refractive index formula derivation

Expressions for the RA and f-number are derived below that are valid for aplanatic lenses. Figure 1.1. After lens focusing movement, the Gaussian image plane will coincide with the sensor plane. In practice, an exposure distribution is required which stimulates a useful response from the imaging sensor. The effective focal length {f}_{{\rm{E}}} is defined as the reciprocal of the total refractive power [7], Combining equations (1.13), (1.14), and (1.15) yields the relation. It can be seen that e satisfies, This can be rearranged in the quadratic form. All powers and focal lengths are measured in diopters and metres, respectively. This will affect the amount of light reaching the sensor plane but will not affect the AFoV. When s = H, the blur spot never reaches the value 0.030 mm behind s, but it reaches 0.030 mm at a distance H/2 in front of s. In reality, the blur spot will never vanish due to the effects of diffraction. This defines the object plane and the near DoF. The generalised formula is. A surface with a positive power (a converging surface) can bend a ray travelling parallel to the optical axis so that it intersects with the optical axis. The entrance pupil is situated in front of the first lens surface in figure 1.16, however in photographic lenses the entrance pupil is usually virtual as in figure 1.17. In this case, rays leave the exit pupil which has diameter {m}_{{\rm{p}}}D and converge at the rear focal point, thus forming a cone as illustrated in figure 1.19. Manual focusing scales on lenses are calibrated using the distance from the sensor plane to the object plane. Substituting the above expression for H into the near DoF equation and working through the algebra yields. With s\to \infty, the physical distance s^{\prime} from the second principal point ({\rm{P}}^{\prime} ) to {\rm{F}}^{\prime} is defined as the second effective focal length [3], posterior focal length [2], or rear effective focal length [7] denoted by {f}_{{\rm{R}}}^{^{\prime} }. This law is implemented in optical devices like in contact lenses. A photographer must balance a variety of technical and aesthetic factors when controlling the nature of the optical image formed at the sensor plane. Example rays for an object point on the optical axis and at the top of the object of height h are shown. Figure 1.33. The magnification and pupil magnification are contained within the bellows factor b. Focus is set at infinity in all cases. Subsequently, an expression will be obtained which is valid within Gaussian optics. The ray will intersect the optical axis at a distance s^{\prime} from the second principal plane, and so the paraxial tangent slope u^{\prime} is given by, where m is the Gaussian magnification at the optical axis. The pupils and principal planes are not required to be in the order shown. It follows that the shape and material of a refractive medium such as a lens can be used to control the direction of light rays. First note that dA and {\rm{d}}A^{\prime} may be expressed as the product of infinitesimal heights in the x and y directions. Photometric or luminous flux {{\rm{\Phi }}}_{{\rm{v}}} is the rate of flow of electromagnetic energy emitted from or received by a specified surface area, appropriately weighted to take into account the spectral sensitivity of the HVS. Here the first medium is described by its refractive index n. where c0 is the speed of light in a vacuum, and c is the speed of light in the medium. The rear focal plane has been denoted by RFP. Writing N={f}_{{\rm{F}}}/D can lead to the incorrect assumption that the f-number can be made arbitrarily small. The quantity h is an approximation to H defined by equation (1.37) earlier, When s is equal to H, the corresponding near DoF is given by H-{s}_{{\rm{n}}}. This limit can be lowered by using an image-space medium with a higher refractive index than object space, n^{\prime} \gt n. This will increase the value of the infinity-focus image-space numerical aperture {\rm{N}}{{\rm{A}}}_{\infty }^{^{\prime} } relative to the value of the object-space refractive index n. For example, if an image-space medium is used with a refractive index n^{\prime} =1.5 and the object-space medium is air, the lowest possible f-number would be N = 0.33 [2]. For the 35 mm full-frame format, the diagonal AFoV for a 50 mm macro lens at 1:1 magnification is seen to decrease from 46.8° to 24.4°. The idealised thin lens with t\to 0. Two new planes need to be defined; the first and second principal planes H and {\rm{H}}^{\prime}. (a) An angle of 1 radian (rad) is defined by an arc length r of a circle with radius r. Since the radius of a circle is 2\pi r, the angle corresponding to a whole circle is 2\pi radians. The Gaussian images of the limiting field stop seen through the front and back of the lens are referred to as the entrance window and exit window, respectively. In particular, it will be shown that the Gaussian expression is exact for an aplanatic lens. The requirement is that the object must lie within a distance from the object plane known as the depth of field (DoF). Lens nodal points; a ray aimed at the first nodal point N emerges from the second nodal point {\rm{N}}^{\prime} at the same angle u. However, the rays involved are paraxial and so strictly speaking the object and image can only be points on or infinitesimally close to the optical axis. Table 1.1. The tangent slope u^{\prime} is seen to be. Published April 2017 Figure 1.26. The nodal points are therefore points of unit angular magnification. In the nineteenth century, the quantity D/{f}_{{\rm{F}}} was defined as the apertal ratio [13], however this term has not come into widespread use. The tangent planes of the two spherical surfaces shown in figure 1.8 are coincident 'at the lens' since t\to 0. Accurate focus is critical when the DoF is narrow. Pages 1-1 to 1-62. In this case b = 2 assuming {m}_{{\rm{p}}}=1. In this case the {s}_{{\rm{e}}{\rm{p}}} term can be dropped. The familiar f-number will emerge as one of the fundamental quantities. The distance {s}_{{\rm{f}}} can be found by rearranging equation (1.33). In Gaussian optics, all rays including the chief and marginal rays are treated mathematically as paraxial ray tangent slopes by extending the paraxial region. In an aplanatic lens, M = m. In other words, the real marginal magnification is equal to the Gaussian magnification and so the position of the image plane defined by the real marginal ray will be identical with the Gaussian image plane position. Luminous intensity takes into account the direction of propagation. This is known as focus breathing. Traditional exposure calculations assume that the lens is focused at infinity so that the working f-number appearing in equation (1.58) is replaced by the f-number. Here the pupil magnification is unity for simplicity. For a thin lens the total refractive power Φ is simply the sum of the individual surface powers [1]. You do not need to reset your password if you login via Athens or an Institutional login. The object plane and image plane have been denoted by OP and IP. Figure 1.12. Examples are illustrated in figure 1.34. This is known as the lensmakers' formula. Exposure is a measure of the amount of light per unit area which reaches the sensor plane while the camera shutter is open.

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