POLYGON MIRROR

Abstract: A polygon mirror has a plurality of elliptically-­ curved convex mirror surfaces (1a to 1f) functioning as light reflection surfaces, each having substantially the same shape as the maximum curvature portion of an ellipse (2) which has long axis b and short axis c. The convex mirror surfaces (1a to 1f) are uniformly arranged on the circumference of an inscribed circle whose center (O) is set on the long axis b, whose radius is Rm, and which is in contact with the maximum curvature portion of the ellipse (2). The polygon mirror (1) reflects an incident light beam while being rotated, and scans the incident light beam in the range of effec­ tive scanning length Lo on a preset scanning plane (SP) which is set at incident light beam travelling distance R from the center (O) of the inscribed circle of the polygon mirror (1).

Description: This invention relates to a rotatable polygon mirrors used in a post-objective type light deflector, according to the preamble of claim 1. Such a mirror is known from DE-A-3 022 365 and DE-A-3 641 038. Optical scanning devices used in optical systems incorporated in laser printers or the like can be divided into two main types: one is the post-objective type, in which light is focused into a beam by a focusing lens and then sent to a light deflector, and the other is the pre-objective type, in which light is first deflected by means of a light deflector and then focused by a focusing lens. Since, in the case of the first type of scanning device, the light beam is focused before reaching the deflector, the focusing lens can therefore be small and simple in construction. However, in this case, since the beam focusing spot is generally set on a curved surface, an error in the scanning linearity and the curvature of image field will increase. In the case of the latter type device, in contrast, the beam is first deflected by the light deflector, before being focused by the focusing lens, so that the beam spot can be easily set on a flat plane and therefore an error in the scanning linearity and the curvature of image field can be easily corrected. However, in this case, it is necessary to use a wide angle lens, such as an fϑ lens, as the focusing lens, in order to obtain a proper deflection angle, making the lens construction complex. As has been made clear above, each of the two types of scanning devices (post-objective type and pre-objective type) has its advantages as well as disadvantages. In the case of the conventional post objective light deflector disclosed in Japanese Patent Disclosure (Kokai) No. 61-156020, for example, an error in the curvature of the image field is corrected by use of a polygon mirror having a spherical or cylindrical convex reflection surface with a radius R of curvature, and the error in scanning linearity is corrected by a succeeding electrical correction processing (the clock frequency of the image signal is changed according to variation in the scanning speed). However, the above conventional device has the disadvantage that it is necessary to effect the electrical correction processing in order to correct the error in the scanning linearity. An object of the present invention is to provide a polygon mirror, for use with a post objective light deflector, which is capable of reducing to a minimum, any error in the scanning linearity and the curvature of image field, without effecting the electrical correction processing and enhancing the accuracy of a pattern obtained by a scanning operation. This object is attained by the features of claim 1. Each of the convex reflection surfaces of the light deflector of the present invention is defined by part of an imaginary ellipse, the major axis of which is so arranged as to pass through the rotation center of the polygon mirror. The ellipse has long and short radii b and c these being the major and minor axes, respectively, and rotation axis O, which passes along radius b and extends in a direction which intersects the major and minor axes (radii b and c) at right angles. Assume that the radius of an inscribed circle of the polygon mirror is Rm, the distance from rotation axis O to scanning plane SP is R, the effective scanning length is Lo, rotation angle ϑ of the polygon mirror is set at "0", as is also beam position y when the major axis (corresponding to long radius b) crosses scanning plane SP at right angles, and that ϑ is set at "ϑo" when beam position y is set at Lo/2. This being so, the following equations can be obtained in the range of rotation angle ϑ (-ϑo≦ϑ≦ϑo) of the polygon mirror: Assuming that a=b-Rm , then the following expressions can be obtained: -0.02Lo ≦ (R - γ)tan2(ϑ-β) - ϑ ϑo Lo ≦ 0.02Lo (error in the scanning linearity) -0.05 R < xc·cosϑ + yc·sinϑ - R ≦ 0.05R (error in the curvature of image field). This invention can be more fully understood from the following detailed description when taken in conjunction with the accompanying drawings, in which: Fig. 1 is a perspective view schematically illustrating the construction of a post objective light deflector using a polygon mirror according to one embodiment of this invention; Fig. 2 is a top plan view of the device shown in Fig. 1; Fig. 3 is a side view of the device shown in Fig. 1; Fig. 4 is a diagram showing the geometrical relation between the elliptically curved surface and rotation center of the polygon mirror and the light beam scanning plane; Fig. 5 is an enlargement of a portion of Fig. 4, showing part of the polygon mirror; Fig. 6 is a diagram showing the geometrical relation between the elliptically curved surface of the polygon mirror and the light beam scanning plane; Fig. 7 is a diagram showing part of Fig. 6, for explaining the aforementioned geometrical relation more in detail; and Figs. 8A, 8B to 16A, 16B are diagrams showing the simulation results indicating the degree of improvement obtained with regard to reducing the error in scanning linearity and the curvature of image field by using the polygon mirror according to this invention. An embodiment of this invention will now be described, with reference to the accompanying drawings. For ease of understanding, corresponding portions shown in several of the drawings are denoted by the same reference numerals, and hence, additional explanation thereof will be omitted. Fig. 1 shows the construction of a post objective light deflector using polygon mirror 1 according to one embodiment of this invention. Figs. 2 and 3 are, respectively, top and side views of Fig. 1. Referring to Figs. 1 to 3, it can be seen that laser beam LB emitted from laser diode 10 is irradiated via light collimating objective lens 12, cylindrical lens 14, and spherical lens 16, onto one of six light reflection surfaces of hexagonal polygon mirror 1, from which it is reflected and projected onto scanning plane SP of photoreceptor drum 20, which is set apart from rotation axis O of polygon mirror 1 by distance R. Motor 18 is used to rotate polygon mirror 1 so that laser beam LB reflected therefrom can be deflected within a range of ±Lo/2 with respect to the center (focusing position) CT of scanning plane SP of drum 20. With the exception of the convex shape of the reflection surface of polygon mirror 1 and the relation between polygon mirror 1 and scanning plane SP, the construction of this invention, as it is shown in Figs. 1 to 3, can adopt other forms known in this art. In Fig. 1, polygon mirror 1 is illustrated in a sliced-pineapple ring form, but it can be obtained by slicing a hexagonal hollow cylindrical body having convex side surfaces or a football-like body having six side surfaces. Further, each of the six side surfaces has a curvature defined by part of the outer surface of an elliptic cylinder. Fig. 4 shows the geometrical relation between elliptic outer surface 2 of an elliptic cylinder, part of which constitutes one of the reflection surfaces of polygon mirror 1, rotation center O of polygon mirror 1, and light beam scanning plane SP, and Fig. 5 is an enlargement of a portion of Fig. 4, showing part of polygon mirror 1. With regard to the x-y plane of Fig. 4, short radius c of elliptic outer surface 2 lies on the y axis, and long radius b thereof lies on the x axis. The curvature of elliptic outer surface 2 continuously varies as the x or y value varies, and reaches a minimum when surface 2 is on the y axis (x=0), and a maximum when it is on the x axis (y=0). Inscribed circle INT of polygon mirror 1 has radius Rm and is in contact with the maximum curvature point CP (y=0) of elliptic outer surface 2. Fig. 5 shows an enlargement of a portion of polygon mirror 1 including maximum curvature point CP. As is shown in Fig. 5, polygon mirror 1 has six elliptic mirror surfaces 1a to 1f, and six edges (i) to (vi) which are in contact with circumscribed circle EXT. Mirror surface 1a, corresponding to part of elliptic outer surface 2, is symmetrical with respect to the major axis (corresponding to long radius b) which passes through maximum curvature point CP. The curvature of mirror surface 1a reaches a maximum at maximum curvature point CP, and gradually decreases at a given rate as the curvature defining point moves from CP to edge (i) or (vi) on mirror surface 1a. This feature is obtained by using part of elliptic surface 2 as the mirror surface of polygon mirror 1, and is utilized by the present invention to reduce the error in the curvature of the image field and in the scanning linearity on scanning plane SP, thereby to dispense with the need for an fϑ lens and electrical error-correction means. The way in which the elliptically curved surface of polygon mirror 1 is used in this invention to attain the above error-reducing effect will now be explained in detail. In this case, center O of inscribed circle INT is used as the origin of the x-y plane. Assume now that, as is shown in Fig. 6, an axis light beam and differentiated light beam from lens 16 of Fig. 1 are projected to be focused on point So and reflected from the reflection surface to form an image on point S. In this case, point So is set in position apart from rotation axis O by distance ℓ. Next, coordinates (xc, yc) of focusing point S are determined as follows, by using x and y axes corresponding to radii b and c of ellipse 2. The formula expressing ellipse 2 is given as follows: the axis light beam and the differentiated light beam are expressed as follows: y (axis light beam) = x·tanϑ (2) y + ℓ · sin (differentiated light beam) = (x+ℓ·cosϑ) · tan(ϑ+dϑ) (3) and the reflected axis and differentiated light beams are expressed as follows: y - yo (reflected axis light beam) = (x-xo) · tan(2β-ϑ) (4) y - (yo+dy) (reflected differentiated light beam) = {x - (xo+dx)} · tan{2(β+dβ) - (ϑ+dϑ)} (5) If secondary and higher order derivatives are neglected in equation (5), the following equation can be obtained: Based on equations (4) and (6), the coordinates of the intersection (focusing point S) of the reflected axis and differentiated light beams can be obtained as follows: In this case, xo and yo can be expressed as a function of ϑ, as follows: Further, β can be expressed as As can be clearly seen from Fig. 7, differential coefficients dx/dϑ and dy/dϑ can be derived as follows: wherein γ can be expressed as From equations (12) and (13), the differential coefficients can be obtained as follows: In equations (7) and (8), dβ/dϑ is the only unknown variable, and can be obtained as follows, based on equations (9), (10), and (11): Equations (7) and (8) can be converted into equations (19) and (20) as follows, by using the rotation formula of the following equation (18) and focusing point S (xc, yc) on the fixed coordinate system: Further, ℓ can be expressed as follows, based on the condition of image formation when ϑ=0: An optical simulation was effected to detect the condition in which the error in the scanning linearity can be reduced within ±2% and the curvature of the image field can be set within 5% of length R of the optical scanning path. In this case, the rotation angle of 60° of polygon mirror 1 is set to correspond to one scanning cycle, in order that the rotation angle of 40° of polygon mirror 1 can cover the image formation spot area of 220 mm, and the error in the scanning linearity at scanning plane center CT is reduced to 0. As a result of the optical simulation, it was found that radii b and c along the major and minor axes of ellipse 2 and distance R of the spot position from rotation center O of polygon mirror 1 can be determined in the above-described condition, so that the error in the scanning linearity can be set within ±2% and the curvature of the image field can be set within ±0.05R (within 5% of length R of the optical scanning path). Next, several results of the simulation carried out to satisfy the condition described above will be explained. Figs. 8A and 8B show the result of simulation in which distance R of focusing point CT from rotation center O of polygon mirror 1 was set to 350 mm, radius Rm of the inscribed circle of polygon mirror 1 was set to 22.65625 mm, and long and short radii b and c of ellipse 2 were respectively set to 264.96875 mm and 105.908431 mm. When polygon mirror 1 was rotated through 40°, the beam was scanned over a range of Lo=220 mm on the surface of photoreceptor drum 20 (the plane perpendicular to the optical axis containing focusing point CT at ϑ=0). In the above case, as is shown in Fig. 8A, the error in scanning linearity was reduced to the extent that its highest value was below -1.47 mm (-0.67%), and as is shown in Fig. 8B, the curvature of the image field was reduced to the extent of having a maximum value below 1.66 mm (0.47%). Figs. 9A and 9B show the result of simulation in which distance R of focusing point CT from rotation center O of polygon mirror 1 was set to 350 mm, radius Rm of the inscribed circle of polygon mirror 1 was set to 24.53125 mm, and long and short radii b and c of ellipse 2 were respectively set to 246.3125 mm and 106.549469 mm. In this case, as is shown in Fig. 9A, the error in scanning linearity was reduced such that its highest value was below -1.45 mm (-0.66%), and as is shown in Fig. 9B, the curvature of the image field was reduced such that its maximum value was below 11.67 mm (3.33%). Figs. 10A and 10B show the result of simulation in which distance R of focusing point CT from rotation center O of polygon mirror 1 was set to 382.65625 mm, radius Rm of the inscribed circle of polygon mirror 1 was set to 25.0 mm, and long and short radii b and c of ellipse 2 were respectively set to 147.671875 mm and 80.2821472 mm. In this case, as is shown in Fig. 10A, the error in scanning linearity was reduced such that its highest value was below -1.30 mm (-0.59%), and as is shown in Fig. 10B, the curvature of the image field was reduced such that its maximum value was below than 0.35 mm (0.09%). Figs. 11A and 11B show the result of simulation in which distance R of focusing point CT from rotation center O of polygon mirror 1 was set to 467.5 mm, radius Rm of the inscribed circle of polygon mirror 1 was set to 25.0 mm, and long and short radii b and c of ellipse 2 were respectively set to 165.46875 mm and 85.6534702 mm. In this case, as is shown in Fig. 11A, the error in scanning linearity was reduced such that its highest value was below -0.86 mm (-0.39%), and as is shown in Fig. 11B, the curvature of the image field was reduced such that its maximum value was below 0.16 mm (0.03%). Figs. 12A and 12B show the result of simulation in which distance R of focusing point CT from rotation center 0 of polygon mirror 1 was set to 455.0 mm, radius Rm of the inscribed circle of polygon mirror 1 was set to 25.0 mm, and long and short radii b and c of ellipse 2 were respectively set to 101.265625 mm and 64.8617095 mm. In this case, as is shown in Fig. 12A, the error in scanning linearity was reduced such that its highest value was below -0.93 mm (-0.42%), and as is shown in Fig. 12B, the curvature of the image field was reduced such that its maximum value was below 8.52 mm (1.87%). Figs. 13A and 13B show the result of simulation in which distance R of focusing point CT from rotation center O of polygon mirror 1 was set to 424.375 mm, radius Rm of the inscribed circle of polygon mirror 1 was set to 25.0 mm, and long and short radii b and c of ellipse 2 were respectively set to 147.671875 mm and 80.3066963 mm. In this case, as is shown in Fig. 13A, the error in scanning linearity was reduced such that its highest value was below -1.05 mm (-0.48%), and as is shown in Fig. 13B, the curvature of the image field was reduced such that its maximum value was below 0.21 mm (-0.05%). Figs. 14A and 14B show the result of simulation in which distance R of focusing point CT from rotation center O of polygon mirror 1 was set to 419.375 mm, radius Rm of the inscribed circle of polygon mirror 1 was set to 25.0 mm, and long and short radii b and c of ellipse 2 were respectively set to 93.53125 mm and 61.8695557 mm. In this case, as is shown in Fig. 14A, the error in scanning linearity was reduced such that its highest value was below -1.10 mm (-0.50%), and as is shown in Fig. 14B, the curvature of the image field was reduced such that its maximum value was below 10.87 mm (2.59%). Figs. 15A and 15B show the result of simulation in which distance R of focusing point CT from rotation center O of polygon mirror 1 was set to 491.875 mm, radius Rm of the inscribed circle of polygon mirror 1 was set to 25.0 mm, and long and short radii b and c of ellipse 2 were respectively set to 118.828125 mm and 71.2061837 mm. In this case, as is shown in Fig. 15A, the error in scanning linearity was reduced such that its highest value was below -0.78 mm (-0.36%), and as is shown in Fig. 15B, the curvature of the image field was reduced such that its maximum value was below 5.80 mm (1.18%). Figs. 16A and 16B show the result of simulation in which distance R of focusing point CT from rotation center O of polygon mirror 1 was set to 443.125 mm, radius Rm of the inscribed circle of polygon mirror 1 was set to 25.0 mm, and long and short radii b and c of ellipse 2 were respectively set to 81.640625 mm and 56.8074427 mm. In this case, as is shown in Fig. 16A, the error in scanning linearity was reduced such that its highest value was below -1.05 mm (-0.48%), and as is shown in Fig. 16B, the curvature of the image field was reduced such that its maximum value was below 4.42 mm (0.99%). As can be clearly understood from the examples described above, the error in scanning linearity and the curvature of the image field can be reduced to sufficiently small values by selectively setting distance R of focusing point CT from rotation center O of polygon mirror 1, radius Rm of the inscribed circle of polygon mirror 1, and long and short radii b and c of ellipse 2 to suitable values. In each of the above examples, long and short radii b and c of ellipse 2 are respectively set on the x and y axes. However, it is possible to set radii b and c on the y and x axes, respectively, and rotation center O can be set to pass through the axis of radius b. As described above, according to this invention, a post objective light deflector can be provided in which the error in scanning linearity and the curvature of the image field can be reduced to be sufficiently small without the need to use electrical correction processing, thus enhancing the pattern forming precision.

Claims: A rotatable polygonal mirror (1) for converting an incident light beam into a reflected light beam which scans a medium (20) along a line (SP) of a given length (L o ), said line being at a given distance (R) from the center (O) of said rotatable polygonal mirror, said rotatable polygonal mirror comprising a plurality of convex surfaces (1a to 1f) functioning as light reflecting surfaces, each having the same shape, and being uniformly arranged on the circumference of an inscribed circle (INT), characterized in that the cross-section of each of said surfaces in the plane of said inscribed circle (INT) has the shape of the maximum curvature portion of an ellipse (2), and said inscribed circle (INT) has its center (O) set on the long axis (b) of said ellipse and is in contact with the maximum curvature point (CP) of said ellipse. A rotatable polygonal mirror according to Claim 1, characterized in that the long (b) and short (c) axes of said ellipse and the quantities: R m : the radius of said inscribed circle (INT), R: the distance from the center (O) of said inscribed circle (INT) of said rotatable polygonal mirror (1) to said line (SP), and L o : the length of the line (SP) along which the light is scanned, are determined to satisfy the algebraic relation: -0.02 L o ≦ (R-γ) tan 2(ϑ-β) - (ϑ/ϑ o ) L o ≦ 0.02 L o for a generic point of said elliptic convex surfaces of said rotatable polygonal mirror from which a light beam is being reflected, said point having the coordinates (x o , y o ) on an x-y coordinate plane having the center (O) of said inscribed circle (INT) as the origin, and having its Ox axis coincident with the long axis of the ellipse, and where: γ is the distance between the center (O) of said inscribed circle (INT) and said generic point (x o , y o ), i.e. γ = ( x o ² + y o ² ) 1/2 , β is the angle between the normal to said elliptic convex surfaces of said rotatable polygonal mirror at said generic point (x o , y o ) and the long axis of the ellipse, i.e. β = arctan ( b² y o / c² (x o + a)), ϑ is the angle of view of said generic point (x o , y o ), as viewed from the center (O) of said inscribed circle and with respect to said long axis b, i.e. ϑ = arctan (y o /x o ), ϑ o is the rotation angle of the rotatable polygonal mirror with respect to said long axis of said ellipse when the y-coordinate of the focusing point (S) of a light beam reflected from said generic point (x o , y o ) is equal to half the scanning length L o , and a is the distance between the center of the ellipse (O*) and the center (O) of said inscribed circle (INT), i.e. a = b - R m . A rotatable polygonal mirror according to any of Claims 1 or 2, characterized in that the long (b) and short (c) axes of the ellipse and the quantities R m and R defined in Claim 2 are determined to satisfy the algebraic relation: -0.05 R < x c cosϑ + y c sinϑ - R ≦ 0.05 R for a generic point of said elliptic convex surface of said rotatable polygonal mirror from which a light beam is being reflected, said point having the coordinates (x o , y o ) on an x-y coordinate plane having the center (O) of said inscribed circle (INT) as the origin, and having its Ox axis coincident with the long axis of the ellipse, and where: x c and y c are the coordinates of the focusing point (S) of a light beam after being reflected by said generic point (x o , y o ) dx/dϑ= -(1+γ)sinβ/cos(ϑ-β) dy/dϑ=(1+γ)cosβ/cos(ϑ-β) dβ/dϑ=(b²/c²)(1+γ/x o +a)[(y o sinβ)/(x o +a) + cosβ] [cos(ϑ-β)[1+(b⁴y o ²)/c⁴(x o +a)²]]⁻¹ 1 = [c²(R-R m )/(2b(R-R m )+c²)]-R m and the quantities a, β, ϑ and γ have been defined in Claim 2.

Inventors: IKUMI, TOMONORI, IWAFUNE, YASUO, MATSUMOTO, YASUO, MURAKAMI, KAZUNORI
Applicants: TOKYO ELECTRIC CO., LTD.
Application year: 1988
Main classification: G02B5
IPC codes: G02B5, G02B26