- In this chapter we develop the relations governing the passage of light rays through imaging elements on the basis of the paraxial approximation using matrix algebra. - Finally we go through some of the standard imaging systems.. - Consider Figure 2.1 where we have a sphere of radius R centred at C and with refractive index n . - The ray is incident on a plane which is normal to the radius R and the angle of incidence θ is the angle between the ray and the radius from C. - Figure 2.1 Refraction at a spherical interface. - is called the power of the surface.. - The spherical surface in Figure 2.1 might be the front surface of a spherical lens. - It is also common to define R as positive when the vertex V of the surface is to the left of the centre C and negative when it is to the right of C.. - At the point A in Figure 2.1 the height is unaltered, and this fact can be expressed as. - After these remarks we proceed by considering the system in Figure 2.2 consisting of two refracting surfaces with radii of curvature R 1 and R 2 separated by a distance D 12 . - Figure 2.2 Ray tracing through a spherical lens. - These equations may be combined to give the overall transformation from a point just to the left of A 1 to a point just to the right of A 2. - in the matrix form. - The determinant of M is the product of all the determinants of the refraction and translation matrices. - Thus the determinant of M is the product of the determinants of the separate refraction matrices and takes the form. - THE GENERAL IMAGE-FORMING SYSTEM 19 where n is the index of the medium to the left of the first refracting surface, and n is the index of the medium to the right of the last refracting surface.. - This means that we have x = x, independently of the value of α.. - The transformation matrix from the H-plane to the H -plane can be written in terms of M by adding translation T and T. - We now equate the elements of the matrices in Equation (2.22) and M 11 + D M 21 = 1 i.e. - It should be noted that for an afocal system like the plane wave set-up in Figure 1.14 where the two focal points coincide, M 21 = 0. - to the left of the first principal plane. - to the right of the second principal plane. - In addition to the lateral (or transversal) magnification m x , one might introduce a longi- tudinal (or axial) magnification defined as b/a. - Figure 2.5 Principal planes with some key rays. - It should be emphasized that the physical location of the principal planes could be inside one of the components of the image-forming system. - There is no a priori reason for the order of the principal planes. - The plane H could be to the right of H . - The plane H will be to the right of F and H to the left of F if f and f are positive.. - In Figure 2.6 a light ray making an angle α with the z-axis is incident on the sphere at a point A at height x and is reflected at an angle α to the z-axis. - Figure 2.6 Reflection at a spherical surface. - at C and therefore the reflection angle θ, equal to the angle of incidence, is as shown in the figure. - Comparing this with the object–image transformation matrix, Equation (2.34), we get for the focal length of the spherical mirror. - Figure 2.7 shows four rays from an object point that can be used to find the location of the image point. - Note that one of the rays goes through C and the image point. - Figure 2.7 Imaging by a reflecting spherical surface. - The mirror in Figure 2.7 is concave. - The equation for a parabola with its vertex at the origin and its focus a distance f to the right (see Figure 2.8) is. - In the paraxial region, i.e. - in the immediate vicinity of the optical axis, these two configurations will be essentially indistinguishable. - We also might quote from Laikin (1991): ‘The author’s best advice concerning aspherics is that unless you have to, don’t be tempted to use an aspheric surface’. - A simple example is shown in Figure 2.9(a) where the hole in the screen limits the solid angle of rays from the object at P o . - If we move the screen to the left of F, we have the situation shown in Figure 2.9(b). - The screen is still the aperture stop, but the images A , B of A and B are now to the right of P o . - A ‘space’ may be defined that contains all physical objects to the right of the lens plus all points conjugate to physical objects that are to the left of the lens. - The image of the aperture stop in image space is called the exit pupil. - Figure 2.9 Illustrations of entrance and exit pupils. - by the exit pupil A B as in Figure 2.9(a) or as if the rays diverging from P o are limited in solid angle by A B as in Figure 2.9(b).. - By analogy to the image space, a space called the object space may be defined that contains all physical objects to the left of the lens plus all points conjugate to any physical object that may be to the right of the lens. - In Figure 2.9(a, b) all unprimed objects are in the object space. - The image of the aperture stop in the object space is defined as the entrance pupil. - The aperture stop in Figure 2.9(a, b) is already in the object space, hence it is itself the entrance pupil.. - pupil is to image all stops and lens rims to the left through all intervening refracting elements of the system into the object space and find the solid angle subtended by each at P o . - Alternatively we may image all stops and lens rims to the right through all intervening refractive elements into the image space and determine the solid angle subtended by each image at P o . - The ray-tracing equations used in the theory of Gaussian optics are correct to first order in the inclination angles of the rays and the normals to refracting or reflecting surfaces.. - When higher-order approximations are used for the trigonometric functions of the angles, departures from the predictions of Gaussian optics will be found. - (For a centred system with rotational symmetry, only odd powers of the angles will appear in the ray-tracing formulas.). - The performance of a lens system must be judged according to the intended use. - Figure 2.10 illustrates spherical aberration, and in Section 10.4.1 distortion is treated in. - Figure 2.10 Spherical aberration. - The focus of the paraxial rays is at P o . - The marginal rays focus at a point closer to the lens. - Because of the complexity of the higher-order aberrations they are usually treated numerically. - We have found that a general imaging system is characterized by the focal length f and the positions of the two principal planes H and H which determine the four cardinal points F, F , H and H : see Figure 2.11. - If the object plane lies to the right of the vertex of the first refracting surface, we have no real object point, but rays that converge to a virtual object point behind the first refracting surface: see Figure 2.12(c). - In the same way we have a virtual image plane if the image lies to the left of the last vertex of the lens system:. - Figure 2.12(b). - Only if rays really intersect at the image point do we have a real image point and that happens only if the image plane lies to the right of the last vertex. - Figure 2.11 Principal points. - Figure 2.12 Real and virtual object (O) and image (I) points: (a) real object, real image. - of the system. - 0 we have a negative lens: see Figure 2.12(b). - For a negative lens, F is to the right of H, while F is to the left of H. - In addition to the above-mentioned cardinal points, we also have the so-called nodal points N and N on the axis: see Figure 2.11. - It should be remembered that the systems described below are visual instruments of which the eye of the observer is an integral part.. - This can be realized by two lenses separated by a distance t equal to the sum of the individual focal lengths, t = f 1 + f 2 : see Figure 2.13.. - Figure 2.13 The telescope. - Computing the transformation from a plane a distance d in front of the first lens to a plane a distance d behind the second lens gives. - is constant and independent of the object and image distances. - Therefore, when judging the quality of a stellar telescope, the diameter of the front lens is a more important parameter than the magnification. - Figure 2.14 shows some of the most common designs.. - 2.10.2 The Simple Magnifier. - In Figure 2.15 the object of height h is placed at a distance a <. - f , where f is the focal length of the magnifier. - The resulting virtual image is located a distance b in front of the lens, given by the lens formula 1/a + 1/b = 1/f . - Figure 2.14 Some common telescope designs: (a) Newtonian. - Figure 2.15 The simple magnifier. - 2.10.3 The Microscope. - A microscope is used for observation of very small objects where the magnification of the viewing angle is so large that the assumptions of paraxial optics are no longer valid.. - The magnification of the objective is therefore given by m ob ≈ T /f ob . - The magnified intermediate image is observed by the ocular, which focuses at infinity, giving a magnification of the viewing angle equal to d o /f oc ≈ 25 cm/f oc . - 2.1 Verify directly by matrix methods that use of the matrix in Equation (2.34) will yield values of (x , α ) for rays in Figure 2.5 so that they behave as shown.. - 2.2 Consider the system shown in Figure P2.1 where the focal lengths of the first system are f 1 , f 1 and those of the second f 2 , f 2 . - 2 between the first principal plane of the first system and the second principal plane of the second system.. - We denote the principal planes of the whole system by H and H and the distances HH 1 = D and H 2 H = D. - (b) Express the transformation matrix M HH between H and H in terms of the total power P and n and m. - Figure P2.2 (c) Find the total power of the system.. - (a) Find the power P of the doublet in terms of P 1 , P 2 and l.. - 2.4 Find the power and the locations of the principal planes for a combination of two thin lenses each with the same focal length f >. - to the left of V 1 ? Is the aperture stop always the same? (No calculation is necessary to solve this problem.). - Find the position and size of (a) the entrance pupil, (b) the exit pupil, (c) the image
Xem thử không khả dụng, vui lòng xem tại trang nguồn hoặc xem
Tóm tắt