Two-element f-θ lens used for micro-electro mechanical system (MEMS) laser scanning unit

A two-element f-θ lens used for a micro-electro mechanical system (MEMS) laser scanning unit includes a first lens and a second lens, the first lens is a biconvex lens and the second lens is a biconcave lens. At least one optical surface is aspherical surface in both main scanning direction and sub scanning direction, and satisfies special optical conditions. The two-element f-θ lens corrects the nonlinear relationship between scanned angle and the time into the linear relationship between the image spot distances and the time. Meanwhile, the two-element f-θ lens focuses the scan light to the target in the main scanning and sun scanning directions, such that the purpose of the scanning linearity effect and the high resolution scanning can be achieved.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a two-element fθ lens used for a micro-electro mechanical system (MEMS) laser scanning unit (LSU), and more particularly to a two-element fθ lens using an angular change varying with time in a sinusoidal relation for correcting a MEMS reflecting mirror having a simple harmonic movement to achieve the scanning linearity effect required by the laser scanning unit.

2. Description of the Related Art

At present, a laser scanning unit (LSU) used by a laser beam printer (LBP) controls a laser beam scanning by a high-speed rotating polygon mirror as disclosed in U.S. Pat. Nos. 7,079,171, 6,377,293 and 6,295,116 or TW Pat No. I198966, and principles of those inventions are described as the following: a semiconductor laser emits a laser beam through a collimator and an aperture to form parallel beams. After the parallel beams pass through a cylindrical lens, the beams are focused at the width of the X-axis in the sub scanning direction and along a direction parallel to the Y-axis of the main scanning direction to form a line image and projected onto a high-speed rotating polygon mirror. The polygon mirror includes a plurality of continuous reflecting mirrors disposed substantially at or proximate to the focusing position of the line image. The polygon mirror is provided for controlling the direction of projecting the laser beam, so that when a plurality of continuous reflecting mirrors are rotated at high speed, the laser beam projected onto a reflecting mirror can be extended in a direction parallel to the main scanning direction (Y-axis) at the same angular velocity and deviated from and reflected onto a fθ linear scanning lens. The fθ linear scanning lens is installed next to the polygon mirror and may be either a single-element lens structure (or a single scanning lens) or a two-element lens structure. The function of this fθ linear scanning lens is to focus a laser beam reflected by the reflecting mirror of the polygon mirror and projected onto the fθ lens into an oval spot that is projected onto a photoreceptor (or a photoreceptor drum, which is an image surface) to achieve the requirement of the scanning linearity. However, the traditional laser scanning unit (LSU) still has the following drawbacks in its practical use.

(1) The manufacture of the rotating polygon mirror incurs a high level of difficulty and a high cost, and thus increasing the manufacturing cost of the LSU.

(2) The polygon mirror requires a function of a high-speed rotation (such as 40000 rpm) and a high precision, and thus a cylindrical lens is required and installed to the traditional LSU since the width of the general polygon mirror along the Y-axis of the reflecting surface of the mirror is very thin, so that the laser beam pass through the cylindrical lens can be focused and concentrated into a line (or a spot on the Y-axis) and projected onto the reflecting mirror of the polygon mirror. Such arrangement increases the number of components and also complicates the assembling operation procedure.

(3) The traditional polygon mirror requires a high-speed rotation (such as 40000 rpm), and thus the noise level is raised. Furthermore, the polygon mirror takes a longer time to be accelerated from a starting speed to an operating speed, and thus increasing the booting time of the laser scanning.

(4) In the fabrication of the traditional LSU, the central axis of a laser beam projected onto the reflecting mirror of the polygon mirror is not aligned precisely with the central rotating axis of the polygon mirror, so that it is necessary to take the off axis deviation of the polygon mirror into consideration for the design of the fθ lens, and thus increasing the difficulty of design and manufacturing the fθ lens.

In recent years, an oscillatory MEMS reflecting mirror is introduced to overcome the shortcomings of the traditional LSU assembly and replace the laser beam scanning controlled by the traditional polygon mirror. The surface of a torsion oscillator of the MEMS reflecting mirror comprises a reflecting layer, and the reflecting layer is oscillated for reflecting the light and further for the scanning. In the future, such arrangement will be applied in a laser scanning unit (LSU) of an imaging system, a scanner or a laser printer, and its scanning efficiency is higher than the traditional rotating polygon mirror. As disclosed in the U.S. Pat. Nos. 6,844,951 and 6,956,597, at least one driving signal is generated, and its driving frequency approaches the resonant frequency of a plurality of MEMS reflecting mirrors, and the driving signal drives the MEMS reflecting mirror to produce a scanning path. In U.S. Pat. Nos. 7,064,876, 7,184,187, 7,190,499, 2006/0033021, 2007/0008401 and 2006/0279826 or TW Pat No. 253133, or JP Pat. No. 2006-201350, a MEMS reflecting mirror installed between a collimator and a fθ lens of a LSU module replaces the traditional rotating polygon mirror for controlling the projecting direction of a laser beam. The MEMS reflecting mirror features the advantages of small components, fast rotation, and low manufacturing cost. However, after the MEMS reflecting mirror is driven by the received voltage for a simple harmonic with a sinusoidal relation of time and angular speed, and a laser beam projected on the MEMS reflecting mirror is reflected with a relation of reflecting angle θ(t) and time as follows
θ(t)=θs·sin(2πf·t)  (1)

wherein, f is the scanning frequency of the MEMS reflecting mirror and θsis the maximum scanning angle at a single side (symmetrical with the optical Z axis) after the laser beam passes through the MEMS reflecting mirror.

In the same time interval Δt, the corresponding variation of the reflecting angle is not the same but decreasing, and thus constituting a sinusoidal relation with time. In other words, the variation of the reflecting angle in the same time interval Δt is Δθ(t)=θs·(sin(2π·f·t1)−sin(2π·f·t2)), which constitutes a non-linear relation with time. If the reflected light is projected onto the target from a different angle, the distance from the spot will be different in the same time interval due to the different angle.

Since the angle of the MEMS reflecting mirror situated at a peak and a valley of a sine wave varies with time, and the rotating movements if a traditional polygon mirror are at a constant angular speed, if a traditional fθ lens is installed on a laser scanning unit (LSU) of the MEMS reflecting mirror, the angle of the MEMS reflecting mirror produced by the sinusoidal relation varied with time cannot be corrected, so that the speed of laser beam projected on an image side will not be an uniform speed, and the image on the image side will be deviated. Therefore, the laser scanning unit or the MEMS laser scanning unit (MEMS LSU) composed of MEMS reflecting mirrors has a characteristic that after the laser beam is scanned by the MEMS reflecting mirror, scan lights at different angles are formed in the same time. Thus, finding a way of developing a fθ lens (some prior art named as f-sin θ lens) for the MEMS laser scanning unit to correct the scan lights, such that a correct image will be projected onto the light, examples as, U.S. Pat. No. 7,184,187 provided a polynomial surface for fθ lens to adjust the angular velocity variation in the main-scanning direction only. However, the laser light beam is essential an oval-like shape of the cross section that corrects the scan lights in the main-scanning direction only may not be achieve the accuracy requirement. Since, a fθ lens with main-scanning direction correcting as well as sub-scanning direction correcting demands immediate attentions and feasible solutions.

SUMMARY OF THE INVENTION

The primary objective of the present invention is to overcome the shortcomings of the prior art by providing a two-element lens used for a micro-electro mechanical system (MEMS) laser scanning unit, the first lens is a biconvex lens and the second lens is a biconcave lens, which counted from the MEMS reflecting mirror, for projecting a scan light reflected by the MEMS reflecting mirror onto the correct image of a target to achieve a scanning linearity effect required by the laser scanning unit.

Another objective of the present invention is to provide a two-element fθ lens used for a micro-electro mechanical system (MEMS) laser scanning unit for reducing the area of a spot projected onto the target to achieve the effect of improving the resolution.

A further objective of the present invention is to provide a two-element fθ lens used for a MEMS laser scanning unit, and the two-element fθ lens can make a distortion correction to correct optical axis caused by the deviation of the scan light resulting in the problems of an increased deviation of the main scanning direction and the sub scanning direction, and a change of a spot of a drum at the image into an oval-like shape, and the two-element fθ lens can unify the size of each image spot to achieve the effect of enhancing the image quality.

Therefore, the two-element lens used for a micro-electro mechanical system (MEMS) laser scanning unit of the invention is applicable for a light source comprising an emitting laser beam, wherein a resonant oscillation is used for reflecting the laser beam of the light source onto MEMS reflecting mirror of the scan light to form an image on the target. As to a laser printer, the target is generally a drum. The spot of the image forms a scan light after the laser beam is emitted from the light source, scanned oscillatory by the MEMS reflecting mirror, and reflected by the MEMS reflecting mirror. After the angle and position of the scan light are corrected by the two-element fθ lens of the invention, a spot may be formed on the drum. Since a photosensitive agent is coated onto the drum, data can be printed out on a piece of paper by the sensing carbon powder centralized on the paper.

The two-element fθ lens of the invention comprises a first lens and a second lens, counted from the MEMS reflecting mirror, wherein the first lens includes a first optical surface and a second optical surface, the second lens includes a third optical surface and a fourth optical surface. These optical surfaces provided the functions of correcting the phenomenon of non-uniform speed scanning which results in decreasing or increasing the distance between spots on an image surface of a MEMS reflecting mirror with a simple harmonic movement with time into a constant speed scanning, so that the projection of a laser beam onto an image side can give a constant speed scanning, and unify the deviation of image formed on the drum which caused by a scan light in the main scanning direction and the sub scanning direction deviated from the optical axis, so as to make a correction to focus the scan light at a target.

To make it easier for our examiner to understand the technical characteristics and effects of the present invention, we use preferred embodiments and related drawings for the detailed description of the present invention as follows:

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Referring toFIG. 1for a schematic view of optical paths of a two-element fθ lens used for micro-electro mechanical system (MEMS) laser scanning unit in accordance with the present invention, the two-element fθ lens used for the micro-electro mechanical system (MEMS) laser scanning unit13comprises: a first lens131having a first optical surface131aand a second optical surface131b, and a second lens132having a third optical surface132aand a fourth optical surface132b. InFIG. 1, the MEMS laser scanning unit comprises a laser source11, a MEMS reflecting mirror10, a cylindrical lens16, two photoelectric sensors14a,14b, and a light sensing target. InFIG. 1, the target is implemented by a drum15. After a beam111produced by the light laser source11is passed through a cylindrical lens16, the beam111is projected onto the MEMS reflecting mirror10. The MEMS reflecting mirror10generates a resonant oscillation to reflect the beam111into scan lights113a,113b,114a,114b,115a,115bat different time frames along the direction of Z, wherein the scan lights113a,113b,114a,114b,115a,115bare projected in a X direction which is called a sub scanning direction, and projected in a Y direction which is called a main scanning direction, and the maximum scanning angle of the MEMS reflecting mirror10is θc.

Since the MEMS reflecting mirror10comes with a simple harmonic movement, and the angle of movement shows a sinusoidal change with time as shown inFIG. 2, therefore the angle and time of reflecting the scan light are in a non-linear relation. The swinging angle of the MEM reflecting mirror10has a wave peak a-a′ and a wave valley b-b′ as shown in the figure, and its swinging angle is significantly smaller than the wave sections a-b and a′-b′, and this non-uniform angular speed may cause an image deviation easily produced on the drum15by the scan light. Therefore, photoelectric sensors14a,14bare installed at the angle ±θp within range below the maximum scanning angle ±θc of the MEMS reflecting mirror10and the laser beam111starts to be reflected by the MEMS reflecting mirror10at the wave peak as shown inFIG. 2, which is equivalent to the scan light115aas shown inFIG. 1. If the photoelectric sensor14adetects a scanned beam, it means that the MEMS reflecting mirror10swings to an angle of ±θp, which is equivalent to the scan light114aas shown inFIG. 1. If the MEMS reflecting mirror10scans point “a” at an angle variation as shown inFIG. 2, such point is equivalent to the position of the scan light113a. Now, the laser source11is controlled to start emitting the laser beam111. When the point “b” as shown in HG2is scanned, such point is equivalent to the position of the scan light113b(which is equivalent to the laser beam111emitted by the laser source11awithin an angle of ±θn). When the MEMS reflecting mirror10swings in an opposite direction to a wave section a′-b′, the laser source11is controlled to start emitting the laser beam111to complete a cycle.

Referring to HG3for an optical path chart of a scan light passing through a first lens and a second lens, in which ±θn is a valid scanning angle. If the MEMS reflecting mirror10is swung to the angle of ±θn, the laser source11starts emitting the desired scanning laser beam111which is reflected into a scan light by the MEMS reflecting mirror10, and the scan light is passed through the first lens131and refracted by the first optical surface and the second optical surface of the first lens131, and the scan light reflected by the MEMS reflecting mirror10with a non-linear relation between distance and time is converted into a scan light with a linear relation between distance and time. After the scan light is passed through the first lens131and the second lens132, the focusing effect of the first optical surface131a, the second optical surface131b, the third optical surface132aand the fourth optical surface132bof the first lens131and the second lens132and the interval of each optical surface can focus the scan light at the drum15and form a column of spots2on the drum15, and the distance between the farthest two spots2projected on the drum15is called an effective scanning window3, wherein along the optical axis Z, d1is the distance between the MEMS reflecting mirror10and the first optical surface, d2is the distance between the first optical surface and the second optical surface, d3is the distance between the second optical surface R2and the third optical surface, d4is the distance between the third optical surface and the fourth optical surface R4, d5is the distance between the fourth optical surface and the drum15, R1is the radius of curvature of the first optical surface, R2is the radius of curvature of the second optical surface, R3is the radius of curvature of the third optical surface, and is the radius of curvature of the fourth optical surface on the optical axis.

Referring toFIG. 4for a spot area varied with a different projecting position after a scan light is projected onto a drum, if the scan light113ais projected in a direction along the optical axis Z and onto the drum15by the first lens131and the second lens132, the incident angles of the first lens131and the second lens132are zero, and thus the deviation of the main scanning direction is minimum (said zero), and the image at the spot2aon the drum15is in an inferenced circle-like shape (same shape as laser light beam). After the scan light113band113cis projected on the drum15by the first lens131and the second lens132, the incident angle of the first lens131and the second lens132with respect to the optical axis is non-zero, and the deviation of the main scanning direction is non-zero, and thus the projection distance of the main scanning direction is longer than the spot formed by the scan light111ais also bigger. Not only has the phenomenon existed in the main scanning direction but also in the sub scanning direction. Therefore, the image at the spot2b,2con the drum15is in an oval-like shape, and the area of2b,2cis greater than the area of2a. Denoted Sa0and Sb0are the lengths of spots of the scan lights in the main scanning direction (Y direction) and the sub scanning direction (X direction) on a reflecting surface of the MEMS reflecting mirror10, and Gaand Gbare the Gaussian beam diameter of scanning light emitted by fθ lens13at the intensity is 13.5% of maximum intensity on Y direction and the X direction, illustrated byFIG. 5. InFIG. 5, only Y direction Gaussian beam is shown.

The two-element fθ lens of the invention can control the spot size in the main scanning direction within a limited range by the distortion correction of the fθ lens13and correct the spot size in the sub scanning direction by the distortion correction of the first lens131and the second lens132of the two-element fθ lens13, such that the spot size is controlled within a limited range, and the distribution of the spot size (or the ratio of largest spots and smallest spots) is controlled within an appropriate range in compliance with the required resolution.

To achieve the foregoing effects, the two-element fθ lens of the invention comes with a first lens having a first optical surface and a second optical surface and a second lens having a third optical surface and a fourth optical surface of a design of with a spherical surface or an aspherical surface. If the aspherical surface is adopted, the aspherical surface is designed with the following equations (2) or (3):

where, Z is the sag of any point on the surface parallel to the Z-axis, Cxand Cyare curvatures in the X direction and the Y direction respectively, Kxand Kyare the conic coefficients in the X direction and the Y direction respectively and correspond to eccentricity in the same way as conic coefficient for the aspherical surface type, AR, BR, CRand DRare deformations from the conic coefficient of rotationally symmetric portions of the fourth order, the sixth order, the eighth order and the tenth order respectively, and Ap, Bp, Cpand Dpare deformation from the conic coefficient of non-rotationally symmetric components to the fourth order, the sixth order, the eighth order and the tenth order respectively. This reduces to aspherical surface type when Cx=Cy, Kx=Kyand Ap=Bp=Cp=Dp=0.

where, Z is the sag of any point on the surface parallel to the Z-axis; Cxand Cyare curvatures in the X direction and the Y direction respectively, Kyis a conic coefficient in the Y direction, B4, B6, B8and B10are deformations from the conic coefficient to the fourth, sixth, eighth and tenth orders respectively. When Cx=Cyand Ky=B4=B6=B8=B10=0 is reduced to a single spherical surface.

To unify the scan speed of the scan light projected onto the image of the target, the invention adopts two equal time intervals and an equal distance between two spots, and the two-element fθ lens of the invention can correct the emergence angle of the scan light between the scan light113ato the scan light113b, so that the first lens131and the second lens132corrects the emergence angle of the scan light to produce two scan lights at the same time interval. After the emergence angle is corrected, the distance between any two spots formed on the drum15of the image is equal. Further, after the laser beam111is reflected by the MEMS reflecting mirror10, the spot is diverged and becomes larger. After the scan light is passed through the distance from the MEMS reflecting mirror10to the drum15, the spot becomes even larger. Such arrangement is incompliance with the actual required resolution. The two-element fθ lens of the invention further focuses from the scan light113ato the scan light113breflected by the MEMS reflecting mirror10at the drum15of the image to form a smaller spot in the main scanning and sub scanning directions. The two-element fθ lens of the invention further uniformizes the spot size of the image on the drum15(to limit the spot size in a range to comply with the required resolution) for the best resolution.

The two-element fθ lens comprises a first lens131and a second lens132, counted from the MEMS reflecting mirror10, the first lens is a biconvex lens and the second lens is a biconcave lens, wherein the first lens131includes a first optical surface131aand a second optical surface131bfor converting a scan spot with a non-linear relation between angle and time and reflected by the MEMS reflecting mirror10into a scan spot with a linear relation between distance and time; and the second lens132includes a third optical surface132aand a fourth optical surface132bfor correcting the focus of the scan light of the first lens131onto target; such that the two-element fθ lens projects a scan light reflected by the MEMS reflecting mirror10onto the image of the drum15. The first optical surface131a, the second optical surface131b, the third optical surface132aand the fourth optical surface132bare optical surfaces are composed of at least one aspherical surface in the main scanning direction. The first optical surface131aand the second optical surface131bare optical surfaces composed of at least one aspherical surface in the sub scanning direction. Further, the assembly of the first lens131and the second lens132of the two-element fθ lens in accordance with the present invention has an optical effect in the main scanning direction that satisfies the conditions of Equations (4) and (5):

or, the main scanning direction satisfies the conditions of Equation (6),

and the sub scanning direction satisfies the conditions of Equation (7).

where, f(1)Yis the focal length of the first lens131in the main scanning direction, f(2)Yis the focal length of the second lens132in the main scanning direction, d3is the distance between an optical surface on a target side of the first lens131when θ=0° and an optical surface on the MEMS reflecting mirror side of the second lens132, d4is the thickness of the second lens when θ=0°, d5is the distance between an optical surface on a target side of the second lens132when θ=0° and the target, f(1)Xis the focal length of the first lens in the sub scanning direction, f(2)Xis the focal length of the second lens in the sub scanning direction, fsis the combined focal length of the two-element fθ lens, Rixis the radius of curvature of the i-th optical surface in the X direction; and nd1and nd2are the refraction indexes of the first lens and the second lens13respectively.

Further, the spot uniformity produced by the two-element fθ lens of the invention can be indicated by the ratio δ of the largest spot and the smallest spot size that satisfies the conditions of Equation (8):

The resolution produced by the two-element fθ lens of the invention can be indicated by the ratio ηmaxof the largest spot on the drum15formed by the scan light on the reflecting surface of the MEMS reflecting mirror10(or the ratio of scanning light of maximum spot) and the ratio ηminof the smallest spot formed by the scan light on the reflecting surface of the MEMS reflecting mirror10(or the ratio of scanning light of minimum spot), and the ratios satisfy the conditions of Equations (9) and (10)

where, Saand Sbare the lengths of any one spot of the scan light formed on the drum in the main scanning direction and the sub scanning direction, δ is the ratio of the smallest spot and the largest spot on the drum15, Sa0and Sb0are the lengths of the spots of the scan light on the reflecting surface of the MEMS reflecting mirror10in the main scanning direction and the sub scanning direction respectively.

The following preferred embodiments of the invention disclose a two-element fθ lens used for a micro-electro mechanical system (MEMS) laser scanning unit by using major elements for the illustration, and thus the preferred embodiments can be applied in a MEMS laser scanning unit including but not limited to the two-element fθ lens with components illustrated in the embodiments only, but any other equivalents are intended to be covered in the scope of the present invention. In other words, any variation and modification of the two-element fθ lens used for a micro-electro mechanical system (MEMS) laser scanning unit can be made by the persons skilled in the art. For example, the radius of curvature of the first lens and the second lens, the design of the shape, the selected material and the distance can be adjusted without any particular limitation.

In a first preferred embodiment, the two-element fθ lens comprises a first lens and a second lens. The first lens is a biconvex lens and the second lens is a biconcave lens. A first optical surface and a second optical surface of the first lens, and a third optical surface and a fourth optical surface of the second lens are all aspherical surfaces designed in accordance with the Equation (2), and the optical characteristics and the aspherical surface parameters are listed in Tables 1 and 2.

TABLE 2Parameters of Aspherical Surface of Optical Surface Parameter for FirstPreferred EmbodimentAnamorphic equation coefficientKy, Conic4th Order6th Order8th Order10th Orderoptical surfaceCoefficientCoefficient (AR)Coefficient (BR)Coefficient (CR)Coefficient (DR)R1*−2.2007E+01−6.5965E−061.0782E−130.0000E+000.0000E+00R2*−7.6338E−013.0048E−08−2.1961E−10−1.2963E−098.1727E−16R3*7.8416E+00−5.0148E−061.1356E−11−8.0222E−15−5.5747E−09R4*−7.8779E+00−2.9418E−072.8385E−110.0000E+000.0000E+00Kx, Conic4th Order6th Order8th Order10th Orderoptical surfaceCoefficientCoefficient (AP)Coefficient (BP)Coefficient (CP)Coefficient (DP)R1*0.0000E+00−1.2187E+001.9174E+010.0000E+000.0000E+00R2*1.5561E+00−3.1946E+00−1.8428E−01−1.0314E+00−2.5299E−02R3*2.6734E+00−6.1096E−011.0979E+004.3448E−01−9.9696E−01R4*0.0000E+00−5.6197E−01−1.6934E−010.0000E+000.0000E+00
The two-element fθ lens13, f(1)Y=67.05, f(2)Y=−93.76, fsX=32.257, fsY=147 (mm), so that the scan light can be converted into a scan spot with a linear relation of distance and time, and the spots with spot Sa0=19.434 and Sb0=3972.24 (μm) on the MEMS reflecting mirror10are scanned into scan lights and focused on the drum15to form a smaller spot6and satisfy the conditions of Equations (4) to (10) as listed in Table 3. The maximum diameter (μm) of geometric spot on the drum surface at distance Y (mm) from the center point along the drum surface is shown in Table 4. The distribution of spot sizes from the central axis to the left side of the scan window3is outlined asFIG. 7, and the right sides of the scan window3is symmetrical to left side essentially, where the diameter of unity circle is 0.05 mm.

TABLE 3Conditions for First Preferred Embodimentd3+d4+d5f(1)⁢Y1.9310d5f(2)⁢Y−1.1678Main scanning directionfsY·((nd⁢⁢1-1)f(1)⁢y+(nd⁢⁢2-1)f(2)⁢y0.3071Sub scanning direction(1R1⁢x-1R2⁢x)+(1R3⁢x-1R4⁢x)⁢fsX1.2693δ=min⁡(Sb·Sa)max⁡(Sb·Sa)0.8884ηmax=max⁡(Sb·Sa)(Sb⁢⁢0·Sa⁢⁢0)0.0574ηmin=min⁡(Sb·Sa)(Sb⁢⁢0·Sa⁢⁢0)0.0510

TABLE 4The maximum diameter (μm) of light spot on the drumY−107.500−98.223−89.648−71.924−53.905−35.870−17.9060.000Max diameter4.42E−033.42E−034.84E−034.79E−034.41E−034.12E−033.39E−032.66E−03

In a second preferred embodiment, the two-element fθ lens comprises a first lens and a second lens. The first lens is a biconvex lens and the second lens is a biconcave lens. A first optical surface and a second optical surface of the first lens, and a third optical surface and a fourth optical surface of the second lens are all aspherical surfaces designed in accordance with the Equation (2), and the optical characteristics and the aspherical surface parameters are listed in Tables 5 and 6.

The two-element fθ lens, f(1)Y=60.299, f(2)Y=−80.169 (mm), fsX=27.399, fsY=145.725 (mm), so that the scan light can be converted into a scan spot with a linear relation of distance and time, and the spots with Sa0=19.434 and Sb0=3972.24 (μm) on the MEMS reflecting mirror10are scanned into scan lights and focused on the drum15to form a smaller spot8and satisfy the conditions of Equations (4) to (10) as listed in Table 7. The maximum diameter (μm) of geometric spot on the drum surface at distance Y (mm) from the center point along the drum surface is shown in Table 8. The distribution of spot sizes from the central axis to the left side of the scan window3is outlined asFIG. 8, and the right sides of the scan window3is symmetrical to left side essentially, where the diameter of unity circle is 0.05 mm.

TABLE 7Conditions for Second Preferred Embodimentd3+d4+d5f(1)⁢Y2.1725d5f(2)⁢Y−1.3746Main scanning directionfsY·((nd⁢⁢1-1)f(1)⁢y+(nd⁢⁢2-1)f(2)⁢y0.2945Sub scanning direction(1R1⁢x-1R2⁢x)+(1R3⁢x-1R4⁢x)⁢fsX1.2444δ=min⁡(Sb·Sa)max⁡(Sb·Sa)0.8510ηmax=max⁡(Sb·Sa)(Sb⁢⁢0·Sa⁢⁢0)0.0789ηmin=min⁡(Sb·Sa)(Sb⁢⁢0·Sa⁢⁢0)0.0671

TABLE 8The maximum diameter (μm) of geometric spot on the drumY−107.545−98.226−89.602−71.765−53.660−35.621−17.7500.000Max diameter1.40E−021.49E−021.99E−023.27E−039.30E−032.04E−022.15E−028.17E−03

In a third preferred embodiment, the two-element fθ lens comprises a first lens and a second lens. The first lens is a biconvex lens and the second lens is a biconcave lens. A first optical surface and a second optical surface of the first lens, and a third optical surface and a fourth optical surface of the second lens are all aspherical surfaces designed in accordance with the Equation (2), and the optical characteristics and the aspherical surface parameters are listed in Tables 9 and 10.

The two-element fθ lens, f(1)Y=66.828, f(2)Y=−93.029 (mm), fsX=31.634, fsY=146.296 (mm), so that the scan light can be converted into a scan spot with a linear relation of distance and time, and the spots with Sa0=19.434 and Sb0=3972.24 (μm) on the MEMS reflecting mirror10are scanned into scan lights and focused on the drum15to form a smaller spot8and satisfy the conditions of Equations (4) to (10) as listed in Table 11. The maximum diameter (μm) of geometric spot on the drum surface at distance Y (mm) from the center point along the drum surface is shown in Table 12. The distribution of spot sizes from the central axis to the left side of the scan window3is outlined asFIG. 9, and the right sides of the scan window3is symmetrical to left side essentially, where the diameter of unity circle is 0.05 mm.

TABLE 11Conditions for Third Preferred Embodimentd3+d4+d5f(1)⁢Y1.9153d5f(2)⁢Y−1.1670Main scanning directionfsY·((nd⁢⁢1-1)f(1)⁢y+(nd⁢⁢2-1)f(2)⁢y0.3249Sub scanning direction(1R1⁢x-1R2⁢x)+(1R3⁢x-1R4⁢x)⁢fsX1.286δ=min⁡(Sb·Sa)max⁡(Sb·Sa)0.8701ηmax=max⁡(Sb·Sa)(Sb⁢⁢0·Sa⁢⁢0)0.0626ηmin=min⁡(Sb·Sa)(Sb⁢⁢0·Sa⁢⁢0)0.0545

TABLE 12The maximum diameter (μm) of geometric spot on the drumY−108.012−98.457−89.700−71.761−53.680−35.677−17.7990.000Max diameter5.57E−037.53E−031.01E−021.39E−021.27E−026.20E−037.21E−037.90E−03

In a fourth preferred embodiment, the two-element fθ lens comprises a first lens and a second lens. The first lens is a biconvex lens and the second lens is a biconcave lens. A first optical surface and a second lens of the first lens, a third optical surface of the second lens are all aspherical surfaces designed in accordance with the Equation (2), a fourth optical surface of the second lens is an aspherical lens designed in accordance with the Equation (3), and the optical characteristics and the aspherical surface parameters are listed in Tables 13 and 14.

The two-element fθ lens, f(1)Y=67.743, f(2)Y=−94.854 (mm), fsX=32.864, fsY=147.91 (mm), so that the scan light can be converted into a scan spot with a linear relation of distance and time, and the spots with Sa0=19.434 and Sb0=3972.24 (μm) on the MEMS reflecting mirror10are scanned into scan lights and focused on the drum15to form a smaller spot8and satisfy the conditions of Equations (4) to (10) as listed in Table 15. The maximum diameter (μm) of geometric spot on the drum surface at distance Y (mm) from the center point along the drum surface is shown in Table 16. The distribution of spot sizes from the central axis to the left side of the scan window3is outlined asFIG. 10, and the right sides of the scan window3is symmetrical to left side essentially, where the diameter of unity circle is 0.05 mm.

TABLE 15Conditions for Fourth Preferred Embodimentd3+d4+d5f(1)⁢Y2.1700d5f(2)⁢Y−1.3845Main scanning directionfsY·((nd⁢⁢1-1)f(1)⁢y+(nd⁢⁢2-1)f(2)⁢y0.3278Sub scanning direction(1R1⁢x-1R2⁢x)+(1R3⁢x-1R4⁢x)⁢fsX1.1290δ=min⁡(Sb·Sa)max⁡(Sb·Sa)0.8634ηmax=max⁡(Sb·Sa)(Sb⁢⁢0·Sa⁢⁢0)0.0044ηmin=min⁡(Sb·Sa)(Sb⁢⁢0·Sa⁢⁢0)0.0038

TABLE 16The maximum diameter (μm) of geometric spot on the drumY−107.545−98.226−89.602−71.765−53.660−35.621−17.7500.000Max diameter4.58E−033.66E−034.84E−034.59E−033.16E−033.62E−033.32E−032.50E−03

In view of the aforementioned preferred embodiments, the present invention at least has the following effects:

(1) With the two-element fθ lens of the invention, the scanning is corrected the phenomenon of non-uniform speed which results in decreasing or increasing the distance between spots on an image surface of a MEMS reflecting mirror with a simple harmonic movement with time into a constant speed scanning, so that the laser beam at the image side is projected for a uniform speed scanning and an equal distance between any two adjacent spots can be achieved for the image on a target.
(2) With the two-element fθ lens of the invention, the distortion correction is provided for correcting the main scanning direction and sub scanning direction of the scan light, so that the image size of the spot focused at the target can be decreased.
(3) With the two-element fθ lens of the invention, the distortion correction is provided for correcting the main scanning direction and the sub scanning direction of the scan light, so as to focus the spot size focused and imaged at the target.