Optics

A

1

Snell’s Law

This important law, named after the Dutch mathematician Willebrord van Roijen Snell, states that the product of the refractive index and the sine of the angle of incidence of a ray in one medium is equal to the product of the refractive index and the sine of the angle of refraction in a successive medium. Algebraically, this can be written n₁ sinθ₁ = n₂ sinθ₂, where n₁, n₂ are the two values of refractive index, and θ₁, θ₂ are the angles of incidence and refraction. The incident ray, the refracted ray, and the normal to the boundary at the point of incidence all lie in the same plane. Generally, the refractive index of a denser transparent substance is higher than that of a less dense material; that is, the speed of light is lower in the denser substance. So if a ray is incident obliquely, then a ray entering a medium with a higher refractive index is bent towards the normal, and a ray entering a medium of lower refractive index is deviated away from the normal. Rays incident along the normal are reflected and refracted along the normal.

To an observer in a less dense medium such as air, an object in a denser medium appears to lie closer to the boundary than is the actual case. A common example is that of an object lying underwater and observed from above the water. An oblique ray from the object is bent away from the normal towards the position of the observer. The object, therefore, appears to lie slightly away from its true position and at a point where a straight from the observer intersects a line normal to the surface of the water and passing through the object. In the case of light passing through more than two media with parallel boundaries, another effect occurs. If the refractive index of the first and last medium is the same, but different from the that of the intermediate medium, the ray emerges parallel to the incident ray, but is displaced laterally.

A

2

Prism

If light passes through a prism, a transparent object with flat, surfaces and a uniform cross-section, the exit ray is no longer parallel to the incident ray. Because the refractive index of a substance varies for the different wavelengths, a prism can spread out the various wavelengths of light contained in an incident beam and form a spectrum. In this, the angle between the path of the incident ray and the path of the emergent ray is the angle of deviation. It can be shown that when the angle of incidence is such that it is equal to the angle made by the emergent ray, the deviation is at a minimum. The refractive index of the prism can be calculated by measuring the angle of minimum deviation and the angle between the faces of the prism.

A

3

Critical Angle

Given that a ray is bent away from the normal when it enters a less dense medium, and that the deviation from the normal increases as the angle of incidence increases, an angle of incidence exists, known as the critical angle, such that the refracted ray makes an angle of 90° with the normal and travels along the boundary between the two media. If the angle of incidence is increased beyond the critical angle, the light rays will be totally reflected. Total reflection cannot occur if light is travelling from a less dense medium to a denser one. In recent years, a new, practical application of total reflection has been found in fibre optics. If light enters a solid glass or plastic tube at one end, it can be totally reflected at the boundary of the tube and, after a number of successive total reflections, emerge from the other end. Glass fibres can be drawn to a very small diameter, coated with a material of lower refractive index, and then assembled into flexible bundles or fused into plates of fibres that are used to transmit images. The flexible bundles, which can be used to provide illumination as well as to transmit images, are valuable in medical examination, as they can be passed along narrow passages or even blood vessels.

B

Spherical and Aspherical Surfaces

Most of the traditional terminology of geometrical optics was developed with reference to spherical reflecting and refracting surfaces. Aspherical surfaces, however, are sometimes involved. The optic axis is a reference line that is an axis of symmetry. The optic axis passes through the centre of a spherical lens or mirror and through its centre of curvature. If a narrow beam of rays travelling along the optic axis is incident on the spherical surface of a mirror or a thin lens, the rays are reflected or refracted so that they intersect or appear to intersect at a point on the optic axis. The distance between this point and the mirror or lens is the focal length. If a lens is thick, calculations are made with reference to planes called principal planes, rather than to the surface of the lens. A lens may have two focal lengths, if the surfaces are not alike, depending on which surface the light strikes first. If an object is at the focal point, the rays emerging from it are parallel to the optic axis after reflection or refraction. If rays are converged by a lens or mirror so that they intersect in front of it, the image is real and inverted (upside down). If the rays diverge after reflection or refraction so that they only appear to come from a point through which they have not actually passed, the image is erect and is described as virtual. The ratio of the height of the image to the height of the object is the lateral magnification.

If it is understood that distances measured from the surface of a lens or mirror to objects or to real images are positive and distances measured to virtual images are negative, then, if u is the object distance, v is the image distance, and f is the focal length of a mirror or of a thin lens, the equation

1/v + 1/u = 1/f

applies to spherical mirrors and spherical lenses. If a simple lens has surfaces with radii r₁ and r₂, and the ratio of its refractive index to that of the medium surrounding it is n, then

1/f = (n - 1) (1/r1 + 1/r2)

The radii r₁, r₂ are taken to be positive or negative, depending on whether the surfaces are convex or concave, respectively.

The focal length of a spherical mirror is equal to half the radius of curvature. A narrow beam of rays travelling along the optic axis and incident on a concave mirror is reflected so that it intersects the radius at the focal point, or principal focus, halfway between the pole, or centre, of the mirror's surface and the mirror's centre of curvature. If the object distance is greater than the distance between the pole and the centre of curvature, the image is real, inverted, and diminished. If the object lies between the centre of curvature and the focal point, the image is real, inverted, and enlarged. If the object is located between the surface of the mirror and the focus, the image is virtual, upright, and enlarged. A convex mirror forms only virtual, erect, and diminished images, unless the mirror is used in conjunction with other optical components.

C

Lenses

Lenses with surfaces of small radii have short focal lengths. A lens with two convex surfaces will always refract rays that are originally parallel to the optic axis so that they converge to a focus on the side of the lens opposite to the object. A concave lens surface will deviate incident rays that are originally parallel to the axis away from the axis. Unless the second surface of the lens is convex and more strongly curved than the first surface, the rays diverge and appear to come from a point on the same side of the lens as the object. Such lenses form only virtual, erect, and diminished images.

If the object distance is greater than the focal length, a converging lens forms a real and inverted image. If the object is sufficiently far away, the image is smaller than the object. If the object distance is smaller than the focal length of this lens, the image is virtual, upright, and larger than the object. The observer is then using the lens as a magnifier or simple microscope. The angle subtended at the eye by this virtual enlarged image (that is, its apparent angular size) is greater than would be the angle subtended by the object if it were at the normal viewing distance. The ratio of these two angles is the magnifying power of the lens. A lens with a shorter focal length would form a virtual image subtending a greater angle and would therefore have a greater magnifying power. The magnifying power of an instrument is a measure of its ability to make the object seem closer to the eye. This is distinct from the lateral magnification of a camera (see Photographic Techniques) or telescope, for example, where the ratio of the actual dimensions of a real image to those of the object increases as the focal length increases.

The amount of light a lens can admit increases with its diameter. Because the area occupied by an image is proportional to the square of the focal length of the lens, the light intensity over the image area is directly proportional to the diameter of the lens and inversely proportional to the square of the focal length. The image produced by a lens of diameter 3 cm and focal length 20 cm would be one-quarter as bright as the image formed by a lens of the same diameter and focal length 10 cm. The ratio of the focal length to the effective diameter of a lens is its focal ratio, the so-called f-number. The reciprocal of this ratio is called the relative aperture. Lenses having the same relative aperture have the same light-gathering power, regardless of the actual diameters and focal lengths.