Basic Lenses

Lenses, like the glasses I wear to see, are built on controlled refraction. They allow us to purposefully change the direction of light rays and as result magnify, shrink, or otherwise alter the size of objects in the world.

Lens Types and Properties

Overall there are two major types of lenses, converging and diverging. Their names sum up what they do with converging lenses focusing light into one spot where, well, all the rays converge. While diverging lenses spread the incoming light rays apart by causing them to diverge. In order to do this each lens has a different shape.

Converging lenses are convex. This shape causes all horizontally oriented light to refract and converge into a single spot called the focal point.

While diverging lenses are concave. This shape causes all incoming light to refract outwards and appear as though it originally emanated from a single spot also called the focal point.

Another property of lenses is their radius of curvature (RoC). RoC always threw me for a bit of a loop but the easiest way to understand what is going on here is by visualizing RoC itself. To do this we will create a circle that matches the curve of the lens in question. As we can see as the lenses curve less and less aggressively their radius of curvature increases massively.

The yellow dot represents the focal point and as seen here the more aggressive the curve the closer the focal point gets to the lens. Specifically, the RoC is two times the size of the focal length (f).

[latexpage]
\[
r=2f
\]
or
\[
f=\frac{r}{2}
\]

Who Cares About Bending Light Anyway

Great, lenses have all these shapes and properties but why do we care? Lenses are all around us and make a ton of awesome things possible. From the lens in my eye to the microscopes, I used during undergraduate research, lenses allow us to get a picture of the world around us.

Lenses accomplish this by forming images or at least giving the illusion of an image. Loosely defined an image is where all of the light from one part of an object converges or would appear to converge. To see how this works let’s think about our eye then more abstractly about the two types of lenses.

In order for us to see anything light must strike the retina (yellow portion) of our eye. However, in order for images to be clear the object, whatever it happens to be, needs to form exactly at the retina. To determine where an object will form we will use a ray drawing or diagram. This tends to be the most confusing subject in all of optics. So long as you remember the following three rules it should be pretty straightforward.

Drawing Ray Diagrams

Rule #1 Always draw your ray from the top of the object.

Rule #2 If a light ray comes in horizontally it always goes through the focal point in a converging lens or appears to come from the focal point in a diverging lens.

Rule #3 If a light ray hits the center of a lens it always passes through straight (or unrefracted).

Where both rays meet the image forms. Specifically the top of the image. The rules are the same with diverging lenses but the outcome is drastically different. We will walk through each of the two types of lenses by themselves to see how this happens.

Notations First

Before we jump straight into those ray diagrams let’s create some shared language for discussing lenses in more depth.

We have already seen the term focal length (f) or the distance from the lens to the focal point come up.

Focal Length defines the strength or the power of a lens (P). Power is measured in diopters and calculated using the equations below:

[latexpage]

\[
P_{lens} = \frac{1}{f}
\]
since

\[P_{lens}\propto\frac{1}{f}
\]

as the focal length decreases the power or strength of the lens will increase.

The object distance (o) is the distance from the object to the lens.

While the image distance (i) spans from the lens edge to where the image is formed.

Alright all together now.

Converging Lenses

Now that we have some shared language around lenses let’s dive into a deeper view of converging lenses. In the above picture we saw the most common scenario for a converging lens, an object positioned outside of the focal length.

Outside of the focal length

Here we can see that the light rays pass through the lens and converge on the other side where they eventually cross forming an image. This image is called a real image because the light rays actually converge at this point. Real images will always be upside down, which is why our brains have to “flip” them around in order for our world to be right side up.

At the focal length

The picture changes when we place our bat at the focal length. Here no image will be formed because the rays that pass through the lens end up coming out parallel to one another destined never to cross.

Within the focal length

Lastly, we can place our object within the focal length right next to the lens. Something strange happens here, the rays actually diverge. So much for being a converging lens. If we trace the rays back to where they would cross we form what is called a virtual image. Virtual because the light rays don’t actually cross they only appear to originate from this point.

At this crossing point, we have a HUGE bat image meaning the image was magnified. Magnification can go both ways though, some images will be bigger and some will be smaller. The key determining factor here is the distance from the lens. The further away the image gets from the lens the bigger it is.

Quantifying The Lens

But how do we know how far away the image will form? The answer is through yet another physics equation. Specifically, the thin lens formula which allows us to determine the focal length, object distance, or image distance so long as you have the other two variables.

[latexpage]
\[
\frac{1}{f}=\frac{1}{o}+\frac{1}{i}
\]

It is a pretty straightforward equation except for the sign conventions. Oh, the sign conventions. Basically, each portion of the thin lens equation can be positive or negative. Focal length is the easiest to keep straight where a converging lens will be + while a diverging lens will be -.

The image and the object sign conventions are a little harder to keep straight, but overall we really only need to care about image conventions. As the object conventions hardly if ever come into play. For this, the distinction is between virtual and real images. Where virtual images will have a – sign while real images have a + sign.

Additionally, we can quantify the degree of magnification (M). There are a couple of equations here, but if we remember that height is proportional to distance we should be able to memorize one and figure out the other. Here h stands for height with the subscripts standing in for the image (i) or object (o).

[latexpage]
\[
M = -\frac{i}{o}\: or \; M=-\frac{h_i}{h_o}
\]

Again magnification also has a sign convention where the – sign means that the image is inverted while a + sign means an image is upright.

In summary

Lens Type	Image Type	Magnification
Converging (+f)	Real (+f)	Upright (+f)
Diverging (-f)	Virtual (-f)	Inverted (-f)

Diverging Lenses

Up last is the diverging lens, which only ever does one thing regardless of where you place the object. It shrinks an object and forms an upright virtual image. All the same equations and sign conventions apply.