Patent Publication Number: US-7898738-B2

Title: Lens array, manufacturing method thereof, LED head having lens array, exposure device having LED head, image forming apparatus having exposure device, and reading apparatus

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to a lens array, a manufacturing method thereof, an LED head having the lens array, an exposure device having the LED head, an image forming apparatus having the exposure device, and a reading apparatus. 
     2. Description of Related Art 
     In an image forming apparatus such as an electrophotographic printer, a rod lens array consisting of multiple rod lenses is used in an exposure device having an LED array consisting of multiple LEDs (light emitting diodes) arranged in array. In a reading apparatus such as an image scanner, a facsimile machine, and the like, the rod lenses array is used in an optical system for forming an optical image of a read document at a light receiving unit having multiple light receiving elements arranged in array. The rod lens is formed by impregnating a glass fiber with ions so that the refraction index becomes lower from the center portion toward the peripheral portion, and is an optical element for forming a same-size erect image of an object. A lens array having multiple rod lenses arranged in array is an optical system for forming an optical image of an object on a line. As disclosed in Japanese Patent Application Publication No. 2003-202411, multiple micro lenses may be integrally formed through resin molding method to constitute an optical system for forming an optical image of an object on a line. 
     There exists a lens array having multiple micro lenses arranged in array to constitute an optical system for forming a same-size erect image of a document on a line on a light receiving unit consisting of multiple light receiving elements arranged in array. As described above, a high resolution can be obtained with the integrally-formed lens array including high-precision micro lenses efficiently manufactured though plastic injection molding. 
     In the optical system in which the micro lenses are arranged in array and the optical image of an object is formed on a line as disclosed in Japanese Patent Application Publication No. 2003-202411, the resolution relies on the shape of the micro lenses constituting the optical system. For example, the resolution of the optical image will not deteriorate in a case where high-precision micro lenses are used in which the central axes on the light emitting side are in line with the central axes on the object plane side in the micro lenses forming the optical system and lens pairs. However, the resolution of the optical image will decrease if low-precision micro lenses are used in which the central axes on the light emitting side are not in line with the central axes on the object plane side in the micro lenses forming the optical system. 
     In a case where the conventional lens array is elongated in an arrangement direction of the micro lenses to extend a printing area, the high-precision micro lenses cannot be manufactured, and the precision of each lens varies depending on a position in the longitudinal direction of the lens array. That is, there exists a problem that each micro lens has different values in the residual aberration, the transmission, the focal length, and the like due to errors of the lens shape and the refraction rate among multiple micro lenses arranged in a row. 
     SUMMARY OF THE INVENTION 
     As a result of intensive research, the inventor of this invention has found out that the deterioration of the resolution of the optical image can be prevented by adjusting an axial off-center distance between the central axis of the micro lens on the light emitting surface side and the central axis of the micro lens on the optical plane side. Furthermore, the inventor has invented a manufacturing method for forming the lens array having the integrally-formed high-precision micro lenses. The object of this invention is to provide a lens array capable of preventing deterioration of the resolution in the optical image, a manufacturing method thereof, an LED head having the lens array, an exposure device having the LED head, an image forming apparatus having the exposure device, and a reading apparatus. 
     The exposure device according to this invention has a light emitting unit having a plurality of light emitting elements and a lens array having a pair of lenses having a first lens and a second lens, the lens array having a shielding unit for shielding a light from any one of the pair of lenses, wherein a formula EC&lt;EP/2 is satisfied, where EP denotes an interval between two adjacent light emitting elements of the plurality of light emitting elements and where EC denotes an off-set between a central axis of the first lens and a central axis of the second lens. 
     The exposure device of this invention satisfies the relationship EC&lt;EP/2, where EP denotes an interval between two adjacent light emitting elements of the plurality of light emitting elements and where EC denotes an off-set between a central axis of the first lens and a central axis of the second lens, thus being able to prevent deterioration of the resolution that would occur in the use of a conventional lens having an axial offset. 
     The image forming apparatus of this invention has the above-described exposure device, thus being able to form an image on a medium faithfully corresponding to image data and being able to prevent deterioration of the quality of the image on the medium such as streaks and uneven density. 
     The reading apparatus of this invention has a light receiving unit having a plurality of light receiving elements and a lens array having a pair of lenses including a first lens and a second lens, the lens array having a shielding unit for shielding a light from any one of the pair of lenses, wherein a formula EC&lt;RP/2 is satisfied, where RP denotes an interval between two adjacent light receiving elements of the plurality of light receiving elements and where EC denotes an off-set between a central axis of the first lens and a central axis of the second lens. 
     The reading apparatus of this invention satisfies the relationship EC&lt;RP/2, where RP denotes an interval between two adjacent light receiving elements of the plurality of light receiving elements and where EC denotes an off-set between a central axis of the first lens and a central axis of the second lens, thus being able to prevent deterioration of the resolution that would occur in the use of a conventional lens having an axial offset and being able to provide image data faithfully reproducing an image on a document. 
     This invention can prevent deterioration of the resolution by using the lens array that satisfies a relationship EC&lt;EP/2, where EC denotes an axial off-center distance between the central axis of the first lens and the central axis of the second lens and where EP denotes an arrangement interval between each of the light emitting elements in the exposure device. That is, the exposure device including the lens array and the image forming apparatus including the exposure device can prevent deterioration of the quality of an image. The reading apparatus including the lens array can provide image data faithfully representing a document. 
    
    
     
       DETAILED DESCRIPTION OF THE DRAWINGS 
       This invention may take physical form in certain parts and arrangements of parts, a preferred embodiment and method of which will be described in detail in this specification and illustrated in the accompanying drawings which form a part hereof, and wherein: 
         FIG. 1  is a figure showing an image forming apparatus of this invention; 
         FIG. 2  is a figure showing an exposure device mounted on the image forming apparatus of this invention; 
         FIG. 3  is a figure showing an LED array used in the exposure device of this invention; 
         FIG. 4  is a figure showing the lens array used in the exposure device of this invention; 
         FIG. 5  is a figure showing the lens array used in the exposure device of this invention; 
         FIG. 6  is a figure showing the shape of an aperture of a shielding unit in the lens array used in the exposure device of this invention; 
         FIG. 7  is a figure showing the shape of a micro lens of the lens array used in the exposure device of this invention; 
         FIG. 8  is a cross section taken along line A-A in  FIG. 4 ; 
         FIG. 9  is a cross section taken along line B-B in  FIG. 4 ; 
         FIG. 10  is a schematic diagram showing the optical system of the exposure device of this invention; 
         FIG. 11  is a figure showing the optical path in the lens array in the exposure device of this invention; 
         FIG. 12  is a figure showing the light distribution in a case where the axial off-center distance EC of the micro lens is less than half of the arrangement interval EP; 
         FIG. 13  is a figure showing the light distribution in a case where the axial off-center distance EC of the micro lens is more than half of the arrangement interval EP; 
         FIG. 14  is a figure showing the relationship between the axial off-center distance EC and MTF in the exposure device of the resolution 600 dpi; 
         FIG. 15  is a figure showing the relationship between the axial off-center distance EC and MTF in the exposure device of the resolution 1200 dpi; 
         FIG. 16  is a figure showing the relationship between the axial off-center distance EC and MTF in the exposure device of the resolution 2400 dpi; 
         FIG. 17  is a figure showing a reading apparatus of this invention; 
         FIG. 18  is a schematic diagram showing the optical system of the reading apparatus of this invention; 
         FIG. 19  is a plan view of the lens array used in the exposure device of this invention; 
         FIG. 20  is a plan view of a shielding unit used in the exposure device of this invention; 
         FIG. 21  is a cross section taken along line C-C in  FIG. 19 ; 
         FIG. 22  is a figure showing the shape of the aperture of the shielding unit in the lens array used in the exposure device of this invention; 
         FIG. 23  is a figure describing the optical path of the lens array in the exposure device of this invention; 
         FIG. 24  is a cross section of the lens array taken along a surface perpendicular to the surface of the lens plate  21  to show the optical axes of the micro lenses  22 ; 
         FIG. 25  is a schematic diagram showing the optical arrangement in a case where the micro lenses integrally formed on the lens array of this invention are arranged in two linear rows; 
         FIG. 26  is a schematic diagram showing the optical arrangement in a case where the micro lenses integrally formed on the lens array of this invention are arranged in multiple linear rows; 
         FIG. 27  is a schematic diagram showing an image in which every two pixels are printed in the entire pixels relating to the evaluation of the lens array of this invention; 
         FIG. 28  is a figure showing a conventional mold used in a conventional manufacturing method for a lens plate of a lens array; 
         FIG. 29  is a schematic diagram showing a process of injecting a resin into the conventional mold; 
         FIG. 30  is a schematic diagram showing the conventional mold when the resin has been injected into the conventional mold; 
         FIG. 31  is a figure showing a mold used in the manufacturing method of the lens plate of the lens array of this invention; 
         FIG. 32  is a figure showing the mold used in the manufacturing method of the lens plate of the lens array of this invention; 
         FIG. 33  is a schematic diagram showing the mold of this invention when a space enclosed by an upper mold and a lower mold is filled with a soften resin; 
         FIG. 34  is a schematic diagram showing the mold of this invention when the upper mold is released from the lower mold; 
         FIG. 35  is a schematic diagram showing the lens plate of the lens array of this invention after the molding is finished; 
         FIG. 36  is a perspective view of the lens plate of the lens array of this invention; 
         FIG. 37  is a figure showing a reading apparatus of this invention; 
         FIG. 38  is a figure showing the reading head arranged on the reading apparatus of this invention; and 
         FIG. 39  is a schematic diagram showing the optical system of the reading head of this invention. 
     
    
    
     PREFERRED EMBODIMENTS 
     An exposure device, an image forming apparatus, and a reading apparatus of this invention will be hereinafter described. This invention is not limited to the embodiments as described below, and the embodiments can be arbitrarily changed without deviating from the gist of this invention. 
     First Embodiment 
       FIG. 1  is a figure describing the structure of the image forming apparatus having the exposure device according to this invention. The image forming apparatus  100  uses toner made of resin including pigments as a color material to form an image on a print medium P based on image data. The image forming apparatus  100  is, for example, an electrophotographic printer, a facsimile, a multi-function apparatus, or the like. In the below description, it is assumed that the image forming apparatus  100  provides a color image, but the image forming apparatus  100  may provide a black and white image. 
     As shown in  FIG. 1 , the image forming apparatus  100  has a paper cassette  60  containing the print medium P on which an image has not yet been formed. A feeding roller  61  rotates to feed the print medium P contained in the paper cassette  60 . Conveyance rollers  62 ,  63  downstream of the feeding roller  61  further convey the print medium P to a transfer belt  81  at a predetermined time interval, and the print medium P is placed on the transfer belt  81 . The transfer belt  81  is rotated by a motor, not shown, at a rotation speed according to a printing speed. 
     The image forming apparatus  100  uses an electrophotographic printing method, and has four image forming units  40 C,  40 M,  40 Y,  40 K respectively corresponding to four colors Cyan (C), Magenta (M), Yellow (Y), Black (K). The four image forming units  40 C,  40 M,  40 Y,  40 K are arranged along a transfer belt  81  from a feeding side to a discharge side of the print medium P. Each of the image forming units  40 C,  40 M,  40 Y,  40 K forms an image using a toner of each color on the print medium P placed on the transfer belt  81  rotated by a gear and the like transmitting a driving force of a motor, not shown. Specifically, the image forming units  40 C,  40 M,  40 Y,  40 K respectively have photosensitive drums  41 C,  41 M,  41 Y,  41 K as electrostatic latent image holders, charging rollers  42 C,  42 M,  42 Y,  42 K respectively charging peripheries of the photosensitive drums  41 C,  41 M,  41 Y,  41 K, exposure devices  3 C,  3 M,  3 Y,  3 K respectively forming electrostatic latent images upon selectively emitting light to peripheries of the photosensitive drums  41 C,  41 M,  41 Y,  41 K based on image data received from an external apparatus via an interface, not shown, developing devices  5 C,  5 M,  5 Y,  5 K respectively developing the electrostatic latent images formed on the peripheries of the photosensitive drums  41 C,  41 M,  41 Y,  41 K using toner to form tone images, toner cartridges  51 C,  51 M,  51 Y,  51 K containing toners to respectively supply the toners to the developing devices  5 C,  5 M,  5 Y,  5 K, transfer rollers  80 C,  80 M,  80 Y,  80 K transferring the toner images to the print medium P, and cleaning blades  43 C,  43 M,  43 Y,  43 K cleaning and collecting residual toners that are not transferred to the print medium P and are remaining on the peripheries of the photosensitive drums  41 C,  41 M,  41 Y,  41 K. The photosensitive drums  41 C,  41 M,  41 Y,  41 K, the charging rollers  42 C,  42 M,  42 Y,  42 K, and the transfer rollers  80 C,  80 M,  80 Y,  80 K are driven and rotated by gears and the like transmitting a driving force of the motor. The exposure devices  3 C,  3 M,  3 Y,  3 K, the developing devices  5 C,  5 M,  5 Y,  5 K, and the motor are connected to a power supply and a control unit, not shown. 
     In the image forming units  40 C,  40 M,  40 Y,  40 K as described above, a power supply, not shown, applies a predetermined voltage to the charging rollers  42 C,  42 M,  42 Y,  42 K, which uniformly charge the peripheries of the photosensitive drums  41 C,  41 M,  41 Y,  41 K to a voltage according to the control of the control unit. When charged portions of the peripheries of the photosensitive drums  41 C,  41 M,  41 Y,  41 K reach the exposure devices  3 C,  3 M,  3 Y,  3 K as the photosensitive drums  41 C,  41 M,  41 Y,  41 K rotate, the exposure devices  3 C,  3 M,  3 Y,  3 K respectively emit lights modulated according to the image to the photosensitive drums  41 C,  41 M,  41 Y,  41 K to form the electrostatic latent images. The image forming units  40 C,  40 M,  40 Y,  40 K attach the toners of each color respectively supplied from the developing devices  5 C,  5 M,  5 Y,  5 K to the formed electrostatic latent images, thus forming the toner images of each color. 
     As the photosensitive drums  41 C,  41 M,  41 Y,  41 K rotate, the portions of the peripheries having the toner images come in contact with the print medium P conveyed on the transfer belt  81  sandwiched between the photosensitive drums  41 C,  41 M,  41 Y,  41 K and the transfer rollers  80 C,  80 M,  80 Y,  80 K arranged opposite to the photosensitive drums  41 C,  41 M,  41 Y,  41 K. When the toner images come in contact with the print medium P, the toner images of each color developed by the image forming units  40 C,  40 M,  40 Y,  40 K are successively transferred to the print medium P according to the control of the control unit so that the toner images are overlapped with each other. At this moment, a power supply, not shown, applies a predetermined voltage to the transfer rollers  80 C,  80 M,  80 Y,  80 K. 
     When the transfer is completed, the image forming units  40 C,  40 M,  40 Y,  40 K use the cleaning blades  43 C,  43 M,  43 Y,  43 K to clean the residual toners remaining on the peripheries of the photosensitive drums  41 C,  41 M,  41 Y,  41 K according to the control of the control unit. The cleaning blade  82  is used to clean the residual toner remaining on the surface of the transfer belt  81  according to the control of the control unit. The image forming apparatus  100  causes the image forming units  40 C,  40 M,  40 Y,  40 K as described above to successively form the images of each color on the print medium P to form a color image. In the image forming apparatus  100 , the print medium P is conveyed to a fixing device  9  while the print medium  9  is held on the transfer belt  81  with electrostatic attraction. 
     The image forming apparatus  100  has the fixing device  9  downstream of the image forming units  40 C,  40 M,  40 Y,  40 K. The fixing device  9  has a fixing roller made by, for example, adhering an elastic member on the outer periphery of a metal hollow roller, and also has a pressurizing roller for pressurizing the print medium P with the fixing roller. The pressurizing roller is arranged opposite to and in contact with the fixing roller, and has a nip portion at which the print medium P is sandwiched. A heater or a halogen lamp, not shown, generating heat or light with the supply of electricity from a power source, not shown, is arranged in the inside of the fixing roller. In the fixing device  9 , the control unit controls the heater to generate heat or the halogen lamp to emit light, so that the fixing roller is heated. The fixing device  9  rotates the fixing roller and the pressurizing roller to pass the print medium P through the nip portion, so that the print medium P is heated and pressurized and the toner thereon is melted. Thus, the toner image is fixed on the print medium with heat. Once the fixing device  9  in the image forming apparatus  100  fixes the image on the print medium P, the print medium P is conveyed and discharged to the outside of the image forming apparatus  100  as a conveyance roller  64  and a discharge roller  65  rotate, and is placed on a paper discharge unit  7 . 
     The image forming apparatus  100  has an external interface, not shown, for receiving print data from an external apparatus connected to and capable of communicating with the image forming apparatus  100 . The image forming apparatus  100  also has a control unit for receiving the print data via the external interface and for controlling the entire image forming apparatus  100 . 
     In the image forming apparatus  100 , the exposure devices  3 C,  3 M,  3 Y,  3 K are configured as described later. In the image forming apparatus  100 , all of the four image forming units  40 C,  40 M,  40 Y,  40 K corresponding to the four colors Cyan (C), Magenta (M), Yellow (Y), and Black (K) are substantially the same with each other. Thus, in the below description, reference numerals are used without the alphabetical portion C, M, Y, and K. For example, “the image forming unit  40 ” is recited instead of “the image forming units  40 C,  40 M,  40 Y,  40 K”. 
       FIG. 2  is a cross section of the exposure device  3  of this invention. As shown in  FIG. 2 , the exposure device  3  faces the photosensitive drum  41 , and is arranged at a predetermined distance from the photosensitive drum  41 . The exposure device  3  has a lens array  1  held by a holding member  34 , and also has an LED (Light Emitting Diode) array  30  as a light-emitting unit having multiple light emitting elements  35  formed on a circuit board  33  in the holding member  34 . The circuit board  33  has a driver IC  31  controlling light emission of the light emitting elements  35  of the LED array  30 . The LED array  30  and the driver IC  31  are connected via a wire  32 . 
     In the first embodiment, the LED array having the light emitting elements  35  is described as an example. But this invention is not limited to the LED array. For example, an organic EL (Electroluminescence) element may be used as a light source. A semiconductor laser generally used as an exposure device of an image forming apparatus may be used. The exposure device may use a light source such as a fluorescent light, a halogen lamp, and the like in combination with a shutter having a liquid crystal element. 
       FIG. 3  is a figure describing the structure and arrangement of the light emitting elements in the LED array. The multiple light emitting elements  35  are arranged on the LED array  30  in a predetermined direction, and are evenly spaced at a predetermined arrangement interval EP, and are connected to electrodes  36 . The light emitting elements  35  are in a rectangular shape, and have a predetermined length EY in a direction parallel to the arrangement direction of the light emitting elements  35  and a predetermined length EX in a direction perpendicular to the arrangement direction of the light emitting elements  35 . The size and the arrangement interval EP of the light emitting elements  35  are changed as necessary according to a resolution of the exposure device. For example, in a case of the exposure device of 600 dpi, the length EY is 21 μm, the length EX is 21 μm, and the arrangement interval EP is 42 μm. In a case of the exposure device of 1200 dpi, the length EY is 10 μm, the length EX is 10 μm, and the arrangement interval EP is 21 μm. In a case of the exposure device of 2400 dpi, the length EY is 5 μm, the length EX is 5 μm, and the arrangement interval EP is 10.6 μm. 
     When the control unit of the image forming apparatus transmits a control signal for the exposure device  3  based on image data, the exposure device  3  as described above causes the light emitting elements  35  to emit light at a light quantity according to the control signal of the driver IC 31 . The light beam from the light emitting element  35  enters into the lens array  1  as described later in detail, and forms an optical image on the photosensitive drum  41 . 
       FIG. 4  is a plan view of the lens array in the LED array.  FIG. 5  is a perspective view of the lens array. As shown in  FIGS. 4 and 5 , the lens array  1  has lens plates 11 as the first and second lens groups, micro lenses  12  as the first and second lenses, and a shielding unit  13  as a shielding unit. The shielding unit  13  has two comb members  13   a  having a length t in a direction of an optical axis and having apertures evenly spaced at an interval PY on one side of the comb member  13   a , and also has a separating plate  13   b  having a thickness Tb and arranged between the two comb members  31   a . As shown in  FIG. 6 , the aperture is in a shape of a circle having a radius RA, a distance PA exists between the center of the separating plate and the center of the circle (hereinafter referred to as the center of the aperture) having the radius RA, and the aperture is in a shape enclosed by the circle having the radius RA and a line arranged at a distance PA-Tb/ 2  from the center of the circle. The apertures of the comb members  13   a  face the apertures of the other comb members  13   a  via the separating plate  13   b . The two comb members  13   a  are arranged so that the distance between the center of the aperture of the comb member  13   a  and the center of the aperture of the other comb member  13   a  on the opposite side over the separating plate  13   b  is PN. The comb members  13   a  and the separating plate  13  are formed with a material shielding the light beams from the light sources. 
     As shown in  FIG. 5 , the shielding unit  13  has the two lens plates  11  to shield the apertures so that the shielding unit  13  is sandwiched between the two lens plates  11 . That is, a lens pair is formed with the two micro lenses  12  facing each other via the aperture of the shielding unit  13 . On the lens plate  11 , multiple micro lenses  12  are arranged in two rows in a staggered arrangement, and portions of the micro lenses  12  are arranged to overlap with the adjacent micro lenses  12 . The micro lenses are arranged so that the interval between the micro lenses  12  on the same row is PY and that the distance between the center of the micro lens  12  and the width-wise center of the lens plate  11  is PA. 
     The micro lens  12  arranged on the lens plate  11  has two rotationally symmetrical high-order aspherical surfaces defined by Formula 1 below. In the cross section of the micro lens  12  taken along parallel to the lens plate  11  as shown in  FIG. 7 , the circle of the micro lens  12  has the radius RL, the distance between the center of the circle of the micro lens  12  and the center of the circle of the adjacent micro lens  12  is PN, and the micro lens  12  has a shape enclosed by the circle having the radius RL and two lines arranged at the distance PN/2 from the center of the circle having the radius RL. 
     
       
         
           
             
               
                 
                   
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     This function z(r) is defined in a rotating coordinate system in which a coordinate in a direction of the radius is r and a direction parallel to the optical axis of the micro lens  12  is z axis. The origin of the coordinate axes is the vertex of the aspherical surface of each micro lens  12 . The coordinate of z axis increases in a direction from the light emitting plane to the imaging plane in the lens array  1 . C denotes a curvature radius, A denotes the fourth-order aspherical surface coefficient, and B denotes the sixth-order aspherical surface coefficient. The micro lens  12  of this invention is not limited to the one having the rotationally symmetrical high-order aspherical surface, but may have a spherical surface. Furthermore, the micro lens  12  may have a conic surface such as a parabola surface, an ellipse surface, a hyperboloid surface, and the like, and may have an aspherical toroid surface and an aspherical cylinder surface asymmetrical with respect to the optical axis. The micro lens  12  may have a publicly-known free-form surface. 
       FIG. 8  is an A-A cross section of  FIG. 4 .  FIG. 9  is a cross section taken along line B-B in  FIG. 4 . As shown in  FIG. 8 , in the micro lens  12  as described above, the central axis of an inner side curved surface  12   b  on a side of the shielding unit  13  is offset by a length ECX in a widthwise direction of the lens plate  11  from the central axis of an outer side curved surface  12   a  opposite to the inner side curved surface  12   b . That is, the central axis of the outer curved surface  12   a  of the micro lens  12  in the upper side of  FIG. 8  is offset to the left by the length ECX from the central axis of the inner side curved surface  12   b . Similarly, the micro lens  12  on the opposite side of the shielding unit  13  has the optical axes offset by the length ECX in the widthwise direction of the lens plate  11 , but the optical axis is offset to the right, which is opposite to the offset direction of the micro lens  12  of the upper side of  FIG. 8 . 
     As shown in  FIG. 9 , the central axis of the inner side curved surface  12   b  of the micro lens  12  is offset by a length ECY in a longitudinal direction of the lens plate  11  from the central axis of the outer side curved surface  12   a . This offset direction in the longitudinal direction of the lens plate  11  is different from the above-described offset direction in the widthwise direction of the lens plate  11 . The central axis of the outer side curved surface  12   a  of each micro lens  12  is offset to the left from the central axis of the inner side curved surface  12   b . The off-center distance of the micro lens  12  having the offsets as described above is a length EC that is an offset between the central axis of the outer side curved surface  12   a  and the central axis of the inner side curved surface  12   b . The off-center distance EC of the micro lens  12  can be calculated from a Pythagorean theorem EC 2 =ECY 2 +ECX 2  using the offset ECY in the longitudinal direction of the lens plate  11  and the offset ECX in the widthwise direction of the lens plate  11 . 
       FIG. 10  is a schematic diagram showing the optical system according to this invention. As shown in  FIG. 2 , the lens array  1  as described above is arranged between the photosensitive drum  41  and the LED array  30  at predetermined distances therefrom. At this moment, the surface of the photosensitive drum  41  is the imaging plane, and the surface of the LED array  30  is the object plane (hereinafter referred to as a light emitting plane in the first embodiment). It is assumed that the thickness of the micro lens  12  is LT, and the distance between each pair of micro lenses  12  is LS. It is assumed that the distance between the light emitting plane (the surface of the LED array  30 ) and the outer side curved surface  12   a  of the micro lens  12  on the light emitting plane side is LO, and that the distance between the imaging plane (the surface of the photosensitive drum  41 ) and the outer side curved surface  12   a  of the micro lens  12  on the imaging plane side is LI. The distance between the imaging plane and the light emitting plane is defined as TC. In this invention, it is assumed that an optical axis S of the micro lens  12  whose central axis is offset as described above is a line connecting the vertexes of each of the inner side curved surfaces  12   b  and the center of the aperture of the shielding unit  13 . 
     The lens plate  11  constituting the lens array  1  as described above is made of, for example, a cycloolefin optical resin (made by ZEON CORPORATION under the trade name of ZEONEX E48R). The lens plate  11  and the multiple micro lenses  12  may be integrally formed through injection molding. For example, the micro lenses  12  have the arrangement intervals PY=1.2 mm, PA=0.2 mm, and PN=0.721 mm. Although the lens plate  11  is integrally formed including the multiple micro lenses  12  in this embodiment, this invention is not limited thereto. The micro lenses  12  may be formed separately and fixed at the predetermined arrangement interval. 
     The shielding unit  13  constituting the lens array  1  uses, for example, a polycarbonate, and can be formed through resin molding. The shielding unit  13  has, for example, the length in the optical axis direction t=2.5 mm, the aperture radius RA=0.45 mm, and the thickness of the separating plate  13   b  Tb=0.2 mm. The shielding unit  13  is not necessarily made though the resin molding, but may be formed through cutting operation. Alternatively, the shielding unit for shielding the light from the light source may be formed with the transparent material that has a shielding pattern thereon. The shielding unit may be formed on portions of the lens plate  11  to form the shielding pattern. Portions of the lens plate  11  may be roughened to shield the light, and furthermore, portions of the lens plate  11  may be cut off to shield the light. 
       FIG. 11  is a figure showing the path of the light in the lens array  1 . Among the members of the exposure device of this invention,  FIG. 11  shows only the light emitting elements, some of the multiple micro lenses, the shielding unit, and the photosensitive drum. In the lens array  1 , the lens plate  11  on the side of the light emitting elements  35  has the multiple micro lenses  12 , namely, micro lenses MLI 1 , MLI 2 , MLI 3 , MLI 4 , . . . , and the lens plate  11  on the side of the imaging plane has the micro lenses  12 , namely, micro lenses MLO 1 , MLO 2 , MLO 3 , MLO 4 , . . . as shown in  FIG. 11 . 
     When the light emitting device  35  emit light, the emitted light R 1 , R 2 , R 3  enters into the micro lens MLI 1  nearest to the light emitting element  35 . The light R 1 , R 2 , R 3  having entered the micro lens MLI 1  is once converged in the aperture of the shielding unit  13 , thereafter enters into the micro lens MLO 1 , and forms an optical image I on the photosensitive drum  41 . A light R′ having entered the micro lens MLI 2  arranged next to the micro lens MLI 1  nearest to the light emitting element  35  is shielded by the shielding unit  13 . Accordingly, an optical image I′ will not be formed on the photosensitive drum  41 . 
     As described above, the light emitted from the light emitting element  35  is converged on the photosensitive drum  41  as an same-size erect image through the lens array, thus forming an exposure image of the light emitting element  35 . The shielding unit  13 A shields a so-called stray light, namely, a light that does not form an exposure image. Thus, the exposure image of the light emitting element  35  becomes clear. 
     Hereinafter described is a light quantity distribution of the exposure image formed by the exposure device of this invention. The lens array used in the exposure device has lengths and coefficients as shown in Table 1. 
     
       
         
           
               
               
               
             
               
                 TABLE 1 
               
               
                   
               
               
                 Variable 
                 Description 
                 Value in the Embodiment 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
            
               
                 PY 
                 Interval between adjacent micro lenses in 
                 1.200 
               
               
                   
                 arrangement direction (mm) 
               
               
                 PN 
                 Interval between adjacent micro lenses in a oblique 
                 0.721 
               
               
                   
                 direction (mm) 
               
               
                 PA 
                 Distance between center of lens and center of 
                 0.200 
               
               
                   
                 separating unit in widthwise direction 
               
               
                 LO 
                 Distance between object and lens surface (mm) 
                 2.300 
               
               
                 RL 
                 Radius of micro lens (mm) 
                 0.500 
               
               
                 CO 
                 Curvature radius of outer side curved surface (mm) 
                 0.6915 
               
               
                 AO 
                 4th-order aspherical surface coefficient of outer 
                 −0.3150 
               
               
                   
                 side curved surface 
               
               
                 BO 
                 6th-order aspherical surface coefficient of outer 
                 −0.3378 
               
               
                   
                 side curved surface 
               
               
                 CI 
                 Curvature radius of inner side curved surface (mm) 
                 −1.3786 
               
               
                 AI 
                 4th-order aspherical surface coefficient of inner 
                 0.6100 
               
               
                   
                 side curved surface 
               
               
                 Bl 
                 6th-order aspherical surface coefficient of inner 
                 1.2575 
               
               
                   
                 side curved surface 
               
               
                 LT 
                 Thickness of lens (mm) 
                 1.000 
               
               
                 LS 
                 Distance between the surfaces of the lenses (mm) 
                 2.520 
               
               
                 RA 
                 Radius of aperture of shielding unit (mm) 
                 0.450 
               
               
                 t 
                 Thickness of shielding unit in optical axis direction 
                 2.500 
               
               
                   
                 (mm) 
               
               
                 LI 
                 Distance between imaging plane and surface of 
                 2.300 
               
               
                   
                 lens 
               
               
                 TC 
                 Distance between imaging plane and object plane 
                 9.120 
               
               
                   
                 (mm) 
               
               
                   
               
            
           
         
       
     
       FIG. 12  is a light quantity distribution in a case where the axial off-center distance EC of the micro lenses is less than half of the arrangement interval EP.  FIG. 13  is a light quantity distribution in a case where the axial off-center distance EC of the micro lenses is more than half of the arrangement interval EP. It should be noted that these are the light quantity distributions in a case where every two light emitting elements  35  are made to emit light, namely, where the interval between the light emitting elements  35  emitting light is 2EP. 
     As shown in  FIG. 12 , in the case where the axial off-center distance EC of the micro lenses is less than half of the arrangement interval EP, the exposure image of the light emitting elements  35  has high contrast. On the other hand, as shown in  FIG. 13 , in the case where the axial off-center distance EC of the micro lenses is more than half of the arrangement interval EP, the maximum light quantity of the exposure image becomes lower, and the light quantity distribution of each light emitting element becomes wider on the lower portion of the  FIG. 13 , and thus, the contrast of the exposure image of the light emitting elements  35  becomes lower. 
     Hereinafter described is a measurement result of an MTF (Modulation Transfer Function) showing the resolution of the optical image. It should be noted that the MTF shows the resolution of the exposure device, and shows the contrast of the light quantity in the optical image made by the LED array emitting light in the exposure device. Where the MTF is 100%, the optical image has the highest contrast, and the exposure device produces a high resolution. As the MTF becomes lower, the optical image has a lower contrast, and the exposure device produces a lower resolution. The MTF (%) is defined as MTF=(I max −I min )/(I max +I min )×100(%), where the maximum light quantity of the optical image is I max , and the minimum light quantity between the two adjacent optical images is I min . 
     To measure the MTF, a microscopic digital camera is used to take a picture of the optical image at a position away by the distance LI (mm) as recited in Table 1 from the end surface of the lens array  1  of the exposure device  3  on the imaging plane side (the side of the photosensitive drum  41 ) and by analyzing the taken picture to figure out the light quantity distribution of the optical image of the light emitting element  35  to calculate the MTF. In the lens array  1  used in this measurement, the lens plate  11  on the light emitting plane side (the side of the LED array  30 ) has the micro lenses  12  whose optical axes are offset only in the arrangement direction of the light emitting element  35  and are not offset in a direction perpendicular to the arrangement direction, and the lens plate  11  on the object plane side (the side of the photosensitive drum  41 ) has the micro lenses  12  whose optical axes are not offset. The lens arrays  1  having various axial off-center distances EC are used to measure the MTF for each resolution. Every two light emitting elements  35  are made to emit light in the exposure device including the LED array  30  having the resolution 600 dpi as shown in  FIG. 14 , the exposure device including the LED array  30  having the resolution 1200 dpi as shown in  FIG. 15 , the exposure device including the LED array  30  having the resolution 2400 dpi as shown in  FIG. 16 . 
     As shown in  FIG. 14 , in the exposure device having the resolution 600 dpi, the MTF becomes 80% or more where EC is less than 21 μm. In the range where EC is more than 21 μm, the MTF curve becomes steeper. This means that the variation of the MTF with respect to the variation of EC is large in the range where EC is more than 21 μm. As shown in  FIG. 15 , the MTF is equal to or more than 80% in the exposure device having the resolution 1200 dpi where EC is less than 10.6 μm. The MTF curve becomes steeper in the range where EC is more than 10.6 μm. This means that the variation of the MTF with respect to the variation of EC is larger in the range where EC is more than 10.6 μm. As shown in  FIG. 16 , the MTF is equal to or more than 80% in the exposure device having the resolution 2400 dpi where EC is less than 5.3 μm. The MTF curve becomes steeper in the range where EC is more than 5.3 μm. This means that the variation of the MTF with respect to the variation of EC is large in the range where EC is more than 6 μm. 
     Specifically, in a case of the exposure device having the resolution 600 dpi as shown in  FIG. 14 , the MTF suddenly drops to less than 80% where the axial off-center distance EC becomes more than 21 μm. The resolution is 600 dpi (600 dots per inch) and 1 inch is 25400 μm, and accordingly, the distance EP between the adjacent light emitting elements is EP=25400/600=42 μm. It is preferable to satisfy EC&lt;21 μm=42 μm/2=EP/2 because a picture of good quality cannot be obtained unless the MTF is equal to or more than 80%. Thus, EC should be less than EP/2. Similarly, in a case of the exposure device having the resolution 1200 dpi as shown in  FIG. 15 , the MTF suddenly drops to less than 80% where the offset EC becomes more than 10.6 μm. The resolution is 1200 dpi and 1 inch=25400 μm, and accordingly, the distance EP between the adjacent light emitting elements is EP=25400/1200≈21.2 μm. Considering that the MTF should be equal to or more than 80%, it is preferable to satisfy EC&lt;10.6 μm=21.2 μm/2=EP/2. Thus, EP should be less than EP/2. Similarly, in a case of the exposure device having the resolution 2400 dpi as shown in  FIG. 16 , the MTF suddenly drops to less than 80% where the offset EC becomes more than 5.3 μm. The resolution is 2400 dpi and 1 inch=25400 μm, and accordingly, the distance EP between the adjacent light emitting elements is EP=25400/2400≈10.6 μm. Considering that the MTF should be equal to or more than 80%, it is preferable to satisfy EC&lt;5.3 μm=10.6 μm/2=EP/2. Thus, EC should be less than EP/2. As described above, EC&lt;EP/2 can be generally derived from the data based on the three resolutions, namely, 600 dpi, 1200 dpi, and 2400 dpi. 
     Subsequently, the lens array according to this embodiment was installed in the image forming apparatus, namely, a color LED printer. The image quality of images formed by the LED color printer having the lens array according to this embodiment was evaluated. The quality of an image formed by each of the image forming apparatuses was evaluated by forming and evaluating an image in which every two pixels are printed in the entire pixels. In the image forming apparatus of the resolution 600 dpi, the lens array including the lens plate whose axial off-center distance EC is less than 21 μm produced an image of good quality. On the other hand, the lens array including the lens plate whose axial off-center distance EC is equal to or more than 22 μm produced streaks and uneven density. In the image forming apparatus of the resolution 1200 dpi, the lens array including the lens plate whose axial off-center distance EC is less than 11 μm produced an image of good quality. On the other hand, the lens array including the lens plate whose axial off-center distance EC is equal to or more than 12 μm produced streaks and uneven density. In the image forming apparatus of the resolution 2400 dpi, the lens array including the lens plate whose axial off-center distance is less than 5 μm produced an image of good quality. On the other hand, the lens array including the lens plate whose axial off-center distance EC is equal to or more than 6 μm produced streaks and uneven density. Thus, an image of good quality without streaks and uneven density can be obtained where the measured MTF value is equal to or more than 80% and where the variation of the MTF is small with respect to EC. 
     Hereinafter described is the reason why an image of good quality can be obtained where the MTF is less than 80%. Essentially, it is necessary to sufficiently raise the potential of a portion of the electrostatic latent image to which the toner should not be attached in a printed image, and such portion should be dark in an image formed by the LED head. However, if the MTF value does not reach 80%, a light from the LED head may be emitted to a portion of an image that should not be exposed to light, thereby reducing the potential of such portion that should have a sufficiently high potential in the electrostatic latent image. As described above, the toner attached to the portion having the high potential produces streaks and dark spots in the printed image obtained though the above-described image evaluation method. Thus, a threshold value should be MTF=80%. 
     From this result and the relationship between the resolution of the exposure device and the arrangement interval of the light emitting elements, it can be understood that the axial off-center distance EC of the micro lens  12  and the arrangement interval EP of the light emitting elements  35  in the exposure device  3  should satisfy a relationship EC&lt;EP/2 in order to obtain good optical characteristics. 
     This embodiment can prevent deterioration of the resolution conventionally occurring with lenses having axial offsets. In the exposure device according to this embodiment, a formula EC&lt;EP/2 is satisfied in the relationship between the axial off-center distance EC and the arrangement interval EP of the light emitting elements  35  in the exposure device  3 . The axial off-center distance EC is an offset distance between the central axis of the micro lens  12  on the light emitting plane side (the side of the LED array  30 ) and the central axis of the micro lens  12  on the imaging plane side (the side of the photosensitive drum  41 ). The image forming apparatus  100  having the exposure device can form an image on a printing medium P according to print data without producing streaks, uneven density, and the like deteriorating the quality of the image printed on the print medium P. 
     The exposure device and the image forming apparatus of this invention does not need the high-precision micro lenses  12  whose axial off-center distance EC is small. The exposure device and the image forming apparatus of this invention can use the lens plates  11  in which the multiple micro lenses  12  are formed integrally to constitute the lens group, and can also use the micro lenses  12  formed though resin molding. That is, an image of good quality can be obtained, even where the lens plate  11  having the integrally-formed multiple micro lenses  12  is used. An image of good quality can also be obtained, even where the micro lenses  12  formed through resin molding is used. 
     Second Embodiment 
     In the second embodiment, an image reading apparatus is described. The image reading apparatus according to the second embodiment uses the lens array used in the exposure device in the image forming apparatus according to the first embodiment. 
       FIG. 17  is a cross section of the reading apparatus according to this invention. As shown in  FIG. 17 , the reading apparatus  101  is held by a holding member  114 . The reading apparatus  101  has the lens array  1  being the same as the first embodiment, light receiving elements  111  formed on a wiring circuit board  112  in the holding member  114  for converting a received optical image into an electric signal, and a light source  113  arranged between the holding member  114  and a document table  115  supporting a document. The light source  113  emits light to the document to be read. The light receiving elements  111  are arranged on a line and are spaced at an interval RP on the wiring circuit board  112 . 
     The arrangement interval RP is changed as necessary according to the resolution of the reading apparatus. For example, the arrangement interval RP is 42 μm in a case of the exposure device of 600 dpi. The arrangement interval RP is 21 μm in a case of the exposure device of 1200 dpi. The arrangement interval RP is 10.6 μm in a case of the exposure device of 2400 dpi. 
       FIG. 18  is a schematic diagram showing the optical system in the reading apparatus of this invention. The lens array  1  in the second embodiment is the same as the lens array  1  according to the first embodiment. The lens array  1  is arranged between the light receiving elements  111  and the document on the document table  115  at predetermined distances therefrom. At this moment, the surfaces of the light receiving elements  111  is an optical plane, and the surface of the document to be read is an object plane. It is assumed here that the thickness of the micro lens  12  is LT, that the distance between the micro lenses  12  is LS, that the distance between the outer side curved surface  12   a  of the micro lens  12  on the light emitting plane side and the outer side curved surface  12   a  of the micro lens  12  on the object plane side (the surface of the document) is LO, and that the distance between the imaging plane (the surface of the light receiving element  111 ) and the outer side curved surface  12   a  of and the object plane is defined as TC. The optical axis S of the micro lens  12  whose central axis is offset as described above is a line connecting the vertexes of the inner side curved surfaces  12   b  and the center of the aperture of the shielding unit  13 . 
     In the reading apparatus as described above, the light emitted from the light source  113  is reflected on the surface of the document, not shown, arranged on the top surface of the document table  115 . Some of the reflected light from the document passes through the lens array  1  and forms an optical image on the surface of the light receiving element  111 . The light receiving element  111  converts the optical image into the electric signal. An image processing unit, not shown, generates image data based on the electric signal corresponding to the optical image of the document. 
     Using the reading apparatus as described above with various lens arrays  1  including the micro lenses  12  of different axial off-center distances EC according to the resolutions, a document was actually read and image data thereof were generated. The document to be read by the reading apparatus has an image in which every two pixels are printed in the entire pixels. Such document was prepared for each resolution. That is, a document having the dots printed at every 84 μm on the print medium P was used for the resolution 600 dpi. A document having the dots printed at every 42 μm on the print medium P was used for the resolution 1200 dpi. A document having the dots printed at every 21 μm on the print medium P was used for the resolution 2400 dpi. 
     First, with the reading apparatus of the resolution 600 dpi, image data of good quality faithfully reproducing the image on the document was obtained using the lens array  1  consisting of the micro lenses  12  whose axial off-center distance EC is less than 21 μm. On the other hand, in a case where the lens array  1  consisting of the micro lenses  12  whose axial off-center distance EC is 22 μm or more, the dots printed at every two pixels were sometimes incorrectly read as two successive dots, and some dots were not read at all. With the reading apparatus of the resolution 1200 dpi, image data of good quality faithfully reproducing the image on the document was obtained using the lens array  1  consisting of the micro lenses  12  whose axial off-center distance EC is less than 11 μm. On the other hand, in a case where the lens array  1  consisting of the micro lenses  12  whose axial off-center distance EC is 11 μm or more, the dots printed at every two pixels were sometimes incorrectly read as two successive dots, and some dots were not read at all. With the reading apparatus of the resolution 2400 dpi, image data of good quality substantially the same as the document was obtained using the lens array  1  consisting of the micro lenses  12  whose axial off-center distance EC is less than 5 μm. On the other hand, in a case where the lens array  1  consisting of the micro lenses  12  whose axial off-center distance EC is 5 μm or more, the dots printed at every two pixels were sometimes incorrectly read as two successive dots, and some dots were not read at all. 
     From this result, it can be understood that a formula EC&lt;RP/2 is satisfied in the relationship between the arrangement interval RP of the light receiving elements  111  and the axial off-center distance EC of the micro lens  12  in the reading apparatus  101  providing good optical characteristics. 
     As hereinabove described, the image data faithfully reproducing the image on the document can be obtained by satisfying a formula EC&lt;RP/2 in the relationship between the arrangement interval EP of the light receiving elements  111  and the axial off-center distance EC in the reading apparatus  101 , namely, the offset distance between the central axis of the micro lens  12  on the object plane side (the side of the document) and the central axis of the micro lens  12  on the imaging plane side (the side of the light receiving element). 
     The reading apparatus according to this invention does not need the high-precision micro lenses  12  whose axial off-center distance EC is small. The reading apparatus according to this invention can use the lens plates  11  in which the multiple micro lenses  12  are formed integrally to constitute the lens group, and can also use the micro lenses  12  formed though resin molding. That is, the reading apparatus according to this invention can provide image data faithfully reproducing the image on the document, even where the reading apparatus uses the lens plates  11  having the integrally-formed multiple micro lenses  12 . Furthermore, the reading apparatus can also provide image data faithfully reproducing the image on the document, even where the reading apparatus uses the micro lenses  12  formed through resin molding. 
     Third Embodiment 
     In the third embodiment, the lens array used in the exposure device in the image forming apparatus will be described with respect to the structure, the operation, the optical characteristics, and the like. 
     The lens array will be specifically described with reference to  FIGS. 19 to 22 .  FIG. 19  is a plan view of the lens array.  FIG. 20  is a plan view of a shielding unit  23 .  FIG. 21  is a cross section taken along line C-C in  FIG. 19 .  FIG. 22  shows the shape of the aperture of the shielding unit  23  of the lens array. The lens array consists of a lens plate  21  and the shielding unit  23 . The lens array is consisted of the lens plates  21  serving as lens aggregations and the shielding unit  23 . The lens array has lens groups, each of which consists of two micro lenses  22  arranged to have the same optical line. The lens groups are arranged in two rows in a direction perpendicular to the optical axes. The lens plate  21  has multiple micro lenses  22  arranged thereon. Any one of the micro lenses  22  has the first curved surface  22   a  and the second curved surface  22   b . The second curved surface  22   b  faces the shielding unit  23 . The first curved surface  22   a  is a surface opposite to the second curved surface  22   b . The vertex of the first curved surface  22   a  and the vertex of the second curved surface  22   b  are arranged on the same optical line. The micro lenses  22  are arranged on the lens plate  21 , and are evenly spaced at an interval PY. The micro lenses  22  are arranged in two rows in the arrangement direction. The two rows are spaced apart by an interval PX. In the third embodiment, PX is less than PY. The micro lens  22  has a radius RL. The distance between the obliquely adjacent micro lenses  22  is PN. The micro lenses  22  are arranged so that portions of the micro lens  22  overlaps with the adjacent micro lenses  22 . The lens plate  21  is made of a transparent material that allows light from the light emitting elements to pass through the lens plate  21 . 
     The shielding unit  23  is made of a material that shields the light emitting elements. The shielding unit  23  is formed with polycarbonate through injection molding. The shielding unit  23  has an aperture  23   a  formed as a diaphragm. An arrangement interval of the apertures  23   a  is PY that is the same as the arrangement interval of the micro lenses  22 . The apertures  23   a  are arranged in two rows in the arrangement direction. The rows are spaced apart by the interval PX. The aperture  23   a  is in a shape enclosed by a cylindrical shape having a radius RA and a surface away by a distance (PX-TB)/2 from the central axis of the cylindrical shape. The aperture  23   a  and the micro lenses  22  are arranged so that the central axis of the cylindrical shape constituting the aperture  23   a  is the same as the optical axis of the micro lens  22 . 
     The structure and the operation of the lens array will be hereinafter specifically described with reference to  FIG. 23 .  FIG. 23  shows an optical path in the lens array. 
     The structure of the lens array will be described.  FIG. 23  is a cross section of the lens array taken along a surface perpendicular to the surface of the lens plate  21  to show the optical axis of the micro lenses  22 . The horizontal direction of  FIG. 23  is the arrangement direction of the micro lenses  22 . The first micro lens  22 - 1  is arranged at a position away by the distance LO from the object plane of the lens array. The second micro lens  22 - 2  is arranged at a position away by the distance LS from the first micro lens  22 - 1  to face the first micro lens  22 - 1 . The first micro lens  22 - 1  and the second micro lens  22 - 2  are arranged so that the optical axis of the first micro lens  22 - 1  is the same as the optical axis of the second micro lens  22 - 2 . The lens array forms the optical image at a position away by the distance LI from the micro lens  22 - 2  in the optical axis direction. The first micro lens  22 - 1  has a thickness LT 1  and a focal length F 1 . Through the first micro lens  22 - 1 , an object at a position away by the distance LO 1  forms an optical image on a surface away by the distance LI 1  in the optical axis direction. The second micro lens  22 - 2  has a focal length F 2 . Through the second micro lens  22 - 2 , an object at a position away by the distance LO 2  forms an optical image at a position away by the distance LI 2  in the optical axis direction. The distance LO between the object plane of the lens array and the first micro lens  22 - 1  is set to be the same as the distance LO 1 . The distance LS between the first micro lens  22 - 1  and the second micro lens  22 - 2  is made to be LS=LI 1 +LO 2 . 
     The distance LI between the second micro lens  22 - 2  and the imaging plane of the lens array is set to be the same as LI 2 . The first micro lens  22 - 1  and the second micro lens  22 - 2  may be the same with each other. Both of the first micro lens  22 - 1  and the second micro lens  22 - 2  have the thickness LT 1  and the focal length F 1 . Where the object at the position away by the distance LO 1  forms the optical image on the surface away by the distance LI 1  in the optical direction, the distance between the object plane of the lens array and the first micro lens  22 - 1  is set to be the same as the distance LO 1 . The interval LS between the first micro lens  22 - 1  and the second micro lens  22 - 2  is set to be LS=2*LI 1 . The first micro lens  22 - 1  and the second micro lens  22 - 2  are arranged to face each other so that the curved surface of the second micro lens  22 - 2  on the imaging plane side has the same shape as the curved surface of the first micro lens  22 - 1  on the object plane side. The distance between the second micro lens  22 - 2  and the imaging plane of the lens array is set to be the same as LO 1 . Thus, LI=LO. The focal length F 2  of the second micro lens  22 - 2  is the same as the focal length F 1  of the first micro lens  22 - 1 . Thus, F 2 =F 1 . Each curved surface of the micro lens  22  is formed with the rotationally symmetrical high-order aspherical surface defined by Formula 1 described above, so that spherical aberration can be corrected and a high resolution can be obtained. 
     The operation of the lens array will be hereinafter described. The light from an LED array  70  as an object  90   a  enters into the first micro lens  22 - 1 , and the first micro lens  22 - 1  forms an intermediate image  90   b  on an intermediate image surface MIP at a position away by the distance LI 1  in the optical axis direction. Through the second micro lens  22 - 2 , the intermediate image  90   b  forms an optical image  90   c . The optical image  90   c  is the same-size erect image of the object  90   a . The intermediate object  90   b  is an inverted reduced image of the object  90   a . The optical image  90   c  is an inverted enlarged image of the intermediate image  90   b  through the second micro lens  22 - 2 . The chief ray of light rays from each point of the surface of an object becomes parallel to the optical axis between the first micro lens  22 - 1  and the second micro lens  22 - 2 , which constitute a so-called telecentric optical system. The lens array arranged as described above produces a same-size erect image of the LED array  70 . Among light rays from the LED array  70 , stray lights not forming an optical image are shielded by the shielding unit  23 . 
     The lens array forms a same-size erect image of the LED array  70 , even where the lens having the same optical characteristics is used in the first micro lens  22 - 1  and the second micro lens  22 - 2 . A light from the LED array  70  as the object  90   a  enters into the first micro lens  22 - 1 , and the first micro lens  22 - 1  forms the intermediate image  90   b  on the intermediate image surface MIP at the position away by the distance LI 1  in the optical axis direction. The second micro lens  22 - 2  forms the optical image  90   c  of the intermediate image  90   b . The optical image  90   b  is the same-size erect image of the object  90   a . The telecentric optical system is constituted between the first micro lens  22 - 1  and the second micro lens  22 - 2 . The above-described optical arrangement enables the lens array to form the same-size erect image of the LED array  70 , even where the lens having the same optical characteristics is used in the first micro lens  22 - 1  and the second micro lens  22 - 2 . 
     Next, the optical characteristics of the micro lenses  22  formed on the lens array are described with reference to  FIG. 24 .  FIG. 24  is a cross section of the lens array taken along a surface perpendicular to the surface of the lens plate  21  to show the optical axis of the micro lenses  22 . 
     The horizontal direction of  FIG. 24  is the arrangement direction of the micro lenses  22 . The first micro lens  22 - 1  has the focal length F 1 . Specifically, the first micro lens  22 - 1  has the focal length F 1 . Specifically, the focal length F 1  is the distance between the first principal plane H 1 - 1  of the first micro lens  22 - 1  to the first focal plane FP 1 - 1 . The distance between the first principal plane H 1 - 1  and the object plane is SO. The second micro lens  22 - 2  has the focal length F 2 . Specifically, the focal length F 2  is the distance between the second principal plane H 2 - 2  of the second micro lens  22 - 2  to the second focal plane FP 2 - 2 . The distance between the second principal plane H 2 - 2  and the image plane is SI. The difference between the distance SO and the distance LO is inversely proportional to the curvature radius of the curved surface of the first micro lens  22 - 1  on the object plane side. Similarly, the difference between the distance SI and the focal length LI is inversely proportional to the curvature radius of the curved surface of the second micro lens  22 - 2  on the imaging plane side. But the difference between the distance SO and the distance LO and the difference between the distance SI and the distance LI can be ignored because the curvature radius of the curved surface of each micro lens  22  is sufficiently large in the lens array according to the third embodiment. Thus, SO is approximately equal to LO, and SI is approximately equal to LI. A vision radius RV of the first micro lens  22 - 1  can be obtained from Formula 2 below because: the chief ray of light rays from each point of the surface of an object becomes parallel to the optical axis between the first micro lens  22 - 1  and the second micro lens  22 - 2 ; the light rays are restricted by the side walls of the shielding unit  23 ; and the two triangles formed by the light ray, the object plane, the first principal plane of the first micro lens  22 - 1 , and the optical axis AXI of the first micro lens  22 - 1  are geometrically similar to each other. 
     
       
         
           
             
               
                 
                   RV 
                   = 
                   
                     RA 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         LO 
                         - 
                         
                           F 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                         
                       
                       
                         F 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                     
                   
                 
               
               
                 
                   Formula 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   2 
                 
               
             
           
         
       
     
     The relationship between the arrangement of the micro lens  22  and the vision radius RV will be specifically hereinafter described with reference to  FIGS. 25 and 26 .  FIG. 25  shows a schematic diagram showing the optical arrangement in a case where the micro lenses  22  integrally formed into the lens array are arranged in two rows.  FIG. 26  shows a schematic diagram showing the optical arrangement in a case where the micro lenses  22  integrally formed into the lens array are arranged in multiple rows. 
       FIG. 25  shows a schematic diagram showing the optical arrangement in a case where the micro lenses  22  integrally formed into the lens array are arranged in two rows.  FIG. 25  shows a case where the micro lenses  22  are arranged in two rows, where all the LED elements in the LED array  70  are covered by the vision of at least one micro lens  22 , and where the vision radius RV of all the optical images of the LED array  70  formed on the photosensitive drum  41  become the smallest. That is,  FIG. 25  is shows the condition where the vision radius RV of the micro lens  22  is the smallest while the lens array can operate. An intersecting point between the optical axis of the micro lens  22  and the object plane is  22   m , and the vision of the micro lens  22  is  22   n . The vision radius RV in this condition is expressed with Formula 3 below, where the interval of the micro lenses  22  in the arrangement direction is PY and where the interval of the micro lenses  22  in a direction perpendicular to the arrangement direction is PX. 
     
       
         
           
             
               
                 
                   RV 
                   = 
                   
                     
                       
                         
                           ( 
                           
                             PX 
                             2 
                           
                           ) 
                         
                         2 
                       
                       + 
                       
                         
                           ( 
                           
                             PY 
                             4 
                           
                           ) 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   Formula 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   3 
                 
               
             
           
         
       
     
     Formula 4 below expressing an operation condition of the lens array can be derived from Formula 2 and Formula 3, where the focal length of the micro lens  22  is F 1 , the maximum value of the distance between the optical axis of the micro lens  22  and where the inner sidewall of the aperture  23   a  of the shielding unit  23  is RA. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           ( 
                           
                             PX 
                             2 
                           
                           ) 
                         
                         2 
                       
                       + 
                       
                         
                           ( 
                           
                             PY 
                             4 
                           
                           ) 
                         
                         2 
                       
                     
                   
                   ≦ 
                   
                     RA 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         LO 
                         - 
                         
                           F 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                         
                       
                       
                         F 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                     
                   
                 
               
               
                 
                   Formula 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   4 
                 
               
             
           
         
       
     
       FIG. 26  shows a schematic diagram showing the optical arrangement in a case where the micro lenses  22  integrally formed into the lens array are arranged in multiple rows. The LED elements are arranged in array to form the LED array  70 .  FIG. 26  shows the positional relationship on the object plane between the LED array  70  and the optical axes of the micro lenses  22 .  FIG. 26  shows a case where all the LED elements in the LED array  70  are covered by the vision of at least one micro lens  22  on the outermost row and where the vision radius RV of the micro lenses  22  on the outermost row is the smallest. The vision radius RV in this condition can be obtained from Formula 5, where XO is the distance between the optical axis of the micro lenses on the outermost row and the LED array  70  in the direction perpendicular to the arrangement direction of the micro lenses  22  and to the optical axis of the micro lenses  22 , where PY is the distance between the micro lenses  22  in the arrangement direction of the micro lenses  22 , and where PX is the distance between the micro lenses in a direction perpendicular to the arrangement direction of the micro lenses  22 . 
     
       
         
           
             
               
                 
                   RV 
                   = 
                   
                     
                       
                         
                           ( 
                           XO 
                           ) 
                         
                         2 
                       
                       + 
                       
                         
                           ( 
                           
                             PY 
                             4 
                           
                           ) 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   Formula 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   5 
                 
               
             
           
         
       
     
     Formula 6 expressing the operation condition of the lens array can be derived from Formula 2 and Formula 5, where the focal length of the micro lens  22  is F 1 , the distance between the lens array and the object plane of the lens array is LO, and the maximum distance between the optical axis of the micro lens  22  and the inner side wall of the aperture  23   a  of the shielding unit  23  is RA. It should be noted that in a case where the micro lenses  22  are arranged in one row, the operating condition of the lens array is obtained by substituting XO=0 into Formula 6. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           ( 
                           XO 
                           ) 
                         
                         2 
                       
                       + 
                       
                         
                           ( 
                           
                             PY 
                             4 
                           
                           ) 
                         
                         2 
                       
                     
                   
                   ≦ 
                   
                     RA 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         LO 
                         - 
                         
                           F 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                         
                       
                       
                         F 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                     
                   
                 
               
               
                 
                   Formula 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   6 
                 
               
             
           
         
       
     
     The evaluation result of the optical characteristics of the lens array according to this embodiment will be hereinafter described with reference to  FIG. 27 .  FIG. 27  shows a schematic diagram showing an image in which every two pixels are printed among the pixels relating to the evaluation of the lens array. 
     The LED head including the lens array according to this embodiment were subjected to the measurement of the MTF (Modulation Transfer Function) representing the resolution of the optical image. The measurement result showed that all of the LED head have the MTF equal to or more than 80%. The MTF represents the resolution of the exposure device and the contrast of the light quantity of the optical image formed by the LED array  70  emitting light in the exposure device. Where the MTF is 100%, the optical image has the highest contrast, and the exposure device produces a high resolution. As the MTF becomes lower, the optical image has a lower contrast, and the exposure device produces a lower resolution. The MTF (%) is defined as MTF=(I max −I min )/(I max +I min )×100(%), where the maximum light quantity of the optical image is I max  and the minimum light quantity between the two adjacent optical images is I min . In this MTF measurement, a microscopic digital camera is used to take a picture of the optical image at a position away by the distance LI (mm) from the end surface of the lens array of the LED head on the side of the photosensitive drum  41  and by analyzing the taken picture to figure out the light quantity distribution of the optical image of the LED array  70  to calculate the MTF. In the measurement of the MTF, the LED head is used that has the LED array  70  whose arrangement interval is PD=0.0423 mm. The resolution of the LED head is 600 dpi. Thus, 600 pieces of the LED elements are arranged per 1 inch, i.e., approximately 25.4 mm, in the LED array  70 . The measurement was performed by causing every two LED elements to emit light in the LED array  70  having the lens array on the LED head. The LED array  70  according to this embodiment was installed in the image forming apparatus, namely, a color LED printer. The image quality of images formed by the LED color printer having the LED array  70  according to this embodiment was evaluated. The evaluation of the images of the image forming apparatus is done by forming an image in which dots are printed at every two pixels on all over a printing area as shown in  FIG. 27  and evaluating the image quality of the formed image. It should be noted that D 1  denotes a printed dot and D 2  denotes a non-printed dot. 
     In the third embodiment, the micro lens  22  is described as a rotationally symmetrical high-order aspherical surface. But this invention is not limited thereto. The micro lens  22  may be formed with surfaces having a spherical surface, an anamorphic aspherical surface, a parabola surface, an ellipse surface, a hyperboloid surface, a conic surface, and the like. In the third embodiment, the lens plate  21  is formed by injecting the resin into the mold and molding the resin into the shape of the mold. But the resin may also be used as the mold, and the lens plate  21  may be formed through cutting operation. In the third embodiment, the resin is used as the material of the lens plate  21 , but glass may also be used as the material of the lens plate  21 . In the third embodiment, the polycarbonate is used as the material of the shielding unit  23 , but other materials may also be used for the shielding unit  23 . In the third embodiment, the shielding unit  23  is formed through injection molding, but other molding and forming methods may be used. In the third embodiment, the LED array  70  having multiple LED elements is used as the light emitting unit. But the light emitting unit may use, for example, organic EL (Electroluminescence) elements and a semiconductor laser. Alternatively, the exposure device may be made that has the light emitting unit having a fluorescent light, a halogen lamp, and the like combined with a shutter consisting of a liquid crystal element. 
     As hereinabove described in the third embodiment, all the micro lenses  22  can be formed with high-precision, even where the lens array is long in the arrangement direction of the micro lenses. The exposure device having the lens array according to this embodiment can provide the optical image of sufficiently high contrast. The image forming apparatus having the lens array according to this embodiment can provide a printed image of high quality without streaks and dark and light spots. 
     Fourth embodiment 
     The fourth embodiment describes a manufacturing method of the lens array in the exposure device in the image forming apparatus according to the third embodiment. 
     For the sake of easy understanding of the manufacturing of the lens plate  21  of this invention, a manufacturing method and a mold for a conventional lens plate  19  are described first. Thereafter, a manufacturing method and a mold for the lens plate  21  of this invention will be described. Hereinafter, the mold for the conventional lens plate  19  will be specifically described with reference to  FIGS. 28 to 30 . 
       FIG. 28  is a schematic diagram showing a conventional mold  700  used for manufacturing the conventional lens plate  19  of the lens array.  FIG. 28  is a cross section parallel to the arrangement direction of multiple movable-side curved surfaces  701 . The horizontal direction of  FIG. 28  is the arrangement direction of the movable-side curved surface  701 .  FIG. 29  is a schematic diagram showing a process of injecting a resin  800  into the conventional mold  700 .  FIG. 30  is a schematic diagram showing the conventional mold when the resin has been injected into the conventional mold. The conventional mold  700  is consisted of an upper mold  703 , a lower mold  704 , and gates  707 . Compared with a mold  600  according to the fourth embodiment of this invention described later, the conventional mold  700  has a different number of gates  707 , described later, serving as inlets for injecting the mold  800  forming the lens plate  19 . Each constituent member of the conventional mold  700  is described later. 
     The upper mold  703  in the conventional mold  700  is a movable mold separating from the lens plate  19  at first. The movable-side curved surface  701  formed on the upper mold  703  faces the lower mold  704 . The movable-side curved surface  701  has a curved shape corresponding to a second surface shape of the lens plate  19 , and forms the second surface shape on the lens plate  19 . It should be noted that the movable-side curved surfaces  701  are arranged in two substantially-linear rows parallel to each other to correspond to the second surface of the lens plate  19 . The lower mold  704  in the conventional mold  700  is separated from the lens plate  19  after the upper mold  703  is separated from the lens plate  19 . A fixed-side curved surface  702  formed on the lower mold  704  faces the upper mold  703 . The fixed-side curved surface  702  has a curved shape corresponding to a first surface shape of the lens plate  19 , and forms the first surface shape on the lens plate  19 . It should be noted that the fixed-side curved surfaces  702  are arranged in two substantially-linear rows parallel to each other to correspond to the first surface of the lens plate  19 . The gates  707  in the conventional mold  700  are inlets for the resin  800 , and are arranged at both end portions of the conventional mold  700  in the arrangement direction of the multiple movable-side curved surfaces  701 . 
     The manufacturing process for forming the lens plate  19  using the conventional mold  700  will be hereinafter specifically described with reference to  FIGS. 28 to 30 . 
     As shown in  FIG. 28 , the resin  800  is softened by heat, and is injected through the gates  707  into a space enclosed by the upper mold  703  and the lower mold  704  of the conventional mold  700 . A flow front  800   a  shown in  FIG. 29  is a forefront portion of the resin  800  injected into the space enclosed by the upper mold  703  and the lower mold  704  of the conventional mold  700 . The flow front  800   a  is a boundary surface between the resin  800  and air. The resin  800  is injected through the gates  707  formed at both end portions of the upper mold  703  of the conventional mold  700 , and the flow front  800   a  moves from the gates  707  to the center in the arrangement direction of the movable-side curved surfaces  701  of the conventional mold  700 . When the resin  800  is sufficiently injected into the space enclosed by the upper mold  703  and the lower mold  704  of the conventional mold  700  as shown in  FIG. 30 , the two flow fronts  800   a  moving from the both end portions of the conventional mold  700  collide with each other in the proximity to the center of the conventional mold  700  in the arrangement direction of the movable-side curved surfaces  701 . After the flow fronts  800   a  move from the both end portions of the conventional mold  700  in the arrangement direction and collide with each other, a stress occurs in the resin  800 . 
     The flow fronts  800   a  colliding with each other as described above form a weld line  800   b  in the proximity to the center of the conventional mold  700  in the arrangement direction of the micro lenses  22  on the lens plate  19  after the lens plate  19  is formed. The weld line  800   b  is a trace of thread-like thin line emerging at a portion where two or more flow fronts  800   a  merge into one and collide with each other within the mold. At the portion in which the weld line  800   b  emerges, mechanical characteristics especially impact tenacity characteristics are greatly deteriorated, and the optical characteristics are also deteriorated. Specifically, at the portion in which the weld line  800   b  emerges, the molecular orientation of the resin  800  changes, and accordingly, the refraction index changes. For example, where the refraction index becomes higher, the transparency is deteriorated, and the focal length of the lens becomes shorter. Thus, a change of the refraction index raises a difference in the focal lengths of the lenses between a portion in which the weld line  800   b  emerges and the other portions. In addition, an internal stress occurring at the portion of the resin  800  in which the weld line  800   b  emerges deteriorates the precision of the shape of the aspherical lens having the first surface and the second surface. If the precision of the shape of the lens deteriorates, the spherical aberration cannot be sufficiently corrected. 
     The manufacturing method, the mold, and the like for the lens array according to the fourth embodiment of this invention will be hereinafter described. Specifically, a mold  600  used for manufacturing the lens plate  21  of the lens array according to the fourth embodiment will be first described. Then, the manufacturing method for forming the lens plate using the mold  600  will be described. Thereafter, the molded lens  11  will be described. 
     The mold  600  used for manufacturing the lens plate  21  of the lens array according to the fourth embodiment will be hereinafter specifically described with reference to  FIGS. 31 and 32 .  FIGS. 31 and 32  are schematic diagrams showing the mold  600  used for manufacturing the lens plate  21  of the lens array according to the fourth embodiment. 
       FIG. 31  shows not only the mold  600  but also reference symbols relating to dimensional precisions, described later. The mold  600  shown in  FIG. 32  has a gate  607  for manufacturing the lens as described later. Both of  FIGS. 31 and 32  are cross sections of the mold  600  taken along parallel to the arrangement direction of multiple movable-side curved surfaces  601  and to a direction of the optical axis of each lens of the lens plate. The horizontal direction of  FIGS. 31 and 32  are the arrangement direction of the movable-side curved surfaces  601 . The mold  600  consists of an upper mold  603 , a lower mold  604 , and the gate  607 . Compared with the conventional mold  700  as described above, the mold  600  has the only one gate  607 , as described later, serving as an inlet for injecting the resin  800  for forming the lens plate  21 . Each constituent member of the mold  600  will be hereinafter described. 
     The upper mold  603  of the mold  600  is a movable mold separating from the lens plate  21  at first. The movable-side curved surfaces  601  formed on the upper mold  603  face the lower mold  604 . The movable-side curved surface  601  has a curved shape corresponding to the shape of a second surface  22   b  of the lens plate  21 , and forms the shape of the second surface  22   b  on the lens plate  21 . It should be noted that the movable-side curved surfaces  601  are arranged in two substantially-linear rows parallel to each other to correspond to the second surface  22   b  of the lens plate  21 . The lower mold  604  of the mold  600  is a mold separating from the lens plate  21  after the upper mold  603  is separated from the lens plate  21 . Fixed-side curved surfaces  602  formed on the lower mold  604  face the upper mold  603 . The fixed-side curved surface  602  has a curved shape corresponding to the shape of the first surface  22   a  of the lens plate  21 , and forms the shape of the first surface  22   a  on the lens plate  21 . It should be noted that the fixed-side curved surfaces  602  are arranged in two substantially-linear rows parallel to each other to correspond to the first surface  22   a  of the lens plate  21 . The gate  607  in the mold  600  is an inlet for the resin  800 , and is arranged at an end portion of the mold  600  in the arrangement direction of the multiple movable-side curved surfaces  601 . 
     As shown in  FIG. 31 , an interval PCM at the end portion of the upper mold  603  in the arrangement direction of the movable-side curved surfaces  601  is set to be different from an interval PCF at the end portion of the lower mold  604  in the arrangement direction of the fixed-side curved surfaces  602 . Specifically, the interval PCM is larger than the interval PCF. The mold  600  is made so that a distance PEM between one of the movable-side curved surfaces  601  at one end in the arrangement direction and one of the movable-side curved surfaces  601  at the other end in the arrangement direction is different from a distance PEF between one of the fixed-side curved surfaces  602  at one end in the arrangement direction and one of the fixed-side curved surfaces  602  at the other end in the arrangement direction. Specifically, the distance PEM is larger than the distance PEF. It should be noted that PEF is, for example, 300 mm, and that the difference between PEM and PEF is, for example, 0.03 mm. The mold  600  as described above forms a space enclosed by the movable-side curved surfaces  601  and the fixed-side curved surfaces  602 , when the upper mold  603  and the lower mold  604  are combined. When the softened resin  800  is injected into the space through the gate  607 , the lens plate  21  is formed. 
     The manufacturing method for forming the lens plate  21  using the mold  600  will be hereinafter specifically described in detail with reference to  FIGS. 33 to 35 .  FIG. 33  is a schematic diagram showing the mold  600  having the soften resin  800  filled in the space formed by the upper mold  603  and the lower mold  604  combined together.  FIG. 34  is a schematic diagram showing the mold  600  when the upper mold  603  is released from the lower mold  604 .  FIG. 35  shows the schematic diagram showing the lens plate  21  after the molding. 
     As shown in  FIG. 33 , the resin  800  softened by heat is injected into the space enclosed by the upper mold  603  and the lower mold  604  of the mold  600 . As shown in  FIG. 34 , the upper mold  603  moves in a direction away from the lens plate  21  at first. Thus, as shown in  FIG. 34 , the upper surface of the lens plate  21  is exposed to open air. Thereby, the temperature of the upper surface of the lens plate  21  decreases, and the lens  11  shrinks. However, the temperature of the lower surface of the lens plate  21  does not decrease as much as the temperature of the upper surface of the lens plate  21 . Accordingly, the lower surface of the lens plate  21  does not shrink as much as the upper surface of the lens plate  21  because the lower surface of the lens plate  21  is still in contact with the lower mold  604 . Subsequently, as shown in  FIG. 35 , the lower mold  604  is released from the lens plate  21 . At this moment, the entire lens plate  21  is exposed to open air, and the temperature of the lens plate  21  decreases, and the entire lens plate  21  shrinks. If the conventional mold is used in the above manufacturing method, the decrease of the temperature would be different between the upper surface and the lower surface of the lens plate  21  because the upper surface of the lens plate  21  is in contact with open air for a longer time than the lower surface. Thus, with the conventional mold, the shrinking ratio of the lens plate  21  would be different between the upper surface and the lower surface. Specifically, the shrinking ratio of the upper surface is larger than the shrinking ratio of the lower surface. Thus, the position of the second surface  22   b  would move toward the center portion of the lens plate  21  in the arrangement direction of the micro lenses  22  from the position corresponding to the first surface  22   a.    
     However, in the mold  600  of this invention, the arrangement interval of the movable-side curved surfaces  601  on the upper mold  603  is larger by a predetermined amount than the arrangement interval of the fixing side curved surfaces  602 . With the mold  600  of this invention, the position of the second surface  22   b  is coincident with the position of the first surface  22   a  in the arrangement direction of the micro lenses  22  after the shrinking of the lens plate  21  is finished. The difference between the arrangement interval of the movable-side curved surfaces  601  formed on the upper mold  603  and the arrangement interval of the fixed-side curved surfaces  602  is derived from experiments. Specifically, in a case where the number of lenses per row is 272 pieces, a molded product as shown in  FIG. 35  has the dimension of: the interval between the lenses at both ends on the first surface of the lens plate PEM 1 =325.2 mm; the interval between the lenses at both ends on the second surface of the lens plate PEF 1 =325.2 mm; the distance between both edges of the lens plate FRY 1 =333 mm; the lens pitch on the first surface of the lens plate PCM 1 =1.2 mm; and the lens pitch of the second surface on the lens plate PCF 1 =1.2 mm. However, if the mold for forming a product has the same dimension as the product to be molded, a desired product dimension cannot be obtained due to molding conditions in the injection molding such as a stretching rate of the resin  800  and the like. The stretching rate of the resin  800  can be obtained by manufacturing the lens plate as a trial at predetermined molding conditions. With the stretching rate of the resin  800  taken into consideration, the dimension of the metal mold capable of absorbing the difference of shrinking between the upper surface and the lower surface becomes as follows: the interval between the lenses at both ends on the first surface of the lens plate PEM=326.8585 mm; the interval between the lenses at both ends on the second surface of the lens plate PEF=326.826 mm; the distance between both edges of the lens plate FRY=334.665 mm; the lens pitch of the first surface of the lens plate PCM=1.20612 mm; and the lens pitch of the second surface of the lens plate PCF=1.206 mm. 
     The influence on the optical performance caused by the difference of the stretching ratio between the first surface and the second surface of the lens plate can be ignored. In a case where the number of lenses per row between both ends of the lens plate  21  is N=272, a formula PCM=PCF+0.00012 can be obtained from the relationship PCM=PCF+0.03/(N−1) as described above. Terminal lenses are lenses which expose images of the LED elements located at the outermost of the LED array. A central lens is a lens located at a position away by a distance PEM/2 from one of the terminal lenses, where the distance between one of the terminal lenses and the other of the terminal lenses is PEM. 
     The molded lens  11  will be specifically described with reference to  FIG. 36 .  FIG. 36  is a perspective view of the lens plate  21 . 
     The micro lenses  22  are integrally formed on one surface of the lens plate  21  made into a plate-like part. There exists a trace of gate  21   a  at one of the end portions of the lens plate  21  in the arrangement direction of the micro lenses  22 . The trace of gate  21   a  is a trace of the gate protruding from the lens plate  21  and cut off by a cutting tool or a laser after the lens plate  21  is molded and cooled and is released from the lens plate  21  in the lens array manufacturing steps. The lens plate  21  is molded by an injection molding machine using a cycloolefin optical resin sold under the trade name ZEONEX E48R by ZEON CORPORATION. Because only one gate is arranged on the mold  600 , the mold  600  can prevent the occurrence of a trace of thread-like thin line that would emerge at a portion in which two or more flow fronts  800   a  merge into one within the mold. Thus, the micro lenses  22  are integrally formed on the lens plate  21  with high-precision. The refraction rate and the shape of all the micro lenses  22  can be uniformly formed. As for the injection molding, a general injection molding method is used here, but an injection compression molding method and the like may also be employed. 
     As hereinabove described, the manufacturing method of the lens array according to the fourth embodiment can prevent the occurrence of the weld line  800   b , and achieves to mold all the micro lenses  22  with high-precision even where the lens array is long in the arrangement direction of the micro lenses  22 . 
     Fifth Embodiment 
     In the fifth embodiment, a reading apparatus for reading a document Q will be described. The reading apparatus according to the fifth embodiment uses the lens array used in the exposure device in the image forming apparatus according to the third embodiment. 
     A reading apparatus  500  is a scanner for reading a printed image printed on the document Q and generating electric data.  FIG. 37  shows a block diagram showing the reading apparatus  500 .  FIG. 38  shows a block diagram showing a reading head  400 . The reading apparatus  500  consists of a reading head  400  and constituent members for operating the reading head  400 . The constituent members for operating the reading head  400  will be described first. Thereafter, the reading head  400  will be described. The constituent members for operating the reading head  400  of the reading apparatus  500  will be specifically described with reference to  FIG. 37 . The constituent members for operating the reading head  400  consists of a light source  501 , a document table  502 , a rail  503 , a pulley  504 , a driving belt  505 , a motor  506 , and a transmitting belt  507 . Each constituent member in the reading apparatus  500  will be hereinafter described. The light source  501  is a illuminating light source emitting an illuminating light to the document Q, and is arranged in proximity to the reading head  400  so that the illuminating light is reflected by the surface of the document Q and the reflected light enters into the reading head  400 . For example, a rare gas fluorescent lamp may be used as the light source  501 . The illuminating light source is not limited to lamps, and for example, a white LED (Light Emitting Diode), a semiconductor laser, and the like may be used. The document table  502  is a table for supporting the document Q from which the electric data is to be generated, and is arranged above the reading head  400 . Specifically, the document table  502  is arranged to pass the illuminating light emitted from the light source  501  and to pass the reflected light reflected by the document Q placed on the document table  502 , so that the reflected light enters into the reading head  400 . The document table  502  as described above is made of glass and the like that can sufficiently pass visible lights. However, the document table  502  is not limited to the glass. For example, a plastic and the like may be used for the document table  502  that has a refraction index for passing the visible light necessary for reading the document Q and has heat and light resistance preventing deterioration caused by ultraviolet rays included in the light source  501 , heat radiated form the light source  501 , and the like. 
     The rail  503  is a constituent member for operating the reading head  400  of the reading apparatus  500 , and is arranged under the reading head  400  to mount the reading head  400  for scanning the document Q. Specifically, the driving belt  505 , described later, connected to multiple rail supporting tables  503 A are driven, so that the reading head  400  moves on the rail  503  and scans the printed image printed on the document Q. The pulley  504  consists of a pair of pulleys  504 A and  504 B, and is arranged on both ends of the endless driving belt  505 , described later, to give a predetermined tension to the driving belt  505 . The pulley  504 A and the pulley  504 B are made of a material having a high friction resistance. The motor  506 , described later, rotates the pulley  504 A to rotate the driving belt  505 . The driving belt  505  is a conveyance means to move the reading unit  400  on the rail  503 , and is made of the endless belt. The motor  506  is arranged adjacent to the pulley  504 A, and is connected to the pulley  504 A via the transmission belt  507 . The motor  506  rotates based on the control of a control unit, not shown, to drive the pulley  504 A to rotate. 
     Next, the constituent members of the reading head  400  of the reading apparatus  500  will be hereinafter specifically described with reference to  FIGS. 37 and 38 . The reading head  400  in the reading apparatus  500  reads the printed image printed on the document Q. The reading head  400  has a mirror  402 , a pair of micro lenses  22 , a lens array including a shielding unit  23  inserted between the micro lenses  22 , and a line sensor  401 . Each constituent member of the reading head  400  will be hereinafter described. The illuminating light reflected by the surface of the document Q passes the document table  502 . The mirror  402  is a reflection member, and reflects the reflected light form the document Q, so that the optical axis of the reflected light turns, for example, at an off-axis angle of 90 degrees, and enters into the micro lens  22 . The mirror  402  is made by evaporating aluminum and the like as a reflection film onto a planar shape substrate made of glass, metal, heat-resistant plastic and the like. It should be noted that it is not necessary to evaporate the reflection film onto the substrate if the substrate has a sufficient reflection rate in the range of visible light. The mirror  402  is not limited to the planar shape. For example, the mirror  402  may be made into a toroidal shape to correct astigmatism caused by the off-axis angle. 
     The lens array consists of the pair of micro lenses  22  and the shielding unit  23  inserted between the micro lenses  22 , which are the constituent members of the reading head  400  of the reading apparatus  500 . The reflection light generated at the surface of the document Q by the illuminating light of the light source  501  forms an optical image on the line sensor  401 , as described later, through the lens array. The lens array according to the fifth embodiment is the same as the lens array according to the third and the fourth embodiments. The line sensor  401  is a sensor arranged at a position of the optical plane of the lens array. Multiple light receiving elements such as CCD and the like evenly spaced at an interval PR in line are used as the line sensor  401 . The resolution of the line sensor  401  is 600 dpi, and 600 pieces of the light receiving elements are arranged per 1 inch, i.e., approximately 25.4 mm. Thus, the interval PR of the light receiving elements is 0.0423 mm. The line sensor  401  reads the printed image printed on the document Q, and generates the electric data by converting the optical image formed by the reflection light from the surface of the document Q into the electric signal. 
     The optical system of the reading head  400  in the reading apparatus  500  will be hereinafter specifically described.  FIG. 39  shows a schematic diagram showing the optical system of the reading head  400 . The illuminating light from the light source  501  is reflected by the surface of the document Q, and becomes the reflection light. The reflection light enters into one of the pair of micro lenses  22  arranged at a position away by the distance LO from the surface of the document Q as the object plane. The micro lens  22  has the lens thickness LT and has a collimator lens effect. The reflection light having entered into the micro lens  22  is collimated by the micro lens  22  and becomes a collimated light. The shielding unit  23  having the length LS sufficiently removes stray lights from the collimated light. The other of the pair of micro lenses  22  arranged on the opposite side has the thickness LT and has a condenser lens effect. The micro lens  22  converges the collimated light, from which the stray lights have been sufficiently removed, on the line sensor  401  arranged on the optical plane at a position away by the distance LI. The total length TC is the total sum of the distance LO, the lens thickness LT, the distance LS, the lens thickness LT, and the distance LI, as described above. The pair of micro lenses  22  consists of the collimator lens and the condenser lens, which are identical to each other and arranged to face to each other on either side of the shielding unit  23 . Thus, the pair of micro lenses  22  has the optical magnification of one. The mirror  402 , not shown, is arranged within the moving distance LO. The reflection light from the object plane on the document Q converges on the optical plane on the line sensor  401  and forms the optical image thereon while the aberration is sufficiently suppressed, because the micro lens  22  is the aspherical lens and the shielding unit  23  can sufficiently remove the stray lights. 
     The operation of the reading apparatus  500  will be hereinafter described. The reading apparatus  500  according to the fifth embodiment of this invention was used under the conditions where the dot interval PD is PD=0.0423 mm and the resolution of the line sensor  401  is 600 dpi, and the reading apparatus  500  generated the electric data of good quality that faithfully reproduces the image on the document Q without causing any reading errors and the like. 
     The reading apparatus  500  according to the fifth embodiment is described as the scanner for converting the printed image printed on the document Q into the electric data. However, the reading apparatus  500  may be applied to a sensor or a switch converting an optical signal into an electric signal, an input and output apparatus using the sensor or the switch converting the optical signal into the electric signal, a biometric authentication apparatus, a communication apparatus, and a dimensional measurement apparatus. 
     In the fifth embodiment, the reading apparatus  500  read the printed data on the document Q and generated the electric data of good quality that faithfully reproduces the printed data on the document Q without causing reading errors and the like. 
     The foregoing description of preferred embodiments of the invention has been presented for purposes of illustration and description, and is not intended to be exhaustive or to limit the invention to the precise form disclosed. The description was selected to best explain the principles of the invention and their practical application to enable others skilled in the art to best utilize the invention in various embodiments and various modifications as are suited to the particular use contemplated. It is intended that the scope of the invention should not be limited by the specification, but be defined by the claims set forth below.