Optics

Written by tutor Andrea L.

In this section we examine several wavelike behaviors of light, specifically: reflection, refraction,
diffraction and interference. Visible light is a type of electromagnetic wave, whose wavelength ranges
between about 7×10^-7 meters (red light) and 4×10^-7 meters (violet light). Like with other types of waves,
the speed (v), frequency (f), and wavelength (λ) of light are related by: v = fλ. Unless otherwise noted,
we may take the speed of light to be equal to its speed in a vacuum, c (c = 3×10⁸ m/s).

Reflection and Refraction

When examining the behavior of light at a boundary surface, there are three types of rays we often
discuss:

1) The original incident ray, traveling at angle θ₁
2) The part of the incident ray which reflects from the surface (the reflected ray, which bounces off
the surface at angle θ₁’)
3) The part of the incident ray that travels into the surface (the refracted ray, traveling at angle θ₂)

The angles of these three rays are always measured with respect to a line called the normal that is
perpendicular to the surface at the point where the incident ray encounters the surface.

Reflection

According to the Law of Reflection, θ₁ = θ₁’. The angle of incidence is always equal to the angle of
reflection, except on the opposite side of the “normal.”

Refraction

To determine the angle of refraction, we need to consider both the angle of incidence and the
speed of light in the two media. Although the frequency of the light remains the same in both media,
the wavelength and velocity may be different, causing the light ray to bend towards or away from the
normal. The speed of light in a medium is described by the index of refraction, n, which is defined as
n = c/v (c = speed of light in a vacuum, 3×10⁸ m/s; v = speed of light in the medium). Note that since v
must be less than c, n is always ≥ 1.

This information may be used to calculate the angle of refraction via Snell’s Law:

n₁sin(θ₁) = n₂sin(θ₂

where n₁ and n₂ are the indices of refraction of the two media. Depending on the incident angle and
the indices of refraction, we may observe different behavior in the refracted ray:

-If θ₁ = 0° (the incident ray is exactly along the normal), θ₂ is also 0°, regardless of n₁ and n₂.
-For all other incident angles, there are three possible cases:
1) If n₂ = n₁, then θ₂ = θ₁, and the light ray continues in its original direction.

2) If n₂ > n₁, then θ₂ <>₁. The ray slows down when it enters the second medium, bending
towards the normal.

3) If n₂ <>₁, then θ₂ > θ₁. The ray speeds up when it enters the second medium, bending
away from the normal.

Total internal reflection

When a light ray encounters a boundary surface such that n₂ <>₁, there is a particular angle of
incidence for which all the light refracts at exactly 90° (parallel to the surface). The angle of incidence
in this situation is called the critical angle, θ_c, and may be calculated with: θ_c = sin^-1(n₂/n₁). At this angle
and any larger angle of incidence, light cannot escape the first medium. When θ₁ > θ_c, all the light is
reflected so there is no refracted ray. This behavior is called total internal reflection.

Chromatic dispersion

Here we briefly consider how the angle of refraction depends on the color of incoming light.
Different colors of light have different wavelengths, and the index of refraction n varies depending
on the wavelength of the light. Generally, n is greater for shorter wavelengths (violet/blue light, for
example) and smaller for longer wavelengths (red light). This means that violet light will be bent more
than red light. Since white light is made up of all different colors, this property causes chromatic
dispersion of incoming white light – the different colors are separated when the light enters a different
medium, because of their different refraction angles. Note that we assume monochromatic (single-
wavelength) light in all other sections.

Reflecting and refracting surfaces

There are specific reflecting and refracting surfaces which we will examine in detail. The reflecting
surfaces fall into three categories: plane mirrors, concave mirrors, and convex mirrors. The refracting
surfaces fall into two categories: convex lenses and concave lenses. As we will see, there are many
similarities between concave mirrors and convex lenses, and between convex mirrors and concave
lenses. In all of the following cases, “in front” of the mirror or lens will mean on the same side as the
incident light and “behind” will mean on the opposite side from the incident light.

Mirrors

An object placed in front of a mirror can be thought of as a source of light. We wish to determine
the location and properties of the object’s image, as created by the mirror. The light from this object
spreads out as it travels towards the mirror, so is incident on the mirror at different angles. We
can locate the image created by this object by drawing a few representative incident rays and their
corresponding reflections (making use of the Law of Reflection, described above). The top of the image
occurs where the reflected rays intersect, and the distance of this point from the mirror is called d i while
the original object distance is called do. Images may be either real or virtual, depending on the type of
mirror and object distance (see specifics below). Real images can be formed onto a surface, such as
film or a screen, while virtual images exist only because in certain cases our brain perceives light rays as
having come from a point where rays do not actually converge.

A couple important points to keep in mind for mirrors:

-Real images always form on the same side of a mirror as the object and are inverted, and virtual
images always form on the opposite side and are the same orientation as the object
-Image distance is positive when the image is real and negative when the image is virtual

Plane mirrors

To see how virtual images work, we can examine the case of a plane mirror. As we see in this
case, the reflected rays only intersect if we extend them backwards (behind the mirror).

Since there are not actually any light rays at the point where the image is created, this is called
a virtual image. We find that the image in this case is created at the same distance behind the mirror as
the object is in front of it. By convention, object distances are taken to be positive, and image distances
for virtual images are taken to be negative. So in the plane mirror case, d_i = -d_o.

Spherical mirrors

Spherical mirrors are mirrors in the shape of a small section of the surface of a sphere and occur
in two types: concave and convex. Before we can locate images created by spherical mirrors, there are a
few important quantities we need to define:

-The center of curvature, C, of a spherical mirror is the distance from the mirror to the center of
the sphere of which the mirror is a part.
-The central axis is a line that extends through the center of curvature and through the center of
the mirror.
-The focal point of a mirror is the point at which rays incident from infinitely far away and
parallel to the central axis would converge upon reflection.
o For concave mirrors, the reflected rays really do converge, at a point in front of the
mirror, so this

is a “real” focal point.
o For convex mirrors, the parallel incident rays are not really reflected through a common
point;
however, the eye interprets the reflections as having come from a single point
behind the mirror. This perceived
point is a “virtual” focal point.
o The focal length is defined as being positive for a concave mirror and negative for a
convex mirror
o For both concave and convex mirrors, the distance to the center of curvature and
distance to the
focal point are related by: f = ½ C

Concave mirrors

To locate images created by spherical mirrors, there are specific incident and reflected rays that
are useful to draw. Which rays we choose to draw depends somewhat on the position of the object with
relation to the focal point and center of curvature. We could draw any two of these four special rays:

1. A ray initially parallel to the principal axis, that then reflects through the focal point
2. A ray that passes through the focal point, then reflects parallel to the central axis
3. A ray that passes through the center of curvature, then reflects along itself
4. A ray incident towards the center of the mirror, that then reflects symmetrically about the
principal axis

The point where the rays intersect indicates the location at which the image appears. Five specific
cases may be considered (in the following figures, note that black arrow represent objects, light blue
arrows represent images, and the numbers next to the rays denote which of the “special rays” is drawn)

As we can see, as the object is moved closer to the mirror, the image gets larger and moves further
from the mirror. If the object is at the focal point, no image appears. When the object moves in closer
than the focal point, the image becomes virtual and on the opposite side of the mirror.

Convex mirrors

To locate images created by convex mirrors, we can draw just two rays:

1. A ray parallel to the principal axis, that then reflects as if it came from the virtual focal point
2. A ray incident towards the virtual focal point, then reflected parallel to the principal axis

For convex mirrors, we need only consider one case:

For both concave and convex mirrors, the focal point (f), image distance (d_i), or object distance (d_o)
can be calculated exactly from the following equation, if the other two quantities are known:

¹/_f = ¹/_di + ¹/_do

We may also calculate a mirror’s magnification (M). This describes the ratio of image height (h i) to
object height (ho) and is given by:

M = ^d_i/_do = ^h_i/_ho

Note that a negative magnification or object height simply means the image is inverted compared to the
original object.

Refracting Surfaces

There are two types of refracting surfaces we will consider: convex lenses and concave lenses. In
both cases, we assume that the light begins to refract at a line drawn vertically through the center of the
lens (this is called the thin lens approximation).

A couple important points to keep in mind for lenses:

-Real images will form on the side of the lens opposite the object, and virtual images form on the
same side of the lens as the object. (this is the opposite to the behavior for mirrors)
-The focal length f is defined as being positive for convex lenses and negative for concave lenses
(also opposite to the convention for mirrors)
-Image distance is positive when the image is real and negative when the image is virtual (same
as for mirrors)

Like with mirrors, we can again relate focal length, the object and image distances, and magnification
with the equations:

¹/_f = ¹/_di + ¹/_do and M = ^d_i/_do = ^h_i/_ho

Convex (“converging”) lenses

Convex lenses have two real focal points, one on either side of the lens. Let us define f₁ as the focal
point in front of the lens, and f₂ as the focal point behind the lens. We can locate the image created by
drawing a ray diagram using the following rays:

1. A ray parallel to the principal axis, which is refracted through f₂
2. A ray through f₁, which is refracted parallel to the principal axis
3. A ray incident towards the center of the lens, which continues through in the same direction

Note that if the object is inside the focal point, we need to extend each of these rays back towards the
object’s side of the lens to see where they intersect to form an image.

Like with concave mirrors, there are five separate cases we can consider for convex lenses:

Concave (“diverging”) lenses

Concave lenses have two virtual focal points, defined f₁ and f₂ as above. In this case, we draw:

1. A ray parallel to the principal axis, which diverges away from f₁
2. A ray aimed towards f₂, which emerges parallel to the axis
3. A ray incident towards the center of the lens, which continues through in the same direction

For concave lenses, like with convex mirrors, we need only consider one case:

Two-lens systems

There are many applications which utilize multiple thin lenses; compound microscopes and
refracting telescopes are two common examples. The final image created by a two-lens system can be
found by splitting the problem into two single-lens problems:

1. Ignoring the presence of lens 2, locate the image distance d_i of the image produced by lens 1.
2. Now ignoring the presence 1, treat the image from step 1 as the object for lens 2 and find the
image distance for this new object.

For multiple-lens systems, the total magnification is simply the product of the magnifications of the
individual lenses.

Diffraction and Interference

When a light wave encounters a barrier with an opening of similar size to the wavelength of
the light, it “diffracts” or spreads out in a semi-circular pattern on the opposite side of the barrier. The
narrower the slit, the more the wave spreads out on the opposite side. When two waves come from
nearby points, such as slits in a screen, they will overlap and “interfere.” Evidence of this overlapping is
observed if a viewing screen is placed some distance away from the slits, since we see an interference
pattern of alternating bright and dark bands (sometimes termed bright and dark “fringes”).

-Bright bands mean:
o Constructive interference (the maxima of the waves line up)
o The interfering beams have traveled along paths with a length difference (ΔL) of 0, or
equal to an
integer number of wavelengths of the light (ΔL = 0, λ, 2λ, …)
-Dark bands mean:
o Destructive interference (one wave’s maximum lines up with the other’s minimum)
o The interfering beams have traveled along paths with a length difference equal to a half-
integer
number of wavelengths (ΔL = ½ λ, 3/2 λ, …)

Double-slit diffraction

In double-slit diffraction, each slit acts as a point source of light, from which semi-circular wave
fronts originate and interfere. In this setup, the central axis is defined as a line straight from a point
halfway between the slits to the screen where the interference pattern is observed, and the locations of
bright and dark fringes on the screen can be identified by their angles θ from this central axis. If the slits
are a distance d apart, the path length difference (ΔL) between the waves from the two slits is given by:
ΔL = d sin(θ)

-Bright fringes are observed at points on the screen where ΔL is an integer number of
wavelengths, so:
ΔL = mλ
mλ = d sin(θ)
-For dark fringes, ΔL must be a half-integer number of wavelengths, so:
ΔL = (m + ½) λ
(m + ½) λ = d sin(θ)

In these equations, m is an integer meaning the number of bright or dark fringes away from the
central axis. Sometimes m is called the “order” of the maximum or minimum, and it also tells by how
many wavelengths the paths of the interfering waves differ. The central maximum is called the “0 order
maximum” because there is no difference in the paths traveled by the two waves at this point.

After calculating the angle that corresponds to a particular maximum or minimum, its distance y
along the screen from the central axis can be determined with the tangent function: tan(θ) = y/D

Single-Slit Diffraction

The methods used in this case are similar for double slit diffraction, though we can only
calculate the locations of the dark fringes due to more involved mathematics. Since there is just one slit
in this case, we look at the path length difference between rays coming from two specific points along
the slit, rather than from two separate slits. In order for there to be a dark fringe at a certain point on
the screen, each ray we could choose must be paired with a ray whose path length is different by (m +
½)λ at that point. Rather than examining each possible pair of rays here, we will simply quote the result.
It is found that dark fringes are located at points along the screen where the path difference, ΔL, for rays
originating at the top and bottom of the slit is an integer number of wavelengths. Although the top and
bottom rays interfere constructively in this case, each experiences destructive interference with a ray
from another part of the slit. So we can locate the dark fringes with the equation:

mλ = a sin(θ)

Note that for a narrower slit width a, the angle θ of the dark fringes must be greater, so the pattern will
become more spread out on the screen. Again m represents the “order,” how many fringes we are away
from the central axis.

Diffraction gratings

This setup is again similar to double slit diffraction, only now we have a much greater number of
slits, N. Similarly for diffraction gratings, the path length difference between adjacent rays is again ΔL =
d sin(θ). If the distance d between adjacent slits is not given directly, we can calculate it with d = w/N, if
N slits occupy a total width w. Like in the double slit case, bright lines will be located at angles from the
central axis given by mλ = d sin(θ).

Optics Quiz