.2).sup.2 ! 
and the following products of even powers of x and the powers of y are 
added to the above basic formulation: "x.sup.2 y", "y.sup.3 ", "x.sup.4 
y", "x.sup.2 y.sup.3 " and "y.sup.5 ". The asymmetrical terms in the y 
direction that does not much affect the uniform linear scan are positively 
added in order to reduce the beam aberration. 
By optimizing the coefficient values of the above phase transfer function 
.phi.(x, y), it becomes possible to construct an optical scanning system 
having the high-performance laser scanning function. This optimizing 
process can be the attenuation least square algorithm (normally called 
DLS) used for the lens design. In the optimizing process, the coefficient 
values are updated so that the function values of the phase transfer 
function become more appropriate than previously. Then, ray tracing is 
performed by using the phase transfer function .phi.(x, y) having the 
coefficient values thus determined in order to evaluate speed uniformity, 
linearity, waveform jitters and beam aberration. Until satisfactory 
evaluation is obtained, the updating process for the coefficient values is 
repeatedly performed. 
FIGS. 5A and 5B show examples of the coefficient values c.sub.i and d.sub.i 
of the above-mentioned phase transfer function .phi.(x, y) determined 
according to the attenuation least square algorithm under the following 
conditions in the structure shown in FIG. 4: 
wavelength of laser beam 785 nm 
radius of curvature of lens 7 100 mm 
distance between mirror 3 and lens 7 60 mm 
distance between lens 7 and element 4 30 mm 
distance between elements 4 and 5 5 mm 
distance between element 5 and drum 6a 200 mm 
In FIGS. 5A and 5B, ab! denotes e.sup.ab. 
FIG. 6 is a graph of the speed uniformity (constant velocity performance) 
of the optical scanning system realized by the coefficient values shown in 
FIGS. 5A and 5B. FIG. 7 is a graph of simulation data of a jitter caused 
in the optical scanning system realized by the coefficient values shown in 
FIGS. 5A and 5B obtained when the wavelength of the laser beam varies by 2 
nm. FIG. 8 is a spot diagram showing simulation data of the beam 
aberration taking place in the optical scanning system realized by the 
coefficient values shown in FIGS. 5A and 5B. 
It can be seen from the graph of FIG. 6 that the optical scanning system 
realized by the coefficient values shown in FIGS. 4A and 4B has a 
satisfactory speed uniformity in which the scanning speed obtained at 
either end of the drum 6a increases by 0.5% with respect to the reference 
scanning speed obtained at the reference center position of the drum 6a. 
It can be seen from the graph of FIG. 7 that the optical scanning system 
realized by the coefficient values shown in FIGS. 4A and 4B has a 
satisfactory jitter characteristic in which a jitter of approximately 
.+-.10 .mu.m in the x direction occurs at either end of the drum 6a with 
respect to the reference center position of the drum 6a. The simulation 
data assumes the in-line type hologram, and hence there is no jitter in 
the y direction. 
It can be seen from FIG. 8 that the optical scanning system realized by the 
coefficient values shown in FIGS. 4A and 4B has a satisfactory beam 
aberration characteristic in which a beam aberration of about 50 .mu.m 
occurs in either end of the drum 6a with respect to the reference center 
position of the drum 6a. The above beam aberration value can achieve a 
resolution of 300 dpi requiring a beam spot size of 80 .mu.m. Further, it 
can be seen from FIG. 8 that the optical scanning system realized by the 
coefficient values shown in FIGS. 4A and 4B can perform sufficient line 
scanning. 
Consequently, it is confirmed from the simulation data shown in FIGS. 5 
through FIG. 7 that the optical scanning system with the coefficient 
values shown in FIGS. 4A and 4B can realize highly-precise, high-speed 
laser scanning. 
In the phase transfer function .phi.(x, y) assumed in the above 
description, the asymmetrical terms related to the y direction are added 
in order to reduce the beam aberration. In order to further reduce the 
beam aberration, it is preferable that one or a plurality of terms of the 
products of even powers of x and even powers of y, namely "cx.sup.2a 
y.sup.2b " are added where c is a coefficient, and a and b are positive 
integers. 
In the embodiment shown in FIG. 3, the flat plane/convex spherical lens 7 
is provided between the polygon mirror 3 and the first-stage hologram 
element 4. Alternatively, it is possible to omit the lens 7, as shown in 
FIG. 9. In the structure shown in FIG. 9, the first-stage hologram element 
4 and the second-stage hologram element 5 are attached to the opposite 
surfaces of the glass plate 8. Alternatively, it is possible to omit the 
glass plate 8, as shown in FIG. 10. In the structure shown in FIG. 10, a 
cylindrical lens 9 is provided between the collimating lens 2 and the 
polygon mirror 3 in order to prevent a variation in the image point 
position with respect to a variation in the angle of the mirror surface 
with respect to the incident laser beam. The cylindrical lens 9 functions 
to gather the laser beam in the vertical direction. 
Instead of the polygon mirror 3 shown in FIG. 10, a sine-wave vibration 
mirror 10 can be used, as shown in FIG. 11. 
The phase transfer function .phi.(x, y) written by the above-mentioned 
mathematical expression assumes an in-line type hologram, as shown in FIG. 
12, in which the direction of the optical axis of the optical system is 
not affected by holograms. However, in practice, it may be difficult to 
form the in-line type hologram and eliminate the oth-order light after 
diffraction. With the above in mind, an off-axis type hologram as shown in 
FIG. 13 can be used in which the direction of the optical axis of the 
optical system is affected by holograms. The off-axis type hologram has a 
phase transfer function .phi.(x, y) in which the term 
"(2.pi./.lambda.).multidot.sin(.alpha.)y" is added to the phase transfer 
function of the in-line type hologram. That is, the phase transfer 
function .phi.(x, y) of the in-line type hologram has the following 
mathematical expression: 
EQU .phi.(x, y)=(2.pi./.lambda.).multidot.(ax.sup.2 +by.sup.2) 
while the phase transfer function .phi.(x, y) of the off-axis type hologram 
has the following mathematical expression: 
EQU .phi.(x, y)=(2.pi./.lambda.).multidot.(ax.sup.2 +by.sup.2 +sin(.alpha.)y) 
From the above expression, the spatial frequency of the in-line type 
hologram with respect to the y direction is written as follows: 
EQU (1/2.pi.).multidot.(.delta./.delta.y).multidot.{(2.pi./.lambda.).multidot.( 
ax.sup.2 +by.sup.2)} 
Hence, the above expression is rewritten as follows: 
EQU (1/.lambda.).multidot.2by 
The spatial frequency of the off-axis type hologram with respect to the y 
direction is written as follows: 
EQU (1/2.pi.).multidot.(.delta./.delta.y).multidot.{(2.pi./.lambda.).multidot.( 
ax.sup.2 +by.sup.2 +sin(.alpha.)y)} 
Hence, the above expression is rewritten as follows: 
EQU (1/.lambda.).multidot.(2by+sin(.alpha.)} 
It can be seen from the above that the optical scanning system using the 
in-line type hologram can achieve linear scan as shown in FIG. 14, while 
the optical scanning system using the off-axis type hologram makes a 
curved scan as shown in FIG. 15. 
In order to eliminate the curved-scan problem, when the first-stage 
hologram element 4 has an off-axis hologram, the second-stage hologram 
element 5 has an off-axis hologram, so that the terms sin(.alpha.) of 
these two holograms have different signs and have adjusted values so that 
the curved scans generated in the two holograms are canceled, whereby 
linear scanning as shown in FIG. 16 can be achieved. When the above 
structure is employed, the optical-axis center of the first-stage hologram 
element 4 and the optical-axis center of the second-stage hologram element 
5 are not located on the same plane. 
FIG. 17 shows an example of simulation data of the beam aberration of the 
present invention optical scanning system in which the first-stage 
hologram element 4 and the second-stage hologram element 5 have off-axis 
holograms. In the simulation performed in order to obtain the simulation 
data shown in FIG. 17, a plurality of terms of products of even powers of 
x and odd powers of y are added to the phase transfer function .phi.(x, y) 
in order to reduce the beam aberration. It can be seen from the simulation 
data shown in FIG. 17 that the sufficiently reduced beam aberration in 
terms of practical use can be obtained. 
Holograms are categorized into a convergency type hologram in which the 
phase transfer function .phi.(x, y) have negative values in the peripheral 
portions of the holograms or a divergency type in which the phase transfer 
function .phi.(x, y) have positive values in the peripheral portions of 
the holograms. The convergency type holograms have the convex lens 
function, and the divergency type holograms have the concave lens 
function. 
With the above in mind, according to the present invention, in order to 
further reduce the size of the beam spot, the coefficient values of the 
phase transfer functions .phi.(x, y) of the hologram elements 4 and 5 are 
determined so that when the phase characteristic of the phase transfer 
function .phi.(x, y) of the first-stage hologram element 4 indicates 
convergency, the phase characteristic of the phase transfer function 
.phi.(x, y) of the second-stage hologram element 5 indicates divergency. 
Alternatively, the coefficient values of the phase transfer functions 
.phi.(x, y) of the hologram elements 4 and 5 are determined so that when 
the phase characteristic of the phase transfer function .phi.(x, y) of the 
first-stage hologram element 4 indicates divergency, the phase 
characteristic of the phase transfer function .phi.(x, y) of the 
second-stage hologram element 5 indicates convergency. 
It has been confirmed from simulation data that less beam aberration can be 
obtained when the phase characteristic of the phase transfer function 
.phi.(x, y) of the first-stage hologram element 4 indicates convergency, 
and the phase characteristic of the phase transfer function .phi.(x, y) of 
the second-stage hologram element 5 indicates divergency. 
As described above, when the distance from the center of the hologram is 
expressed with r, namely (x.sup.2 +y.sup.2).sup.0.5, the present invention 
has the basic structure in which the phase transfer functions .phi.(x, y) 
of the first-stage hologram element 4 and the second-stage hologram 
element 5 are expanded with even powers of r. Then, the coefficient values 
of the above phase transfer functions of the phase transfer functions 
.phi.(x, y) are determined so that the constant-velocity linear scan, 
cancellation of waveform jitters and reduced beam aberration can be 
achieved. 
As a result, it becomes possible to use the high-revolution polygon mirror 
located in front of the first-stage hologram element 4 and to achieve 
high-speed laser scanning. However, the present invention is not limited 
to use of the polygon mirror but can use a hologram disk 11, as shown in 
FIG. 18. When the hologram disk 11 is used, the coefficient values of the 
phase transfer functions .phi.(x, y) of the hologram disk 11, the 
first-stage hologram element 4, and the second-stage hologram element 5 
are optimized in order to achieve the constant-velocity linear scan, 
cancellation of waveform jitters, and reduced beam aberration. 
The present invention is not limited to the specifically disclosed 
embodiment. For example, the present invention is not limited to the 
optical scanning system having the coefficient values shown in FIGS. 4A 
and 4B. 
According to the present invention, it becomes possible to provide an 
optical scanning system capable of performing highly-precise, high-speed 
laser scanning. The present invention does not need any f-.theta. lens 
system, and a compact and less-expensive optical scanning system can be 
provided and practically used.