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We are going to start drawing some ray diagrams, this you may recall from earlier courses, it may be review for you guys
but knowing what we've carefully considered up until now
we know that if we only consider paraxial rays
point object forms a point image, and that all these different rays go to the same point
if we know where two such ways intersects to form an intersection
we can assume that all the rays that emanates out would go through that same point and form an image there
so that's why ray diagrams are a quick shorthand to quickly figure out
where the image is
of course, if we need more quantitative results, we can always fall back to the thin lens formula
so how you draw a ray diagram for a single lens, you first mark out where
the focal point of the lens is, probably on both sides as it could be useful
and hopefully you know where the object is
we always then to draw the object
coming from a point a little off the axis, because that'll help us determine also the
size of resulting image and whether it is inverted or erect
the continual approximation here is of course that
of the extent of this height
call this y of the object is still small
because we are only concerned with paraxial rays
and that typically works quite well
and if this is a small, then we can kind of ignore its existence
and we can assume that all the derivation we have done so far
up to this point still applies
now there are a few rays that's easy to draw
so that's what we will focus on drawing, so ray 1
comes in at parallel
goes through the center of the lens
because we are not considering the two interfaces separately, we are just considering the lens as a whole
characterized by the focal length, and as you know, by definition
a ray that's coming in parallel must come out and go through the focal length
so that's parallel
and then this goes through the focal point
conversely, if you have a point that goes through the focal point
it has to come out parallel
so that's two rays that are easy to draw. Another ray that is probably easier to draw, and we often draw this one
instead of one of the other two
is this point that goes right through the middle
and the assumption here is because we have a very thin lens
near the middle, both these points are flat
and it's so thin
that it does not deflect by very much even with Snell's law, that it just goes through undeflected
so if you go through the middle
then you have no deflection at all
and you notice that if you draw them properly, all three of these goes through the same point
as well as any other one, but we don't know how to draw those exactly. These three lines are the line that to count on to get you
where you meet. You only need two of the three though, of course
and then this image as you can see
is formed over here and is upside down
and you have a different size
and so we can talk about
various things, you can talk about
this thing called magnification
so magnification we define to be the size of the
image vs. the size of the objects as a ratio
so we can be magnifying 2 times or half a time
we can of course have negative magnification, in which case
which is the case here, where the image gets inverted
but more directly and to the point, because we call this
s_o and we call this s_i
we can also redefine
knowing that these are similar triangles, we can redefined this ratio to be negative s_i/s_o
negative here because, as you notice, with a negative y_i, we have a positive s_i, so that's where the negative comes in
so when you have a negative magnification then you have an inverted image
and then when you have a positive, you have an erect image or upright
another issue that you may run into is real vs. virtual images, and I will draw another
ray diagram to show you the example of a virtual image
so same type of lens
we have the two focus
and if you place the object really close to the lens
we get one ray goes through the focus, one ray goes straight, look, they don't meet on
this side over here, you have to extrapolate backwards and you form an image way back here
so there is your i and your o to make the distinction
and so, first of all, you have a negative image distance
but more importantly, you have what's known as a virtual image because you have to extrapolate the lines backwards
for them to join to make a point, if you have to do that for one or more of the rays
then your image is virtual, there is no actual
physical point in space where all the rays cross
you have a virtual image when you have to extrapolate back
by contrast, if you have a physical point where the rays goes through, that's called a real image
so you can actually put a pinhole there and all the rays goes right through it
just a couple more iterations for you
so you can see how these gets drawn
for a diverging lens
you still have your f, but of course, these are negative f's, so you have
a thing here
it actually goes out
from the focus like that, and the other way of course goes down here, and the criss-cross happens here which is still a virtual image
but it's a virtual upright image in this case. It is virtual because one of the line you still have to extrapolate backwards
by contrast if I put it much closer
same f
I still get a virtual image over here
so basically with these ray diagrams, you can figure out whether you get a virtual or real image
or a upright erect or inverted image and roughly where it is
now you may a summarizing table that talks about if you have this type of lens
if you have certain distance with relation to the focal length and you will get certain types of image
you can memorize all that, or you can just learn how to draw these diagram and it'll quickly pop out at you
what type you have instead of having to memorize this tricky table
because for the converging lens, there is a couple entry that has to do with twice the focal length
so the division is not as straight forward as you would think
so I would definitely suggest learning how to draw these ray diagrams quickly as rough sketches
of course if you draw them to scale, you can get quantitative results from it, but it is probably easier to do the quantitative bit
with the thin lens formula, of course, while being careful with the signs
so let's show you a couple examples where we actually solve some problems