Patent Publication Number: US-11640046-B2

Title: Lens system and electronic device

Description:
RELATED APPLICATIONS 
     This application is a continuation patent application of U.S. application Ser. No. 16/734,134, filed on Jan. 3, 2020, which claims priority to Taiwan Application 108143778, filed on Nov. 29, 2019, which is incorporated by reference herein in its entirety. 
    
    
     BACKGROUND 
     Technical Field 
     The present disclosure relates to a lens system and an electronic device, more particularly to a lens system applicable to an electronic device. 
     Description of Related Art 
     With the development of semiconductor manufacturing technology, the performance of image sensors has improved, and the pixel size thereof has been scaled down. Therefore, featuring high image quality becomes one of the indispensable features of an optical system nowadays. 
     Furthermore, due to the rapid changes in technology, electronic devices equipped with optical systems are trending towards multi-functionality for various applications, and therefore the functionality requirements for the optical systems have been increasing. However, it is difficult for a conventional optical system to obtain a balance among the requirements such as high image quality, low sensitivity, a proper aperture size, miniaturization and a desirable field of view. 
     SUMMARY 
     According to one aspect of the present disclosure, a lens system includes four lens elements. The four lens elements are, in order from an object side to an image side, a first lens element, a second lens element, a third lens element and a fourth lens element. Each of the four lens elements has an object-side surface facing toward the object side and an image-side surface facing toward the image side. 
     At least one of the object-side surface and the image-side surface of the first lens element has at least one inflection point. The fourth lens element has positive refractive power, the object-side surface of the fourth lens element is convex in a paraxial region thereof, and at least one of the object-side surface and the image-side surface of the fourth lens element has at least one inflection point. 
     When an axial distance between the object-side surface of the first lens element and an image surface is TL, a focal length of the lens system is f, an Abbe number of the fourth lens element is Vd4, a curvature radius of the object-side surface of the fourth lens element is R7, and a curvature radius of the image-side surface of the fourth lens element is R8, the following conditions are satisfied:
         4.85&lt;TL/f&lt;15.0;   10.0&lt;Vd4&lt;38.0; and   −3.50&lt;(R7+R8)/(R7−R8)&lt;0.85.       

     According to another aspect of the present disclosure, a lens system includes four lens elements. The four lens elements are, in order from an object side to an image side, a first lens element, a second lens element, a third lens element and a fourth lens element. Each of the four lens elements has an object-side surface facing toward the object side and an image-side surface facing toward the image side. 
     The object-side surface of the first lens element is concave in a paraxial region thereof, and at least one of the object-side surface and the image-side surface of the first lens element has at least one inflection point. The fourth lens element has positive refractive power. 
     When an axial distance between the object-side surface of the first lens element and an image surface is TL, a focal length of the lens system is f, a central thickness of the second lens element is CT2, a central thickness of the third lens element is CT3, a curvature radius of the object-side surface of the fourth lens element is R7, a curvature radius of the image-side surface of the fourth lens element is R8, and an entrance pupil diameter of the lens system is EPD, the following conditions are satisfied: 
     4.85&lt;TL/f&lt;15.0; 
     0.10&lt;CT2/CT3&lt;0.95; 
     (R7+R8)/(R7−R8)&lt;0.35; and 
     1.25&lt;f/EPD&lt;2.50. 
     According to another aspect of the present disclosure, an electronic device includes an identification module and a plate. The identification module includes one of the aforementioned lens systems and an image sensor. The image sensor is disposed on the image surface of the lens system. The plate is disposed on an object side of the identification module, and the plate is a module having a display function. In addition, an axial distance between the plate and the first lens element is smaller than 1.50 mm. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The disclosure can be better understood by reading the following detailed description of the embodiments, with reference made to the accompanying drawings as follows: 
         FIG.  1    is a schematic view of an identification module and a plate according to the 1st embodiment of the present disclosure; 
         FIG.  2    is a schematic view of the identification module in  FIG.  1   ; 
         FIG.  3    shows spherical aberration curves, astigmatic field curves and a distortion curve of the identification module according to the 1st embodiment; 
         FIG.  4    is a schematic view of an identification module and a plate according to the 2nd embodiment of the present disclosure; 
         FIG.  5    is a schematic view of the identification module in  FIG.  4   ; 
         FIG.  6    shows spherical aberration curves, astigmatic field curves and a distortion curve of the identification module according to the 2nd embodiment; 
         FIG.  7    is a schematic view of an identification module and a plate according to the 3rd embodiment of the present disclosure; 
         FIG.  8    is a schematic view of the identification module in  FIG.  7   ; 
         FIG.  9    shows spherical aberration curves, astigmatic field curves and a distortion curve of the identification module according to the 3rd embodiment; 
         FIG.  10    is a schematic view of an identification module and a plate according to the 4th embodiment of the present disclosure; 
         FIG.  11    is a schematic view of the identification module in  FIG.  10   ; 
         FIG.  12    shows spherical aberration curves, astigmatic field curves and a distortion curve of the identification module according to the 4th embodiment; 
         FIG.  13    is a schematic view of an identification module and a plate according to the 5th embodiment of the present disclosure; 
         FIG.  14    is a schematic view of the identification module in  FIG.  13   ; 
         FIG.  15    shows spherical aberration curves, astigmatic field curves and a distortion curve of the identification module according to the 5th embodiment; 
         FIG.  16    is a schematic view of an identification module and a plate according to the 6th embodiment of the present disclosure; 
         FIG.  17    is a schematic view of the identification module in  FIG.  16   ; 
         FIG.  18    shows spherical aberration curves, astigmatic field curves and a distortion curve of the identification module according to the 6th embodiment; 
         FIG.  19    is a schematic view of an identification module and a plate according to the 7th embodiment of the present disclosure; 
         FIG.  20    is a schematic view of the identification module in  FIG.  19   ; 
         FIG.  21    shows spherical aberration curves, astigmatic field curves and a distortion curve of the identification module according to the 7th embodiment; 
         FIG.  22    is a schematic view of an identification module and a plate according to the 8th embodiment of the present disclosure; 
         FIG.  23    is a schematic view of the identification module in  FIG.  22   ; 
         FIG.  24    shows spherical aberration curves, astigmatic field curves and a distortion curve of the identification module according to the 8th embodiment; 
         FIG.  25    is a schematic view of an electronic device according to the 9th embodiment of the present disclosure; 
         FIG.  26    is a schematic view of the electronic device in  FIG.  25    identifying a fingerprint; 
         FIG.  27    is a schematic view of an electronic device according to the 10th embodiment of the present disclosure identifying a fingerprint; 
         FIG.  28    is a front view of an electronic device according to the 11th embodiment of the present disclosure; 
         FIG.  29    is a rear view of the electronic device in  FIG.  28   ; 
         FIG.  30    shows a schematic view of Y 12 , Yc 11 , Yc 41 , Yc 42  and inflection points and critical points of the first and fourth lens elements according to the 1st embodiment of the present disclosure; and 
         FIG.  31    shows a schematic view of SAG 21 , SAG 22  and the second lens element according to the 1st embodiment of the present disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     An electronic device includes an identification module and a plate. The identification module includes a lens system and an image sensor. The lens system includes four lens elements, and the four lens elements are, in order from an object side to an image side, a first lens element, a second lens element, a third lens element and a fourth lens element. Each of the four lens elements of the lens system has an object-side surface facing toward the object side and an image-side surface facing toward the image side. Moreover, the image sensor is disposed on an image surface of the lens system, and the plate is disposed on an object side of the identification module. 
     The first lens element can have negative refractive power. Therefore, it is favorable for enlarging the viewing angle of the lens system so as to increase the image identification area. The object-side surface of the first lens element can be concave in a paraxial region thereof. Therefore, it is favorable for the lens system to have a retrofocus configuration so as to increase the detection range thereof. The object-side surface of the first lens element can have at least one convex critical point in an off-axis region thereof. Therefore, it is favorable for controlling the incident angles of peripheral light on the lens surfaces so as to prevent total reflection, and it is also favorable for controlling the size of the first lens element so as to minimize the overall size of the lens system. At least one of the object-side surface and the image-side surface of the first lens element has at least one inflection point. Therefore, it is favorable for controlling the size of the first lens element while receiving light with a large incident angle so as to prevent the object-side size of the lens system from being overly large. Please refer to  FIG.  30   , which shows a schematic view a convex critical point C and an inflection point P of the object-side surface  111  of the first lens element  110  according to the 1st embodiment of the present disclosure. 
     The image-side surface of the third lens element can be concave in a paraxial region thereof. Therefore, it is favorable for reducing the back focal length so as to control the total track length of the lens system. The image-side surface of the third lens element can have at least one convex critical point in an off-axis region thereof. Therefore, it is favorable for balancing the shape of the image-side surface of the third lens element so as to correct off-axis aberrations, thereby meeting the requirement of compactness. 
     The fourth lens element has positive refractive power. Therefore, it is favorable for providing significant light converging capability so as to control the size of the lens system, and it&#39;s also favorable for the lens system to have a retrofocus configuration. The object-side surface of the fourth lens element can be convex in a paraxial region thereof. Therefore, it is favorable for arranging the surface curvatures of the fourth lens element so as to enhance light converging capability. The object-side surface of the fourth lens element can have at least one concave critical point in an off-axis region thereof. Therefore, it is favorable for reducing the incident angle of peripheral light on the fourth lens element so as to correct field curvature. The image-side surface of the fourth lens element can be convex in a paraxial region thereof. Therefore, it is favorable for controlling the incident angle of light on the image sensor so as to prevent vignetting in the peripheral region of images. The image-side surface of the fourth lens element can have a convex-to-concave-to-convex shape change in order from the paraxial region thereof to an off-axis region thereof. Therefore, it is favorable for correcting off-axis aberrations and reducing the size of the lens system. At least one of the object-side surface and the image-side surface of the fourth lens element can have at least one inflection point. Therefore, it is favorable for reducing the back focal length of the lens system for controlling the total track length. Please refer to  FIG.  30   , which shows a schematic view of critical points C and an inflection point P of the fourth lens element  140  according to the 1st embodiment of the present disclosure. The critical points and some of the inflection points on the object-side surface of the first lens element, the object-side surface and image-side surface of the fourth lens element in  FIG.  30    are only exemplary. Other lens surfaces of the four lens elements may also have one or more critical points and inflection points. 
     When an axial distance between the object-side surface of the first lens element and the image surface is TL, and a focal length of the lens system is f, the following condition is satisfied: 4.85&lt;TL/f&lt;15.0. Therefore, it is favorable for balancing the total track length and the field of view for meeting the specifications. Moreover, the following condition can also be satisfied: 5.50&lt;TL/f&lt;12.0. 
     When an Abbe number of the fourth lens element is Vd4, the following condition can be satisfied: 10.0&lt;Vd4&lt;38.0. Therefore, it is favorable for increasing the refractive capability of the fourth lens element so as to effectively reduce the total track length of the lens system. Moreover, the following condition can also be satisfied: 13.0&lt;Vd4&lt;30.0. According to the present disclosure, the Abbe number Vd of one lens element is obtained from the following equation: Vd=(N−1)/(NF−NC), wherein Nd is the refractive index of said lens element at the wavelength of helium d-line (587.6 nm), NF is the refractive index of said lens element at the wavelength of hydrogen F-line (486.1 nm), and NC is the refractive index of said lens element at the wavelength of hydrogen C-line (656.3 nm). 
     When a curvature radius of the object-side surface of the fourth lens element is R7, and a curvature radius of the image-side surface of the fourth lens element is R8, the following condition is satisfied: −3.50&lt;(R7+R8)/(R7−R8)&lt;0.85. Therefore, it is favorable for controlling light path on the object side of the fourth lens element so as to improve image quality by correcting aberrations on the image side thereof. Moreover, the following condition can also be satisfied: −1.80&lt;(R7+R8)/(R7−R8)&lt;0.50. Moreover, the following condition can also be satisfied: (R7+R8)/(R7−R8)&lt;0.35. Moreover, the following condition can also be satisfied: −50.0&lt;(R7+R8)/(R7−R8)&lt;0.25. 
     When a central thickness of the second lens element is CT2, and a central thickness of the third lens element is CT3, the following condition can be satisfied: 0.10&lt;CT2/CT3&lt;0.95. Therefore, it is favorable for properly allocating the space in the lens system so as to increase manufacturing yield rate and image quality as well as to enhance the controlling capability of the third lens element. Moreover, the following condition can also be satisfied: 0.20&lt;CT2/CT3&lt;0.65. Moreover, the following condition can also be satisfied: 0.40&lt;CT2/CT3&lt;0.63. 
     When the focal length of the lens system is f, and an entrance pupil diameter of the lens system is EPD, the following condition can be satisfied: 1.25&lt;f/EPD&lt;2.50. Therefore, it is favorable for adjusting the entrance pupil so as to provide sufficient incident light, thereby increasing image brightness. Moreover, the following condition can also be satisfied: 1.0&lt;f/EPD&lt;2.05. Moreover, the following condition can also be satisfied: 1.0&lt;f/EPD&lt;2.0. Moreover, the following condition can also be satisfied: 1.20&lt;f/EPD&lt;1.80. 
     When the focal length of the lens system is f, and a focal length of the second lens element is f2, the following condition can be satisfied: −0.15&lt;f/f2&lt;0.15. Therefore, the second lens element configured as a correction lens is favorable for correcting spherical aberration and coma generated by the first lens element and improving image quality at the off-axis region. Moreover, the following condition can also be satisfied: −0.10&lt;f/f2&lt;0.10. 
     When a minimum value among Abbe numbers of all lens elements of the lens system is Vdmin, the following condition can be satisfied: 13.0&lt;Vdmin&lt;21.0. Therefore, it is favorable for better controlling the light path so as to increase the design flexibility for satisfying high-end product specifications. 
     When a maximum effective radius of the image-side surface of the first lens element is Y12, and a curvature radius of the image-side surface of the first lens element is R2, the following condition can be satisfied: 1.02&lt;Y 12 /R2&lt;4.50. Therefore, it is favorable for controlling the size of the first lens element so as to meet the requirements of a large field of view and compactness. Moreover, the following condition can also be satisfied: 1.02&lt;Y 12 /R2&lt;2.50. Please refer to  FIG.  30   , which shows a schematic view of Y 12  according to the 1st embodiment of the present disclosure. 
     When a vertical distance between a non-axial critical point on the object-side surface of the first lens element and an optical axis is Yc 11 , and the focal length of the lens system is f, the object-side surface of the first lens element can have at least one critical point in the off-axis region thereof satisfying the following condition: 0.50&lt;Yc 11 /f&lt;5.0. Therefore, it is favorable for correcting off-axis aberrations and reducing the total track length of the lens system so as to meet the requirements of compactness and a large field of view. Moreover, the following condition can also be satisfied: 0.80&lt;Yc 11 /f&lt;3.0. Please refer to  FIG.  30   , which shows a schematic view of Yc 11  according to the 1st embodiment of the present disclosure. 
     When a maximum image height of the lens system (half of a diagonal length of an effective photosensitive area of the image sensor) is ImgH, and the focal length of the lens system is f, the following condition can be satisfied: 2.0&lt;ImgH/f&lt;8.0. Therefore, it is favorable for providing a better viewing angle of the lens system for various applications. Moreover, the following condition can also be satisfied: 2.3&lt;ImgH/f&lt;4.0. 
     When half of a maximum field of view of the lens system is HFOV, and an f-number of the lens system is Fno, the following condition can be satisfied: 1.50&lt;tan(HFOV)/Fno. Therefore, it is favorable for retrieving more light while satisfying the requirement of a large field of view. Moreover, the following condition can also be satisfied: 1.70&lt;tan(HFOV)/Fno&lt;5.50. 
     According to the present disclosure, the lens system further includes an aperture stop, and the aperture stop can be disposed between the second lens element and the third lens element. Therefore, it is favorable for positioning the aperture stop so as to balance the field of view and total track length for meeting the requirements of a large field of view and compactness. 
     When an axial distance between the aperture stop and the image-side surface of the fourth lens element is SD, and an axial distance between the object-side surface of the first lens element and the image-side surface of the fourth lens element is TD, the following condition can be satisfied: 0.40&lt;SD/TD&lt;0.95. Therefore, it is favorable for balancing the position of the aperture stop in the lens system so as to control the field of view and the total track length. 
     When the axial distance between the object-side surface of the first lens element and the image surface is TL, and the maximum image height of the lens system is ImgH, the following condition can be satisfied: 2.0&lt;TL/ImgH&lt;3.50. Therefore, it is favorable for providing a sufficient light-receiving area while obtaining compactness so as to maintain sufficient image brightness. Moreover, the following condition can also be satisfied: 2.0&lt;TL/ImgH&lt;3.0. Moreover, the following condition can also be satisfied: 2.30&lt;TL/ImgH&lt;3.0. 
     When a curvature radius of the object-side surface of the first lens element is R1, and the focal length of the lens system is f, the following condition can be satisfied: −12.0&lt;R1/f&lt;0. Therefore, it is favorable for the first lens element to have a proper light divergence capability so as to obtain a wide field of view. Moreover, the following condition can also be satisfied: −8.0&lt;R1/f&lt;−2.0. 
     When a displacement in parallel with the optical axis from an axial vertex to a maximum effective radius position on the object-side surface of the second lens element is SAG 21 , a displacement in parallel with the optical axis from an axial vertex to a maximum effective radius position on the image-side surface of the second lens element is SAG 22 , and the focal length of the lens system is f, the following condition can be satisfied: 0≤(|SAG 21 |+|SAG 22 |)/f&lt;0.50. Therefore, it is favorable for avoiding excessive surface shape changes of the second lens element so as to maintain sufficient peripheral image quality. Moreover, the following condition can also be satisfied: 0≤(|SAG 21 |+|SAG 22 |)/f&lt;0.35. Please refer to  FIG.  31   , which shows a schematic view of SAG 21 , SAG 22  and the second lens element  120  according to the 1st embodiment of the present disclosure. Furthermore, each of SAG 21  and SAG 22  is a directional displacement. When the displacement from the axial vertex to the maximum effective radius position on one surface is facing towards the image side of the lens system, the value of displacement is positive; when the displacement from the axial vertex to the maximum effective radius position on one surface is facing towards the object side of the lens system, the value of displacement is negative. 
     According to the present disclosure, at least one lens element with positive refractive power of the lens system can have an Abbe number smaller than 30.0. Therefore, it is favorable for increasing the refractive capability of the lens system so as to reduce the size thereof, thereby achieving compactness. Moreover, at least one lens element with positive refractive power of the lens system can have an Abbe number smaller than 28.0. 
     When an Abbe number of the first lens element is Vd1, an Abbe number of the second lens element is Vd2, an Abbe number of the third lens element is Vd3, the Abbe number of the fourth lens element is Vd4, an Abbe number of the i-th lens element is Vdi, a refractive index of the first lens element is N1, a refractive index of the second lens element is N2, a refractive index of the third lens element is N3, a refractive index of the fourth lens element is N4, and a refractive index of the i-th lens element is Ni, at least one lens element of the lens system can satisfy the following condition: Vdi/Ni&lt;13.5, wherein i=1, 2, 3 or 4. Therefore, it is favorable for increasing the refractive capability of lens elements so as to improve design specifications and image quality. Moreover, at least one lens element of the lens system can also satisfy the following condition: Vdi/Ni&lt;12.0, wherein i=1, 2, 3 or 4. 
     When the axial distance between the object-side surface of the first lens element and the image surface is TL, the following condition can be satisfied: 1.0 [mm]&lt;TL&lt;3.0 [mm]. Therefore, it is favorable for controlling the total track length so as to meet the requirement of compactness. 
     According to the present disclosure, the central thickness of the third lens element can be a maximum among central thicknesses of all lens elements of the lens system. Therefore, it is favorable for balancing the refractive power distribution of the lens system. 
     When a vertical distance between a non-axial critical point on the image-side surface of the fourth lens element and the optical axis is Yc 42 , and the focal length of the lens system is f, the image-side surface of the fourth lens element can have at least one critical point in the off-axis region thereof satisfying the following condition: 0.20&lt;Yc 42 /f&lt;2.0. Therefore, it is favorable for correcting field curvature, reducing the total track length and flattening the Petzval surface of the lens system. Please refer to  FIG.  30   , which shows a schematic view of Yc 42  according to the 1st embodiment of the present disclosure. 
     When a vertical distance between a non-axial critical point on the object-side surface of the fourth lens element and the optical axis is Yc 41 , and the focal length of the lens system is f, the object-side surface of the fourth lens element can have at least one critical point in the off-axis region thereof satisfying the following condition: 0.30&lt;Yc 41 /f&lt;2.0. Therefore, it is favorable for reducing the back focal length so as to reduce the overall size of the lens system. Please refer to  FIG.  30   , which shows a schematic view of Yc 41  according to the 1st embodiment of the present disclosure. 
     According to the present disclosure, an axial distance between the plate and the first lens element of the lens system can be smaller than 1.50 mm. Therefore, it is favorable for controlling the thickness of the electronic device so as to meet the requirements of becoming lighter and thinner. 
     According to the present disclosure, the lens system can be operated with infrared light, visible light or monochromatic light, but the present disclosure is not limited thereto. 
     According to the present disclosure, the aforementioned features and conditions can be utilized in numerous combinations so as to achieve corresponding effects. 
     According to the present disclosure, the lens elements of the lens system can be made of either glass or plastic material. When the lens elements are made of glass material, the refractive power distribution of the lens system may be more flexible, and the influence on imaging caused by external environment temperature change may be reduced. The glass lens element can either be made by grinding or molding. When the lens elements are made of plastic material, the manufacturing costs can be effectively reduced. Furthermore, surfaces of each lens element can be arranged to be spherical or aspheric, wherein the former reduces manufacturing difficulty, and the latter allows more control variables for eliminating aberrations thereof, the required number of the lens elements can be reduced, and the total track length of the lens system can be effectively shortened. Furthermore, the aspheric surfaces may be formed by plastic injection molding or glass molding. 
     According to the present disclosure, when a lens surface is aspheric, it means that the lens surface has an aspheric shape throughout its optically effective area, or a portion(s) thereof. 
     According to the present disclosure, one or more of the lens elements&#39; material may optionally include an additive which alters the lens elements&#39; transmittance in a specific range of wavelength for a reduction in unwanted stray light or colour deviation. For example, the additive may optionally filter out light in the wavelength range of 600 nm to 800 nm to reduce excessive red light and/or near infrared light; or may optionally filter out light in the wavelength range of 350 nm to 450 nm to reduce excessive blue light and/or near ultraviolet light from interfering the final image. The additive may be homogeneously mixed with a plastic material to be used in manufacturing a mixed-material lens element by injection molding. 
     According to the present disclosure, each of an object-side surface and an image-side surface has a paraxial region and an off-axis region. The paraxial region refers to the region of the surface where light rays travel close to the optical axis, and the off-axis region refers to the region of the surface away from the paraxial region. Particularly, unless otherwise stated, when the lens element has a convex surface, it indicates that the surface is convex in the paraxial region thereof; when the lens element has a concave surface, it indicates that the surface is concave in the paraxial region thereof. Moreover, when a region of refractive power or focus of a lens element is not defined, it indicates that the region of refractive power or focus of the lens element is in the paraxial region thereof. 
     According to the present disclosure, when the parameters (e.g., refractive index and focal length) of the lens system, the identification module and the electronic device are not specifically defined, these parameters may be determined according to the operating wavelength range. 
     According to the present disclosure, an inflection point is a point on the surface of the lens element at which the surface changes from concave to convex, or vice versa. A critical point is a non-axial point of the lens surface where its tangent is perpendicular to the optical axis. 
     According to the present disclosure, the image surface of the lens system, based on the corresponding image sensor, can be flat or curved, especially a curved surface being concave facing towards the object side of the lens system. 
     According to the present disclosure, an image correction unit, such as a field flattener, can be optionally disposed between the lens element closest to the image side of the lens system and the image surface for correction of aberrations such as field curvature. The optical properties of the image correction unit, such as curvature, thickness, index of refraction, position and surface shape (convex or concave surface with spherical, aspheric, diffractive or Fresnel types), can be adjusted according to the design of the identification module or the image capturing unit. In general, a preferable image correction unit is, for example, a thin transparent element having a concave object-side surface and a planar image-side surface, and the thin transparent element is disposed near the image surface. 
     According to the present disclosure, the lens system can include at least one stop, such as an aperture stop, a glare stop or a field stop. Said glare stop or said field stop is set for eliminating the stray light and thereby improving image quality thereof. 
     According to the present disclosure, an aperture stop can be configured as a front stop or a middle stop. A front stop disposed between an imaged object and the first lens element can provide a longer distance between an exit pupil of the lens system and the image surface to produce a telecentric effect, and thereby improves the image-sensing efficiency of an image sensor (for example, CCD or CMOS). A middle stop disposed between the first lens element and the image surface is favorable for enlarging the viewing angle of the lens system and thereby provides a wider field of view for the same. 
     According to the present disclosure, the lens system can include an aperture control unit. The aperture control unit may be a mechanical component or a light modulator, which can control the size and shape of the aperture through electricity or electrical signals. The mechanical component can include a movable member, such as a blade assembly or a light shielding sheet. The light modulator can include a shielding element, such as a filter, an electrochromic material or a liquid-crystal layer. The aperture control unit controls the amount of incident light or exposure time to enhance the capability of image quality adjustment. In addition, the aperture control unit can be the aperture stop of the present disclosure, which changes the f-number to obtain different image effects, such as the depth of field or lens speed. 
     According to the above description of the present disclosure, the following specific embodiments are provided for further explanation. 
     1st Embodiment 
       FIG.  1    is a schematic view of an identification module and a plate according to the 1st embodiment of the present disclosure.  FIG.  2    is a schematic view of the identification module in  FIG.  1   .  FIG.  3    shows, in order from left to right, spherical aberration curves, astigmatic field curves and a distortion curve of the identification module according to the 1st embodiment. In  FIG.  1    and  FIG.  2   , the identification module includes the lens system (its reference numeral is omitted) of the present disclosure and an image sensor  170 . The lens system includes, in order from an object side to an image side, a first lens element  110 , a second lens element  120 , an aperture stop  100 , a third lens element  130 , a fourth lens element  140 , a filter  150  and an image surface  160 . The lens system includes four lens elements ( 110 ,  120 ,  130  and  140 ) with no additional lens element disposed between each of the adjacent four lens elements. 
     The first lens element  110  with negative refractive power has an object-side surface  111  being concave in a paraxial region thereof and an image-side surface  112  being concave in a paraxial region thereof. The first lens element  110  is made of plastic material and has the object-side surface  111  and the image-side surface  112  being both aspheric. The object-side surface  111  of the first lens element  110  has at least one inflection point. The image-side surface  112  of the first lens element  110  has at least one inflection point. The object-side surface  111  of the first lens element  110  has at least one convex critical point in an off-axis region thereof. 
     The second lens element  120  with positive refractive power has an object-side surface  121  being convex in a paraxial region thereof and an image-side surface  122  being concave in a paraxial region thereof. The second lens element  120  is made of plastic material and has the object-side surface  121  and the image-side surface  122  being both aspheric. 
     The third lens element  130  with positive refractive power has an object-side surface  131  being convex in a paraxial region thereof and an image-side surface  132  being convex in a paraxial region thereof. The third lens element  130  is made of plastic material and has the object-side surface  131  and the image-side surface  132  being both aspheric. 
     The fourth lens element  140  with positive refractive power has an object-side surface  141  being convex in a paraxial region thereof and an image-side surface  142  being convex in a paraxial region thereof. The fourth lens element  140  is made of plastic material and has the object-side surface  141  and the image-side surface  142  being both aspheric. The object-side surface  141  of the fourth lens element  140  has at least one inflection point. The image-side surface  142  of the fourth lens element  140  has at least one inflection point. The object-side surface  141  of the fourth lens element  140  has at least one concave critical point in an off-axis region thereof. The image-side surface  142  of the fourth lens element  140  has at least one critical point in an off-axis region thereof. In addition, the image-side surface  142  of the fourth lens element  140  has a convex-to-concave-to-convex shape change in order from the paraxial region thereof to the off-axis region thereof. 
     The filter  150  is made of glass material and located between the fourth lens element  140  and the image surface  160 , and will not affect the focal length of the lens system. The image sensor  170  is disposed on or near the image surface  160  of the lens system. 
     The plate  180  is made of glass material and located on an object side of the identification module, and will not affect the focal length of the lens system. An axial distance between the plate  180  and the first lens element  110  is 1.000 mm. 
     The equation of the aspheric surface profiles of the aforementioned lens elements of the 1st embodiment is expressed as follows: 
                 X   ⁡     (   Y   )       =         (       Y   2     ⁢     /     ⁢   R     )     ⁢     /     ⁢     (     1   +     s   ⁢   q   ⁢   r   ⁢     t   ⁡     (     1   -       (     1   +   k     )     ×       (     Y   ⁢     /     ⁢   R     )     2         )           )       +       ∑   i     ⁢       (   Ai   )     ×     (     Y   i     )             ,         
where,
 
     X is the relative distance between a point on the aspheric surface spaced at a distance Y from an optical axis and the tangential plane at the aspheric surface vertex on the optical axis; 
     Y is the vertical distance from the point on the aspheric surface to the optical axis; 
     R is the curvature radius; 
     k is the conic coefficient; and 
     Ai is the i-th aspheric coefficient, and in the embodiments, i may be, but is not limited to, 4, 6, 8, 10, 12, 14, 16 and 18. 
     In the lens system of the identification module according to the 1st embodiment, when a focal length of the lens system is f, an f-number of the lens system is Fno, and half of a maximum field of view of the lens system is HFOV, these parameters have the following values: f=0.37 millimeters (mm), Fno=1.63, HFOV=73.3 degrees (deg.). 
     When an Abbe number of the fourth lens element  140  is Vd4, the following condition is satisfied: Vd4=19.4. 
     When a minimum value among Abbe numbers of all lens elements of the lens system is Vdmin, the following condition is satisfied: Vdmin=19.4. In this embodiment, among the four lens elements ( 110 ,  120 ,  130  and  140 ) of the lens system, the Abbe number of the fourth lens element  140  is smaller than Abbe numbers of the other lens elements, and Vdmin is equal to the Abbe number of the fourth lens element  140 . 
     When an Abbe number of the first lens element  110  is Vd1, and a refractive index of the first lens element  110  is N1, the following condition is satisfied: Vd1/N1=36.18. When an Abbe number of the second lens element  120  is Vd2, and a refractive index of the second lens element  120  is N2, the following condition is satisfied: Vd2/N2=23.79. 
     When an Abbe number of the third lens element  130  is Vd3, and a refractive index of the third lens element  130  is N3, the following condition is satisfied: Vd3/N3=36.18. 
     When the Abbe number of the fourth lens element  140  is Vd4, and a refractive index of the fourth lens element  140  is N4, the following condition is satisfied: Vd4/N4=11.53. 
     When a central thickness of the second lens element  120  is CT2, and a central thickness of the third lens element  130  is CT3, the following condition is satisfied: CT2/CT3=0.50. 
     When a curvature radius of the object-side surface  111  of the first lens element  110  is R1, and the focal length of the lens system is f, the following condition is satisfied: R1/f=−6.00. 
     When a curvature radius of the object-side surface  141  of the fourth lens element  140  is R7, and a curvature radius of the image-side surface  142  of the fourth lens element  140  is R8, the following condition is satisfied: (R7+R8)/(R7−R8)=−0.45. 
     When the focal length of the lens system is f, and a focal length of the second lens element  120  is f2, the following condition is satisfied: f/f2=0.002. 
     When an axial distance between the object-side surface  111  of the first lens element  110  and the image surface  160  is TL, the following condition is satisfied: T=2.81 [mm]. 
     When an axial distance between the aperture stop  100  and the image-side surface  142  of the fourth lens element  140  is SD, and an axial distance between the object-side surface  111  of the first lens element  110  and the image-side surface  142  of the fourth lens element  140  is TD, the following condition is satisfied: SD/TD=0.45. 
     When a maximum image height of the lens system is ImgH, and the focal length of the lens system is f, the following condition is satisfied: ImgH/f=2.71. When the axial distance between the object-side surface  111  of the first lens element  110  and the image surface  160  is TL, and the maximum image height of the lens system is ImgH, the following condition is satisfied: TL/ImgH=2.81. 
     When the axial distance between the object-side surface  111  of the first lens element  110  and the image surface  160  is TL, and the focal length of the lens system is f, the following condition is satisfied: TL/f=7.60. 
     When the focal length of the lens system is f, and an entrance pupil diameter of the lens system is EPD, the following condition is satisfied: f/EPD=1.63. 
     When a maximum effective radius of the image-side surface  112  of the first lens element  110  is Y12, and a curvature radius of the image-side surface  112  of the first lens element  110  is R2, the following condition is satisfied: Y 12 /R2=1.45. 
     When half of the maximum field of view of the lens system is HFOV, and the f-number of the lens system is Fno, the following condition is satisfied: tan(HFOV)/Fno=2.04. 
     When a displacement in parallel with the optical axis from an axial vertex to a maximum effective radius position on the object-side surface  121  of the second lens element  120  is SAG 21 , a displacement in parallel with the optical axis from an axial vertex to a maximum effective radius position on the image-side surface  122  of the second lens element  120  is SAG 22 , and the focal length of the lens system is f, the following condition is satisfied: (|SAG 21 |+|SAG 22 |)/f=0.23. 
     When a vertical distance between the critical point on the object-side surface  111  of the first lens element  110  and the optical axis is Yc 11 , and the focal length of the lens system is f, the following condition is satisfied: Yc 11 /f=1.93. 
     When a vertical distance between the critical point on the object-side surface  141  of the fourth lens element  140  and the optical axis is Yc 41 , and the focal length of the lens system is f, the following condition is satisfied: Yc 41 /f=1.33. 
     When a vertical distance between the critical point on the image-side surface  142  of the fourth lens element  140  and the optical axis is Yc 42 , and the focal length of the lens system is f, the following condition is satisfied: Yc 42 /f=0.75 and 1.40. 
     The detailed optical data of the 1st embodiment are shown in Table 1 and the aspheric surface data are shown in Table 2 below. 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 1st Embodiment 
               
               
                 f = 0.37 mm, Fno = 1.63, HFOV = 73.3 deg. 
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 Surface # 
                   
                 Curvature Radius 
                 Thickness 
                 Material 
                 Index 
                 Abbe # 
                 Focal Length 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 0 
                 Object 
                 Plano 
                 0.000 
                   
                   
                   
                   
               
               
                 1 
                 Plate 
                 Plano 
                 1.465 
                 Glass 
                 1.517 
                 64.2 
                 — 
               
               
                 2 
                   
                 Plano 
                 1.000 
                   
                   
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                 3 
                 Lens 1 
                 −2.215 
                 (ASP) 
                 0.220 
                 Plastic 
                 1.548 
                 56.0 
                 −1.04 
               
               
                 4 
                   
                 0.795 
                 (ASP) 
                 0.691 
                   
                   
                   
                   
               
               
                 5 
                 Lens 2 
                 8.074 
                 (ASP) 
                 0.220 
                 Plastic 
                 1.572 
                 37.4 
                 171.37 
               
               
                 6 
                   
                 8.711 
                 (ASP) 
                 0.093 
                   
                   
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 7 
                 Ape. Stop 
                 Plano 
                 0.041 
                   
                   
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                 8 
                 Lens 3 
                 2.359 
                 (ASP) 
                 0.444 
                 Plastic 
                 1.548 
                 56.0 
                 0.81 
               
               
                 9 
                   
                 −0.513 
                 (ASP) 
                 0.221 
                   
                   
                   
                   
               
               
                 10 
                 Lens 4 
                 1.517 
                 (ASP) 
                 0.307 
                 Plastic 
                 1.682 
                 19.4 
                 1.65 
               
               
                 11 
                   
                 −3.999 
                 (ASP) 
                 0.300 
                   
                   
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 12 
                 Filter 
                 Plano 
                 0.145 
                 Glass 
                 1.517 
                 64.2 
                 — 
               
               
                 13 
                   
                 Plano 
                 0.125 
                   
                   
                   
                   
               
               
                 14 
                 Image 
                 Plano 
                 — 
               
               
                   
               
               
                 Note: Reference wavelength is 525.0 nm. 
               
            
           
         
       
     
     
       
         
           
               
             
               
                 TABLE 2 
               
               
                   
               
               
                 Aspheric Coefficients 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 Sur- 
                   
                   
                   
                   
               
               
                 face # 
                 3 
                 4 
                 5 
                 6 
               
               
                   
               
               
                 k = 
                 −4.5457E+01 
                 −1.0647E+00 
                 −5.0000E+01 
                 −9.0000E+01 
               
               
                 A4 = 
                 1.4243E−01 
                 −4.0076E−01 
                 −1.8165E+00 
                 4.4097E−01 
               
               
                 A6= 
                 −8.3307E−02 
                 2.2158E+00 
                 1.5111E+01 
                 3.9396E+01 
               
               
                 A8 = 
                 3.3028E−02 
                 −5.7827E+00 
                 −5.3375E+01 
                 −1.1634E+02 
               
               
                 A10 = 
                 −8.1611E−03 
                 6.6905E+00 
                 1.1047E+02 
                 −2.9746E+03 
               
               
                 A12 = 
                 1.0366E−03 
                 −3.6908E+00 
                 −1.2252E+02 
                 5.0345E+04 
               
               
                 A14 = 
                 −3.9220E−05 
                 8.0346E−01 
                 5.5911E+01 
                 −2.0658E+05 
               
               
                   
               
               
                 Sur- 
                   
                   
                   
                   
               
               
                 face # 
                 8 
                 9 
                 10 
                 11 
               
               
                   
               
               
                 k = 
                 8.6931E+00 
                 −6.1802E+00 
                 1.6016E+00 
                 −8.7791E+01 
               
               
                 A4 = 
                 −1.6223E+00 
                 −4.2958E+00 
                 1.4004E+00 
                 6.8232E−01 
               
               
                 A6 = 
                 4.2607E+01 
                 −4.6582E+00 
                 −4.4578E+01 
                 8.6627E−01 
               
               
                 A8 = 
                 −9.4874E+02 
                 3.8271E+02 
                 6.0375E+02 
                 −5.6698E+00 
               
               
                 A10 = 
                 1.3100E+04 
                 −5.0858E+03 
                 −4.6820E+03 
                 −5.3071E+01 
               
               
                 A12 = 
                 −8.3196E+04 
                 3.7405E+04 
                 2.1154E+04 
                 3.4494E+02 
               
               
                 A14 = 
                 2.0002E+05 
                 −1.5407E+05 
                 −5.5624E+04 
                 −8.5359E+02 
               
               
                 A16 = 
                 — 
                 2.8561E+05 
                 7.8566E+04 
                 1.0023E+03 
               
               
                 A18 = 
                 — 
                 — 
                 −4.5886E+04 
                 −4.6127E+02 
               
               
                   
               
            
           
         
       
     
     In Table 1, the curvature radius, the thickness and the focal length are shown in millimeters (mm). Surface numbers 0-14 represent the surfaces sequentially arranged from the object side to the image side along the optical axis. In Table 2, k represents the conic coefficient of the equation of the aspheric surface profiles. A4-18 represent the aspheric coefficients ranging from the 4th order to the 18th order. The tables presented below for each embodiment are the corresponding schematic parameter and aberration curves, and the definitions of the tables are the same as Table 1 and Table 2 of the 1st embodiment. Therefore, an explanation in this regard will not be provided again. 
     2nd Embodiment 
       FIG.  4    is a schematic view of an identification module and a plate according to the 2nd embodiment of the present disclosure.  FIG.  5    is a schematic view of the identification module in  FIG.  4   .  FIG.  6    shows, in order from left to right, spherical aberration curves, astigmatic field curves and a distortion curve of the identification module according to the 2nd embodiment. In  FIG.  4    and  FIG.  5   , the identification module includes the lens system (its reference numeral is omitted) of the present disclosure and an image sensor  270 . The lens system includes, in order from an object side to an image side, a first lens element  210 , a second lens element  220 , an aperture stop  200 , a third lens element  230 , a fourth lens element  240 , a filter  250  and an image surface  260 . The lens system includes four lens elements ( 210 ,  220 ,  230  and  240 ) with no additional lens element disposed between each of the adjacent four lens elements. 
     The first lens element  210  with negative refractive power has an object-side surface  211  being concave in a paraxial region thereof and an image-side surface  212  being concave in a paraxial region thereof. The first lens element  210  is made of plastic material and has the object-side surface  211  and the image-side surface  212  being both aspheric. The object-side surface  211  of the first lens element  210  has at least one inflection point. The image-side surface  212  of the first lens element  210  has at least one inflection point. The object-side surface  211  of the first lens element  210  has at least one convex critical point in an off-axis region thereof. 
     The second lens element  220  with negative refractive power has an object-side surface  221  being convex in a paraxial region thereof and an image-side surface  222  being concave in a paraxial region thereof. The second lens element  220  is made of plastic material and has the object-side surface  221  and the image-side surface  222  being both aspheric. 
     The third lens element  230  with positive refractive power has an object-side surface  231  being convex in a paraxial region thereof and an image-side surface  232  being convex in a paraxial region thereof. The third lens element  230  is made of plastic material and has the object-side surface  231  and the image-side surface  232  being both aspheric. 
     The fourth lens element  240  with positive refractive power has an object-side surface  241  being convex in a paraxial region thereof and an image-side surface  242  being convex in a paraxial region thereof. The fourth lens element  240  is made of plastic material and has the object-side surface  241  and the image-side surface  242  being both aspheric. The object-side surface  241  of the fourth lens element  240  has at least one inflection point. The image-side surface  242  of the fourth lens element  240  has at least one inflection point. The object-side surface  241  of the fourth lens element  240  has at least one concave critical point in an off-axis region thereof. In addition, the image-side surface  242  of the fourth lens element  240  has a convex-to-concave-to-convex shape change in order from the paraxial region thereof to an off-axis region thereof. 
     The filter  250  is made of glass material and located between the fourth lens element  240  and the image surface  260 , and will not affect the focal length of the lens system. The image sensor  270  is disposed on or near the image surface  260  of the lens system. 
     The plate  280  is made of glass material and located on an object side of the identification module, and will not affect the focal length of the lens system. An axial distance between the plate  280  and the first lens element  210  is 1.000 mm. 
     The detailed optical data of the 2nd embodiment are shown in Table 3 and the aspheric surface data are shown in Table 4 below. 
     
       
         
           
               
             
               
                 TABLE 3 
               
             
            
               
                   
               
               
                 2nd Embodiment 
               
               
                 f = 0.37 mm, Fno = 1.60, HFOV = 74.1 deg. 
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 Surface # 
                   
                 Curvature Radius 
                 Thickness 
                 Material 
                 Index 
                 Abbe # 
                 Focal Length 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 0 
                 Object 
                 Plano 
                 0.000 
                   
                   
                   
                   
               
               
                 1 
                 Plate 
                 Plano 
                 1.465 
                 Glass 
                 1.517 
                 64.2 
                 — 
               
               
                 2 
                   
                 Plano 
                 1.000 
                   
                   
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                 3 
                 Lens 1 
                 −2.051 
                 (ASP) 
                 0.220 
                 Plastic 
                 1.548 
                 56.0 
                 −1.01 
               
               
                 4 
                   
                 0.790 
                 (ASP) 
                 0.711 
                   
                   
                   
                   
               
               
                 5 
                 Lens 2 
                 6.535 
                 (ASP) 
                 0.221 
                 Plastic 
                 1.548 
                 55.9 
                 −31.62 
               
               
                 6 
                   
                 4.688 
                 (ASP) 
                 0.088 
                   
                   
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 7 
                 Ape. Stop 
                 Plano 
                 0.020 
                   
                   
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                 8 
                 Lens 3 
                 2.203 
                 (ASP) 
                 0.388 
                 Plastic 
                 1.548 
                 56.0 
                 0.80 
               
               
                 9 
                   
                 −0.512 
                 (ASP) 
                 0.248 
                   
                   
                   
                   
               
               
                 10 
                 Lens 4 
                 2.444 
                 (ASP) 
                 0.343 
                 Plastic 
                 1.682 
                 19.4 
                 1.45 
               
               
                 11 
                   
                 −1.576 
                 (ASP) 
                 0.300 
                   
                   
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 12 
                 Filter 
                 Plano 
                 0.145 
                 Glass 
                 1.517 
                 64.2 
                 — 
               
               
                 13 
                   
                 Plano 
                 0.119 
                   
                   
                   
                   
               
               
                 14 
                 Image 
                 Plano 
                 — 
               
               
                   
               
               
                 Note: Reference wavelength is 525.0 nm. 
               
            
           
         
       
     
     
       
         
           
               
             
               
                 TABLE 4 
               
               
                   
               
               
                 Aspheric Coefficients 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 Sur- 
                   
                   
                   
                   
               
               
                 face # 
                 3 
                 4 
                 5 
                 6 
               
               
                   
               
               
                 k = 
                 −3.4078E+01 
                 −1.0651E+00 
                 −5.0000E+01 
                 −9.0000E+01 
               
               
                 A4 = 
                 1.3944E−01 
                 −3.4253E−01 
                 −2.3722E+00 
                 −2.5390E+00 
               
               
                 A6 = 
                 −8.3120E−02 
                 1.9039E+00 
                 1.9022E+01 
                 2.9509E+02 
               
               
                 A8 = 
                 3.3272E−02 
                 −5.0818E+00 
                 −6.6898E+01 
                 −1.0012E+04 
               
               
                 A10 = 
                 −8.1494E−03 
                 5.8984E+00 
                 1.3993E+02 
                 2.0923E+05 
               
               
                 A12 = 
                 1.0302E−03 
                 −3.2468E+00 
                 −1.6027E+02 
                 −2.2203E+06 
               
               
                 A14 = 
                 −4.3149E−05 
                 7.0417E−01 
                 8.0510E+01 
                 9.6743E+06 
               
               
                   
               
               
                 Sur- 
                   
                   
                   
                   
               
               
                 face # 
                 8 
                 9 
                 10 
                 11 
               
               
                   
               
               
                 k = 
                 2.0000E+01 
                 −9.2931E+00 
                 2.4663E+00 
                 −8.7791E+01 
               
               
                 A4 = 
                 −7.9083E−01 
                 −5.1565E+00 
                 1.5778E+00 
                 −1.6408E+00 
               
               
                 A6= 
                 5.5186E+01 
                 −5.6144E+01 
                 −6.2815E+01 
                 2.4261E+01 
               
               
                 A8 = 
                 −1.5179E+03 
                 2.8914E+03 
                 1.0899E+03 
                 −1.5908E+02 
               
               
                 A10 = 
                 2.6684E+04 
                 −5.1054E+04 
                 −1.0562E+04 
                 6.1629E+02 
               
               
                 A12 = 
                 −2.1501E+05 
                 4.7200E+05 
                 5.9448E+04 
                 −1.5075E+03 
               
               
                 A14 = 
                 6.4436E+05 
                 −2.2451E+06 
                 −1.9502E+05 
                 2.1990E+03 
               
               
                 A16 = 
                 — 
                 4.4246E+06 
                 3.4559E+05 
                 −1.6914E+03 
               
               
                 A18 = 
                 — 
                 — 
                 −2.5641E+05 
                 4.9403E+02 
               
               
                   
               
            
           
         
       
     
     In the 2nd embodiment, the equation of the aspheric surface profiles of the aforementioned lens elements is the same as the equation of the 1st embodiment. Also, the definitions of these parameters shown in the following table are the same as those stated in the 1st embodiment with corresponding values for the 2nd embodiment, so an explanation in this regard will not be provided again. 
     Moreover, these parameters can be calculated from Table 3 and Table 4 as the following values and satisfy the following conditions: 
     
       
         
           
               
             
               
                   
               
               
                 2nd Embodiment 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                 f [mm] 
                 0.37 
                 f/f2 
                 −0.01 
               
               
                 Fno 
                 1.60 
                 TL [mm] 
                 2.80 
               
               
                 HFOV [deg.] 
                 74.1 
                 SD/TD 
                 0.45 
               
               
                 Vd4 
                 19.4 
                 ImgH/f 
                 2.60 
               
               
                 Vdmin 
                 19.4 
                 TL/ImgH 
                 2.92 
               
               
                 Vd1/N1 
                 36.18 
                 TL/f 
                 7.61 
               
               
                 Vd2/N2 
                 36.11 
                 f/EPD 
                 1.60 
               
               
                 Vd3/N3 
                 36.18 
                 Y12/R2 
                 1.46 
               
               
                 Vd4/N4 
                 11.53 
                 tan(HFOV)/Fno 
                 2.19 
               
               
                 CT2/CT3 
                 0.57 
                 (|SAG21| + |SAG22|)/f 
                 0.25 
               
               
                 R1/f 
                 −5.56 
                 Yc11/f 
                 2.11 
               
               
                 (R7 + R8)/(R7 − R8) 
                 0.22 
                 Yc41/f 
                 1.17 
               
               
                   
               
            
           
         
       
     
     3rd Embodiment 
       FIG.  7    is a schematic view of an identification module and a plate according to the 3rd embodiment of the present disclosure.  FIG.  8    is a schematic view of the identification module in  FIG.  7   .  FIG.  9    shows, in order from left to right, spherical aberration curves, astigmatic field curves and a distortion curve of the identification module according to the 3rd embodiment. In  FIG.  7    and  FIG.  8   , the identification module includes the lens system (its reference numeral is omitted) of the present disclosure and an image sensor  370 . The lens system includes, in order from an object side to an image side, a first lens element  310 , a second lens element  320 , an aperture stop  300 , a third lens element  330 , a fourth lens element  340 , a filter  350  and an image surface  360 . The lens system includes four lens elements ( 310 ,  320 ,  330  and  340 ) with no additional lens element disposed between each of the adjacent four lens elements. 
     The first lens element  310  with negative refractive power has an object-side surface  311  being concave in a paraxial region thereof and an image-side surface  312  being concave in a paraxial region thereof. The first lens element  310  is made of plastic material and has the object-side surface  311  and the image-side surface  312  being both aspheric. The object-side surface  311  of the first lens element  310  has at least one inflection point. The image-side surface  312  of the first lens element  310  has at least one inflection point. The object-side surface  311  of the first lens element  310  has at least one convex critical point in an off-axis region thereof. 
     The second lens element  320  with positive refractive power has an object-side surface  321  being concave in a paraxial region thereof and an image-side surface  322  being convex in a paraxial region thereof. The second lens element  320  is made of plastic material and has the object-side surface  321  and the image-side surface  322  being both aspheric. 
     The third lens element  330  with positive refractive power has an object-side surface  331  being convex in a paraxial region thereof and an image-side surface  332  being convex in a paraxial region thereof. The third lens element  330  is made of plastic material and has the object-side surface  331  and the image-side surface  332  being both aspheric. 
     The fourth lens element  340  with positive refractive power has an object-side surface  341  being convex in a paraxial region thereof and an image-side surface  342  being convex in a paraxial region thereof. The fourth lens element  340  is made of plastic material and has the object-side surface  341  and the image-side surface  342  being both aspheric. The object-side surface  341  of the fourth lens element  340  has at least one inflection point. The image-side surface  342  of the fourth lens element  340  has at least one inflection point. The object-side surface  341  of the fourth lens element  340  has at least one concave critical point in an off-axis region thereof. The image-side surface  342  of the fourth lens element  340  has at least one critical point in an off-axis region thereof. In addition, the image-side surface  342  of the fourth lens element  340  has a convex-to-concave-to-convex shape change in order from the paraxial region thereof to the off-axis region thereof. 
     The filter  350  is made of glass material and located between the fourth lens element  340  and the image surface  360 , and will not affect the focal length of the lens system. The image sensor  370  is disposed on or near the image surface  360  of the lens system. 
     The plate  380  is made of glass material and located on an object side of the identification module, and will not affect the focal length of the lens system. An axial distance between the plate  380  and the first lens element  310  is 1.000 mm. 
     The detailed optical data of the 3rd embodiment are shown in Table 5 and the aspheric surface data are shown in Table 6 below. 
     
       
         
           
               
             
               
                 TABLE 5 
               
             
            
               
                   
               
               
                 3rd Embodiment 
               
               
                 f = 0.37 mm, Fno = 1.72, HFOV = 73.4 deg. 
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 Surface # 
                   
                 Curvature Radius 
                 Thickness 
                 Material 
                 Index 
                 Abbe # 
                 Focal Length 
               
               
                   
               
               
                 0 
                 Object 
                 Plano 
                 0.000 
                   
                   
                   
                   
               
               
                 1 
                 Plate 
                 Plano 
                 1.465 
                 Glass 
                 1.517 
                 64.2 
                 — 
               
               
                 2 
                   
                 Plano 
                 1.000 
                   
                   
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                 3 
                 Lens 1 
                 −2.204 
                 (ASP) 
                 0.220 
                 Plastic 
                 1.548 
                 56.0 
                 −1.03 
               
               
                 4 
                   
                 0.786 
                 (ASP) 
                 0.686 
                   
                   
                   
                   
               
               
                 5 
                 Lens 2 
                 −125.000 
                 (ASP) 
                 0.220 
                 Plastic 
                 1.572 
                 37.4 
                 59.21 
               
               
                 6 
                   
                 −26.659 
                 (ASP) 
                 0.098 
                   
                   
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 7 
                 Ape. Stop 
                 Plano 
                 0.049 
                   
                   
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                 8 
                 Lens 3 
                 2.278 
                 (ASP) 
                 0.436 
                 Plastic 
                 1.548 
                 56.0 
                 0.82 
               
               
                 9 
                   
                 −0.524 
                 (ASP) 
                 0.221 
                   
                   
                   
                   
               
               
                 10 
                 Lens 4 
                 1.423 
                 (ASP) 
                 0.307 
                 Plastic 
                 1.682 
                 19.4 
                 1.69 
               
               
                 11 
                   
                 −5.569 
                 (ASP) 
                 0.300 
                   
                   
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 12 
                 Filter 
                 Plano 
                 0.145 
                 Glass 
                 1.517 
                 64.2 
                 — 
               
               
                 13 
                   
                 Plano 
                 0.126 
                   
                   
                   
                   
               
               
                 14 
                 Image 
                 Plano 
                 — 
               
               
                   
               
               
                 Note: Reference wavelength is 525.0 nm. 
               
            
           
         
       
     
     
       
         
           
               
             
               
                 TABLE 6 
               
               
                   
               
               
                 Aspheric Coefficients 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 Sur- 
                   
                   
                   
                   
               
               
                 face # 
                 3 
                 4 
                 5 
                 6 
               
               
                   
               
               
                 k = 
                 −4.7006E+01 
                 −1.0653E+00 
                 2.0000E+01 
                 −9.0000E+01 
               
               
                 A4 = 
                 1.4297E−01 
                 −3.5248E−01 
                 −1.6504E+00 
                 6.2143E−01 
               
               
                 A6 = 
                 −8.3248E−02 
                 1.9555E+00 
                 1.5807E+01 
                 4.8222E+01 
               
               
                 A8 = 
                 3.2919E−02 
                 −5.2831E+00 
                 −6.2990E+01 
                 −4.8688E+02 
               
               
                 A10 = 
                 −8.1371E−03 
                 6.2161E+00 
                 1.4712E+02 
                 2.4893E+03 
               
               
                 A12 = 
                 1.0411E−03 
                 −3.4651E+00 
                 −1.8722E+02 
                 7.0599E+03 
               
               
                 A14 = 
                 −4.0065E−05 
                 7.6096E−01 
                 9.9840E+01 
                 −7.2868E+04 
               
               
                   
               
               
                 Sur- 
                   
                   
                   
                   
               
               
                 face # 
                 8 
                 9 
                 10 
                 11 
               
               
                   
               
               
                 k = 
                 2.3660E+00 
                 −9.0567E+00 
                 9.6319E−01 
                 −8.7791E+01 
               
               
                 A4 = 
                 −1.6024E+00 
                 −6.4937E+00 
                 9.7404E−01 
                 6.1170E−01 
               
               
                 A6= 
                 4.0340E+01 
                 3.5061E+01 
                 −3.8910E+01 
                 6.4322E−01 
               
               
                 A8 = 
                 −7.8820E+02 
                 −1.0003E+02 
                 5.4071E+02 
                 −2.4932E+00 
               
               
                 A10 = 
                 1.0663E+04 
                 −9.6486E+02 
                 −4.2919E+03 
                 −8.2097E+01 
               
               
                 A12 = 
                 −6.8155E+04 
                 1.4152E+04 
                 1.9790E+04 
                 4.6259E+02 
               
               
                 A14 = 
                 1.6617E+05 
                 −7.5774E+04 
                 −5.3233E+04 
                 −1.0889E+03 
               
               
                 A16 = 
                 — 
                 1.7056E+05 
                 7.7253E+04 
                 1.2396E+03 
               
               
                 A18 = 
                 — 
                 — 
                 −4.6542E+04 
                 −5.5972E+02 
               
               
                   
               
            
           
         
       
     
     In the 3rd embodiment, the equation of the aspheric surface profiles of the aforementioned lens elements is the same as the equation of the 1st embodiment. Also, the definitions of these parameters shown in the following table are the same as those stated in the 1st embodiment with corresponding values for the 3rd embodiment, so an explanation in this regard will not be provided again. 
     Moreover, these parameters can be calculated from Table 5 and Table 6 as the following values and satisfy the following conditions: 
     
       
         
           
               
             
               
                   
               
               
                 3rd Embodiment 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                 f [mm] 
                 0.37 
                 TL [mm] 
                 2.81 
               
               
                 Fno 
                 1.72 
                 SD/TD 
                 0.45 
               
               
                 HFOV [deg.] 
                 73.4 
                 ImgH/f 
                 2.72 
               
               
                 Vd4 
                 19.4 
                 TL/ImgH 
                 2.81 
               
               
                 Vdmin 
                 19.4 
                 TL/f 
                 7.63 
               
               
                 Vd1/N1 
                 36.18 
                 f/EPD 
                 1.72 
               
               
                 Vd2/N2 
                 23.79 
                 Y12/R2 
                 1.45 
               
               
                 Vd3/N3 
                 36.18 
                 tan(HFOV)/Fno 
                 1.95 
               
               
                 Vd4/N4 
                 11.53 
                 (|SAG21| + |SAG22|)/f 
                 0.18 
               
               
                 CT2/CT3 
                 0.50 
                 Yc11/f 
                 1.92 
               
               
                 R1/f 
                 −5.99 
                 Yc41/f 
                 1.27 
               
               
                 (R7 + R8)/(R7 − R8) 
                 −0.59 
                 Yc42/f 
                 0.70; 1.30 
               
               
                 f/f2 
                 0.01 
                 — 
                 — 
               
               
                   
               
            
           
         
       
     
     4th Embodiment 
       FIG.  10    is a schematic view of an identification module and a plate according to the 4th embodiment of the present disclosure.  FIG.  11    is a schematic view of the identification module in  FIG.  10   .  FIG.  12    shows, in order from left to right, spherical aberration curves, astigmatic field curves and a distortion curve of the identification module according to the 4th embodiment. In  FIG.  10    and  FIG.  11   , the identification module includes the lens system (its reference numeral is omitted) of the present disclosure and an image sensor  470 . The lens system includes, in order from an object side to an image side, a first lens element  410 , a second lens element  420 , an aperture stop  400 , a third lens element  430 , a fourth lens element  440 , a filter  450  and an image surface  460 . The lens system includes four lens elements ( 410 ,  420 ,  430  and  440 ) with no additional lens element disposed between each of the adjacent four lens elements. 
     The first lens element  410  with negative refractive power has an object-side surface  411  being concave in a paraxial region thereof and an image-side surface  412  being concave in a paraxial region thereof. The first lens element  410  is made of plastic material and has the object-side surface  411  and the image-side surface  412  being both aspheric. The object-side surface  411  of the first lens element  410  has at least one inflection point. The image-side surface  412  of the first lens element  410  has at least one inflection point. The object-side surface  411  of the first lens element  410  has at least one convex critical point in an off-axis region thereof. 
     The second lens element  420  with positive refractive power has an object-side surface  421  being convex in a paraxial region thereof and an image-side surface  422  being concave in a paraxial region thereof. The second lens element  420  is made of plastic material and has the object-side surface  421  and the image-side surface  422  being both aspheric. 
     The third lens element  430  with negative refractive power has an object-side surface  431  being convex in a paraxial region thereof and an image-side surface  432  being concave in a paraxial region thereof. The third lens element  430  is made of plastic material and has the object-side surface  431  and the image-side surface  432  being both aspheric. The image-side surface  432  of the third lens element  430  has at least one convex critical point in an off-axis region thereof. 
     The fourth lens element  440  with positive refractive power has an object-side surface  441  being convex in a paraxial region thereof and an image-side surface  442  being concave in a paraxial region thereof. The fourth lens element  440  is made of plastic material and has the object-side surface  441  and the image-side surface  442  being both aspheric. The object-side surface  441  of the fourth lens element  440  has at least one inflection point. The image-side surface  442  of the fourth lens element  440  has at least one inflection point. The object-side surface  441  of the fourth lens element  440  has at least one concave critical point in an off-axis region thereof. The image-side surface  442  of the fourth lens element  440  has at least one critical point in an off-axis region thereof. 
     The filter  450  is made of glass material and located between the fourth lens element  440  and the image surface  460 , and will not affect the focal length of the lens system. The image sensor  470  is disposed on or near the image surface  460  of the lens system. 
     The plate  480  is made of glass material and located on an object side of the identification module, and will not affect the focal length of the lens system. An axial distance between the plate  480  and the first lens element  410  is 1.000 mm. 
     The detailed optical data of the 4th embodiment are shown in Table 7 and the aspheric surface data are shown in Table 8 below. 
     
       
         
           
               
             
               
                 TABLE 7 
               
             
            
               
                   
               
               
                 4th Embodiment 
               
               
                 f = 0.37 mm, Fno = 1.71, HFOV = 72.2 deg. 
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 Surface # 
                   
                 Curvature Radius 
                 Thickness 
                 Material 
                 Index 
                 Abbe # 
                 Focal Length 
               
               
                   
               
               
                 0 
                 Object 
                 Plano 
                 0.000 
                   
                   
                   
                   
               
               
                 1 
                 Plate 
                 Plano 
                 1.465 
                 Glass 
                 1.517 
                 64.2 
                 — 
               
               
                 2 
                   
                 Plano 
                 1.000 
                   
                   
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                 3 
                 Lens 1 
                 −2.784 
                 (ASP) 
                 0.299 
                 Plastic 
                 1.548 
                 56.0 
                 −1.04 
               
               
                 4 
                   
                 0.747 
                 (ASP) 
                 0.596 
                   
                   
                   
                   
               
               
                 5 
                 Lens 2 
                 7.594 
                 (ASP) 
                 0.221 
                 Plastic 
                 1.548 
                 56.0 
                 14.16 
               
               
                 6 
                   
                 353.912 
                 (ASP) 
                 0.131 
                   
                   
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 7 
                 Ape. Stop 
                 Plano 
                 0.036 
                   
                   
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                 8 
                 Lens 3 
                 1.554 
                 (ASP) 
                 0.526 
                 Plastic 
                 1.548 
                 56.0 
                 −12.25 
               
               
                 9 
                   
                 1.111 
                 (ASP) 
                 0.034 
                   
                   
                   
                   
               
               
                 10 
                 Lens 4 
                 0.346 
                 (ASP) 
                 0.323 
                 Plastic 
                 1.682 
                 19.4 
                 0.51 
               
               
                 11 
                   
                 25.773 
                 (ASP) 
                 0.300 
                   
                   
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 12 
                 Filter 
                 Plano 
                 0.145 
                 Glass 
                 1.517 
                 64.2 
                 — 
               
               
                 13 
                   
                 Plano 
                 0.191 
                   
                   
                   
                   
               
               
                 14 
                 Image 
                 Plano 
                 — 
               
               
                   
               
               
                 Note: Reference wavelength is 525.0 nm. 
               
            
           
         
       
     
     
       
         
           
               
             
               
                 TABLE 8 
               
               
                   
               
               
                 Aspheric Coefficients 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 Sur- 
                   
                   
                   
                   
               
               
                 face # 
                 3 
                 4 
                 5 
                 6 
               
               
                   
               
               
                 k = 
                 −8.3824E+01 
                 −1.0869E+00 
                 1.7980E+01 
                 −9.0000E+01 
               
               
                 A4 = 
                 1.3519E−01 
                 1.3847E−01 
                 −1.5809E+00 
                 −3.0159E−02 
               
               
                 A6 = 
                 −7.9878E−02 
                 −6.0705E−01 
                 1.3290E+01 
                 3.9173E+01 
               
               
                 A8 = 
                 3.3334E−02 
                 2.6941E−01 
                 −5.0492E+01 
                 −3.7311E+02 
               
               
                 A10 = 
                 −8.1270E−03 
                 1.4541E−01 
                 1.0898E+02 
                 1.7343E+03 
               
               
                 A12 = 
                 1.0255E−03 
                 −1.4691E−01 
                 −1.2581E+02 
                 −1.7990E+03 
               
               
                 A14 = 
                 −5.3685E−05 
                 3.2435E−02 
                 6.0059E+01 
                 −4.7706E+03 
               
               
                   
               
               
                 Sur- 
                   
                   
                   
                   
               
               
                 face # 
                 8 
                 9 
                 10 
                 11 
               
               
                   
               
               
                 k = 
                 −6.5861E+01 
                 −4.1354E+01 
                 −1.9803E+00 
                 −9.0000E+01 
               
               
                 A4 = 
                 1.0885E+00 
                 −1.9547E+01 
                 −1.1745E+01 
                 3.8825E+00 
               
               
                 A6 = 
                 −8.3575E+00 
                 2.2396E+02 
                 1.6826E+02 
                 −3.3267E+01 
               
               
                 A8 = 
                 6.0670E+01 
                 −1.8875E+03 
                 −1.7139E+03 
                 1.6695E+02 
               
               
                 A10 = 
                 −7.6099E+02 
                 1.0243E+04 
                 1.1748E+04 
                 −6.0361E+02 
               
               
                 A12 = 
                 5.7142E+03 
                 −3.2917E+04 
                 −5.3406E+04 
                 1.4596E+03 
               
               
                 A14 = 
                 −1.4740E+04 
                 5.5281E+04 
                 1.5215E+05 
                 −2.2018E+03 
               
               
                 A16 = 
                 — 
                 −3.5639E+04 
                 −2.4473E+05 
                 1.8572E+03 
               
               
                 A18 = 
                 — 
                 — 
                 1.6832E+05 
                 −6.6814E+02 
               
               
                   
               
            
           
         
       
     
     In the 4th embodiment, the equation of the aspheric surface profiles of the aforementioned lens elements is the same as the equation of the 1st embodiment. Also, the definitions of these parameters shown in the following table are the same as those stated in the 1st embodiment with corresponding values for the 4th embodiment, so an explanation in this regard will not be provided again. 
     Moreover, these parameters can be calculated from Table 7 and Table 8 as the following values and satisfy the following conditions: 
     
       
         
           
               
             
               
                   
               
               
                 4th Embodiment 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                 f [mm] 
                 0.37 
                 TL [mm] 
                 2.80 
               
               
                 Fno 
                 1.71 
                 SD/TD 
                 0.42 
               
               
                 HFOV [deg.] 
                 72.2 
                 ImgH/f 
                 2.71 
               
               
                 Vd4 
                 19.4 
                 TL/ImgH 
                 2.80 
               
               
                 Vdmin 
                 19.4 
                 TL/f 
                 7.59 
               
               
                 Vd1/N1 
                 36.18 
                 f/EPD 
                 1.71 
               
               
                 Vd2/N2 
                 36.18 
                 Y12/R2 
                 1.68 
               
               
                 Vd3/N3 
                 36.18 
                 tan(HFOV)/Fno 
                 1.82 
               
               
                 Vd4/N4 
                 11.53 
                 (|SAG21| + |SAG22|)/f 
                 0.28 
               
               
                 CT2/CT3 
                 0.42 
                 Yc11/f 
                 1.72 
               
               
                 R1/f 
                 −7.54 
                 Yc41/f 
                 1.19 
               
               
                 (R7 + R8)/(R7 − R8) 
                 −1.03 
                 Yc42/f 
                 1.32 
               
               
                 f/f2 
                 0.03 
                 — 
                 — 
               
               
                   
               
            
           
         
       
     
     5th Embodiment 
       FIG.  13    is a schematic view of an identification module and a plate according to the 5th embodiment of the present disclosure.  FIG.  14    is a schematic view of the identification module in  FIG.  13   .  FIG.  15    shows, in order from left to right, spherical aberration curves, astigmatic field curves and a distortion curve of the identification module according to the 5th embodiment. In  FIG.  13    and  FIG.  14   , the identification module includes the lens system (its reference numeral is omitted) of the present disclosure and an image sensor  570 . The lens system includes, in order from an object side to an image side, a first lens element  510 , a second lens element  520 , an aperture stop  500 , a third lens element  530 , a fourth lens element  540 , a filter  550  and an image surface  560 . The lens system includes four lens elements ( 510 ,  520 ,  530  and  540 ) with no additional lens element disposed between each of the adjacent four lens elements. 
     The first lens element  510  with negative refractive power has an object-side surface  511  being concave in a paraxial region thereof and an image-side surface  512  being concave in a paraxial region thereof. The first lens element  510  is made of plastic material and has the object-side surface  511  and the image-side surface  512  being both aspheric. The object-side surface  511  of the first lens element  510  has at least one inflection point. The image-side surface  512  of the first lens element  510  has at least one inflection point. The object-side surface  511  of the first lens element  510  has at least one convex critical point in an off-axis region thereof. 
     The second lens element  520  with positive refractive power has an object-side surface  521  being convex in a paraxial region thereof and an image-side surface  522  being convex in a paraxial region thereof. The second lens element  520  is made of plastic material and has the object-side surface  521  and the image-side surface  522  being both aspheric. 
     The third lens element  530  with negative refractive power has an object-side surface  531  being convex in a paraxial region thereof and an image-side surface  532  being concave in a paraxial region thereof. The third lens element  530  is made of plastic material and has the object-side surface  531  and the image-side surface  532  being both aspheric. The image-side surface  532  of the third lens element  530  has at least one convex critical point in an off-axis region thereof. 
     The fourth lens element  540  with positive refractive power has an object-side surface  541  being convex in a paraxial region thereof and an image-side surface  542  being convex in a paraxial region thereof. The fourth lens element  540  is made of plastic material and has the object-side surface  541  and the image-side surface  542  being both aspheric. The object-side surface  541  of the fourth lens element  540  has at least one inflection point. The image-side surface  542  of the fourth lens element  540  has at least one inflection point. The object-side surface  541  of the fourth lens element  540  has at least one concave critical point in an off-axis region thereof. The image-side surface  542  of the fourth lens element  540  has at least one critical point in an off-axis region thereof. In addition, the image-side surface  542  of the fourth lens element  540  has a convex-to-concave-to-convex shape change in order from the paraxial region thereof to the off-axis region thereof. 
     The filter  550  is made of glass material and located between the fourth lens element  540  and the image surface  560 , and will not affect the focal length of the lens system. The image sensor  570  is disposed on or near the image surface  560  of the lens system. 
     The plate  580  is made of glass material and located on an object side of the identification module, and will not affect the focal length of the lens system. An axial distance between the plate  580  and the first lens element  510  is 1.050 mm. 
     The detailed optical data of the 5th embodiment are shown in Table 9 and the aspheric surface data are shown in Table 10 below. 
     
       
         
           
               
             
               
                 TABLE 9 
               
             
            
               
                   
               
               
                 5th Embodiment 
               
               
                 f = 0.37 mm, Fno = 1.73, HFOV = 72.3 deg. 
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 Surface # 
                   
                 Curvature Radius 
                 Thickness 
                 Material 
                 Index 
                 Abbe # 
                 Focal Length 
               
               
                   
               
               
                 0 
                 Object 
                 Plano 
                 0.000 
                   
                   
                   
                   
               
               
                 1 
                 Plate 
                 Plano 
                 1.465 
                 Glass 
                 1.517 
                 64.2 
                 — 
               
               
                 2 
                   
                 Plano 
                 1.050 
                   
                   
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                 3 
                 Lens 1 
                 −1.789 
                 (ASP) 
                 0.230 
                 Plastic 
                 1.548 
                 56.0 
                 −1.15 
               
               
                 4 
                   
                 1.013 
                 (ASP) 
                 0.619 
                   
                   
                   
                   
               
               
                 5 
                 Lens 2 
                 488.903 
                 (ASP) 
                 0.234 
                 Plastic 
                 1.548 
                 56.0 
                 22.81 
               
               
                 6 
                   
                 −12.818 
                 (ASP) 
                 0.097 
                   
                   
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 7 
                 Ape. Stop 
                 Plano 
                 0.043 
                   
                   
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                 8 
                 Lens 3 
                 2.745 
                 (ASP) 
                 0.459 
                 Plastic 
                 1.548 
                 56.0 
                 −1.17 
               
               
                 9 
                   
                 0.487 
                 (ASP) 
                 0.030 
                   
                   
                   
                   
               
               
                 10 
                 Lens 4 
                 0.254 
                 (ASP) 
                 0.353 
                 Plastic 
                 1.614 
                 27.7 
                 0.39 
               
               
                 11 
                   
                 −2.274 
                 (ASP) 
                 0.300 
                   
                   
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 12 
                 Filter 
                 Plano 
                 0.145 
                 Glass 
                 1.517 
                 64.2 
                 — 
               
               
                 13 
                   
                 Plano 
                 0.243 
                   
                   
                   
                   
               
               
                 14 
                 Image 
                 Plano 
                 — 
               
               
                   
               
               
                 Note: Reference wavelength is 525.0 nm. 
               
            
           
         
       
     
     
       
         
           
               
             
               
                 TABLE 10 
               
               
                   
               
               
                 Aspheric Coefficients 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 Sur- 
                   
                   
                   
                   
               
               
                 face # 
                 3 
                 4 
                 5 
                 6 
               
               
                   
               
               
                 k = 
                 −3.6729E+01 
                 −8.6698E−01 
                 9.0000E+01 
                 2.7682E+01 
               
               
                 A4 = 
                 1.6527E−01 
                 2.1621E−01 
                 −1.4216E+00 
                 −3.0212E−02 
               
               
                 A6 = 
                 −1.0659E−01 
                 −7.2554E−01 
                 1.3397E+01 
                 6.8497E+01 
               
               
                 A8 = 
                 4.6575E−02 
                 5.9931E−01 
                 −5.1107E+01 
                 −9.5022E+02 
               
               
                 A10 = 
                 −1.1981E−02 
                 −2.5028E−01 
                 1.0534E+02 
                 6.7862E+03 
               
               
                 A12 = 
                 1.5612E−03 
                 5.3422E−02 
                 −1.1086E+02 
                 −2.2732E+04 
               
               
                 A14 = 
                 −7.8372E−05 
                 −4.5867E−03 
                 4.6013E+01 
                 2.7728E+04 
               
               
                   
               
               
                 Sur- 
                   
                   
                   
                   
               
               
                 face # 
                 8 
                 9 
                 10 
                 11 
               
               
                   
               
               
                 k = 
                 −6.5280E+01 
                 −4.4662E+01 
                 −7.3331E+00 
                 −9.0000E+01 
               
               
                 A4 = 
                 −6.9509E−01 
                 −2.2288E+01 
                 −1.4129E+00 
                 2.2936E+00 
               
               
                 A6 = 
                 −2.0024E+01 
                 3.4416E+02 
                 1.6207E+01 
                 −1.0133E+01 
               
               
                 A8 = 
                 7.0752E+02 
                 −4.2446E+03 
                 −1.3574E+02 
                 1.4340E+01 
               
               
                 A10 = 
                 −7.5057E+03 
                 3.4823E+04 
                 4.2383E+02 
                 −7.3571E+01 
               
               
                 A12 = 
                 3.4287E+04 
                 −1.8347E+05 
                 −5.6317E+02 
                 5.0684E+02 
               
               
                 A14 = 
                 −5.3021E+04 
                 5.8999E+05 
                 3.8313E+02 
                 −1.5528E+03 
               
               
                 A16 = 
                 — 
                 −1.0312E+06 
                 −3.3995E+02 
                 2.3916E+03 
               
               
                 A18 = 
                 — 
                 7.3956E+05 
                 2.5568E+02 
                 −1.8386E+03 
               
               
                 A20 = 
                 — 
                 — 
                 — 
                 5.6087E+02 
               
               
                   
               
            
           
         
       
     
     In the 5th embodiment, the equation of the aspheric surface profiles of the aforementioned lens elements is the same as the equation of the 1st embodiment. Also, the definitions of these parameters shown in the following table are the same as those stated in the 1st embodiment with corresponding values for the 5th embodiment, so an explanation in this regard will not be provided again. 
     Moreover, these parameters can be calculated from Table 9 and Table 10 as the following values and satisfy the following conditions: 
     
       
         
           
               
             
               
                   
               
               
                 5th Embodiment 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                 f [mm] 
                 0.37 
                 TL [mm] 
                 2.75 
               
               
                 Fno 
                 1.73 
                 SD/TD 
                 0.43 
               
               
                 HFOV [deg.] 
                 72.3 
                 ImgH/f 
                 2.64 
               
               
                 Vd4 
                 27.7 
                 TL/ImgH 
                 2.81 
               
               
                 Vdmin 
                 27.7 
                 TL/f 
                 7.43 
               
               
                 Vd1/N1 
                 36.18 
                 f/EPD 
                 1.73 
               
               
                 Vd2/N2 
                 36.18 
                 Y12/R2 
                 1.35 
               
               
                 Vd3/N3 
                 36.18 
                 tan(HFOV)/Fno 
                 1.81 
               
               
                 Vd4/N4 
                 17.16 
                 (|SAG21| + |SAG22|)/f 
                 0.18 
               
               
                 CT2/CT3 
                 0.51 
                 Yc11/f 
                 1.94 
               
               
                 R1/f 
                 −4.82 
                 Yc41/f 
                 1.16 
               
               
                 (R7 + R8)/(R7 − R8) 
                 −0.80 
                 Yc42/f 
                 0.61; 1.12 
               
               
                 f/f2 
                 0.02 
                 — 
                 — 
               
               
                   
               
            
           
         
       
     
     6th Embodiment 
       FIG.  16    is a schematic view of an identification module and a plate according to the 6th embodiment of the present disclosure.  FIG.  17    is a schematic view of the identification module in  FIG.  16   .  FIG.  18    shows, in order from left to right, spherical aberration curves, astigmatic field curves and a distortion curve of the identification module according to the 6th embodiment. In  FIG.  16    and  FIG.  17   , the identification module includes the lens system (its reference numeral is omitted) of the present disclosure and an image sensor  670 . The lens system includes, in order from an object side to an image side, a first lens element  610 , a second lens element  620 , an aperture stop  600 , a third lens element  630 , a fourth lens element  640 , a filter  650  and an image surface  660 . The lens system includes four lens elements ( 610 ,  620 ,  630  and  640 ) with no additional lens element disposed between each of the adjacent four lens elements. 
     The first lens element  610  with negative refractive power has an object-side surface  611  being concave in a paraxial region thereof and an image-side surface  612  being concave in a paraxial region thereof. The first lens element  610  is made of plastic material and has the object-side surface  611  and the image-side surface  612  being both aspheric. The object-side surface  611  of the first lens element  610  has at least one inflection point. The image-side surface  612  of the first lens element  610  has at least one inflection point. The object-side surface  611  of the first lens element  610  has at least one convex critical point in an off-axis region thereof. 
     The second lens element  620  with positive refractive power has an object-side surface  621  being convex in a paraxial region thereof and an image-side surface  622  being convex in a paraxial region thereof. The second lens element  620  is made of plastic material and has the object-side surface  621  and the image-side surface  622  being both aspheric. 
     The third lens element  630  with negative refractive power has an object-side surface  631  being convex in a paraxial region thereof and an image-side surface  632  being concave in a paraxial region thereof. The third lens element  630  is made of plastic material and has the object-side surface  631  and the image-side surface  632  being both aspheric. The image-side surface  632  of the third lens element  630  has at least one convex critical point in an off-axis region thereof. 
     The fourth lens element  640  with positive refractive power has an object-side surface  641  being convex in a paraxial region thereof and an image-side surface  642  being convex in a paraxial region thereof. The fourth lens element  640  is made of plastic material and has the object-side surface  641  and the image-side surface  642  being both aspheric. The object-side surface  641  of the fourth lens element  640  has at least one inflection point. The image-side surface  642  of the fourth lens element  640  has at least one inflection point. The object-side surface  641  of the fourth lens element  640  has at least one concave critical point in an off-axis region thereof. The image-side surface  642  of the fourth lens element  640  has at least one critical point in an off-axis region thereof. In addition, the image-side surface  642  of the fourth lens element  640  has a convex-to-concave-to-convex shape change in order from the paraxial region thereof to the off-axis region thereof. 
     The filter  650  is made of glass material and located between the fourth lens element  640  and the image surface  660 , and will not affect the focal length of the lens system. The image sensor  670  is disposed on or near the image surface  660  of the lens system. 
     The plate  680  is made of glass material and located on an object side of the identification module, and will not affect the focal length of the lens system. An axial distance between the plate  680  and the first lens element  610  is 1.050 mm. 
     The detailed optical data of the 6th embodiment are shown in Table 11 and the aspheric surface data are shown in Table 12 below. 
     
       
         
           
               
             
               
                 TABLE 11 
               
             
            
               
                   
               
               
                 6th Embodiment 
               
               
                 f = 0.37 mm, Fno = 1.75, HFOV = 72.2 deg. 
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 Surface # 
                   
                 Curvature Radius 
                 Thickness 
                 Material 
                 Index 
                 Abbe # 
                 Focal Length 
               
               
                   
               
               
                 0 
                 Object 
                 Plano 
                 0.000 
                   
                   
                   
                   
               
               
                 1 
                 Plate 
                 Plano 
                 1.465 
                 Glass 
                 1.517 
                 64.2 
                 — 
               
               
                 2 
                   
                 Plano 
                 1.050 
                   
                   
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                 3 
                 Lens 1 
                 −1.797 
                 (ASP) 
                 0.230 
                 Plastic 
                 1.548 
                 56.0 
                 −1.15 
               
               
                 4 
                   
                 1.020 
                 (ASP) 
                 0.615 
                   
                   
                   
                   
               
               
                 5 
                 Lens 2 
                 58.598 
                 (ASP) 
                 0.233 
                 Plastic 
                 1.548 
                 56.0 
                 37.92 
               
               
                 6 
                   
                 −32.131 
                 (ASP) 
                 0.097 
                   
                   
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 7 
                 Ape. Stop 
                 Plano 
                 0.046 
                   
                   
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                 8 
                 Lens 3 
                 2.435 
                 (ASP) 
                 0.455 
                 Plastic 
                 1.548 
                 56.0 
                 −1.87 
               
               
                 9 
                   
                 0.674 
                 (ASP) 
                 0.037 
                   
                   
                   
                   
               
               
                 10 
                 Lens 4 
                 0.288 
                 (ASP) 
                 0.351 
                 Plastic 
                 1.623 
                 26.0 
                 0.44 
               
               
                 11 
                   
                 −3.034 
                 (ASP) 
                 0.300 
                   
                   
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 12 
                 Filter 
                 Plano 
                 0.145 
                 Glass 
                 1.517 
                 64.2 
                 — 
               
               
                 13 
                   
                 Plano 
                 0.242 
                   
                   
                   
                   
               
               
                 14 
                 Image 
                 Plano 
                 — 
               
               
                   
               
               
                 Note: Reference wavelength is 525.0 nm. 
               
            
           
         
       
     
     
       
         
           
               
             
               
                 TABLE 12 
               
               
                   
               
               
                 Aspheric Coefficients 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 Sur- 
                   
                   
                   
                   
               
               
                 face # 
                 3 
                 4 
                 5 
                 6 
               
               
                   
               
               
                 k = 
                 −3.6682E+01 
                 −8.6293E−01 
                 9.0000E+01 
                 8.6350E+01 
               
               
                 A4 = 
                 1.6037E−01 
                 2.1426E−01 
                 −1.5829E+00 
                 −1.0422E+00 
               
               
                 A6 = 
                 −1.0115E−01 
                 −7.3192E−01 
                 1.5003E+01 
                 1.0490E+02 
               
               
                 A8 = 
                 4.3622E−02 
                 6.1356E−01 
                 −5.8377E+01 
                 −1.4855E+03 
               
               
                 A10 = 
                 −1.1124E−02 
                 −2.5986E−01 
                 1.2318E+02 
                 1.0936E+04 
               
               
                 A12 = 
                 1.4341E−03 
                 5.6113E−02 
                 −1.3309E+02 
                 −3.7735E+04 
               
               
                 A14 = 
                 −7.0613E−05 
                 −4.8467E−03 
                 5.6818E+01 
                 4.7492E+04 
               
               
                   
               
               
                 Sur- 
                   
                   
                   
                   
               
               
                 face # 
                 8 
                 9 
                 10 
                 11 
               
               
                   
               
               
                 k = 
                 7.4255E+00 
                 −8.8078E+01 
                 −8.5801E+00 
                 7.8746E+00 
               
               
                 A4 = 
                 −1.5875E+00 
                 −2.0904E+01 
                 −8.4213E−01 
                 1.8153E+00 
               
               
                 A6 = 
                 −8.4295E+00 
                 3.2448E+02 
                 1.0476E+01 
                 2.9286E+00 
               
               
                 A8 = 
                 6.1154E+02 
                 −4.0558E+03 
                 −9.8718E+01 
                 −1.0801E+02 
               
               
                 A10 = 
                 −7.6856E+03 
                 3.3673E+04 
                 2.7181E+02 
                 6.0527E+02 
               
               
                 A12 = 
                 4.2362E+04 
                 −1.7938E+05 
                 −1.3958E+02 
                 −1.8158E+03 
               
               
                 A14 = 
                 −1.0523E+05 
                 5.8338E+05 
                 −4.0156E+02 
                 3.3429E+03 
               
               
                 A16 = 
                 9.6587E+04 
                 −1.0316E+06 
                 5.0250E+02 
                 −3.7497E+03 
               
               
                 A18 = 
                 — 
                 7.4868E+05 
                 −1.2665E+02 
                 2.3338E+03 
               
               
                 A20 = 
                 — 
                 — 
                 — 
                 −6.1384E+02 
               
               
                   
               
            
           
         
       
     
     In the 6th embodiment, the equation of the aspheric surface profiles of the aforementioned lens elements is the same as the equation of the 1st embodiment. Also, the definitions of these parameters shown in the following table are the same as those stated in the 1st embodiment with corresponding values for the 6th embodiment, so an explanation in this regard will not be provided again. 
     Moreover, these parameters can be calculated from Table 11 and Table 12 as the following values and satisfy the following conditions: 
     
       
         
           
               
             
               
                   
               
               
                 6th Embodiment 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                 f [mm] 
                 0.37 
                 TL [mm] 
                 2.75 
               
               
                 Fno 
                 1.75 
                 SD/TD 
                 0.43 
               
               
                 HFOV [deg.] 
                 72.2 
                 ImgH/f 
                 2.57 
               
               
                 Vd4 
                 26.0 
                 TL/ImgH 
                 2.87 
               
               
                 Vdmin 
                 26.0 
                 TL/f 
                 7.37 
               
               
                 Vd1/N1 
                 36.18 
                 f/EPD 
                 1.75 
               
               
                 Vd2/N2 
                 36.18 
                 Y12/R2 
                 1.39 
               
               
                 Vd3/N3 
                 36.18 
                 tan(HFOV)/Fno 
                 1.82 
               
               
                 Vd4/N4 
                 16.02 
                 (|SAG21| + |SAG22|)/f 
                 0.19 
               
               
                 CT2/CT3 
                 0.51 
                 Yc11/f 
                 1.95 
               
               
                 R1/f 
                 −4.81 
                 Yc41/f 
                 1.15 
               
               
                 (R7 + R8)/(R7 − R8) 
                 −0.83 
                 Yc42/f 
                 0.61; 1.11 
               
               
                 f/f2 
                 0.01 
                 — 
                 — 
               
               
                   
               
            
           
         
       
     
     7th Embodiment 
       FIG.  19    is a schematic view of an identification module and a plate according to the 7th embodiment of the present disclosure.  FIG.  20    is a schematic view of the identification module in  FIG.  19   .  FIG.  21    shows, in order from left to right, spherical aberration curves, astigmatic field curves and a distortion curve of the identification module according to the 7th embodiment. In  FIG.  19    and  FIG.  20   , the identification module includes the lens system (its reference numeral is omitted) of the present disclosure and an image sensor  770 . The lens system includes, in order from an object side to an image side, a first lens element  710 , a second lens element  720 , an aperture stop  700 , a third lens element  730 , a fourth lens element  740 , a filter  750  and an image surface  760 . The lens system includes four lens elements ( 710 ,  720 ,  730  and  740 ) with no additional lens element disposed between each of the adjacent four lens elements. 
     The first lens element  710  with negative refractive power has an object-side surface  711  being concave in a paraxial region thereof and an image-side surface  712  being concave in a paraxial region thereof. The first lens element  710  is made of plastic material and has the object-side surface  711  and the image-side surface  712  being both aspheric. The object-side surface  711  of the first lens element  710  has at least one inflection point. The image-side surface  712  of the first lens element  710  has at least one inflection point. The object-side surface  711  of the first lens element  710  has at least one convex critical point in an off-axis region thereof. 
     The second lens element  720  with negative refractive power has an object-side surface  721  being convex in a paraxial region thereof and an image-side surface  722  being concave in a paraxial region thereof. The second lens element  720  is made of plastic material and has the object-side surface  721  and the image-side surface  722  being both aspheric. 
     The third lens element  730  with positive refractive power has an object-side surface  731  being convex in a paraxial region thereof and an image-side surface  732  being concave in a paraxial region thereof. The third lens element  730  is made of plastic material and has the object-side surface  731  and the image-side surface  732  being both aspheric. The image-side surface  732  of the third lens element  730  has at least one convex critical point in an off-axis region thereof. 
     The fourth lens element  740  with positive refractive power has an object-side surface  741  being convex in a paraxial region thereof and an image-side surface  742  being convex in a paraxial region thereof. The fourth lens element  740  is made of plastic material and has the object-side surface  741  and the image-side surface  742  being both aspheric. The object-side surface  741  of the fourth lens element  740  has at least one inflection point. The image-side surface  742  of the fourth lens element  740  has at least one inflection point. The object-side surface  741  of the fourth lens element  740  has at least one concave critical point in an off-axis region thereof. The image-side surface  742  of the fourth lens element  740  has at least one critical point in an off-axis region thereof. In addition, the image-side surface  742  of the fourth lens element  740  has a convex-to-concave-to-convex shape change in order from the paraxial region thereof to the off-axis region thereof. 
     The filter  750  is made of glass material and located between the fourth lens element  740  and the image surface  760 , and will not affect the focal length of the lens system. The image sensor  770  is disposed on or near the image surface  760  of the lens system. 
     The plate  780  is made of glass material and located on an object side of the identification module, and will not affect the focal length of the lens system. An axial distance between the plate  780  and the first lens element  710  is 1.000 mm. 
     The detailed optical data of the 7th embodiment are shown in Table 13 and the aspheric surface data are shown in Table 14 below. 
     
       
         
           
               
             
               
                 TABLE 13 
               
             
            
               
                   
               
               
                 7th Embodiment 
               
               
                 f = 0.37 mm, Fno = 1.70, HFOV = 72.0 deg. 
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 Surface # 
                   
                 Curvature Radius 
                 Thickness 
                 Material 
                 Index 
                 Abbe # 
                 Focal Length 
               
               
                   
               
               
                 0 
                 Object 
                 Plano 
                 0.000 
                   
                   
                   
                   
               
               
                 1 
                 Plate 
                 Plano 
                 1.465 
                 Glass 
                 1.517 
                 64.2 
                 — 
               
               
                 2 
                   
                 Plano 
                 1.000 
                   
                   
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                 3 
                 Lens 1 
                 −2.530 
                 (ASP) 
                 0.270 
                 Plastic 
                 1.548 
                 56.0 
                 −1.01 
               
               
                 4 
                   
                 0.738 
                 (ASP) 
                 0.598 
                   
                   
                   
                   
               
               
                 5 
                 Lens 2 
                 4.715 
                 (ASP) 
                 0.250 
                 Plastic 
                 1.548 
                 56.0 
                 −30.67 
               
               
                 6 
                   
                 3.613 
                 (ASP) 
                 0.093 
                   
                   
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 7 
                 Ape. Stop 
                 Plano 
                 0.019 
                   
                   
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                 8 
                 Lens 3 
                 2.079 
                 (ASP) 
                 0.497 
                 Plastic 
                 1.548 
                 56.0 
                 6.61 
               
               
                 9 
                   
                 4.465 
                 (ASP) 
                 0.035 
                   
                   
                   
                   
               
               
                 10 
                 Lens 4 
                 0.441 
                 (ASP) 
                 0.347 
                 Plastic 
                 1.682 
                 19.4 
                 0.59 
               
               
                 11 
                   
                 −3.442 
                 (ASP) 
                 0.300 
                   
                   
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 12 
                 Filter 
                 Plano 
                 0.145 
                 Glass 
                 1.517 
                 64.2 
                 — 
               
               
                 13 
                   
                 Plano 
                 0.246 
                   
                   
                   
                   
               
               
                 14 
                 Image 
                 Plano 
                 — 
               
               
                   
               
               
                 Note: Reference wavelength is 525.0 nm. 
               
            
           
         
       
     
     
       
         
           
               
             
               
                 TABLE 14 
               
               
                   
               
               
                 Aspheric Coefficients 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 Sur- 
                   
                   
                   
                   
               
               
                 face # 
                 3 
                 4 
                 5 
                 6 
               
               
                   
               
               
                 k = 
                 −8.9853E+01 
                 −1.0727E+00 
                 −8.8904E+00 
                 1.4857E+01 
               
               
                 A4 = 
                 1.3661E−01 
                 2.0834E−01 
                 −1.5601E+00 
                 −1.1951E+00 
               
               
                 A6 = 
                 −7.9865E−02 
                 −1.0025E+00 
                 1.1929E+01 
                 1.0819E+02 
               
               
                 A8 = 
                 3.3432E−02 
                 9.7442E−01 
                 −3.8953E+01 
                 −1.4375E+03 
               
               
                 A10 = 
                 −8.0930E−03 
                 −4.2814E−01 
                 6.8881E+01 
                 1.0325E+04 
               
               
                 A12 = 
                 1.0227E−03 
                 7.3999E−02 
                 −6.2772E+01 
                 −3.1223E+04 
               
               
                 A14 = 
                 −5.5131E−05 
                 −1.2846E−04 
                 2.3644E+01 
                 3.2193E+04 
               
               
                   
               
               
                 Sur- 
                   
                   
                   
                   
               
               
                 face # 
                 8 
                 9 
                 10 
                 11 
               
               
                   
               
               
                 k = 
                 −8.9627E+01 
                 −7.9270E+01 
                 −2.1590E+00 
                 1.3881E+01 
               
               
                 A4 = 
                 6.5579E−01 
                 −1.6587E+01 
                 −1.0498E+01 
                 1.4346E+00 
               
               
                 A6 = 
                 −1.2779E+01 
                 1.6295E+02 
                 1.3490E+02 
                 1.9663E+00 
               
               
                 A8 = 
                 2.5956E+02 
                 −1.1772E+03 
                 −1.1520E+03 
                 −7.3633E+01 
               
               
                 A10 = 
                 −2.3584E+03 
                 5.0866E+03 
                 6.1740E+03 
                 4.1537E+02 
               
               
                 A12 = 
                 8.1257E+03 
                 −1.2265E+04 
                 −2.0598E+04 
                 −1.2504E+03 
               
               
                 A14 = 
                 −8.8409E+03 
                 1.5335E+04 
                 4.0067E+04 
                 2.1572E+03 
               
               
                 A16 = 
                 — 
                 −7.7031E+03 
                 −4.0713E+04 
                 −1.9975E+03 
               
               
                 A18 = 
                 — 
                 — 
                 1.6495E+04 
                 7.6689E+02 
               
               
                   
               
            
           
         
       
     
     In the 7th embodiment, the equation of the aspheric surface profiles of the aforementioned lens elements is the same as the equation of the 1st embodiment. Also, the definitions of these parameters shown in the following table are the same as those stated in the 1st embodiment with corresponding values for the 7th embodiment, so an explanation in this regard will not be provided again. 
     Moreover, these parameters can be calculated from Table 13 and Table 14 as the following values and satisfy the following conditions: 
     
       
         
           
               
             
               
                   
               
               
                 7th Embodiment 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                 f [mm] 
                 0.37 
                 TL [mm] 
                 2.80 
               
               
                 Fno 
                 1.70 
                 SD/TD 
                 0.43 
               
               
                 HFOV [deg.] 
                 72.0 
                 ImgH/f 
                 2.72 
               
               
                 Vd4 
                 19.4 
                 TL/ImgH 
                 2.80 
               
               
                 Vdmin 
                 19.4 
                 TL/f 
                 7.62 
               
               
                 Vd1/N1 
                 36.18 
                 f/EPD 
                 1.70 
               
               
                 Vd2/N2 
                 36.18 
                 Y12/R2 
                 1.69 
               
               
                 Vd3/N3 
                 36.18 
                 tan(HFOV)/Fno 
                 1.82 
               
               
                 Vd4/N4 
                 11.53 
                 (|SAG21| + |SAG22|)/f 
                 0.33 
               
               
                 CT2/CT3 
                 0.50 
                 Yc11/f 
                 1.71 
               
               
                 R1/f 
                 −6.88 
                 Yc41/f 
                 1.16 
               
               
                 (R7 + R8)/(R7 − R8) 
                 −0.77 
                 Yc42/f 
                 0.66; 1.19 
               
               
                 f/f2 
                 −0.01 
                 — 
                 — 
               
               
                   
               
            
           
         
       
     
     8th Embodiment 
       FIG.  22    is a schematic view of an identification module and a plate according to the 8th embodiment of the present disclosure.  FIG.  23    is a schematic view of the identification module in  FIG.  22   .  FIG.  24    shows, in order from left to right, spherical aberration curves, astigmatic field curves and a distortion curve of the identification module according to the 8th embodiment. In  FIG.  22    and  FIG.  23   , the identification module includes the lens system (its reference numeral is omitted) of the present disclosure and an image sensor  870 . The lens system includes, in order from an object side to an image side, a first lens element  810 , a second lens element  820 , an aperture stop  800 , a third lens element  830 , a fourth lens element  840 , a filter  850  and an image surface  860 . The lens system includes four lens elements ( 810 ,  820 ,  830  and  840 ) with no additional lens element disposed between each of the adjacent four lens elements. 
     The first lens element  810  with negative refractive power has an object-side surface  811  being concave in a paraxial region thereof and an image-side surface  812  being concave in a paraxial region thereof. The first lens element  810  is made of plastic material and has the object-side surface  811  and the image-side surface  812  being both aspheric. The object-side surface  811  of the first lens element  810  has at least one inflection point. The image-side surface  812  of the first lens element  810  has at least one inflection point. The object-side surface  811  of the first lens element  810  has at least one convex critical point in an off-axis region thereof. 
     The second lens element  820  with positive refractive power has an object-side surface  821  being concave in a paraxial region thereof and an image-side surface  822  being convex in a paraxial region thereof. The second lens element  820  is made of plastic material and has the object-side surface  821  and the image-side surface  822  being both aspheric. 
     The third lens element  830  with positive refractive power has an object-side surface  831  being convex in a paraxial region thereof and an image-side surface  832  being convex in a paraxial region thereof. The third lens element  830  is made of plastic material and has the object-side surface  831  and the image-side surface  832  being both aspheric. 
     The fourth lens element  840  with positive refractive power has an object-side surface  841  being convex in a paraxial region thereof and an image-side surface  842  being concave in a paraxial region thereof. The fourth lens element  840  is made of plastic material and has the object-side surface  841  and the image-side surface  842  being both aspheric. The object-side surface  841  of the fourth lens element  840  has at least one inflection point. The image-side surface  842  of the fourth lens element  840  has at least one inflection point. The object-side surface  841  of the fourth lens element  840  has at least one concave critical point in an off-axis region thereof. The image-side surface  842  of the fourth lens element  840  has at least one critical point in an off-axis region thereof. 
     The filter  850  is made of glass material and located between the fourth lens element  840  and the image surface  860 , and will not affect the focal length of the lens system. The image sensor  870  is disposed on or near the image surface  860  of the lens system. 
     The plate  880  is made of glass material and located on an object side of the identification module, and will not affect the focal length of the lens system. An axial distance between the plate  880  and the first lens element  810  is 1.039 mm. 
     The detailed optical data of the 8th embodiment are shown in Table 15 and the aspheric surface data are shown in Table 16 below. 
     
       
         
           
               
             
               
                 TABLE 15 
               
             
            
               
                   
               
               
                 8th Embodiment 
               
               
                 f = 0.43 mm, Fno = 2.02, HFOV = 69.9 deg. 
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 Surface # 
                   
                 Curvature Radius 
                 Thickness 
                 Material 
                 Index 
                 Abbe # 
                 Focal Length 
               
               
                   
               
               
                 0 
                 Object 
                 Plano 
                 0.000 
                   
                   
                   
                   
               
               
                 1 
                 Plate 
                 Plano 
                 1.465 
                 Glass 
                 1.517 
                 64.2 
                 — 
               
               
                 2 
                   
                 Plano 
                 1.039 
                   
                   
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                 3 
                 Lens 1 
                 −4.767 
                 (ASP) 
                 0.229 
                 Plastic 
                 1.548 
                 56.0 
                 −1.31 
               
               
                 4 
                   
                 0.859 
                 (ASP) 
                 1.035 
                   
                   
                   
                   
               
               
                 5 
                 Lens 2 
                 −16.699 
                 (ASP) 
                 0.269 
                 Plastic 
                 1.649 
                 23.5 
                 8.49 
               
               
                 6 
                   
                 −4.170 
                 (ASP) 
                 0.126 
                   
                   
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 7 
                 Ape. Stop 
                 Plano 
                 0.127 
                   
                   
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                 8 
                 Lens 3 
                 1.909 
                 (ASP) 
                 0.574 
                 Plastic 
                 1.548 
                 56.0 
                 1.00 
               
               
                 9 
                   
                 −0.691 
                 (ASP) 
                 0.221 
                   
                   
                   
                   
               
               
                 10 
                 Lens 4 
                 0.610 
                 (ASP) 
                 0.220 
                 Plastic 
                 1.705 
                 17.0 
                 4.31 
               
               
                 11 
                   
                 0.649 
                 (ASP) 
                 0.300 
                   
                   
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 12 
                 Filter 
                 Plano 
                 0.145 
                 Glass 
                 1.517 
                 64.2 
                 — 
               
               
                 13 
                   
                 Plano 
                 0.221 
                   
                   
                   
                   
               
               
                 14 
                 Image 
                 Plano 
                 — 
               
               
                   
               
               
                 Note: Reference wavelength is 525.0 nm. 
               
            
           
         
       
     
     
       
         
           
               
             
               
                 TABLE 16 
               
               
                   
               
               
                 Aspheric Coefficients 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 Sur- 
                   
                   
                   
                   
               
               
                 face # 
                 3 
                 4 
                 5 
                 6 
               
               
                   
               
               
                 k = 
                 −7.0457E+01 
                 −8.3912E−01 
                 9.0000E+01 
                 7.4420E+01 
               
               
                 A4 = 
                 1.3030E−01 
                 −7.6617E−02 
                 −7.3143E−01 
                 2.2897E−01 
               
               
                 A6 = 
                 −8.0761E−02 
                 3.9883E−01 
                 4.2453E+00 
                 1.0641E+01 
               
               
                 A8 = 
                 3.3374E−02 
                 −4.2531E−01 
                 −1.3516E+01 
                 −1.1945E+02 
               
               
                 A10 = 
                 −8.1109E−03 
                 −1.0749E−01 
                 3.2061E+01 
                 9.0049E+02 
               
               
                 A12 = 
                 1.0285E−03 
                 2.4456E−01 
                 −4.7654E+01 
                 −3.5135E+03 
               
               
                 A14 = 
                 −5.0584E−05 
                 −6.8813E−02 
                 2.9058E+01 
                 5.3541E+03 
               
               
                   
               
               
                 Sur- 
                   
                   
                   
                   
               
               
                 face # 
                 8 
                 9 
                 10 
                 11 
               
               
                   
               
               
                 k = 
                 −7.0572E+01 
                 −1.8152E+01 
                 −1.7304E+00 
                 −2.7680E+00 
               
               
                 A4 = 
                 2.3469E−01 
                 −6.4238E+00 
                 −6.3220E−01 
                 8.5102E−01 
               
               
                 A6 = 
                 −1.6198E−01 
                 4.9050E+01 
                 −3.3728E+00 
                 −1.2106E+01 
               
               
                 A8 = 
                 −1.4523E+01 
                 −3.0635E+02 
                 1.5473E+01 
                 5.2403E+01 
               
               
                 A10 = 
                 1.2536E+02 
                 1.3075E+03 
                 −7.4374E+01 
                 −1.3772E+02 
               
               
                 A12 = 
                 −3.3566E+02 
                 −3.4715E+03 
                 3.2435E+02 
                 2.3231E+02 
               
               
                 A14 = 
                 3.2263E+02 
                 5.0194E+03 
                 −8.9229E+02 
                 −2.4800E+02 
               
               
                 A16 = 
                 — 
                 −2.7876E+03 
                 1.2409E+03 
                 1.5266E+02 
               
               
                 A18 = 
                 — 
                 — 
                 −6.6180E+02 
                 −4.1048E+01 
               
               
                   
               
            
           
         
       
     
     In the 8th embodiment, the equation of the aspheric surface profiles of the aforementioned lens elements is the same as the equation of the 1st embodiment. Also, the definitions of these parameters shown in the following table are the same as those stated in the 1st embodiment with corresponding values for the 8th embodiment, so an explanation in this regard will not be provided again. 
     Moreover, these parameters can be calculated from Table 15 and Table 16 as the following values and satisfy the following conditions: 
     
       
         
           
               
             
               
                   
               
               
                 8th Embodiment 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                 f [mm] 
                 0.43 
                 TL [mm] 
                 3.47 
               
               
                 Fno 
                 2.02 
                 SD/TD 
                 0.41 
               
               
                 HFOV [deg.] 
                 69.9 
                 ImgH/f 
                 2.44 
               
               
                 Vd4 
                 17.0 
                 TL/ImgH 
                 3.30 
               
               
                 Vdmin 
                 17.0 
                 TL/f 
                 8.04 
               
               
                 Vd1/N1 
                 36.18 
                 f/EPD 
                 2.02 
               
               
                 Vd2/N2 
                 14.25 
                 Y12/R2 
                 1.37 
               
               
                 Vd3/N3 
                 36.18 
                 tan(HFOV)/Fno 
                 1.36 
               
               
                 Vd4/N4 
                 9.97 
                 (|SAG21| + |SAG22|)/f 
                 0.06 
               
               
                 CT2/CT3 
                 0.47 
                 Yc11/f 
                 1.44 
               
               
                 R1/f 
                 −11.05 
                 Yc41/f 
                 1.25 
               
               
                 (R7 + R8)/(R7 − R8) 
                 −32.17 
                 Yc42/f 
                 1.53 
               
               
                 f/f2 
                 0.05 
                 — 
                 — 
               
               
                   
               
            
           
         
       
     
     9th Embodiment 
       FIG.  25    is a schematic view of an electronic device according to the 9th embodiment of the present disclosure.  FIG.  26    is a schematic view of the electronic device in  FIG.  25    identifying a fingerprint. 
     In this embodiment, an electronic device  20   a  is a smartphone having a biometric identification function. The electronic device  20   a  includes an image capturing unit  10   a , an identification module  30   a  and a plate  50   a . The image capturing unit  10   a  is a front-facing camera of the electronic device  20   a  for taking selfies, and the image capturing unit  10   a  includes the lens system of the present disclosure and an image sensor. The identification module  30   a  has a fingerprint identification function, and the identification module  30   a  includes the lens system disclosed in the 1st embodiment and an image sensor, but the present disclosure is not limited thereto. In some other embodiments, the identification module may include the lens system disclosed in another embodiment. In this embodiment, each of the image capturing unit  10   a  and the identification module  30   a  includes the lens system of the present disclosure, but the present disclosure is not limited thereto. For example, in some configurations, only one of the image capturing unit  10   a  and the identification module  30   a  includes the lens system of the present disclosure. 
     The plate  50   a  is disposed on an object side of the identification module  30   a , and the plate  50   a  is a module having a display function. Therefore, the plate  50   a  provides protection to the device and also helps reducing the use of components. Furthermore, the plate  50   a  can be a light-permeable plate, so light rays can travel through the plate  50   a  into the lens system of the identification module  30   a  for wider applications. Moreover, the plate  50   a  can have a touch-screen function, such that there is no need of additional input devices, and it&#39;s favorable for making the operation more intuitive. Additionally, the plate  50   a  can have a light-emitting function, which may include an organic light-emitting diode (OLED) display layer or an active-matrix organic light-emitting diode (AMOLED) display layer, such that the plate  50   a  can be a light source for illuminating an imaged object, thereby saving additional light sources. 
     10th Embodiment 
       FIG.  27    is a schematic view of an electronic device according to the 10th embodiment of the present disclosure identifying a fingerprint. In this embodiment, an electronic device  20   b  is a smartphone having a biometric identification function. The electronic device  20   b  includes an identification module  30   b , a light source S and a plate  50   b . The identification module  30   b  has a fingerprint identification function, and the identification module  30   b  includes the lens system disclosed in the 1st embodiment and an image sensor, but the present disclosure is not limited thereto. In some other embodiments, the identification module may include the lens system disclosed in another embodiment. The plate  50   b  is disposed on an object side of the identification module  30   b , and the plate  50   b  is, for example, a light-permeable glass substrate. The light source S is disposed on one side of the lens system for illuminating an imaged object, such that light rays from the imaged object can travel through the plate  50   b  into the lens system of the identification module  30   b . In this embodiment, the identification module  30   b  is the identification module disclosed in the 1st embodiment, but the present disclosure is not limited thereto. 
     The plate being a module having a display function or a glass substrate in the above embodiments is only exemplary, and the present disclosure is not limited thereto. In other embodiments, the plate can be, for example, a filter. 
     According to the present disclosure, the lens system of the identification module features good capability in aberration corrections and high image quality, and the identification module can be applied to smartphones for under-display fingerprint identification, but the present disclosure is not limited thereto. For example, the identification module can be applied to other biometric identification applications such as iris and face identifications. 
     11th Embodiment 
       FIG.  28    is a front view of an electronic device according to the 11th embodiment of the present disclosure.  FIG.  29    is a rear view of the electronic device in  FIG.  28   . 
     In this embodiment, an electronic device  20   c  is a smartphone including an image capturing unit  21   c , an image capturing unit  22   c , an image capturing unit  23   c , an image capturing unit  24   c  and a display unit  25   c . The image capturing unit  21   c  includes the lens system disclosed in the 1st embodiment and an image sensor, but the present disclosure is not limited thereto. The image capturing unit may include the lens system disclosed in another embodiment of the present disclosure. 
     The image capturing units  22   c ,  23   c  and  24   c  have different fields of view. In detail, the image capturing unit  22   c  is a telephoto image capturing unit, the image capturing unit  23   c  is a wide-angle image capturing unit and the image capturing unit  24   c  is an ultra-wide-angle image capturing unit. As such, the electronic device  20   c  has various magnification ratios so as to meet the requirement of optical zoom functionality for various applications with different requirements. 
     In this embodiment, the image capturing units  22   c ,  23   c  and  24   c  are all disposed on one side of the electronic device  20   c , while the image capturing unit  21   c  and display unit  25   c  are disposed on the opposite side of the electronic device  20   c . The image capturing unit  21   c  is a front-facing camera of the electronic device  20   c  for taking selfies, but the present disclosure is not limited thereto. 
     The lens system of the present disclosure can be installed in various electronic devices and can be applicable to applications such as image capturing and image identification. For example, the lens system can be applied to biometric identification and 3 D (three-dimensional) image capturing applications, in products such as digital cameras, mobile devices, digital tablets, smart televisions, network surveillance devices, dashboard cameras, vehicle backup cameras, multi-camera devices, motion sensing input devices, wearable devices and other electronic imaging devices. 
     The foregoing description, for the purpose of explanation, has been described with reference to specific embodiments. It is to be noted that TABLES 1-16 show different data of the different embodiments; however, the data of the different embodiments are obtained from experiments. The embodiments were chosen and described in order to best explain the principles of the disclosure and its practical applications, to thereby enable others skilled in the art to best utilize the disclosure and various embodiments with various modifications as are suited to the particular use contemplated. The embodiments depicted above and the appended drawings are exemplary and are not intended to be exhaustive or to limit the scope of the present disclosure to the precise forms disclosed. Many modifications and variations are possible in view of the above teachings.