Module 5:  Focal length of a lens
PAGE References to Optical Formulas Tutorial: (first reference is to edition 1 / second reference is to edition 2)

Focal length of a lens

Almost everything that we are going to do this week we are going to do in meters. Remember them? A little longer than a yard? 39.37 inches, in fact. Lets just use 40 inches as an approximation. So, we agree, one meter is about 40 inches.

Well, we also need to be able to convert meters to centimeters (cm) and millimeters (mm). Remember them? 100 cm in a meter, 1,000 mm in a meter? If I have 35 mm, how much of a meter do I have? What is 540 cm in m? In mm? How many cm are there in 0.333 m?

OK, 35 mm is 0.035 m.
And 540 cm = 5.40 m.
540 cm is also 5,400 mm.
There are 33.3 cm in 0.333 m.

You need to be able to do this. If you did not get these answers, go to the math lesson and review it. You only need the metric-to-metric conversions right now. There is also a review of this in the first section of the text book, page (5 / 8-9).
 
 

So now we have talked about what the ray of light does when it is travelling through air and suddenly encounters a lens surface.  It changes speed and direction.  Then, when it exits the lens again it changes back to its original speed, but usually not its original direction.

What direction it will be travelling in will depend on the relationship of the two sides of the lens to each other and to the travel of the ray in the first place.  Lets take a whole 'bunch' of rays, all travelling parallel to each other.

Oops, I guess we have to back up again a bit.  Light always comes from somewhere.  Back in the first lesson we discussed the fact that light rays diverge from their source.  How can we have rays that are parallel to each other?  Do you remember?

They have to be coming from a long distance away.  They are still diverging, but so little that we can ignore what little divergence is left.  If they were coming from an infinite distance away they would be exactly parallel.  Imagine two rays coming from the sun.  By the time they get to your eye, you can imagine that they are essentially parallel, can't you?

Well, we define light rays that are coming from 20 feet away as being parallel to each other.  We will call 20 ft optical infinity.

Lets see. 20 feet is 20x12 = 240 inches. 240 / 40 = 6 meters, since I just 5 minutes ago said that we would use 40 inches as an approximation for one meter. So 6 meters is also optical infinity. OK?

That is a test question. What is 20 feet away called? What is optical infinity in meters? Got that written down on your 'to memorize' flash cards?

So, here is a lens with one flat side just to make drawing it easy for me. The ray that I marked as #1 is going through a place on the lens where the two sides are parallel to each other, and it is going through perpendicular to the two sides, so we have learned that the ray slows down and then speeds up again, but it does not change direction. We are going to call this ray the optical axis of the lens. We are going to call the place where the sides are parallel to each other the optical center of the lens.

("Which point, exactly, is the optical center" asks one of the members of the class.

"Well," I respond, "it is sort of on the axis, somewhere."

"No, I mean, is it on the front surface or the back surface?" the questioner continues. 

"Well, neither, exactly, usually. In this case it is on the flat side, but this is an unusual case." 

"Why? Where is it the rest of the time? Inside the lens?" 

"Well, not really, most of the time. We call it at this point where the sides are parallel, but it is not really there." 

Most of you are now looking a bit confused, and you would rather we just went on with the lecture.

"If you really want to know where the optical center REALLY is, turn to Section VII in the Optical Formulas Tutorial. We do this in the Geometrical Optics course" I really put it in the text book to give the one person in every class who asks the question something to look at.

 

Back to the drawing. Look at ray #2. It is parallel to the axis, and not very far away from it. The sides of the lens are tilted a little toward each other, and ray #2 comes out of the lens heading toward ray #1. They were parallel before they went through the lens. Now they are converging. We called converging 'positive vergence', didn't we? So the lens has added positive vergence to the rays.
 
 

Ray #3 is a little farther from ray #1, but still parallel to it. The sides are tilted a little more for ray #3 than they were for ray #2, so ray #3 bends more. The same happens for ray #4. Each ray is refracted more than the one closer to the axis. They all refract just enough to cross the axis at the same point. Then, they start diverging as if they had COME from the point where they crossed. If I had a little tiny light source at that exact point and facing to the right, the rays from that tiny light would be doing EXACTLY the same thing that these formerly parallel rays are doing!
 
 

We call that point where the rays all cross the focal point of the lens.

"That is not what you said in the book." Now someone in the class wants to make sure that I know that he read the section BEFORE the lecture this time.

"What did I say in the book? I wrote it so long ago that I do not remember." I'm hoping to turn a few of the groans that I just heard into smiles. It rarely works.

"You said that it is really the secondary focal point."

"Well, it is. By definition, the secondary focal point is where rays that are parallel when they enter the lens cross or appear to have crossed." 

"But you just leave it there in the book. If there is a secondary focal point, then there must be another one." 

The primary focal point would be the point on the axis where I might place an infinitely small light so that when the rays have passed through the lens they will come out parallel." 

(See, I could do that without threatening to make him read pages in Section VII again.)

We call the distance that the focal point is from the lens the focal length of the lens.

Yup, up just went his hand again. (By the way, in last years class it was a woman. No, I am not being sexist.)

"What point on the lens do you measure from to get the focal length? Is it the same distance as the other focal length, the one that you would get from the primary focal point? Do you measure them from the same point?"

"Well, we won't answer the first one yet, the second answer is sometimes, and no. Except sometimes." 

Shall I threaten him with Section VII again?

"Actually, we will get into some of this in module 11. Is it OK to postpone the rest of the answer until then?"

So, we now know what the focal point is, and the focal length is how far the focal point is from the lens. Remember starting a fire out in the sun with a magnifying lens? You positioned the lens over the piece of paper until you had the smallest, brightest spot that you could get focused on the paper, and it would start to smoke and curl, and if you had it perfect and the sun was nice and strong, the paper caught on fine. Well, the distance that the lens was from the paper at that moment was the focal length of the magnifier. Sort of.

If you had used your meter stick, (not your yardstick) to measure the distance that the magnifier was from the paper, you could have found out what the power of the plus lens was without using that instrument in the lab that we all use for such purposes. See what lengths opticians had to go to before we had these nice instruments?

So here I am, out in the sun, and I am holding the lens one meter from the paper, and that is when I get the smallest, roundest, brightest image of the sun on the paper. By definition, a one Diopter lens is a lens that focuses parallel light rays at one meter from the lens.

We want the way we define the power of the lens to be a high number if the rays are bent a lot, and a low number if they are only bent a little. So, if a one diopter lens brings parallel rays to a point at one meter, the lens that has a focal length of two meters bends the rays LESS and the lens that brings the rays to a point at 1/2 meter bends the rays MORE. So, a diopter is defined as 1/focal length in meters. The focal length of 1 meter gives 1/1=1 diopter, a focal length of 2 meters gives 1/2=0.5 diopters, and a focal length of 1/2 meter gives a focal length of 1/0.5 = 2 diopters.

What is the power of a lens that has a focal length of 0.25 m? How about 3 m?

What is the focal length of a 5D (read "5 diopter") lens?

This is another of that type of problem where you divide by what you know, and you have what you do not know. If you know the diopters you punch "1" "" diopters "=" into your calculator and out will come the focal length in meters. If you know the focal length in meters you punch "1" "" focal length "=" into your calculator and out will come the dioptric value of the lens.

So, for the questions above, a lens that has a focal length of 0.25 m has a power of 4 D.
A lens that has a focal length of 3 m has a power of 0.33D.
A lens with a power of 5D will have a focal length of 0.20 m.

So much for lenses that look like the one we started with this week.

Recap:

Now for lenses that add negative vergence to the incident rays.


 
 
 

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