A fisheye lens generally refers to a lens where the Field of View (FOV) is 160° or more, and the incidence angle of an incident ray is approximately proportional to the image height on the image plane. There are many application examples where a fisheye lens with FOV of 180° or more is required such as security-surveillance and entertainment. However, fisheye lenses of prior arts often contain more than 10 pieces of lens elements to achieve 180° or more FOV, or the fisheye lenses were very difficult to manufacture because the shape of some of the lens surfaces of lens elements are close to hemispherical surfaces. Also, some lenses use relatively small number of lens elements between 6 and 8. However, the modulation transfer function characteristics are not good, and consequently the lenses do not have enough resolution to obtain sharp images. Also, optical glasses with high refractive indexes are often used to keep the number of lens elements small, and the production cost arises as the result.
Other point of consideration is about projection schemes. Desirable projection schemes of a fisheye lens include an equidistance projection scheme. In an equidistance projection scheme, the incidence angle δ of an incident ray, the effective focal length f of the fisheye lens, and the image height red on the image plane satisfy a proportionality relation given in Eq. 1.red(δ)=fδ  [math figure 1]
Real projection scheme of a lens shows certain amount of deviation from the theoretical projection scheme given in Eq. 1. Although the real projection scheme of a lens can be experimentally measured, it can be theoretically predicted using dedicated lens design software once given the complete lens prescription. For example, image height in the y-axis direction for an incident ray having a given incident angle can be obtained using ‘Reay’ perator in ‘Zemax’ which is dedicated lens design software. Similarly, image height in the x-axis direction can be obtained using ‘Reax’ operator. If the real image height on the image plane for a lens is given as rrp, then the error between the real projection scheme of a lens and an ideal equidistance projection scheme can be calculated as in Eq. 2.
                              distortion          ⁢                                          ⁢                      (            δ            )                          =                                                                              r                  ed                                ⁡                                  (                  δ                  )                                            -                                                r                  rp                                ⁡                                  (                  δ                  )                                                                                    r                ed                            ⁡                              (                                  δ                  _                                )                                              ×          100          ⁢          %                                    [                  math          ⁢                                          ⁢          figure          ⁢                                          ⁢          2                ]            
The distortion of a fisheye lens is generally measured as an f-O distortion given in Eq. 2, and a high-end fisheye lens faithfully follows the equidistance projection scheme given in Eq. 1. It is relatively easy to design a fisheye lens simply having a FOV of 180° or more, but it is considerably more difficult to design a lens that has a FOV of 180° or more and the discrepancy from an equidistance projection scheme is less than 10%.
However, what is important in the industrial use of a fisheye lens is the fact that the incidence angle of an incident ray is proportional to the image height on the image plane, and it is not necessary that the proportionality constant is the effective focal length. Therefore, calibrated distortion, which involves a fictitious focal length fc that minimizes the f-O distortion given by Eq. 2 over the entire range of incidence angle, is often used as a measure of lens performance. Here, the fictitious focal length fc is not related to the actual effective focal length of the lens, and given as an optimum fitting constant by least square error method. In other words, calibrated distortion indicates how close is the functional relation between the incidence angle of an incident ray and the image height on the image plane to a first order equation passing through the origin given by Eq. 1.
Another point of consideration is to secure enough back focal length while keeping the overall length of the lens short. Furthermore, another difficulty is to keep the relative illumination difference between the center and the periphery of the image plane small. If the relative illumination differs greatly, then brightness at the center and at the periphery of the image plane is significantly different.
Even though all these requirements are satisfied, still it is difficult to obtain a design that has enough manufacturing tolerance so that neither fabrication is too difficult nor production cost is overly excessive.
To take a specific example, reference 1 discloses a fisheye lens with 262° FOV.
However, since this is a dark lens with F-number of 14.94, it cannot be used unless the surrounding is brightly lit. Reference 2 discloses a fisheye lens with 170.8° FOV. However, this is also a dark lens with F-number of 7.98. Further, the lens structure makes this lens difficult to be mass produced because the shape of the second lens surface of the first lens element is nearly hemi-spherical. Reference 3 discloses fisheye lenses with 220° and 270° FOV. These lenses are relatively dark with F-number of 5.6, the shapes of the second lens surfaces of the first lens elements are nearly hemispherical, and modulation transfer function characteristics are not good enough to obtain high-resolution images. Reference 4 discloses a fisheye lens with F-number of 2.8 and 180° FOV. Although this lens has relatively high resolution, the calibrated distortion is higher than 15%, and consequently distortion is severe. Reference 5 discloses a fisheye lens with F-number of 2.8, and 220° FOV. However, the shape of the second lens surface of the first lens element is also close to hemi-spherical surface, and modulation transfer function characteristic is not sufficiently good. Reference 6 discloses a fisheye lens for projector with F-number of 2.4, and 163° FOV. However, relative illumination at the maximum incidence angle is low around 60%. Reference 7 provides a remarkable infrared fisheye lens with F-number of 0.7 and 270° FOV. Still, the number of lens element is only 4. Such an astonishing characteristic is partly due to the high refractive index of Germanium that is employed as the lens material in the infrared wavelength region. However, the shape of the second lens surface of the first lens element is hyper-hemispherical, and it is very difficult to be mass produced. Reference 8 concisely summarizes characteristic features of various commercial fisheye lenses. For most of the fisheye lenses, however, it can be seen that relative illuminations at the maximum incidence angles are 60% or less, and calibrated distortions are high, typically 10% or more. Reference 9 discloses an extraordinary fisheye lens with F-number of 2.0, and 180° FOV, and still using only 6 pieces of lens elements. However, this fisheye lens uses ultra high refractive index glass with a refractive index of 1.91, and consequently production cost is high. Furthermore, modulation transfer function characteristic is not sufficiently good. Reference 10 discloses a fisheye lens with F-number of 2.8 and 182° FOV, and following a projection scheme described by a special functional relation. However, this lens employs 11 pieces of lens elements, and therefore structure is complicated and production cost is high. Furthermore, modulation transfer function characteristic is not sufficiently good. Reference 11 discloses a fisheye lens with F-number of 2.8, and 180° FOV. This lens also uses only 6 pieces of lens elements, but production cost is high because aspherical lens element is used. Furthermore, modulation transfer function characteristic is not sufficiently good, and relative illumination at the maximum field angle is relatively low around 70%. On the other hand, reference 12 provides various embodiments of wide-angle lenses satisfying desirable projection schemes which can be implemented by wide-angle lenses.    [Reference 1] A. C. S. van Heel, G. J. Beernink, and H. J. Raterink, “Wide-angle objective lens”, U.S. Pat. No. 2,947,219, date of registration Aug. 2, 1960.    [Reference 2] K. Miyamoto, “Fish eye lens”, J. Opt. Soc. Am., vol. 54, pp. 1060-1061 (1964).    [Reference 3] M. Isshiki and K. Matsuki, “Achromatic super wide-angle lens”, U.S. Pat. No. 3,524,697, date of registration Aug. 18, 1970.    [Reference 4] T. Ogura, “Wide-angle lens system with corrected lateral aberration”, U.S. Pat. No. 3,589,798, date of registration Jun. 29, 1971.    [Reference 5] Y. Shimizu, “Wide-angle fisheye lens”, U.S. Pat. No. 3,737,214, date of registration Sep. 29, 1971.    [Reference 6] R. Doshi, “Fisheye projection lens for large format film”, Proc. SPIE, vol. 2000, pp. 53-61 (1993).    [Reference 7] J. B. Caldwell, “Fast IR fisheye lens with hyper-hemispherical field of view”, Optics & Photonics News, p. 47 (July, 1999).    [Reference 8] J. J. Kumler and M. Bauer, “Fisheye lens designs and their relative performance”, Proc. SPIE, vol. 4093, pp. 360-369 (2000).    [Reference 9] A. Ning, “Compact fisheye objective lens”, U.S. Pat. No. 7,023,628, date of registration Apr. 4, 2006.    [Reference 10] K. Yasuhiro and Y. Kazuyoshi, “Fisheye lens and photographing apparatus with the same”, Japanese patent publication no. 2006-098942, date of publication Apr. 13, 2006.    [Reference H] M. Kawada, “Fisheye lens unit”, U.S. Pat. No. 7,283,312, date of registration Oct. 16, 2007.    [Reference 12] G. Kweon, and M. Laikin, “Wide-angle lens”, Korean patent application no. 10-2007-0106725, date of application Oct. 23, 2007.