# Geometrical Optics

When light travels through a homogeneous medium, it travels in a straight line. This is known as rectilinear propagation. The behavior of light at the boundary of a medium or interface between two media is described by the theory of geometrical optics. Geometrical optics explains reflection and refraction, as well as the applications of mirrors and lenses. Reflection is the rebounding of incident light waves at the boundary of a medium.5 Light waves that are reflected are not absorbed into the second medium; rather, they bounce off of the boundary and travel back through the first medium. The law of reflection is

## θ1 = θ2

where θ1 is the incident angle and θ2 is the reflected angle, both measured from the normal. The normal is a line drawn perpendicular to the boundary of a medium; all angles in optics are measured from the normal, not the surface of the medium.5

Refraction is the bending of light as it passes from one medium to another and changes speed. The speed of light through any medium is always less than its speed through a vacuum. Remember that the speed of light in a vacuum, c, is equal to 3 x 108 m/s. When a pencil (or any straight object) is dipped into a glass of water at an angle, it looks impossibly bent where it intersects the surface of the water because the light reflecting off of the portion of the pencil under water is refracted. When light is in any medium besides a vacuum, its speed is less than c. For a given medium:

## n = c/v

where c is the speed of light in a vacuum, ν is the speed of light in the medium, and n is a dimensionless quantity called the index of refraction of the medium. The index of refraction of a vacuum is 1, by definition; for all other materials, the index of refraction will be greater than 1. For air, n is essentially equal to 1 because the speed of light in air is extremely close to c. where n1 and θ1 refer to the medium from which the light is coming and n2 and θ2 refer to the medium into which the light is entering. Note that θ is once again measured with respect to the normal. From Snell’s law, we can see that when light enters a medium with a higher index of refraction (n2 > n1), it bends toward the normal (sin θ2 < sin θ1; therefore, θ2 < θ1). Conversely, if the light travels into a medium where the index of refraction is smaller (n2 <n1), the light will bend away from the normal (sin θ2 > sin θ1; therefore, θ2 > θ1).

As discussed earlier, the speed of light in a vacuum is the same for all wavelengths. However, when light travels through a medium, different wavelengths travel at different speeds. This fact implies that the index of refraction of a medium affects the wavelength of light passing through the medium because the index of refraction is related to the speed of the wave. When various wavelengths of light separate from each other, this is called dispersion. The most common example of dispersion is the splitting of white light into its component colors using a prism. If a source of white light is incident on one of the faces of a prism, the light emerging from the prism is spread out into a fan-shaped beam. Violet light has a smaller wavelength than red light and this results in violet light being bent to a greater extent. Red experiences the least amount of refraction and so it is always on top of the spectrum while violet, having experienced the greatest amount of refraction, is found at the bottom of the spectrum. When light enters a medium with a different index of refraction, the wavelength changes but the frequency of the light remains the same.

When light enters a medium with a higher index of refraction, it bends toward the normal. When light enters a medium with a lower index of refraction, it bends away from the normal. When light travels from a medium with a higher index of refraction (such as water) to a medium with a lower index of refraction (such as air), the refracted angle is larger than the incident angle (θ 2 > θ1); that is, the refracted light ray bends away from the normal. As the incident angle is increased, the refracted angle also increases, and eventually, a special incident angle called the critical angle (θc) is reached, for which the refracted angle θ2 equals 90 degrees. At the critical angle, the refracted light ray passes along the interface between the two media. The critical angle can be derived from Snell’s law if θ2 = 90°, such that:

Total internal reflection, a phenomenon in which all the light incident on a boundary is reflected back into the original material, results with any angle of incidence greater than the critical angle, θc.

Spherical mirrors come in two types: convex and concave. The mirror can be considered as a spherical cap or section if the image taken from a much larger spherically-shaped mirror.6 Spherical mirrors have an associated center of curvature (C) and a radius of curvature (r). The center of curvature is a point on the optical axis located at a distance equal to the radius of curvature from the vertex of the mirror; in other words, the center of curvature would be the center of the spherically-shaped mirror if it were a complete sphere. If we were to look from the inside of a sphere to its surface, we would see a concave surface. On the other hand, if we were to look from outside the sphere, we would see a convex surface. For a concave surface, the center of curvature and the radius of curvature are located in front of the mirror. For a convex surface, the center of curvature and the radius of curvature are behind the mirror. Concave mirrors are called converging mirrors and convex mirrors are called diverging mirrors because they cause parallel incident light rays to converge and diverge after they reflect, respectively.6 Concave mirrors are converging mirrors. Convex mirrors are diverging mirrors. The reverse is true for lenses. There are several important lengths associated with mirrors. The focal length (f) is the distance between the focal point (F) and the mirror. Note that for all spherical mirrors,

## f = r/2

where the radius of curvature (r) is the distance between C and the mirror.6 The distance between the object and the mirror is o; the distance between the image and the mirror is i. While it is not important which units of distance are used in this equation, it is important that all values used have the same units as each other. A positive distance (i > 0) for an image has, tells us that it is a real image. This implies that the image is in front of the mirror. A negative distance (i < 0), means it is virtual and its location is behind the mirror. Plane mirrors can be considered as spherical mirrors with infinitely large focal distances. As such, for a plane mirror:

## r = f = ∞

A converging lens is always thicker at the center, while a diverging lens is always thinner at the center. The sign conventions changes slightly for lenses. For both lenses and mirrors, positive magnification represents upright images, and negative magnification means inverted images. Also, for both lenses and mirrors, a positive image distance means that the image is real and is located on the real (R) side, whereas a negative image distance means that the image is virtual and located on the virtual (V) side.

 Symbol Positive Negative o Object is on same side of lens as light source Object is on opposite side of lens from light source (extremely rare) i Image is on opposite side of lens from light source (real) Image is on same side of lens as light source (virtual) r Lens is convex (converging) Lens is concave (diverging) f Lens is convex (converging) Lens is concave (diverging) m Image is upright (erect) Image is inverted

Focal lengths and radii of curvature have a simpler sign convention. For both mirrors and lenses, converging species have positive focal lengths and radii of curvature, and diverging species have negative focal lengths and radii of curvature. Remember that lenses have two focal lengths and two radii of curvature because they have two surfaces. For a thin lens where thickness is negligible, the sign of the focal length and radius of curvature are generally given based on the first surface the light passes through. The power of a lens is measured in diopters, where f (the focal length) is in meters and is given by the equation:

## P = 1/f

P has the same sign as f and is, therefore, positive for a converging lens and negative for a diverging lens. Diverging lens are needed for people who are near-sighted (can see near objects clearly), while people who are farsighted (can see distant objects clearly) require converging lenses. Far-sightedness is also called hyperopia and near-sightedness is also known as myopia.  Bifocal lenses are corrective lenses that have two distinct regions: one that utilises convergence of light to correct for far-sightedness and a second region that utilises divergence of light to correct for near-sightedness. Both regions are in the same lens.

Lenses in contact are a series of lenses with negligible distances between them. These systems behave as a single lens with equivalent focal length given by

Because power is the reciprocal of focal length, the equivalent power is

## P = P1 + P2 + P3 + ··· + Pn

Spherical mirrors and lenses are imperfect. They are therefore subject to specific types of errors or aberrationsSpherical aberration is a blurring of the periphery of an image as a result of inadequate reflection of parallel beams at the edge of a mirror or inadequate refraction of parallel beams at the edge of a lens. This creates an area of multiple images with very slightly different image distances at the edge of the image, which appears blurry.

Chromatic aberration is a dispersive effect within a spherical lens. Depending on the thickness and curvature of the lens, there may be significant splitting of white light, which results in a rainbow halo around images. This phenomenon is corrected for in visual lenses like eyeglasses and car windows with special coatings that have different dispersive qualities than the lens itself.

The eye is a complex refractive instrument that uses real lenses. The cornea acts as the primary source of refractive power because the change in refractive index from air is so significant. Then, light is passed through an adaptive lens that can change its focal length before reaching the vitreous humor. It is further diffused through layers of retinal tissue to reach the rods and cones. At this point, the image has been focused and minimized significantly, but is still relatively blurry. Our nervous system processes the remaining errors to provide a crisp view of the world.

References

1) Handel, S. (1995). Timbre perception and auditory object identiﬁcation. Hearing, 425-461.

2) Strutt (Lord Rayleigh), John William (1896). MacMillan & Co, ed. The Theory of Sound. 2 (2 ed.). p. 154.

3) Halliday, David; Resnick, Robert; Walker, Jerl (2005), Fundamental of Physics (7th ed.), USA: John Wiley and Sons, Inc.,

4) James D. Ingle, Jr. and Stanley R. Crouch, Spectrochemical Analysis, Prentice Hall, 1988

5) Lekner, John (1987). Theory of Reflection, of Electromagnetic and Particle Waves. Springer.

6) Hecht, Eugene (1987). “5.4.3”. Optics (2nd ed.). Addison Wesley. pp. 160–1