Unit 7: Optics
An optical phenomenon involves an interaction between electromagnetic waves and matter. The complete study of optics involves complex mathematics, a thorough understanding of both classical and quantum optical effects, and a great deal of engineering.
We begin by applying simplified models to develop a basic understanding of how optics works, without having to remember all of Maxwell's equations. In geometric optics, we ignore all the effects that are germane to waves, such as interference. This will nevertheless allow us to understand a large variety of phenomena. We will return to the complexities of wave physics as related to optics in the section on wave optics at the end of this unit.
Completing this unit should take you approximately 11 hours.
Upon successful completion of this unit, you will be able to:
- determine the size, location, and nature of images by using the mirror and lens equations;
- solve problems using the law of refraction;
- explain the interference pattern seen in a double-slit experiment and what these results mean;
- explain how rainbows are produced;
- explain how the Huygens principle leads to diffraction at a single slit;
- use the Rayleigh criterion for the resolution limit at different wavelengths; and
- explain how polarized light is created and detected.
7.1: Geometric Optics – The Ray Aspect of Light
Light can travel from a source to another location via three ways:
- It can travel directly from the source through empty space, such as from the sun to the earth;
- It can travel through various media, such as through air or glass;
- It can be reflected, such as by a mirror.
In all of these cases, it is possible to almost completely ignore the fact that light is a wave – in particular, we never have to mention the concept of frequency (except indirectly in a quantity called the refractive index, which characterizes the medium).
Geometric optics is essentially the study of all optical phenomena that are independent of light frequency. In an even more simplified version of geometric optics, we also assume that all media through which the light travels are uniform. For example, air is assumed to have uniform density and temperature. Then Maxwell's wave theory predicts light waves transport energy in travels in straight lines, or rays.
Our textbook uses this narrow definition of geometric optics – but you should be aware that light rays can in fact bend, for example when forming a mirage in non-uniform layers of air. Light rays that can bend are part of geometric optics, too! After all, geometry is literally named after a curved object (the Earth).
However, we will only be discussing the geometric optics of light rays that are mostly straight. Such a ray can only change direction when it encounters an object, such as a mirror, or when it passes from one material to another, such as from air to glass, but it then continues in a straight line.
When you see drawings of light rays, do not confuse them with pictures of electric or magnetic field lines. We will not be talking about these fields at all in the context of geometric optics. Light rays always indicate the direction in which the corresponding electromagnetic wave is traveling, and that direction happens to be precisely perpendicular to the orientation of the magnetic and electric field lines at any given point!
Watch this video, which covers the material in this section and introduces the next three sections of our textbook which we will discuss next.
7.2: Geometric Optics – The Law of Reflection
When light hits a smooth reflective surface, it bounces off like a tennis ball, making the same angle on the way out as on the way in, measured relative to a line perpendicular to the surface.
Read this text, which explains why it is important to have a smooth surface to make a good mirror.
7.3: Geometric Optics – The Law of Refraction
You may remember that the speed of light has its own symbol, c, because we said it is a universal speed for all electromagnetic waves. However, that was only true as long as the light waves propagate in a medium that does not interact in any significant way with the electromagnetic fields that make that wave. More precisely, c is the value of the speed of light in a vacuum! It is c = 299,792,458 m/s (meters per second): a very fast speed indeed.
However, just like sound waves have a different speed in air and in water, so does light. Refraction of light occurs when it crosses the interface between two different materials where the speed of light is different. These material-dependent speeds have been tabulated for many materials.
The way one usually quotes the material-dependent speed of light, v, is by giving its value in relation to the vacuum speed of light, c. That ratio is called the refractive index, n = c/v.
Watch this video to see some animations of reflection and refraction.
In the material we just reviewed, we saw that the refractive index change between two materials affects how the direction of a light ray changes. The law that describes this change is Snell's Law. It is similar to the law of refraction in that it describes the angles relative to the perpendicular at the interface. But it differs in that this law refers to transmitted light and not to reflection. Also, the outgoing angle is usually different from the incoming angle. When light hits an interface between two media, there is typically a partially reflected ray as well – and that reflection still obeys the law of equal incident and outgoing angles.
Watch this video, which shows different ways of writing Snell's Law, either with or without making use of the wave speeds directly.
Watch this video, which explains the familiar illusion of the "broken straw" in water.
Watch these two videos for some examples of numerical calculations. The second example is more challenging.
Read this text where we learn that under certain conditions, Snell's Law for the direction of the refracted ray can give us an angle of 90º with respect to the perpendicular. This corresponds to a transmitted ray that is directed exactly parallel to the interface between the two media, and therefore is not actually transmitted across the interface at all. This happens when the index of refraction is larger on the incoming side than on the side we are trying to enter, and if the incident angle is made large enough to reach a critical value. This angle of incidence is called the critical angle for total internal reflection, because for any angle equal to, or larger than, this value the light cannot be transmitted.
The law of energy conservation implies that all of the incident light must be reflected back into the medium it came from. This effect of total internal reflection is at the heart of fiber optics, the technology on which most of our long-distance internet communication is based. The idea is to send light rays into a thin strand of glass surrounded by a material of lower index of refraction (typically just a different type of glass). This optical glass fiber is so thin that it becomes flexible and can bend around corners. The light follows the bending of the fiber because it zig-zags along the fiber at such grazing angles that it always meets the condition for total internal reflection.
Watch this video, which discusses the conditions for total internal reflection in water.
7.4: Dispersion and Prisms
The speed of light is dependent not only on the material, but also on the frequency of the light in that material. Really the only way to avoid a frequency-dependent speed of light is to have a complete vacuum – which is unrealistic in everyday life.
Read this section which discusses the effects that arise when the propagation speed depends on the frequency. This is called dispersion.
A rainbow is the most familiar natural phenomenon that demonstrates dispersion. The light from the sun contains waves with all frequencies of the visible spectrum. The combination of all these waves is perceived as white light. If there are water droplets in the air, the light refracts when it enters the droplets, and then refracts again as it leaves the droplets. Since the angle of refraction is different for light of different frequencies, waves of different colors separate. We perceive this as a rainbow.
Watch this short video on differently-colored light rays as they get refracted by different amounts.
Watch this video for an overview of dispersion and the topics covered in the next section.
7.5: Image Formation by Lenses
Geometric optics is the backbone on which our own visual system is built. Seeing an object requires forming a "copy" of the object on the retina of your eye, where light-sensitive cells record the intensity of the light. So the first thing that needs to happen is that light rays emanating from any single point on the object that we can see should correspond to light rays converging to one single point on the retina. If that does not happen, then you see a blurry image, and your perception of reality is not as faithful as it should be.
The way your eye achieves this primary objective is by refraction through the converging lens at the front of the eye. All lenses are based on refraction, but their specific properties depend on the way their surfaces are curved. Without curved surfaces, you cannot make a lens – but not all curved surfaces make a good lens. Read this text to investigate some typical geometries for a lens, and characterize their light-bending properties.
The idea of diverging and converging bundles of rays, as created by lenses, is also related to the concept of intensity which we encountered earlier. Recall that the intensity of an electromagnetic wave decreases as we move away from the source. This is because the energy in the wave gets "diluted" over a larger area.
Compare this to the light rays fanning out from a point-like source in geometric optics. Here it is the density of the rays that gets diluted. So there is a correspondence between the density of rays and the intensity of the light. Converging a bundle of rays onto a focal point is the same as increasing the intensity of the light.
We use the shape and material of the lens to determine its focal point, not by the way we send in the light rays. The distance of the focal point from the lens is not the same as the distance at which the image of a given object forms. That depends on how far away the object is. Equations that relate the distance of the object and the distance at which the image forms can be given in especially simple form if you assume that the lens is very thin.
Watch this video, which illustrates the geometric constructions behind the thin-lens approximation. All the constructions are drawn with a horizontal axis that goes straight through the center of the lens, while the lens itself is drawn vertically upright. We call the axis in this drawing the optical axis. For symmetric lenses of the type we are considering here, a light ray coming in precisely on the optical axis would hit all the interfaces between glass and air in a perpendicular direction. That would mean the ray would not change its direction because it will not experience any refraction.
Lenses work because light rays are refracted once on the way into the glass, and a second time on the way back out. But in the thin-lens approximation, the two surfaces are so close together that you do not have room to draw the two refraction events individually. Instead, one uses a trick.
Watch this video, which details how this trick works. It is based on the idea that rays going through the exact center of a lens do not suffer any refraction at all, whereas light rays coming into the lens parallel to the optical axis must be refracted precisely through the focal point. This is why in the ray diagrams for a lens, we typically draw only two rays (out of the infinitely many rays coming from the object) to completely characterize what the lens does.
In addition to the image distance, another characteristic feature of a lens is the relation between the size of the image and the size of the object. This is called the magnification of the lens, and it also depends on the object distance.
The next two videos are optional information for those of you who would like to understand how to get the formulas in the text by analyzing the geometry of the ray diagrams.
The image that needs to form on our retina to have clear vision is called a real image because it has rays that originally diverged from one point on the object coming together at one single point in the image. But depending on the object distance and the shape of the lens, one can also observe virtual images. A virtual image is formed when light rays emitted from a single point on the object come out the other side of a lens in a diverging fan which would meet at a single point if traced back to the input side of the lens along straight lines.
Watch this view which explains how virtual images occur in concave lenses.
However, you can also get virtual images in convex lenses. See the video below for a step-by-step explanation of virtual images.
In our final video for this section, we see examples of how to apply the lens formulas, paying special attention to the signs of the quantities. They will tell you if you are going to get an upright or inverted image, and whether it is real or virtual.
It is a little unfortunate that optical scientists use the terminology "image distance" and "object distance" for quantities that can be positive or negative: what these numbers really are is coordinates measured relative to an origin centered on the lens. Distances in geometry are usually understood to be positive, but in optics they can also be negative. So you should think of the quantities appearing in the thin-lens formula as coordinates instead, even though we keep using the traditional terminology in what follows.
7.6: Image Formation by Mirrors
We just saw how virtual images can appear in the context of lenses. We can also use curved mirrors to create the same types of focussing effects that we just discussed for lenses. We can also identify the same characteristics: focal point, image distance and magnification.
Read this text, which illustrates how flat mirrors, like the one in your bathroom, produce virtual images.
The formulas in this section are similar to those for thin lenses; they are also based on an approximation. For mirrors, the approximation is that the mirror diameter that captures the light is much smaller than the radius that characterizes the curvature of the mirror. This means we assume the mirror is nearly flat, but not quite.
When a mirror is flat like this, it is impossible to distinguish whether its shape is actually part of a sphere or part of a different curved form, such as a parabolic cross section. To draw ray diagrams, it is actually more convenient to assume we have a mirror of parabolic shape, because these mirrors reflect all rays that come in parallel to the optical axis back into a single focal point.
In an approximate way, we can apply what we learn about parabolic mirrors in the next video to all weakly curved mirrors because they are indistinguishable in shape from a parabolic cross section.
Watch this next video, which provides additional examples of ray diagrams for parabolic mirrors.
In the next video, we explain why you can see a tiny reflection of yourself in a teaspoon when you look at it with the bottom side facing you. That is a concave mirror.
7.7: Optical Instruments
Now, let's apply the optical design principles of the previous sections to the human eye, and to some of the important optical instruments that enhance the capabilities of our eyes. The lens of our eye is of course not made of glass, but the material does not really matter for the optical principles we have learned so far.
In geometric optics, the material is taken into account entirely within the index of refraction. Most of the human lens is composed of transparent cells, which means the shape of the entire lens is flexible. This gives us the ability to change the focal length of the lens in a process called accommodation, to bring objects at different distances into focus on the retina.
Watch this video, which also discusses how we can correct our vision, such as by using eye glasses or by performing surgery on the eye itself.
LASIK is the most common procedure used to surgically correct the way light is refracted onto the retina. It reshapes the cornea that covers the lens on the outside, not the lens itself. The cornea works together with the lens to refract light, and the corrections required for improved vision are usually so small that we only need to adjust less than half a micrometer of thickness (a micrometer is a thousandth of a millimeter).
For comparison, this is about the same as the wavelength of visible light. But this is a problem because the spot size to which a laser beam can be focused is limited by the wavelength of the light. This is called the resolution limit, which we will discuss in the section on wave optics. The solution is to use laser light which has a wavelength much shorter than visible light – ultraviolet light. This provides the required higher resolution.
The resolution limit for visible light is about a micrometer, but that does not mean we are able to perceive objects that are that small with the naked eye. The problem is that the human lens is not perfect in the sense that rays emitted from a single point of light will not recreate a single bright point of light on the retina, but a somewhat blurred spot instead. Moreover, the light-sensitive cells in our retina have their own size, so we cannot distinguish between closely spaced points of focused light that overlap the same cell.
Read this text, which describes two of the most common conditions that may require corrective eyewear – near and farsightedness.
In a microscope, we guide light rays emitted from two very closely spaced points on the object (for example a microorganism) so they converge back together in two spots on the retina that are spaced much farther apart than on the object itself. That allows our eye to distinguish the two points clearly, and as a result we can see details that would have been completely washed out for the unaided eye.
Watch this lecture, which draws the ray diagrams that explain this.
Read this text, which shows an example of the calculation of the magnification which is achieved by a compound microscope with two lenses.
While microscopes allow us to look more deeply into the microcosmos, telescopes let us see into the vast distances of the cosmos on the largest scales. The problem we face when looking at distant objects is mainly related to the concept of intensity. Because light rays coming from distant stars or planets spread out on their way to us, the intensity of the light that can enter our eye from those objects is very low. When the intensity is too low for our retinal cells, they will not tell our brain that there is a point of light there at all.
The main point of a telescope is to simply catch more light by collecting the rays that each star emits with a large-diameter lens or mirror. After collecting the light, the trick is to then bend the rays in just the right way so they re-converge onto a single spot on our retina. When this is achieved, the intensity of the spot on your retina is much larger than before, when only the limited amount of light entering the small opening in your iris contributed to that spot.
And just as for a microscope, the optics in a telescope must also ensure that spots corresponding to different stars will appear at large enough spacings from each other on the retina, so that they can be perceived as individual points of light instead of just a blur. Watch this lecture, which details this process.
Of course, we use telescopes not only in astronomy, but also to observe distant objects on Earth. There are slightly different requirements for different applications – for example you do not want your binoculars (which are essentially paired telescopes) to flip the image upside down. Note that this is perfectly acceptable to astronomers, whose primary objective is not to lose any light intensity, such as from absorption in the lens material. Read this text to see how to achieve these different goals.
Large modern telescopes, including the famous Hubble Space telescope, use mirrors instead of lenses to collect light. Our textbook shows the ray paths in this type of reflecting telescope. Reflectors are used to make astronomical images using electromagnetic waves at invisible wavelengths, such as radio waves, where it is not possible in practice to make refracting lenses.
It is interesting to note that we can also apply the ray-based methods of geometric optics when designing antenna dishes for radio telescopes, which are essentially mirrors, too. This works because the wavelength, as measured in meters (or micrometers), does not limit the applicability of geometric optics. What matters is the wavelength relative to the size of the mirrors, openings and distances encountered by the electromagnetic waves. A ray description is possible whenever the wavelength is short by comparison to these other length scales.
Radio waves are of a length that puts them right at the borderline where ray optics breaks down. Radio dishes are usually designed to operate at wavelengths in the range of centimeters or even millimeters. This is indeed much shorter than the size of the dish mirrors which can be tens of meters wide. But radio waves can also have much longer wavelengths of several hundred meters, and then it is not possible to treat them with the methods of geometric optics.
7.8: Wave Optics: Interference
The approximations of geometric optics can also break down for light waves, but this happens at a small size scale because the wavelength of, say, visible light is shorter than about 0.8 micrometers. In this regime, we have to return to Maxwell's insight that light is actually a wave phenomenon. Because of its wave nature, light can therefore show all the effects that other waves show, too.
First, let's read this text to remind ourselves of the fundamental relationship between frequency, wavelength and speed of a light wave.
Watch this video, which introduces us to the most important effects that all waves can produce. The discussion talks specifically about light even though the explanations apply to all waves. The reason why light is our main example here has to do with the ambiguous nature of light which was mentioned when we introduced the concept of the photon: light is different from water waves because it does not require a medium to propagate. This makes it much harder to prove that light is actually a wave, because we cannot directly observe the ripples that make a light wave, as we see the ripples on a pond when a water wave passes through.
What we have to do to prove that light is a wave is: verify that light shows all the same effects that only waves (not particles) can produce. Geometric optics cannot help us here, because the rays we have been talking about could still in principle be interpreted as the trajectories of the little particles we called photons.
The main thing waves can do and particles (in the classical sense) cannot is for two of them to occupy the same region of space. When this happens with waves, we get the effect called interference. Building on this, we can observe the additional effects discussed in this video.
7.9: Wave Optics: Diffraction
The idea of interference is also behind Huygens' principle, a construct that allows us to build up wave fronts of practically-arbitrary shapes from the interference of many circular elementary waves. This interpretation is of value because it contains one of the most fundamental ideas in physics: the principle of causality. What a wave does at a given point in space is caused in a simple way by what the wave was doing at an earlier time in a neighboring region of space.
As the reading illustrates, Huygens' principle is not just a philosophical interpretation – it is also a computational tool. In particular, the idea of circular (or spherical) elementary waves makes it relatively easy to explain how a wave can bend around corners and spread out after passing through a constriction. This is called diffraction because it allows wave energy to go around corners in directions that the rays of geometric optics (or the trajectories of classical particles) would not be permitted to go.
Read about the proof that light is a wave in this experiment Thomas Young gave using diffraction by a pair of closely spaced slits.
What makes the double slit experiment better than diffraction by a single slit, if you want to prove the wave nature of light?
It may still be possible to explain why you can see a light beam spreading out behind a single slit, by photon particles ricocheting off the edges of the slit. But when the spreading light waves coming from two slits overlap, they form the characteristic pattern of constructive and destructive interference that only waves can produce.
Moreover, because this interference is created by two waves that are, at any given point, almost parallel, the corresponding interference patterns are spread out spatially. This is important because it magnifies the interference pattern from a microscopic to a macroscopic scale that can be seen by the naked eye.
You can see an analogy of the double-slit interference pattern when you hold two combs on top of each other against the light. The teeth (or tines) of the comb are like the wavelength-spaced ripples of a wave. When the combs are perfectly aligned, you can see through the spaces between both sets of teeth, but when you rotate one comb ever so slightly, you will see the light being blocked in regular patterns that form the analogue of destructive interference between two waves. The more you rotate one comb, the closer the spacing of the dark pattern becomes.
Rotate the combs back to a nearly parallel angle and you see the spacing between the dark lines grow. The latter is what happens in the double-slit experiment if the separation of the slits is decreased, because it brings the diffracted waves from each slit closer and closer into alignment.
Watch this video for a step-by-step construction of the interference pattern.
Watch this video for a discussion of the equation that predicts how the interference pattern depends on the wavelength and the spacing of the two slits.
Watch this video for additional practice with the double-slit setup.
Watch this video, which takes us one step further, from two closely-spaced slits to many closely-spaced slits.
Because diffraction and interference are wave effects, they depend on the wavelength of the light. In particular, the spacings of the multiple-slit diffraction pattern become wider when the wavelength becomes longer.
In Young's double-slit experiment, this actually posed a challenge because you cannot easily see the interference pattern if you shine white light through the slits. Recall that white light is actually a mixture of light with all the colors of the rainbow. If each color shows slightly different interference patterns, the end result will be a washed out intensity distribution where the destructive interference for one color overlaps with constructive interference for another color.
This situation is improved if we use a lot more than two slits to produce the interference pattern. Here is the reason: to get constructive interference between the waves coming from a large number of slits simultaneously, the condition on the angle where this happens becomes more strict, which makes the bright regions of the interference patterns narrower. Then the regions of highest intensity produced by different colors show less overlap.
Read this text, which provides a worked example that calculates the angles where the diffraction pattern shows maximal intensity, for light of different colors.
Diffraction gratings offer a highly-effective method of routing light of different color (wavelength) in different directions. This way, we can split a beam of white light into the rainbow colors, much like what a prism does due to dispersion. But diffraction gratings do not depend on dispersion, the wavelength-dependence of the light speed in a material – they only use the fact that light can diffract and form interference patterns.
Some subtle interference patterns appear when light shines through just a single slit. However, it is more difficult to see them because the contrast between this pattern, and the central part of the light beam that passes straight through, is quite low. However, we have now assembled the tools necessary to calculate this effect as well.
Read more about single slit diffraction in this section of our textbook.
In the context of the microscope, we have briefly mentioned the resolution limit, which makes it impossible to form arbitrarily-sharp image spots on the retina. Read this text as we return to this concept and arrive at the fundamental criterion on which the resolution limit is based – the Rayleigh criterion. The heart of our limit to resolve small objects is the same physics we just covered in the single-slit diffraction pattern.
Read this text to learn more about how reflections at closely-spaced interfaces of a thin layer create the rainbow colors you see on soap bubbles or oil slicks. As with the diffraction grating, the interference pattern depends on the wavelength, leading to constructive interference in different directions for different colors of light.
7.10: Polarization
In the previous video, Greg Clements also covered a property of light that truly distinguishes it from sound waves or even water waves: polarization. This property is directly related to what we learned in the study of Maxwell's equations and electromagnetic waves: like all traveling waves, light has a direction of propagation in which it transports energy, but it also has another direction we do not usually perceive: the orientation of the electric field, and therefore the orientation of the forces that this wave would exert on a movable charge such as an electron.
At any given point that the wave passes through, the electric field is perpendicular to the propagation direction, and we know that the electric field also oscillates in time. The amplitude of the oscillation defines how much energy the wave carries, but we have not said much about the direction in which this oscillation happens.
While the electric field oscillates, its maxima also moves forward at the speed of light. A graph of the wave, traced out by the electric field, looks a lot like a wave on a rope, or a slinky. That is, the entire wave motion occurs in a plane that contains the direction of propagation as one axis, and the direction of the electric field as another axis.
Let's say the wave is traveling in the horizontal direction, from left to right. Imagine the plane of the electric-field wave as a knife blade, with the long edge of the blade pointing in the propagation direction. Then you can still tilt the blade in many different ways, such as vertically, horizontally, or at some angle in between (without changing the direction in which the long edge points). The tilt angle defines the direction of polarization.
Read this text, which discusses several different ways to create light of a specific polarization direction.