Geometrical Optics Study Guide (page 3)
This lesson introduces you to the concept of optics and the geometrical analysis of optical phenomena. We will study the concept of light rays, the laws of reflection, the laws of refraction, and the total internal reflection.
The study of light encompasses many fields of physics, from mechanics to quantum mechanics and from classical to modern physics. Isaac Newton explained reflection and refraction phenomena based on particle motion, and Max Planck's particle theory explained radiation. On the other hand, the theories of electricity, magnetism, and waves explained light behavior as a wave, as Christian Huygens and James Maxwell did in their theories. A dual explanation of the properties and processes in which light is involved emerged through history. In this lesson, we will address the particle explanation of light.
The geometrical analysis of light propagation assumes that light travels in a straight line through a medium and between two points. If the optical properties of the medium change, light changes the path of propagation. This is called geometrical optics and the propagation of light in a straight line is called ray approximation.
Reflection of Light
At the boundary between two media, the light ray will change its direction of propagation depending on the properties of the second medium and on the relative position of the light with respect to the surface of the interface. The light propagating from a source toward a smooth surface will bounce from the surface and return to the medium that contains the source. This phenomenon is called reflection of light, the incoming light ray is called the incident ray, and the returning light ray is called the reflected ray. A diagram of the propagation of light at the interface between two media also defines the important angles we will use to set the laws of reflection: the incident and reflection angles. Both angles are defined relative to the perpendicular to the interface called the normal. The two media are shown differently in Figure 18.1 to enhance the surface of separation where reflection occurs.
Experimental work shows that reflection is governed by the following laws:
- The incident, reflected ray, and the normal are in the same plane.
- The angle of incidence and the angle of reflection are equal:
θi = θr
If several rays of light are parallel and are traveling toward a smooth surface, after reflection, they will be once again propagating in parallel directions, as in Figure 18.2. This is called specular reflection and the reason is that each of the beams will reflect from the same plane surface.
If the surface is not smooth, the outgoing set of rays will no longer be parallel because the roughness of the surface means that different rays have different incident and respective reflection angles. This is called a diffuse reflection, and the way light is reflected at different angles from the rough surface is shown in Figure 18.3.
The rays represented by thicker lines represent the reflected rays and one can see that their direction is dictated by the normal to the surface, which differs between the three cases (the normal is perpendicular on the tangent to the curve at a certain position) .
Two planes are set such that they form an angle of 100° with each other. A light is incident on the first surface at an angle of 60°. Find the angle of the reflected light from the second plane with respect to the normal to that plane.
First, we need to set our angle diagram and use the laws of reflection to determine the angle of the outgoing ray (see Figure 18.4).
In the ABC triangle, the angle at A can be found from the first law of reflection:
θi = θr = 60°
And since the triangle totals 180° with A at 30°, C is 100° and B is 50°.
θi = 90° – B = 90° – 50° = 40°
θ* i = 40°
Again, we use the law of reflection on the second surface:
θ* i = θ* r = 40°
Refraction of Light
The examples we studied previously are probably easy to understand if you have performed experiments such as shining a light on a mirror or a shiny metal surface. But what happens when light meets a transparent medium and not an opaque one? We all know that light is transmitted beyond the surface of separation. Light goes through the water in your glass and even through the glass itself. This phenomenon of light propagation from a transparent medium to another transparent medium is called refraction. But refraction is affected by the properties of the medium itself, and the direction of propagation is changed by it. The corresponding rays are called incident and refracted rays, as are the angles. Figure 18.6 shows an incident wave and the refraction on the separation surfaces.
On the far left, we have the incident ray refracting on the surface and propagating inside Medium 2 at an angle smaller than the incident angle:
θi < θr
This behavior happens when light comes from a medium where the speed of light is larger and propagates into a medium with reduced speed. The light bends toward the normal when vl > v2 where vl and v2 are the speed of light in Medium 1 and Medium 2.
On the right side of the picture, we see the light reflecting from the bottom of Medium 2 and refracting at the surface of separation a second time. This time, the angle of refraction is larger than the angle of incidence because the light returns to Medium 1 where the speed of propagation is larger.
θ* i > θ* r
The light bends away from the normal when vl < v2
Analyzing the geometry of the figure and relating to the speed in respective media, one can determine after some calculations a law of refraction relating the angles of incidence and refraction.
The laws of refraction are as follows:
- The incident, refracted ray, and normal are in the same plane.
- The angle of incidence and the angle of refraction are related by
In optics, we also work with a coefficient that relates the speed of light in the specific medium with the speed in vacuum, and it is called the index of refraction:
The index of refraction is a unitless quantity and it is material dependent. A few values for the most common substances are given in Table 18.l.
Because the index of refraction is so close to 1 for air, we will consider n = 1. One note on propagation in a different medium is that the frequency of the wave will remain constant while the wavelength changes with speed.
Returning to the laws of refraction, we will find a second version of the law of the angles by multiplying the right-hand-side fraction with the speed oflight in a vacuum:
and with the definition of the index of refraction:
or separating on the same side factors related to the incident and refracted rays, respectively:
n1 · sin θi = n2 · sin θr
This is also known as Snell's law of refraction (Willebroad van Roijen Snell, 1580-1626).
For the purpose of this lesson, we will consider the index of refraction constant, although in reality, n is wavelength dependent. This dependence is the explanation for phenomena such as the dispersion of light through a prism.
Light travels from a vacuum through the medium of an index of refraction of 1.55. Find the speed of light in the second medium and the wavelength in each medium if the frequency is 6 · 1014 Hz. If the light comes at an incident angle of 30°, what is the refraction angle?
We want to set all known data and then determine what expressions are useful to determine the unknowns. We know that the first medium is a vacuum and the second is a material that has an index of refraction of 1.55.
n1 = 1
n1 = 1.55
θ1 = 30°
f = 6 · 1014 Hz
v = ?
λ1 = ?
λ2 = ?
θ2 = ?
From the definition of the index of refraction, we can determine v2:
We remember that v = λ · f, and that frequency stays the same with refraction. Then:
v1 = λ · f
And since the first medium is a vacuum, the speed v1 equals c:
λ1 = 600 nm
And in the second medium,
λ2 = 323 nm
Then to determine the refraction angle, we apply Snell's law:
n1 · sin θi = n2 · sin θr
1 · sin 30° = 1.55 · n1 · sin θi
sin θr = 0.32
Let's summarize with a specific case: light rays going from a medium with a large optical density (large n) to a smaller n. In this case, the angle of refraction is larger than the angle of incidence, and so the light will move away from the normal. What happens when the light is at 90° from the normal? In this case, the light is no longer transmitted into the second medium but is returned to the first medium. This is the case of total internal reflection, and the limit angle of internal reflection can be determined by applying Snell's law (see Figure 18.12).
n1 · sin θi = n2 · sin θr
n1 · sin θi = n2 · sin 90°
n1 · sin θi = n2
We will call the angle of incidence for which total internal reflection happens a critical angle θc.
n1 · sin θi = n2
Can a light ray propagating from air to water suffer total internal reflection at any angle?
This is a general question that addresses the expression of the critical angle of total internal reflection:
and n > 1 for any of the two media in the experiment.
If we have Medium 1 to be air, then nl = 1 and the equation becomes
But n2 > 1 and because the limits for the sine function are between –1 and + 1, the above equation has no solution. Hence, no critical angle exists for total internal reflection to happen.
Practice problems of this concept can be found at: Geometrical Optics Practice Questions
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