Geometric Optics 1.Refraction of Light Refraction is the bending of a wave when it enters a medium where its speed is different. The refraction of light when it passes from a fast medium to a slow medium bends the light ray toward the normal to the boundary between the two media. The amount of bending depends on the indices of refraction of the two media and is described quantitatively by Snell's Law. Refraction is responsible for image formation by lenses and the eye. The bending of refraction can be visualized in terms of Huygens' principle. As the speed of light is reduced in the slower medium, the wavelength is shortened proportionately. The frequency is unchanged; it is a characteristic of the source of the light and unaffected by medium changes. 1.1 Index of Refraction The index of refraction is defined as the speed of light in vacuum divided by the speed of light in the medium. The indices of refraction of some common substances are given below with a more complete description of the indices for optical glasses given elsewhere. The values given are approximate and do not account for the small variation of index with light wavelength which is called dispersion. Material n Material n Vacuum 1.000 Ethyl alcohol 1.362 Air 1.000277 Glycerine 1.473 Water 4/3 Ice 1.31 Carbon disulfide 1.63 Polystyrene 1.59 Methylene iodide 1.74 Crown glass 1.50-1.62 Diamond 2.417 Flint glass 1.57-1.75 1.2 Snell's Law Snell's Law relates the indices of refraction n of the two media to the directions of propagation in terms of the angles to the normal. Snell's law can be derived from Fermat's Principle or from the Fresnel Equations. If the incident medium has the larger index of refraction, then the angle with the normal is increased by refraction. The larger index medium is commonly called the "internal" medium, since air with n=1 is usually the surrounding or "external" medium. You can calculate the condition for total internal reflection by setting the refracted angle = 90° and calculating the incident angle. Since you can't refract the light by more than 90°, all of it will reflect for angles of incidence greater than the angle which gives refraction at 90°. 1.3 Total Internal Reflection When light is incident upon a medium of lesser index of refraction, the ray is bent away from the normal, so the exit angle is greater than the incident angle. Such reflection is commonly called "internal reflection". The exit angle will then approach 90° for some critical incident angle θc, and for incident angles greater than the critical angle there will be total internal reflection. The critical angle can be calculated from Snell's law by setting the refraction angle equal to 90°. Total internal reflection is important in fiber optics and is employed in polarizing prisms. 1.4 Law of Reflection A light ray incident upon a reflective surface will be reflected at an angle equal to the incident angle. Both angles are typically measured with respect to the normal to the surface. This law of reflection can be derived from Fermat's principle. The law of reflection gives the familiar reflected image in a plane mirror where the image distance behind the mirror is the same as the object distance in front of the mirror. 1.5 Fermat's Principle:Reflection Fermat's Principle: Light follows the path of least time. Of course the straight line from A to B is the shortest time, but suppose it has a single reflection. The law of reflection can be derived from this principle as follows: The pathlength from A to B is Since the speed is constant, the minimum time path is simply the minimum distance path. This may be found by setting the derivative of L with respect to x equal to zero. This reduces to which is 1.6 Fermat's Principle and Refraction Fermat's Principle: Light follows the path of least time. Snell's Law can be derived from this by setting the derivative of the time =0. We make use of the index of refraction, defined as n=c/v. 1.7 Fresnel's Equations Fresnel's equations describe the reflection and transmission of electromagnetic waves at an interface. That is, they give the reflection and transmission coefficients for waves parallel and perpendicular to the plane of incidence. For a dielectric medium where Snell's Law can be used to relate the incident and transmitted angles, Fresnel's Equations can be stated in terms of the angles of incidence and transmission. Fresnel's equations give the reflection coefficients: and The transmission coefficients are and 2 Principal Focal Length For a thin double convex lens, refraction acts to focus all parallel rays to a point referred to as the principal focal point. The distance from the lens to that point is the principal focal length f of the lens. For a double concave lens where the rays are diverged, the principal focal length is the distance at which the back-projected rays would come together and it is given a negative sign. The lens strength in diopters is defined as the inverse of the focal length in meters. For a thick lens made from spherical surfaces, the focal distance will differ for different rays, and this change is called spherical aberration. The focal length for different wavelengths will also differ slightly, and this is called chromatic aberration. The principal focal length of a lens is determined by the index of refraction of the glass, the radii of curvature of the surfaces, and the medium in which the lens resides. It can be calculated from the lensmaker's formula for thin lenses. This shows parallel beams from two helium-neon lasers converging to the principal focal point of a 30 cm double convex lens. The rays then enter a diverging lens of focal length -10cm on the right. The laser beams were made visible with a spray can of artificial smoke. 2.1 Focal Length and Lens Strength The most important characteristic of a lens is its principal focal length, or its inverse which is called the lens strength or lens "power". Optometrists usually prescribe corrective lenses in terms of the lens power in diopters. The lens power is the inverse of the focal length in meters: the physical unit for lens power is 1/meter which is called diopter. 2.2 Spherical Aberration球差 For lenses made with spherical surfaces, rays which are parallel to the optic axis but at different distances from the optic axis fail to converge to the same point. For a single lens, spherical aberration can be minimized by bending the lens into its best form. For multiple lenses, spherical aberrations can be canceled by overcorrecting some elements. The use of symmetric doublets like the orthoscopic doublet greatly reduces spherical aberration. When the concept of principal focal length is used, the presumption is that all parallel rays focus at the same distance, which is of course true only if there are no aberrations. The use of the lens equation likewise presumes an ideal lens, and that equation is practically true only for the rays close to the optic axis, the socalled paraxial rays. For a lens with spherical aberration, the best approximation to use for the focal length is the distance at which the difference between the paraxial and marginal rays is the smallest. It is not perfect, but the departure from perfect focus forms what is called the "circle of least confusion". Spherical aberration is one of the reasons why a smaller aperture (larger f-number) on a camera lens will give a sharper image and greater depth of field since the difference between the paraxial and marginal rays is less. 2.3 Chromatic Aberration 色差 A lens will not focus different colors in exactly the same place because the focal length depends on refraction and the index of refraction for blue light (short wavelengths) is larger than that of red light (long wavelengths). The amount of chromatic aberration depends on the dispersion of the glass. One way to minimize this aberration is to use glasses of different dispersion in a doublet or other combination. Another approach uses spaced doublets. Doublet for Chromatic Aberration The use of a strong positive lens made from a low dispersion glass like crown glass coupled with a weaker high dispersion glass like flint glass can correct the chromatic aberration for two colors, e.g., red and blue. Such doublets are often cemented together (called achromat doublets) and may be used in compound lenses such as the orthoscopic doublet. Achromat Doublets消色差双片 An achromat doublet does not completely eliminate chromatic aberration, but can eliminate it for two colors, say red and blue. The idea is to use a lens pair with the strongest lens of low dispersion coupled with a weaker one of high dispersion calculated to match the focal lengths for two chosen wavelengths. Cemented doublets of this type are a mainstay of lens design. If the powers of the lenses of the doublet give a focus point in front of the doublet as shown above, it is said to be a positive achromat. Chromatic aberration for three colors can be eliminated with and apochromat triplet. Real Image Formation If a luminous object is placed at a distance greater than the focal length away from a convex lens, then it will form an inverted real image on the opposite side of the lens. The image position may be found from the lens equation or by using a ray diagram provided that it can be considered a "thin lens". If the lens equation yields a negative image distance, then the image is a virtual image on the same side of the lens as the object. If it yields a negative focal length, then the lens is a diverging lens rather than the converging lens in the illustration. The lens equation can be used to calculate the image distance for either real or virtual images and for either positive on negative lenses. The linear magnification relationship allows you to predict the size of the image. Virtual Image Formation Diverging lenses form reduced, erect, virtual images. Using the common form of the lens equation, f, P and i are negative quantities. If a negative focal length is entered to agree with the illustration, then the image is a virtual image on the same side of the lens as the object and will give a negative image distance. If a calculation yields a positive focal length, then the lens is a converging lens rather than the diverging lens in the illustration. The lens equation can be used to calculate the image distance for either real or virtual images and for either positive on negative lenses. The linear magnification relationship allows you to predict the size of the image. Virtual Image Formation A virtual image is formed at the position where the paths of the principal rays cross when projected backward from their paths beyond the lens. Although a virtual image does not form a visible projection on a screen, it is no sense "imaginary", i.e., it has a definite position and size and can be "seen" or imaged by the eye, camera, or other optical instrument. A reduced virtual image if formed by a single negative lens regardless of the object position. An enlarged virtual image can be formed by a positive lens by placing the object inside the principal focal point. The Camera and the Eye Images are formed in a camera by refraction in a manner similar to image formation in the eye. However, accommodation to image closer objects is done differently in the eye and camera. Accommodation in Eye and Camera Accommodation is the process of adjusting the focus distance of an optical instrument to the object which is to be viewed. This process is done very differently by the eye and a camera. Accommodation from distant to close objects for the eye is done by rounding out the lens to shorten its focal length, since the image distance to the retina is essentially fixed. Accommodation from distant to close objects for the camera is done by moving the lens further from the film since the focal length is fixed. aspherical surfaces mirrors Note: An image that is real is always inverted! An image that is virtual is always upright! Derivatives of Common Functions In this table, a is a constant, while u, v, w are functions. The derivatives are expressed as derivatives with respect to an arbitrary variable x.