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Here we have one such situation where we have a piece of lens
it's got a spherical section on the one side and a different spherical section on the other
the object emits light rays coming out
it gets bent through the first interface and gets bent again through the second interface
into our image point back here
to analyze this, we just have to break it down into the individual
into two individually surfaces and apply our formula in a successive manner
basically, we draw there's two different spheres here, we have the first sphere
for the first interface and a difference sphere for the second interface
Naming a bunch of things, the first sphere is centered here and the second sphere is centered here
we extrapolate these lines inside the medium backwards to find out where the first image point is
which then serves as the object for our next interface
naming a few more things, the lens itself is as thick as the distance d, we still got the image distance 1
giving us the image distance 1 which serves as the object of the second interface, giving us the second image distance
so breaking this down we have one lens that's got a big sphere on it and we enter into
glass, we have n_l and n_m on the outside
and we have our object and image distances. In our case, ...
...
and the image distance back here to the left of the lens is going to be negative
and n_1 of the the media we are entering from
is going to be the medium
which is often air and then we have the lens for n_2
on the other side, we are exiting the glass
also another spherical section, in this case
the sphere is now curved the other way
we are not so much looking at what is the glass and what is the medium
it's where the light ray originates from and goes to
so we're going towards an interface where the center of the circle is on the left, so the radius is less than zero
object distance is greater than 0
images distance is to the rights, so it is also greater than 0 and in this case the
n_1 is going to be the lens because that's where we're coming from
and n_2 is the medium because that's where we are going to
so putting it all back, we can start to analyze the system with the formula we just had
let's sub in everything for the first interface. For the first interface, we have
...
...
...
then we also have the second interface
in a very similar manner
n_lens and n_medium swaps around
looking back at the diagram itself, we can see that
...
...
...
and because s_i1 is negative, to make it positive
we have to introduce a negative
...
so we can sub this in back here
this gives us
...
...
...
...
and I will swap them around
and this is equation 3, if we take equation 1 plus equation 3, we end up with
...
...
...
...
...
...
...
...
then we have another lovely approximation, it's called the "thin lens approximation"
where d is very close to zero, or strictly speaking d is much less then the
both the object distance and the image distance as well as the radii
if that is the case then we snub that away and those two terms go bye bye
what we end up with is quite a bit simpler now
...
...
...
...
keeping in mind that in our case R_2 is negative, dividing the n_m over, we have
...
...
and because remembering we have
our d is basically 0
we can call s_o1 to be just the s_o
and then s_i2 is just the s_i, so we can
destroy these subscripts and we end up with
this lovely overall result which we call "the lens maker's formula"
it's useful in the sense that, given the radii of our two spherical sections and the type of material we are working with
we can then work out what is the
image distance for any given object distance. Furthermore, we can charaterize the
lens using what's known as the "focal length", because you see everything on the right hand side here
doesn't really depend on s_o or s_i
all this is just dependent on the characteristic of the lens, so we can use this to define constant to encapsulate all of these
different variables and just use this thing called the focal length
so the focal length is the distance to the focal point
and how the focal points is defined is what happens when you send a bunch of parallel rays
through a lens, where do they come together
to form that image, so you start parallel and come to this point called the focal point
and this length is the focal length
similarly, you can also have
an object start at a particular distance to form these
parallel light beams coming out, that's also the focal point and
for thin lenses, it is perfectly symmetric
so in this case, we use our
the one side of our lens maker's formula we see that
...
...
it's going to be 1/infinity because the object distance of parallel rays is
infinity, so 1/infinity gives you 0, plus 1/f
similarly, we can look at this case
...
so for both cases, it's the same thing
and because that's constant
for a given lens, we can characterize the lens by its focal length ...
and this is
another good famous "Gaussian lens formula"
it allows us for, given the focal length of the lens, we can work out for any particular object distance where the image is going to form
or given basically two of those three things, you can figure out the third one
once again, there's a certain sign convention that we have to be careful about
basically for lenses, the thinking is
you expect to have an object going through a lens and then forming an image of the other side
so this is the case where everything is positive
so therefore if I have
a lens like this
s_o will be greater than 0 on this side
s_o will be negative on that side because we expect the object to come before the lens
we are always going that way by convention
the image also plays a role the focus so
if the image ends up on this side, less than 0, if the image ends up on the right side
then it is greater than 0, because we expect the image to be after the lens
same thing for the focus
we think of the focus has the image of infinite beam, so follows the same convention
couple other words for you, in terms of describing these lenses, each interface can be
surely we've heard these before:
"concave", where the arc caves in
and then "convex" where you have the arc pops out
now you can have any combinations of course
you can have concave-convex lens, or concave-concave lens, or you can have
a convex-planar lens
so concave and convex just describes the looks, but what's more important is we have words like these
we have a "converging lens"
that's a more functional description, if you have a converging lens
the light rays comes together and you have f > 0
and then you can also have a diverging lens
which then the light rays go away and then your focus is actually extrapolated backwards back here to have a negative focus
we usually draw the converging as convex- convex, and the diverging as concave- concave, but that is definitely not
absolutely necessary, you can have
diverging lenses that are
concave and convex by different amounts of each side and typically that's what our eyeglasses look like
you don't see eyeglasses that caves in on both sides, they are just curved outwards by different amounts of the two sides
so next let's put this all together to see how we can predict where images of form and
what kind of images are formed using our ray diagrams