III.

Geometrical Optics

This area of optical science concerns the application of laws of reflection and refraction of light in the design of lenses (see Lenses below) and other optical components of instruments. If a light ray that is traveling through one homogeneous medium is incident on the surface of a second homogeneous medium, part of the light is reflected and part may enter the second medium as the refracted ray and may or may not undergo absorption in the second medium.

A.

Reflection and Refraction

The amount of light reflected depends on the ratio of the refractive indexes for the two media. The plane of incidence contains the incident ray and the normal (line perpendicular) to the surface at the point of incidence (see Fig. 1). The angle of incidence (reflection or refraction) is the angle between the incident (reflected or refracted) ray and this normal. The laws of reflection state that the angle of incidence is equal to the angle of reflection and that the incident ray, the reflected ray, and the normal to the surface at the point of incidence all lie in the same plane. If the surface of the second medium is smooth or polished, it may act as a mirror and produce a reflected image. If the mirror is flat, or plane, the image of the object appears to lie behind the mirror at a distance equal to the distance between the object and the surface of the mirror. The light source in Fig. 2 is the object A, and a point on A sends out rays in all directions. The two rays that strike the mirror at B and C, for example, are reflected as the rays BD and CE. To an observer in front of the mirror, these rays appear to come from the point F behind the mirror. It follows from the laws of reflection that CF and BF form the same angle with the surface of the mirror as do AC and AB. If the surface of the second medium is rough, then normals to various points of the surface lie in random directions. In that case, rays that may lie in the same plane when they emerge from a point source nevertheless lie in random planes of incidence, and therefore of reflection, and are scattered and cannot form an image.

Not all of the light that strikes a mirror is reflected; some of the light can pass through the mirror or be absorbed by the mirror. Many scientists thought a perfect mirror—one that reflects 100 percent of the light that strikes it—could not exist. However, in 1998, scientists made a perfect mirror by stacking up microscopic layers of tellurium and the plastic polystyrene.

A.1.

Snell’s Law

This important law, named after Dutch mathematician Willebrord 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. Also, 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 velocity of light is lower in the denser substance. If a ray is incident obliquely, then a ray entering a medium with a higher refractive index is bent toward 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. In making calculations, the optical path, which is defined as the product of the distance a ray travels in a given medium and the refractive index of that medium, is the important consideration. 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, that of an object lying underwater observed from above water, is shown in Fig. 3. Oblique rays are chosen only for ease of illustration. The ray DB from the object at D is bent away from the normal to A. The object, therefore, appears to lie at C, where the line ABC intersects a line normal to the surface of the water and passing through D.

The path of light passing through several media with parallel boundaries is shown in Fig. 4. The refractive index of water is lower than that of glass. Because the refractive index of the first and last medium is the same, the ray emerges parallel to the incident ray AB, but it is displaced.

Until 2001, all known substances had a positive refractive index. In that year physicist Sheldon Schultz and his colleagues at the University of California at San Diego created a composite from fiberglass and copper wire that refracts microwaves in the direction opposite that in which all other materials refract light. This unusual refraction indicates that the material has a negative refractive index. Since microwaves, like visible light, are a type of electromagnetic radiation, scientists predict that it will eventually be possible to produce a material that refracts visible light in the same way.

A.2.

Prism

If light passes through a prism, a transparent object with flat, polished surfaces at angles to one another, 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 Fig. 5, the angle CBD between the path of the incident ray and the path of the emergent ray is the angle of deviation. If the angle the incident ray makes with the normal 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 to the surface 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 back into the incident medium. Total reflection cannot occur if light is traveling from a less dense medium to a denser one. The three drawings in Fig. 6 show ordinary refraction, refraction at the critical angle, and total reflection. In the late 20th century, a new, practical application of total reflection was found in the use of fiber optics. If light enters a solid glass or plastic tube obliquely, the light can be totally reflected at the boundary of the tube and, after a number of successive total reflections, emerge from the other end. Glass fibers 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 fibers 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 inserted into various openings.

B.

Spherical and Aspherical Surfaces

Traditionally, most of the 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. If the optical component is spherical, the optic axis passes through the center of a lens or mirror and through the center of curvature. Light rays from a very distant source are considered to travel parallel to one another. If rays parallel to the optic axis are incident on a spherical surface, they 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 vertex of a mirror or a thin 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, depending on which surface (if the surfaces are not alike) the light strikes first. If an object is at the focal point, the rays emerging from it are made parallel to the optic axis after reflection or refraction. If rays from an object are converged by a lens or mirror so that they actually intersect in front of a mirror or behind a lens, the image is real and inverted, or upside down. If the rays diverge after reflection or refraction so that the light only appears to converge, the image is virtual and erect. 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 in the direction in which light is traveling are positive and distances measured in the opposite direction are negative, then if u is the object distance, v 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 the equation

1/v - 1/u = 1/f

applies to 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 focal length of a spherical mirror is equal to half the radius of curvature. As is shown in Fig. 7, rays parallel to the optic axis that are incident on a concave mirror with its center of curvature at C are reflected so that they intersect at B, halfway between A and C. If the object distance is greater than the distance AC, then the image is real, inverted, and diminished. If the object lies between the center 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. Convex mirrors form only virtual, erect, and diminished images.

C.

Lenses

Lenses made with surfaces of small radii have the shorter focal lengths. A lens with two convex surfaces will always refract rays 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 parallel to the axis away from the axis, so that even if the second surface of the lens is convex, the rays diverge and only appear to come to a focus on the same side of the lens as the object. Concave 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, erect, 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 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 cause the angle subtended by the virtual image to increase and thus cause the magnifying power to increase. The magnifying power of an instrument is a measure of its ability to bring the object apparently closer to the eye. This is distinct from the lateral magnification of a camera (see Photography) 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. See Lens.

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 1-in diameter and 8-in focal length would be one-fourth as bright as the image formed by a lens of 1-in diameter and 4-in focal length. The ratio of the focal length to the effective diameter of a lens is its focal ratio or 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.

D.

Aberration

Geometrical optics predicts that rays of light emanating from a point are imaged by spherical optical elements as a small blur. The outer parts of a spherical surface have a different focal length than does the central area, and this defect would cause a point to be imaged as a small circle. The difference in focal length for the various parts of the spherical section is called spherical aberration. If, instead of being a portion of a sphere, a concave mirror is a section of a paraboloid (see Parabola) of revolution, parallel rays incident on all areas of the surface are reflected to a point without spherical aberration. Combinations of convex and concave lenses can help to correct spherical aberration, but this defect cannot be eliminated from a single spherical lens for a real object and image.

The manifestation of differences in lateral magnification for rays coming from an object point not on the optic axis is called coma. If coma is present, light from a point is spread out into a family of circles that fit into a cone, and in a plane perpendicular to the optic axis, the image pattern is comet shaped. Coma may be eliminated for a single object-image point pair, but not for all such points, by a suitable choice of surfaces. Corresponding or conjugate object and image points, free from both spherical aberration and coma, are known as aplanatic points, and a lens having such a pair of points is called an aplanatic lens. Astigmatism is the defect in which the light coming from an off-axis object point is spread along the direction of the optic axis. If the object is a vertical line, the cross section of the refracted beam is an ellipse that collapses first into a horizontal line, spreads out again, and later becomes a vertical line. If a flat object has any extent, the surface of best focus is curved, or curvature of field results. Distortion arises from a variation of magnification with axial distance and is not caused by a lack of sharpness in the image. Because the index of refraction varies with wavelength, the focal length of a lens also varies and causes longitudinal or axial chromatic aberration. Magnification of different image sizes by various wavelengths is known as lateral chromatic aberration. Converging and diverging lenses grouped together, and combinations of glasses with different dispersions, help to minimize chromatic aberration. Mirrors are free of this defect. In general, achromatic lens combinations are corrected for chromatic aberration for two or three colors.