Geometrical Optics
Wave fronts and rays
For a one-dimensional wave, for example on a guitar string, all points on the string experience the "passage" of the wave [an exception is the stationary points along the string when a standing wave is produced]. This is not necessarily the case for two or three-dimensional waves. For example, when we talk with our hands cupped around our mouth we send out sound waves directly forward so that any person standing behind us will not hear what we say. So, the air molecules behind us do not get affected by our sound waves, but the ones in front of us do. In this way different points in space experience different wave "passages".
Remember that all points in space that share the same amount of disturbance (i.e. have equal amplitudes) at any given time are said to have equal phase. Points on the wave that have different values of the disturbance then have different phases, or are said to be out of phase. This concept is analogous to the notion that two ideas that are in-step (synch) with each other are in-phase and those that are out-of-step with each other are out-of-phase (like people's thoughts or wants). For periodic waves, such as regular ocean waves traveling toward the beach, all points on the crest of a wave are in-phase with each other; as are all points on the trough. As the wave moves, these points of equal phase move with it at the wave speed.
Wave fronts in the Mediterranean Sea off the coast of Sicily
All points in space that are in a given neighborhood and have equal phase are said to belong to a wave front. It doesn't matter what the phase value is, so long as these points have equal phase they constitute a wave front. In two dimensions a wave front is a curve representing points with equal phase, while in three dimensions a wave front is a surface. For example, in the case of idealized two-dimensional ocean waves, a wave front could be thought to be all the points at the crest of the wave as it is traveling toward the beach. For a long ocean wave. such a wave front is a straight line that approaches the beach. In contrast to this, when you drop a stone in a still pond you create circular wave fronts, again in two dimensions. In the case of a point source of light (electromagnetic waves), in three-dimensions, like a flash created by an explosion, the wave fronts are spherical traveling outward. The sun is so far away from earth that we can think of it as a point source too. Even though sun light begins as spherical wave fronts, by the time the light reaches us at the surface of the earth these "spheres" are so large that they are locally flat planes when we detect them. So, for all practical purposes sun light is made of plane waves at the surface of the earth, i.e. its wave fronts are parallel planes.
About thousand years ago Al Hazem made small holes in the curtains that had darkened his room from daylight to create rays of light. He discovered that these rays followed parallel paths. However, at night, this same situation showed that the rays generated by candle light were not parallel, but their extensions appeared to meet at the candle flame. Rays of light are not just the path that the light takes, but they also indicate the direction that the wave fronts travel in space. Point sources of light (which generate spherical wave fronts) produce rays that begin at the point source and diverge in all directions. But sources that create plane waves produce parallel rays. Laser light that forms a collimated pencil of light clearly is made of parallel rays. So, laser light is made of plane waves (i.e. light with parallel plane wave fronts).
Typically, when we want to investigate the wave nature of light we have to consider how its wave fronts propagate in space. If, on the other hand, we are interested in image formation, then an easier way of dealing with light is to treat it as if it is made of rays. This method of image formation is called geometrical optics or ray optics. It is possible to deal with image formation using the wave properties of light, but it is much more complex.
Questions on Wave Fronts and Rays
Reflection and imaging using mirrors
Light reflects from objects in two distinct ways: specular or diffuse. Well polished surfaces, particularly shiny metals, cause specular reflection. Rough surfaces lead to diffuse reflections. When light strikes an object it gets scattered. In the case of a diffuse reflection the light ray's scattering direction depends on which part of the object the ray strikes. So, a collection of parallel rays falling on a piece of paper, say, will reflect in different directions and light will reach your eye looking at the paper no matter where you stand. But when the surface is smooth so that there is no difference between different sections of the surface, then parallel rays are all reflected in the same direction and your eye will only see the rays if you stand at the right location to receive those parallel reflected rays
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Mirrors are not only specular reflectors, they also reflect rays (laser beams) in a very well ordered way, known as the law of reflection: Angle of incidence = Angle of reflection. This simple law is all that we need, along with the technique of ray tracing, to understand how mirrors form images and to determine the location, size, and orientation of images formed by mirrors. Mirrors differ from ordinary plate glass in that they have a thin metal coating - usually on the back surface opposite to the incident side - that is highly reflective. The metal coating can be designed to be partially reflective, so that it is partially transmissive as well, as, for example, "mirrors" on the "walls" of many stores that allow one-way vision into the store. Plane window glass reflects only about 4% of the incident light intensity at each of its two surfaces when viewing the reflection "head-on".
The simplest mirror to study is a flat mirror. Consider a point source of light located in front of a flat mirror, as shown below. In terms of wave fronts, this source sends out spherical fronts centered at the source. In terms of rays, the source sends out rays in all directions. Those rays that strike the mirror reflect according to the law of reflection. It is simple geometry to show that all of these rays reflect as if they originated from a single point behind the mirror. This apparent "source" is the virtual image of the original point source. It is called "virtual" because no light actually emanates from it, but it appears as a source of light.
In the above diagram we see three arbitrary rays emanating from the point source that strike the mirror, each at a different angle. There are of course many other rays that are sent out from the source, but not drawn. Some of these do not get reflected from the mirror and so do not contribute to the apparent luminosity of the image. What is interesting is that all of the rays that strike the mirror are reflected so that their extensions meet at one single point. To see this, we draw a line (the black dashed line) from the point source perpendicular to the mirror. Evidently, the reflection of all rays appears to originate from a point on this line. This point is located as far behind the mirror as the source is in front of the mirror. The line that we drew is in fact the perpendicular bisector of the triangle that is formed by the incident ray and the extension of the reflected ray (virtual ray). Therefore, from the rules of simple geometry, the angle of incidence will be equal to the angle of reflection provided that the extension of the reflected ray originates from this point.
Any object (source of light) can be thought to be a collection of point sources. So, we could now determine the location, orientation, and sizes of the image of any object simply by repeated application of the above simple geometry. In the case of a straight object, say a stick or you, standing in front of the mirror, the image stands behind the mirror, an equal distance, and with the same orientation (upright). Furthermore, this image has the same size as the object as shown below:
This image is said to be virtual, erect, and of magnification equal to 1.0. Magnification is just the ratio of the image size to object size. In ray optics, again using simple geometric relations, we find that another way to determine magnification is to divide the image distance, i, by the object distant, o. Because, by convention, virtual images (objects) are assigned negative values of distance, then flat mirrors always produce an image distance to object distance ratio of -1. Magnification is defined as the negative of this ratio, so we end up with a magnification value of +1, which signifies that the image has the same orientation as the object, i.e. it is erect.
Summarizing these rules of ray optics:
o = object distance = distance from the object to the mirror
i = image distance = distance from the image to the mirror
m = magnification = - (image size)/ (object size) = - i / o
If object is virtual, then o is assigned a negative value; if it is real, then a positive value
If image is virtual, then i is assigned a negative value; if it is real, then a positive value
When m is positive, the image is erect; otherwise, it is inverted.
Real objects emanate real rays of light; virtual objects emanate extensions of real rays (i.e. virtual rays)
Real images are where real rays come together to intersect; virtual images are where extensions of real rays meet.
It turns out that simple ray tracing can easily show us how images are formed even in rather complex situations. For example, to determine the location of all the images formed by two or more mirrors all we need to do is to find the position of the image formed by one mirror just as we did above and then treat this image as a new (virtual) object for the second mirror, and so on. In the following diagram we see that a total of three separate images are formed by a point object placed in front of two mirrors that are perpendicular to one another.
Reflection of a ball in two mirrors at right angles to each other, just as in the sketch above. Note that the lettering on the central image is correct while those on the other two are mirror-reversed. Why is that? Use the above sketch to help decipher why.
Questions on Reflection and imaging using mirrors
Refraction and imaging using lenses
When light travels from one medium into another of different index of refraction it refracts, i.e. bends. This happens as the light goes from air to glass or from glass to water, etc. In this section we will see how lenses, which are made of glass or other transparent material with an index of refraction that is larger than air's, can be used to bend light rays so as to generate images of objects. In particular, we will use the technique of ray tracing to learn some basic, yet powerful, rules that allow us to determine the location, size, and nature of these images. For simplicity, we will only consider two types of lenses: double convex, and double concave. These lenses are made of two spherical surfaces of equal radii that intersect each other in either a convex or concave manner. Convex lenses typically bend light rays "in" toward the lens axis and in this sense are converging, while the concave shaped lenses bend light rays away from their axis and are thus diverging.
Light refracts on passing through a diverging (top) and converging (bottom) lens, bending away or toward, respectively, the lens optic axis
I - Converging Lenses:
To locate the image of a point source we have to determine the single point where the rays leaving the source are brought back to an intersection by the refraction of the light at the lens. To do this, all we need to do is to find where two separate rays leaving the source intersect each other. Then any other ray that leaves the point source has to pass though this point too (why?). Two very special rays for converging lenses are:
These two types of rays are all we need to locate the images formed by a converging lens. From any point source of light we can draw these two rays and see where they intersect. The point of intersection is then the image. Another rather clever idea is to use the law of reciprocity! Remember that according to this law light rays do not care which direction they travel in space. The material through which they travel matters, but the direction (sense) doesn't. In other words the ray diagram for the refraction that occurs at each boundary works no matter which way the light is actually traveling. So, in the above ray diagram we could think that the light is striking the lens from the left. In this perception the blue rays come to a focus at the focal point, F, on the right of the lens. Or, we could equally think that the light rays are striking the lens from the right. Then the significance of the blue rays is that a point source placed at the focal point on the right has its image appear at infinity (why?). In short, light rays don't have directional arrows, according to the law of reciprocity.
In the above ray diagram we have drawn our two special rays for one point source that is at the "head" of our object. These rays intersect at the other side of the lens creating a real, but inverted image. (Notice that by the law of reciprocity we could equally say that the real inverted object on the right creates a real erect - not inverted - image on the left of the lens.) If we move the object ( the one on the left) closer to the lens, the blue colored ray still refracts the same as before, but the red colored one tilts to intersect the refracted blue ray further away from the lens than before. So, as this real image gets closer to the lens its image gets larger and larger, remains inverted, and gets further and further from the lens. A critical point occurs when the object reaches a focal length distance away from the lens, still on the left side. In that case, as we saw above, its image will be at infinity! What happens when the object moves even closer to the lens than the focal point? That case is shown below where you can see that no real image forms. These virtual rays intersect to form a larger size, erect, but virtual image behind the object (to its left). It is in this configuration that we use magnifying lenses to see a small object (why? How do we see the virtual image?).
Simple cameras have a converging lens for image formation. The real image that falls on the photosensitive film creates a chemical reaction that later on can be developed into different shades and/or colors of the negative. The pictures that we see are prints of these negatives. In today's digital and video cameras, instead of film there is an array of photosensitive detectors. When light falls on one of these detectors, this information is stored digitally in the memory of the camera's computer. But the rest of a digital camera's operations are the same as the conventional camera's. To properly place the image on the film, the camera must be "focused" before taking the picture, either manually or automatically. [Note, as an aside, that many cameras send out an infrared pulse whose reflection from the object to be photographed is then detected. The time delay is used to adjust the focusing automatically.] In effect the camera lens is moved away or toward the film (or photosensitive array) in order to change the image distance. If the image is formed behind or in front of the film, then the light that falls on the film itself will produce a blurred image. In such instances we say that the camera was not properly focused.
The human eye works very much like a camera in terms of imaging. Similar to a digital camera, instead of film there are photosensitive detectors on the retina, in the back of the eye to record the image. But unlike cameras the lens of the human eye is not moved in relation to the retina. Instead, muscles pull on the lens of the eye to change its shape and in this way alter its effective focal length and thus position the sharpest possible image on the retina.
Take a look at this web site for an applet on imaging with lenses
II - Diverging Lenses:
The two special rays for the diverging lens are similar to those for the converging lens. Here again the rays striking the lens center continue on without refracting (red colored rays shown below). The ray that strikes the lens parallel to the lens axis refracts so that its extension passes through the focal point on the same side of the lens (blue colored rays shown below). In this case the refracted rays do not intersect at all, so there is no real image formed by a diverging lens. However, the extensions of the refracted rays do intersect on the same side as the object. Now a real object infinitely far away has a point size virtual image at the focal point on the same side of the lens!
Again, we use these two types of rays to locate the image of real objects, as shown below:
Evidently, in this case a real object cannot have a real image! As the object approaches the lens its virtual and erect image moves closer to the lens and gets larger (although never larger than the object itself). We use diverging lenses in combination with converging lenses for most practical applications. For example, myopic eyes are prescribed correcting lenses which are diverging. The un-corrected myopic eye forms an image of distant objects closer to the eye lens than it should. So, the light (image) that falls on the retina is blurry and not sharp. This image can be brought further back onto the retina using a diverging correcting lens.
How about the size of a lens and its quality? What are the important effects here? Simple ray tracing shows us that the primary effect of lens size is its light gathering feature. The larger the lens, the more light it can bring to the image. So, telescopes need large lenses to let us see far and faint stars. Image blurriness due to lens imperfections can have many different causes. The simplest of these is lack of uniformity of the lens surface or material. If the curvature of the lens surface changes over its different parts, then clearly the lens ends up with an imperfect image forming capability. These imperfections are called lens aberrations.
Questions on Refraction and imaging using lenses
Another key question of geometrical optics has to do with how great a magnification can be achieved. Said another way, using visible light, what is the best possible microscope that could be made in order to image very small objects. From what was said above, you could imagine, incorrectly, that all you need to do is have a series of magnifying glasses that keep enlarging the image until it is as large as you want it to be. Unfortunately, the wave aspects of light begin to complicate matters. Diffraction effects really limit the overall magnification possible with visible light.
Suppose we have two point sources of light that lie very close to one another. We can ask the question 'how close together can they be and, when magnified as much as possible, be distinguished as two neighboring point sources of light rather than one blurred together source?' This question has to do with the resolution achievable with visible light. Any optical system that tries to magnify these objects will have some sort of aperture - whether it be the size of the lenses used or some pinhole aperture used. The problem is that light from each of the point sources will be diffracted at the aperture and spread out so that the image will be blurred. As the two point sources are made closer, the blurring gets worse, until the two images overlap and you would not be able to resolve the two point sources of light. It turns out that the minimum separation of these point sources is roughly half the wavelength of light, and this represents the best possible resolution achievable. For visible light this is about 200 nm. This is roughly a factor of 2000 better than our eyes can see (our eyes can resolve about 1 cm at a distance of 20 m amounting to a closest separation distance of about 0.15 mm when the objects are very close to us). So, you see, the best optical microscope need only have an overall magnification of several thousand since beyond that the images will be blurred by the diffraction effects of light. To improve on this limit, the electron microscope was developed where the wavelength of the electron (think about this!) can be made much shorter and so the resolution is much greater, meaning that two point sources of light can be fractions of a nanometer apart and imaged. Individual atoms can then be seen.
We happen to be in a position now to answer an interesting question about the eye. When two point objects are as close together as physically resolvable by our eyes and we image them on our retina, how close together are the two images? If you take the minimum separation to be 0.15 mm and calculate the separation of the two images on the retina using the picture below, the images are about 15 mm apart. It turns out that the cone cells of our eye, located at the macula - the most sensitive region of our eye - each with its own connection to a single nerve cell coming out of the retina to the optic nerve, are separated by about 15 mm! What does this mean? Suppose our cone cells were spaced much more closely together. Would our eyes have better resolution? The answer is no - diffraction limits our ultimate vision and in that case the image would be blurred over a number of different cones. On the other hand if our cones had greater spacing then our resolution would be limited by the separation of the cone cells rather than by diffraction. So, it appears that our vision is as good as is allowed by physics - and our retina is not over-designed with more cone cells than needed.
Since the distance from the eye lens to the real objects is about ten times larger than the distance to the retina, the image on the retina is about ten times smaller - so a 0.15 mm or 150 mm separation of the objects leads to a 15 mm separation of the images.
