1060
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`L E T T E RS
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`TO T HE
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`E D I T OR
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`Vol. 54
`
`FIG. 2. A fish eye lens
`with corrected color aber­
`ration.
`
`adding a diverging meniscus lens before the hemisphere lens
`[Hill,3 Fig. 1 (c)4], which is a prototype of the present fish eye
`lenses. Fish eye lenses have also been improved by Schultz6
`[Fig. 1 (d)], Merté,6 and recently by van Heel.7
`However, large lateral color aberration cannot be corrected by
`these lenses, and color filters must be employed in order to take
`photographs. The lateral color aberration can be corrected by
`adding a doublet (positive flint glass lens and negative crown
`glass lens) before the pupil as is shown in Figs. 2 and 3 (see also
`Table I). Now all the wavelengths to which the photographic
`material is sensitive can be utilized and color photographs can
`also be taken. When the fish eye lens is pointed toward the zenith,
`the image of the subject of principal interest may appear at the
`edge of the field more frequently than at the center. Therefore
`correction of aberrations at the edge is very important.
`The fish eye lens has inherent distortion; this distortion should
`not be considered as an aberration but as a result of projection
`of a hemisphere on a plane. Let the angle of an incident ray from
`an infinite object be φ and the coordinates of the image be (r',θ).
`The following projections are considered in this section:
`
`where ƒ is the focal length.
`Projection 1 is that of camera lenses. Projection 2 is called
`stereographic projection. From Eq. (2), we have
`
`This result shows that a small circle on a hemisphere having its
`center at the lens is projected as a circle on the image plane, but
`the diameter of the image of a circle at the horizon, φ=90°,
`is
`twice as large as the image of an equally large circle at the pole,
`φ=0°. This projection is very similar to our psychological percept
`of whole sky.3
`Projection 3 is called equidistance projection. It is represented
`at 3 in Fig. 4. This is preferable for measurement of zenith angles
`and azimuth angles. The effect of error of lens position is small,
`and the linear relation of r' and φ is convenient to analyze. An
`attempt was made to accomplish equidistance projection with the
`fish eye lens shown in Fig. 2. However, the image of a small circle
`
`FIG. 3. Aberration curves of fish
`eye lens shown in Fig. 2. Curves a
`show spherical aberration of d and
`g lines and sine condition of d line,
`(lotted. Curves b show astigma­
`tism, and curves c show
`lateral
`color aberration for c and g lines.
`
`Fish Eye Lens
`KΠNRO MIYAMOTO*
`Institute of Plasma Physics, Nagoya University, Nagoya, Japan
`(Received 19 February 1964)
`
`A LENS which covers a hemispherical field (2ω=180°) is
`
`usually called a fish eye lens. This lens is not an extension of
`a wide angle lens. It has inherent large distortion because it is not
`possible to form an image of a hemispheric field on a plane without
`distortion. The classical example of this type of image formation
`is a fish eye under water. Wood1 took a photograph with a pinhole
`camera filled with water [Fig. 1(a)]. Bond2 substituted a hemis­
`pheric lens with a pupil at the center of curvature [Fig. 1 (b)] in
`place of the water in the pinhole camera. However, this lens has a
`large Petzval sum. The field curvature was greatly improved by
`
`FIG. 1. Development of fish eye lens.
`
`APPLE 1007
`
`1
`
`

`

`August 1964
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`L E T T E RS
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`TO T HE
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`E D I T OR
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`1061
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`TABLE I. Lens data of fish eye lens shown in Fig. 2. r, d, nd, and vd are
`radius of curvature, distance, index of refraction
`(d line), and Abbe number,
`respectively.
`
`FIG. 5. Explanation of notations used in Eqs. (5) and (6).
`
`is not a circle because
`
`and
`
`For the fish eye lens shown in Fig. 2, a cosφ is nearly constant and
`W is finally reduced to W = (n/n')2N(π/4F2)(sinφ/φ), where F
`is the ƒ/number. Therefore, the radial flux density is relatively
`uniform, as compared to ordinary lenses, which obey the cos4φ
`law.
`* This work was done when the author was in Nippon Kogaku K. K.
`(Japan Optical Industry Company).
`1 R. W. Wood, Physical Optics (Macmillan and Company, Ltd., London,
`1919), p. 67.
`2 W. N. Bond, Phil. Mag. 44, 999 (1922).
`3 R. Hill, Proceedings of the Optical Convention (1926) 878.
`4 C. Bech, J. Sci. Instr. 2, 135 (1925).
`5 H. Schulz, D. R. Patent No. 620538 (1932).
`6 W. Merté, D. R. Patent No. 672393 (1935).
`7 A. C. S. van Heel et al., U. S. Patent No. 2947219 (1960).
`
`Projection 4 may be called equisolid angle projection, because
`the element of solid angle dΩ is expressed by
`
`and the solid angle Ω is proportional to the corresponding area S'
`in the image plane. This projection is convenient for measuring
`the percentage of the sky covered by clouds, or obstructed by
`buildings. For the equidistance projection, we have
`
`When the radiant flux from a small area dS of the object plane
`(r,θ) in a small solid angle dΩ in the direction φ is concentrated to
`the small area dS' of the image plane (r',θ') within the solid angle
`dΩ' of the direction φ', then the following relation holds:
`
`where n and n' are the indices of refraction of object and image
`space, respectively. When the radiance of object is N, the radiant
`flux density W is
`
`Let the area of entrance pupil and the distance of object plane
`and entrance pupil be a and l (see Fig. 5). When equidistance
`projection is employed we have
`
`FIG. 4. Curves of various projections: (1) ordinary projection, (2) stereo-
`graphic projection,
`(3) equidistance projection,
`(4) equisolid angle
`projection.
`
`2
`
`