Synchronizing apparatus of a cascade scanning optical system having tilting measurement of reflecting surfaces

A cascade scanning optical system which includes: a first laser scanning optical system having a first polygon mirror, provided with a plurality of first reflecting surfaces, for deflecting a first scanning laser beam to scan a part of a scanning surface to generate a first scanning line; a second laser scanning optical system having a second polygon mirror, provided with a plurality of second reflecting surfaces, for deflecting a second scanning laser beam to scan another part of the scanning surface to generate a second scanning line, wherein the first and second laser scanning optical systems are arranged so as to align the first scanning line with the second scanning line at a point of contact therebetween in a main scanning direction to form a single scanning line; means for measuring a degree of tilt of each of the plurality of first reflecting surfaces and the plurality of second reflecting surfaces; and means for determining combinations of the plurality of first reflecting surfaces with the plurality of second reflecting surfaces in accordance with results of measurements of the measuring means so that the single scanning line is formed by any one of the combinations while minimizing a phase difference between a first phase formed by degrees of tilt of the plurality of first reflecting surfaces and a second phase formed by degrees of tilt of the plurality of the second reflecting surfaces.

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
The present invention relates to a cascade scanning optical system having a 
pair of laser scanning optical systems which are arranged along the main 
scanning direction and controlled to operate in synchronization with each 
other so as to realize a wide scanning line. More specifically the present 
invention relates to an apparatus of such a cascade scanning optical 
system, having a pair of laser scanning optical systems, for synchronizing 
the rotation of a polygon mirror of one laser scanning optical system with 
the rotation of a polygon mirror of the other laser scanning optical 
system, to prevent a pair of scanning lines that are to be aligned, 
respectively generated by the pair of laser scanning optical systems, from 
being deviated from each other in the sub-scanning direction. 
2. Description of the Related Art 
A cascade scanning optical system having a plurality of laser scanning 
optical systems arranged along the main scanning direction to realize a 
wide scanning line is known. Such a type of scanning optical system is 
disclosed in Japanese Laid-Open Patent Publication No. 61-11720, published 
on Jan. 20, 1986. This publication discloses a cascade scanning optical 
system having a pair of laser scanning optical systems each having a laser 
beam emitter, a polygon mirror serving as a deflecting device, an f.theta. 
lens, etc. The pair of laser scanning optical systems are synchronously 
driven to emit respective scanning laser beams to a photoconductive 
surface (scanning surface) of a photoconductive drum on a common line 
thereon extending in parallel to the axial direction of the 
photoconductive drum. The pair of scanning laser beams respectively scan 
two adjacent ranges of the common line on the photoconductive surface so 
as to scan the photoconductive surface of the photoconductive drum in the 
main scanning direction in a wide range. 
There is a fundamental problem to be overcome in such a type of cascade 
scanning optical system. Namely, how can a scanning line, made on the 
photoconductive drum by the scanning laser beam emitted from one laser 
scanning optical system of the cascade scanning optical system, be 
accurately aligned with another scanning line, made on the photoconductive 
drum by the scanning laser beam emitted from another laser scanning 
optical system of the cascade scanning optical system, so that the 
scanning lines are not apart from each other in either the main scanning 
direction or the sub-scanning direction, i.e., so as to form a straight 
and continuous scanning line through the combination of the separate 
scanning lines. 
It is sometimes the case that each reflecting surface (scanning laser beam 
deflecting surface) of a polygon mirror used in the cascade scanning 
optical system slightly tilts from its original position. In the case 
where the angle of each reflecting surface of the polygon mirror of one 
laser scanning optical system is different from that of the other 
corresponding laser scanning optical system, the pair of scanning lines, 
which are respectively generated by the aforementioned corresponding 
reflecting surfaces forming a straight and continuous scanning line, will 
deviate from each other in the sub-scanning direction on the 
photoconductive drum. This results in a gap or deviation occurring between 
the two scanning lines in the sub-scanning direction, so that a straight 
and continuous scanning line will not be formed. A similar problem will 
arise in the case where one or both of the polygon mirrors rotate with a 
tremor or oscillation. 
SUMMARY OF THE INVENTION 
The primary object of the present invention is to provide a synchronizing 
apparatus of a cascade scanning optical system which can prevent a 
scanning line, made by the scanning laser beam emitted from one laser 
scanning optical system, and another scanning line, made by the scanning 
laser beam emitted from the other laser scanning optical system, from far 
deviating from each other in the sub-scanning direction on a scanning 
surface. 
To achieve the object mentioned above, according to an aspect of the 
present invention, there is provided a cascade scanning optical system 
which includes: a first laser scanning optical system having a first 
polygon mirror, provided with a plurality of first reflecting surfaces, 
for deflecting a first scanning laser beam to scan a part of a scanning 
surface to generate a first scanning line; a second laser scanning optical 
system having a second polygon mirror, provided with a plurality of second 
reflecting surfaces, for deflecting a second scanning laser beam to scan 
another part of the scanning surface to generate a second scanning line, 
wherein the first and second laser scanning optical systems are arranged 
so as to align the first scanning line with the second scanning line at a 
point of contact therebetween in a main scanning direction to form a 
single scanning line; means for measuring a degree of tilt of each of the 
plurality of first reflecting surfaces and the plurality of second 
reflecting surfaces; and means for determining combinations of the 
plurality of first reflecting surfaces with the plurality of second 
reflecting surfaces in accordance with results of measurements of the 
measuring means so that the single scanning line is formed by any one of 
the combinations while minimizing a phase difference between a first phase 
formed by degrees of tilt of the plurality of first reflecting surfaces 
and a second phase formed by degrees of tilt of the plurality of the 
second reflecting surfaces. 
Preferably, the determining means includes means for comparing the degrees 
of tilt of the plurality of first reflecting surfaces with the degrees of 
tilt of the plurality of the second reflecting surfaces to judge which 
reflecting surface of the plurality of first reflecting surfaces has the 
closest degree of tilt to a reflecting surface of the plurality of the 
second reflecting surfaces. 
Preferably, the measuring means includes: a first position sensitive device 
for detecting a position of the first scanning laser beam in a 
sub-scanning direction perpendicular to the main scanning direction to 
determine the degree of tilt of each of the plurality of first reflecting 
surfaces; and a second position sensitive device for detecting a position 
of the second scanning laser beam in the sub-scanning direction to 
determine the degree of tilt of each of the plurality of second reflecting 
surfaces. 
Preferably, the first position sensitive device is positioned outside a 
first optical path through which the first scanning laser beam passes to 
form the first scanning line, and wherein the second position sensitive 
device is positioned outside a second optical path through which the 
second scanning laser beam passes to form the second scanning line. 
Preferably, the cascade scanning optical system further includes means for 
storing the degree of tilt of each of the plurality of first reflecting 
surfaces and the plurality of second reflecting surfaces. 
Preferably, the storing means includes: a first memory for storing the 
degree of tilt of each of the plurality of first reflecting surfaces; and 
a second memory for storing the degree of tilt of each of the plurality of 
second reflecting surfaces. 
Preferably, the determining means includes means for comparing the degrees 
of tilt of the plurality of first reflecting surfaces which are stored in 
the first memory with the degrees of tilt of the plurality of the second 
reflecting surfaces which are stored in the second memory to judge which 
reflecting surface of the plurality of first reflecting surfaces has the 
closest degree of tilt to a reflecting surface of the plurality of the 
second reflecting surfaces so as to determine the combinations. 
Preferably, the measuring means and the determining means each start 
operating each time a power switch of the cascade scanning optical system 
is turned ON. 
Preferably, the cascade scanning optical system further includes a drum 
having the scanning surface on a periphery of the drum. 
Preferably, the first and second laser scanning optical systems are 
composed of the same optical elements. 
Preferably, the first and second laser scanning optical systems are 
symmetrically arranged. 
According to another aspect of the present invention, there is provided a 
synchronizing apparatus of a cascade scanning optical system, the cascade 
scanning optical system including a pair of laser scanning optical systems 
each having a polygon mirror provided with a plurality of reflecting 
surfaces, the pair of laser scanning optical systems being arranged to 
form a single scanning line, wherein the synchronizing apparatus includes: 
means for measuring a degree of tilt of each of the plurality of 
reflecting surfaces of the polygon mirrors; and means for determining 
combinations of the plurality of reflecting surfaces of one of the polygon 
mirrors with the plurality of reflecting surfaces of the other of the 
polygon mirrors in accordance with results of measurements of the 
measuring means so that the single scanning line is formed by any one of 
the combinations while minimizing a phase difference between a first phase 
formed by degrees of tilt of the plurality of reflecting surfaces of the 
one of the polygon mirrors and a second phase formed by degrees of tilt of 
the plurality of the reflecting surfaces of the other of the polygon 
mirrors. 
The present disclosure relates to subject matter contained in Japanese 
Patent Application No. 8-348106 (filed on Dec. 26, 1996) which is 
expressly incorporated herein by reference in its entirety.

DESCRIPTION OF THE PREFERRED EMBODIMENT 
FIG. 1 shows an embodiment of a cascade scanning optical system for 
scanning the photoconductive surface of a photoconductive drum (rotating 
member) 10 provided in a laser-beam printer. The cascade scanning optical 
system is provided with a pair of laser scanning optical systems, i.e., a 
first scanning optical system 20A and a second scanning optical system 
20B. The first and second optical systems 20A and 20B are each designed as 
a non-telecentric system, so that the incident angle of a scanning laser 
beam emitted from each of the first and second optical systems 20A and 20B 
relative to the photoconductive surface of the drum 10 varies in 
accordance with a variation in the position of the scanning spot of the 
scanning laser beam on the photoconductive surface in the main scanning 
direction. The first and second scanning optical systems 20A and 20B are 
provided with the same optical elements or parts, that is, the first 
scanning optical system 20A is provided with a laser collimating unit 21A 
serving as a laser beam emitter, a cylindrical lens 23A, a polygon mirror 
(scanning laser beam deflector) 24A, an f.theta. lens group 25A, an 
auxiliary lens 26A and a mirror 27A, while the second scanning optical 
system 20B is provided with a laser collimating unit 21B serving as a 
laser beam emitter, a cylindrical lens 23B, a polygon mirror (scanning 
laser beam deflector) 24B, an f.theta. lens group 25B, an auxiliary lens 
26B and a mirror 27B. Each of the f.theta. lens groups 25A and 25B 
consists of two lens elements as can be seen from FIG. 1. The first and 
second scanning optical systems 20A and 20B are arranged side by side in a 
direction parallel to the axial direction of the drum 10 and are supported 
by a common casing 35 on an inner flat surface thereof. 
The laser collimating units 21A and 21B are identical. Each of the laser 
collimating units 21A and 21B is provided therein with a laser diode LD 
and a collimating lens group (not shown) for collimating a laser beam 
emitted from the laser diode LD. In each of the first and second scanning 
optical systems 20A and 20B, the laser beam emitted from the laser diode 
LD is collimated through the collimating lens group. Thereafter this 
collimated laser beam is incident upon the cylindrical lens 23A or 23B 
positioned in front of the corresponding laser collimating unit 21A or 
21B. Each cylindrical lens 23A or 23B has a power in the sub-scanning 
direction, so that the spot of the laser beam incident thereon is 
elongated therethrough in the main scanning direction to be incident upon 
the corresponding polygon mirror 24A or 24B. The polygon mirrors 24A and 
24B are each rotated, so that laser beams incident thereon are deflected 
in the main scanning direction to proceed toward the mirrors 27A and 27B 
through the f.theta. lens groups 25A and 25B and the auxiliary lenses 26A 
and 26B, respectively. Subsequently, the laser beams incident upon the 
mirrors 27A and 27B are reflected thereby towards the photoconductive drum 
10, to thereby scan the same in the main scanning direction. 
Each of the auxiliary lenses 26A and 26B has a power mainly in the 
sub-scanning direction. In order to reduce the size of the cascade 
scanning optical system, it is possible to omit each of the auxiliary 
lenses 26A and 26B. In such a case, the design of the f.theta. lens groups 
25A and 25B would be modified in such a way that they would have the 
equivalent power to that of the combined power of the original f.theta. 
lens groups 25A and 25B and the auxiliary lenses 26A and 26B, 
respectively. In FIG. 2, "X" represents an optical axis of the f.theta. 
lens group 25A or 25B. The optical axis X extends perpendicular to the 
main scanning direction. "S" represents the photoconductive surface of the 
drum 10. The auxiliary lenses 26A and 26B and the mirrors 27A and 27B are 
not illustrated in either FIG. 2 or 4. 
The polygon mirror 24A rotates in a clockwise direction while the polygon 
mirror 24B rotates in a counterclockwise direction, as viewed in FIG. 2. 
Namely, the polygon mirrors 24A and 24B rotate in opposite rotational 
directions to scan the photoconductive surface of the drum 10 from its 
approximate center toward respective opposite ends in opposite directions. 
A mirror 28A is fixedly provided in the casing 35 at a position to receive 
the scanning laser beam emitted from the f.theta. lens group 25A before 
the scanning laser beam is incident on the photoconductive surface of the 
drum 10 through the auxiliary lens 26A and the mirror 27A at each scanning 
sweep while the polygon mirror 24A rotates. The laser beam reflected by 
the mirror 28A is incident on a laser beam detector (BD) 29A fixedly 
provided in the casing 35 at a position opposite to the mirror 28A. 
Likewise, a mirror 28B is fixedly provided in the casing 35 at a position 
to receive the scanning laser beam emitted from the f.theta. lens group 
25B before the scanning laser beam is incident on the photoconductive 
surface of the drum 10 through the auxiliary lens 26B and the mirror 27B 
at each scanning sweep while the polygon mirror 24B rotates. The laser 
beam reflected by the mirror 28B is incident on a laser beam detector (BD) 
29B fixedly provided in the casing 35 at a position opposite to the mirror 
28B. 
The laser diodes LD of the laser collimating units 21A and 21B are each 
controlled to turn its laser emission ON or OFF in accordance with given 
image data to draw a corresponding image (charge-latent image) on the 
photoconductive surface of the drum 10, and subsequently this image drawn 
on the photoconductive surface of the drum 10 is transferred to plain 
paper according to a conventional electrophotographic method. The polygon 
mirrors 24A and 24B are controlled synchronously with the use of the laser 
beam detectors 29A and 29B such that on the photoconductive surface of the 
drum 10 the scanning starting point of a spot of the scanning laser beam 
emitted from the first scanning optical system 20A is properly and 
precisely adjacent to the scanning starting point of a spot of the 
scanning laser beam emitted from the second scanning optical system 20B, 
and that those two spots move in opposite directions apart from each other 
in the main scanning direction to thereby form a wide scanning line on the 
photoconductive surface of the drum 10. With the rotational movement of 
the photoconductive drum 10 which is synchronized to the rotational 
movement of each of the polygon mirrors 24A and 24B, a series of wide 
scanning lines are made on the photoconductive surface of the drum 10 to 
thereby obtain a certain image (charge-latent image) on the 
photoconductive surface of the drum 10. 
The polygon mirror 24A has a regular hexagonal cross section and is 
provided along a circumference thereof with six reflecting surfaces 
(scanning laser beam deflecting surfaces) A, B, C, D, E and F. Likewise, 
the polygon mirror 24B has a regular hexagonal cross section and is 
provided along a circumference thereof with corresponding six reflecting 
surfaces (scanning laser beam deflecting surfaces) a, b, c, d, e and f. In 
either polygon mirror 24A or 24B, there is a possibility of each 
reflecting surface tilting from its original position. Such tilt causes 
the position of the spot of the corresponding scanning laser beam to 
deviate on the photoconductive surface in the sub-scanning direction. In 
the case where the degree (amount) of tilt of one reflecting surface of 
the polygon mirror 24A is different from that of a corresponding 
reflecting surface of the polygon mirror 24B, opposing ends of two 
scanning lines to be combined which are formed by a pair of scanning laser 
beams on the photoconductive surface will be apart from each other in the 
sub-scanning direction. With a synchronizing apparatus which will be 
hereinafter discussed such a problem of deviation of opposing ends of the 
two scanning lines in the sub-scanning direction is effectively prevented 
from occurring. 
A first PSD (semiconductor position sensitive device) 31A is fixedly 
provided in the casing 35 at a position in the vicinity of the laser beam 
detector 29A to receive the scanning laser beam emitted from the f.theta. 
lens group 25A after the scanning laser beam has completed a single 
scanning at each scanning sweep while the polygon mirror 24A rotates. 
Likewise, a second PSD (semiconductor position sensitive device) 31B is 
fixedly provided in the casing 35 at a position in the vicinity of the 
laser beam detector 29B to receive a laser beam emitted from the f.theta. 
lens group 25B after the scanning laser beam has completed a single 
scanning at each scanning sweep while the polygon mirror 24B rotates. Each 
PSD 31A, 31B detects the position of a scanning laser beam received, 
emitted from the corresponding polygon mirror 24A or 24B, in the 
sub-scanning direction so as to determine the degree of tilt of each 
reflecting surface of the corresponding polygon mirror 24A or 24B. FIG. 3A 
is a graph showing the degree of tilt of each reflecting surface (A, B, C, 
D, E and F) of the polygon mirror 24A while FIG. 3B is a graph showing the 
degree of tilt of each reflecting surface (a, b, c, d, e and f) of the 
polygon mirror 24B, in an example. As can be seen from FIGS. 3A and 3B, in 
either polygon mirror 24A or 24B the degree of tilt periodically varies to 
substantially form a sine curve. In the present embodiment a point at 
which the phases of the two sine curves coincides with each other most is 
determined to synchronize the rotation of the polygon mirror 24A with the 
rotation of the polygon mirror 24B so as to form a wide scanning line on 
the photoconductive surface of the drum 10 by a corresponding pair 
(determined pair) of reflecting surfaces of the polygon mirrors 24A and 
24B. In the illustrated particular example shown in FIGS. 3A and 3B, a 
deviation of a pair of scanning lines respectively generated by the 
polygon mirrors 24A and 24B on the photoconductive surface in the 
sub-scanning direction will be greatly reduced or substantially eliminated 
if the rotation of the polygon mirror 24A is synchronized with the 
rotation of the polygon mirror 24B with the reflecting surface `A` of the 
polygon mirror 24A coincident with the reflecting surface `e` of the 
polygon mirror 24B, as will be appreciated from FIGS. 3A and 3B. 
FIG. 4 shows a block diagram of the synchronizing apparatus of the cascade 
scanning optical system which realizes the aforementioned synchronizing 
process. The first and second polygon mirrors 24A and 24B are rotated by 
first and second motor units 55A and 55B, respectively. When the first and 
second motor units 55A and 55B start operating upon the power switch 
turned ON, the motor units 55A and 55B are each controlled, rotating with 
common clock pulses output from a frequency divider 53 which receives 
clock pulses from a clock pulse generator 51. After the rotation of each 
motor unit 55A, 55B has become stable and the PLL (phase-lock loop) 
starts, the rotational speed of the second polygon mirror 24B, i.e., the 
rotational speed of the second motor unit 55B, is controlled in accordance 
with signals which are output from the second laser beam detector 29B each 
time the first laser beam detector 29A detects the laser beam emitted from 
the first polygon mirror 24A. 
The first laser beam detector 29A outputs a signal to both a first phase 
detecting circuit 57A and a phase-difference detector 59 at the time the 
first laser beam detector 29A detects a scanning laser beam. The second 
laser beam detector 29B outputs a signal to each of: a second phase 
detecting circuit 57B, the phase-difference detector 59, and a delay 
circuit (time-delay circuit) 81 at the time the second laser beam detector 
29B detects a scanning laser beam. The phase-difference detector 59 
determines a phase difference between the phase of signals output from the 
first laser beam detector 29A and the phase of signals output from the 
second laser beam detector 29B in accordance with the signals input from 
the first and second laser beam detectors 29A and 29B to output a phase 
difference indicating voltage to an LPF (low pass filter) 61. The terms 
"phase difference indicating voltage" herein used mean a voltage which 
indicates the magnitude of a phase difference. In the case where the phase 
of signals output from the second laser beam detector 29B follows the 
phase of signals output from the first laser beam detector 29A, the 
phase-difference detector 59 outputs a positive phase difference 
indicating voltage. Conversely, in the case where the phase of signals 
output from the second laser beam detector 29B precedes the phase of 
signals output from the first laser beam detector 29A, the 
phase-difference detector 59 outputs a negative phase difference 
indicating voltage. 
Inputting a phase difference indicating voltage output from the 
phase-difference detector 59, the LPF 61 converts the phase difference 
indicating voltage into a DC voltage corresponding to the magnitude of the 
input phase difference indicating voltage. Subsequently, the LPF 61 
outputs the DC voltage to a VCO (voltage controlled oscillator) 63. The 
VCO 63 changes the frequency of clock pulses output therefrom in 
accordance with the DC voltage input from the LPF 61 In this particular 
embodiment, the VCO 63 outputs clock pulses having a high frequency to a 
multiplexer 67 when the DC voltage input from the LPF 61 is a high 
voltage, while the VCO 63 outputs clock pulses having a low frequency to 
the multiplexer 67 when the DC voltage input from the LPF 61 is a low 
voltage. The multiplexer 67 adjusts clock pulses input from the frequency 
divider 53 in accordance with clock pulses input from the VCO 63 to output 
the adjusted clock pulses to the second motor unit 55B. Accordingly, in 
the case where the phase of signals output from the second laser beam 
detector 29B follows that of the first laser beam detector 29A, the 
rotational speed of the second motor unit 55B increases. Conversely, in 
the case where the phase of signals output from the second laser beam 
detector 29B precedes that of the first laser beam detector 29A, the 
rotational speed of the second motor unit 55B decreases. 
When detecting a scanning laser beam, each PSD 31A, 31B outputs a voltage 
corresponding to the detected position of the received scanning laser 
beam. The voltage output from the first PSD 31A is converted into digital 
signals by an A/D converter 71A to be stored in a data memory 73A as data 
(first data group) representing the degrees of tilt of the reflecting 
surfaces A, B, C, D, E and F of the first polygon mirror 24A. Similarly, 
the voltage output from the second PSD 31B is converted into digital 
signals by an A/D converter 71B to be stored in a data memory 73B as data 
(second data group) representing the degrees of tilt of the reflecting 
surfaces a, b, c, d, e and f of the second polygon mirror 24B. It is 
preferable to measure each of the aforementioned first and second data 
groups more than once and store the average values of the first data group 
and the average values of the second data group in the data memories 73A 
and 73B, respectively, so as to improve the reliability of each of the 
first and second data groups. 
After the degrees of tilt of the reflecting surfaces of the first polygon 
mirror 24A have all been stored in the data memory 73A, a 
reflecting-surface position detecting circuit 75A firstly detects the 
degree of tilt of any one of the reflecting surfaces of the first polygon 
mirror 24A, which rotates at a fixed rotational speed, by inputting a 
signal from the first PSD 31A through the A/D converter 71A. Subsequently 
the reflecting-surface position detecting circuit 75A inputs the values 
from the first data group stored in the data memory 73A and compares each 
stored degree of tilt in the first data group with the detected degree of 
tilt of the aforementioned reflecting surface of the first polygon mirror 
24A to determine which one of the reflecting surfaces A, B, C, D, E or F 
of the first polygon mirror 24A is the aforementioned reflecting surface 
of the first polygon mirror 24A. Thereafter the reflecting-surface 
position detecting circuit 75A outputs the data (first surface data) 
representing one of the reflecting surfaces A, B, C, D, E or F of the 
first polygon mirror 24A which is the aforementioned reflecting surface of 
the first polygon mirror 24A, to a motor speed controller 77. 
Likewise, after the degrees of tilt of the reflecting surfaces of the 
second polygon mirror 24B have all been stored in the data memory 73B, a 
reflecting-surface position detecting circuit 75B firstly detects the 
degree of tilt of any one of reflecting surfaces of the second polygon 
mirror 24B, which rotates at a fixed rotational speed, by inputting a 
signal from the second PSD 31B through the A/D converter 71B. Subsequently 
the reflecting-surface position detecting circuit 75B inputs the values of 
the second data group stored in the data memory 73B and compares each 
stored degree of tilt in the second data group with the detected degree of 
tilt of the aforementioned reflecting surface of the second polygon mirror 
24B to determine which one of the reflecting surfaces a, b, c, d, e or f 
of the second polygon mirror 24B is the aforementioned reflecting surface 
of the second polygon mirror 24B. Thereafter the reflecting-surface 
position detecting circuit 75B outputs the data (second surface data) 
representing one of the reflecting surfaces a, b, c, d, e or f of the 
second polygon mirror 24B which is the aforementioned reflecting surface 
of the second polygon mirror 24B, to the motor speed controller 77. 
The motor speed controller 77 compares the first surface data input from 
the reflecting-surface position detecting circuit 75A and the 
corresponding data stored in the data memory 73A which represents the 
degree of tilt of the aforementioned reflecting surface of the first 
polygon mirror 24A with the second surface data input from the 
reflecting-surface position detecting circuit 75B and the corresponding 
data stored in the data memory 73B which represents the degree of tilt of 
the aforementioned reflecting surface of the second polygon mirror 24B to 
determine a phase difference between the phase of the degrees of tilt of 
reflecting surfaces of the first polygon mirror 24A and the phase of the 
degrees of tilt of reflecting surfaces of the second polygon mirror 24B. 
Namely, it is determined which degree of tilt of the reflecting surfaces 
A, B, C, D, E or F of the first polygon mirror 24A is closest to which 
degree of tilt of the reflecting surfaces a, b, c, d, e or f of the second 
polygon mirror 24B. Thereafter, with the reflecting surfaces A, B, C, D, E 
and F of the first polygon mirror 24A regarded as reference surfaces, the 
motor speed controller 77 converts the phase difference into a voltage 
(phase difference indicating voltage) to be output to the LPF 61. 
The LPF 61 converts the voltage input from the motor speed controller 77 
into a DC voltage corresponding to the magnitude of the input voltage and 
outputs the DC voltage to the VCO 63. The VCO 63 varies the frequency of 
clock pulses output therefrom in accordance with the DC voltage input from 
the LPF 61. Due to the variation in frequency of clock pulses output from 
the VCO 63, the rotational speed of the second motor unit 55B is 
controlled to increase or decrease so as to match the phase of a sine 
curve formed by the degrees of tilt of the reflecting surfaces of the 
first polygon mirror 24A with the phase of a sine curve formed by the 
degrees of tilt of the reflecting surfaces of the second polygon mirror 
24B, so that the combinations of the reflecting surfaces A, B, C, D, E and 
F with the reflecting surfaces a, b, c, d, e and f change, i.e., the 
correspondence of each of the reflecting surfaces A, B, C, D, E and F with 
a corresponding reflecting surface a, b, c, d, e or f changes. At the time 
the motor speed controller 77 detects a condition that the data 
representing the degree of tilt of any one of the reflecting surfaces A, 
B, C, D, E and F substantially corresponds to the data representing the 
degree of tilt of a corresponding reflecting surfaces a, b, c, d, e or f 
which is currently synchronized with the aforementioned any one of the 
reflecting surfaces A, B, C, D, E and F, the motor speed controller 77 
stops outputting the voltage (phase difference indicating voltage) to the 
LPF 61 so as to maintain the current phase (correspondence of reflecting 
surfaces), which completes the synchronizing process of the present 
embodiment. Thereafter the synchronization of rotation of the first and 
second polygon mirrors 24A and 24B is maintained according to the phase 
difference indicating voltage output from the phase-difference detector 
59. 
In the illustrated particular example shown in FIGS. 3A and 3B, after the 
above synchronizing process has been completed, the reflecting surface A 
of the first polygon mirror 24A corresponds to the reflecting surface e of 
the second polygon mirror 24B. Accordingly, the motor speed controller 77 
adjusts the rotational speed of the second motor unit 55B to synchronize 
the reflecting surface A of the first polygon mirror 24A with the 
reflecting surface e of the second polygon mirror 24B. 
The motor speed controller 77 can determine the phase difference between 
the sine curve of the degrees of tilt of reflecting surfaces of the first 
polygon mirror 24A and the sine curve of the degrees of tilt of reflecting 
surfaces of the second polygon mirror 24B, using all the aforementioned 
data input from each of the data memories 73A and 73B and the 
reflecting-surface position detecting circuits 75A and 75B, in accordance 
with either one of the following two practical methods. 
[First method] 
Regarding each of the first and second polygon mirrors 24A and 24B, among 
the data representing the degrees of tilts of the six reflecting surfaces 
a specific reflecting surface whose degree of tilt is the largest is 
ranked as Level 3. The other five reflecting surfaces which follow the 
specific reflecting surface in time order are ranked Levels 4, 5, 6, 1 and 
2, respectively. In the example shown in FIG. 3A the reflecting surface B 
is ranked as Level 3. In the example shown in FIG. 3B the reflecting 
surface f is ranked as Level 3. 
Thereafter, one of the reflecting surfaces of the second polygon mirror 24B 
which is currently synchronized with the aforementioned specific 
reflecting surface of the first polygon mirror 24A whose degree of tilt is 
the largest is detected. Subsequently the Level value of the detected one 
of the reflecting surfaces of the second polygon mirror 24B is subtracted 
from the Level value of the aforementioned specific reflecting surface. 
The larger the absolute value of the result of such a subtraction is, the 
larger the phase difference is. Therefore, an amount of variation in the 
number of revolutions of the second polygon mirror 24B per a certain 
period of time can be determined based on the result of the aforementioned 
subtraction. At the same time, by knowing whether the result of the 
subtraction is a negative value or a positive value, it can be judged 
whether the phase of the sine curve representing the degrees of tilt of 
reflecting surfaces of the second polygon mirror 24B precedes or follows 
the phase of the sine curve representing the degrees of tilt of reflecting 
surfaces of the first polygon mirror 24A, i.e., whether the number of 
revolutions of the second polygon mirror 24B per a certain period of time 
should be increased or decreased can be determined. This operation is 
completed when the result of the aforementioned subtraction becomes zero 
(0). In this first method, although a specific reflecting surface whose 
degree of tilt is the largest is ranked as Level 3, a specific reflecting 
surface whose degree of tilt is the smallest may be ranked as Level 3 
(reference level). 
[Second Method] 
The value of the degree of tilt of the reflecting surface `a` is subtracted 
from the value of degree of tilt of one of the reflecting surfaces A, B, 
C, D, E and F which is currently synchronized with the reflecting surface 
`a`, and the result of that subtraction is stored in memory. Similarly, 
the value of the degree of tilt of the reflecting surface `b` is 
subtracted from the value of the degree of tilt of another one of the 
reflecting surfaces A, B, C, D, E and F which is currently synchronized 
with the reflecting surface `b`, and the result of that subtraction is 
stored in memory. A similar operation is performed for each of the 
remaining four reflecting surfaces c, d, e and f. After all the six 
results have been obtained, the number of revolution of the second polygon 
mirror 24B per a certain period of time is adjusted such that the sum of 
the absolute values of the six results will be minimal. 
The processing in either the first or second method can start to be 
performed each time the power switch of the apparatus is turned ON, i.e. 
each time the first and second motor units 55A and 55B start operating, or 
during the idle of each motor units 55A, 55B at the time a certain period 
of time elapses after the power switch of the apparatus is turned ON. 
In FIG. 4, "HSYNC 1" and "HSYNC 2" shown on the left side of the drawing 
each represent a reference signal for commencing an operation of writing 
main scanning data. A delay circuit (time-delay circuit) 81 delays the 
output signal by a specific time interval with respect to the input 
signal, so that the commencement of each scanning sweep made by the second 
scanning optical system 20B is delayed by the aforementioned specified 
time interval with respect to the commencement of each scanning sweep made 
by the first scanning optical system 20A. The data of the specified time 
interval (delay-time data) is prestored in memory 79, so that the delay 
circuit 81 inputs the delay-time data from the memory 79 and outputs the 
reference signal HSYNC 2 in accordance with the delay-time data. 
As can be understood from the foregoing, according to the present 
embodiment of the cascade scanning optical system, a deviation between a 
scanning line made by the scanning laser beam emitted from one laser 
scanning optical system and another scanning line made by the other laser 
scanning optical system, which are to be aligned to form a straight and 
continuous scanning line, can be fallen into an acceptable range of 
deviation. 
Obvious changes may be made in the specific embodiments of the present 
invention described herein, such modifications being within the spirit and 
scope of the invention claimed. It is indicated that all matter contained 
herein is illustrative and does not limit the scope of the present 
invention.