- Double slits are cool because they show
definitively that light can have
wave like interference patterns
and if you shine a green laser through here
what you'd see on the screen
would be something like this.
You'd have these bright spots
but they'd kind of blend in to dark spots
which blend into bright spots
which is why when we draw a graphical
representation of this, it kind of
looks like this, where these spots
are blending into each other
which is cool but it also kind of sucks
because if you were to go actually
try to do this experiment, you'd want to
measure some angles and that means
that you'd have to measure some distances.
I'd measure distance from the screen to the wall.
That would give me this side of the triangle
and then I would also probably want to measure
distance between two of these
bright spots because that's what I can see
but because they're smudgy, it's like
is that the center?
Is this the center?
Sometimes the lights not so strong
and it's hard to tell and what's worse
is these kind of die off
and so there's another problem.
These die off pretty quick.
Sometimes you're lucky to even see
the 5th or 6th bright spot down the line.
So, my question is, is there a better way?
Is there a way to make these spots
more defined and so you can see more of them
so they're brighter and the answer
is definitively yes and we figured out
how to do it and the way you do it,
is you just make more holes.
So you come over to here and if these
are spaced to distance D, I'm just going to
make another hole at distance D
and then I'm going to make another hole
at distance D and then I'm going to make
another hole at distance D.
I'm going to make 1,000's of these holes
extremely close together as long as they're
all at distance D apart,
something magical happens.
So if these are all D apart, what happens is
on the wall over here, instead of getting
this smudgy pattern, you'll get, you really will
just get, a dot right there and then darkness
and then another dot and then darkness
and another dot and you'll see this
continue out much further than you could previously.
Why?
Well let's talk about why.
So let's talk about this.
How can you see this pattern
over here like this?
So, the 1st wave from this 1st hole.
Let's imagine this 1st wave
from this 1st hole, it's going to travel
a certain distance to the wall.
Let's say we look at a point over here
where it is constructive.
Let's say we just had these two holes
to start off with right?
Two holes, double slit, disregard
all this stuff for a minute.
Two waves coming in from two holes
get over to here.
Let's say this is a bright spot
but say it's the bright spot
that corresponds to, delta x equals one wavelength.
In other words, this would be
the constructive point where the 2nd wave
from the 2nd hole travels one wavelength further
than the wave from the 1st hole
and again what that means is if I were
to carefully draw a line from here
at a right angle right there, that means
that this wave, from the 2nd hole,
this is the extra part so that would be
one extra wavelength and because this
2nd wave is traveling one extra wavelength,
it's going to be constructive
because if I draw my wave, they're going to
match up perfectly there so if I draw my wave,
let's say the wave from the 1st hole
happened to be at this
particular point on it's cycle.
It doesn't have to be but let's
just say it was there.
The 1st wave hit that point
at this point in it's cycle.
Well, the 2nd wave, since it's traveling
one wavelength further, is going to hit
at this point in it's cycle so it would be here.
Now these are both hitting there
at the same point so the 1st wave
gets there hitting right here.
The 2nd wave get there hitting right here.
These overlap because these are
two different waves overlapping at this point
it's going to be constructive because
if a peak matches a peak, constructive.
If a valley matches a valley, constructive.
How about the 3rd hole?
Here's where it gets interesting.
This 3rd hole, the wave from the 3rd hole
is going to have to travel this far to get there.
Well, let's see.
From the 2nd, how much further does it travel
compared to the 2nd hole?
I'm going to do the same game
I did just a minute ago.
It travels this much further which again,
since these are the same angle,
this is going to be the same distance here.
This is a also going to be one wavelength.
Remember we derived at this.
That d sin theta, is the path length difference
and the theta is the same for all of them
and so I could just look at these two
consider these two as the double slit.
This one travels one wavelength
further than the 2nd.
How much further does it travel than the 1st?
Well I'm going to continue that line down.
It's just going to travel two wavelengths
farther than the 1st.
So the wave coming out of the 3rd hole
travels two wavelengths farther
than the 1st hole.
The 2nd and the 3rd are going to be
constructive because they're one wavelength apart
and the 3rd and the 1st are going to be
constructive because they're two wavelengths apart.
That's okay, two wavelengths doesn't matter.
Look at, if the 1st wavelength hits here,
the wave from the 2nd one hits here,
the wave from the 3rd hole travels
two extra wavelengths.
When that 3rd wave gets to this point
it will be at this point on it's cycle
there all going to by overlapping
at the same point on their cycle,
it's going to be constructive.
You can keep doing this.
You can come down to this hole
and it will also be constructive interference.
In other words, this 4th hole travels
one wavelength further than the 3rd,
two wavelengths further than the 2nd hole,
three wavelengths further than the 1st hole
but there still all going to overlap perfectly.
You will get an extremely bright spot here
because you have even more light overlapping
and it's all perfectly constructive bright spot.
Now here's where it gets strange
so you've got to be careful.
This is the part of the explanation
I hated as a student.
I thought this made no sense whatsoever
so pay close attention at this point.
Here's the weird thing.
If you deviate slightly from this constructive point
if I go up just a little bit over here
to some point right here.
Let's just see what happens.
This wave from the 1st hole would travel
that far to get there okay and the wave
from the 2nd hole travels that far to get there.
The path length difference isn't
going to be one wavelength.
Let's say the path length difference
happens to be 1.1 wavelengths.
So let's say the wave from the 2nd hole
happened to be traveling not one wavelength
further anymore, it's not traveling
one wavelength further, it's going to be
traveling 1 and .1 wavelengths further.
So if the 1st wave happened to hit
at this point, the 2nd wave would be hitting
at not exactly one wavelength but 1.1.
Now if I keep drawing them over here
I'm going to run out of room.
So since they're all the same points
on the cycle, the cycle's
the same over and over.
One wavelength .1, I'm just going
to draw that right here.
So the 2nd wave would hit right there.
Those would overlap you know,
that's partially constructive.
I mean, just looking at these, you'd think
you'd get a bright spot but if you keep going
let's see what happens.
The wave from the 3rd hole
also travels to get there.
How much farther does it travel?
Well, it travels 1.1 wavelengths further
than the 2nd hole but it travels 1.1
plus 1.1 wavelengths further than
the 1st hole, so it travels 2.2 wavelengths
further than the 1st hole.
So to make this clear, let me just be clear here.
So if I do the same trick right?
I draw this down to a right angle.
It's a little farther so I have to draw
the line a little further out.
1st wave travels one wavelength
and .1 wavelengths further.
How about this one here?
Well, the wave from the 3rd hole
travels one wavelength .1,
1.1 wavelengths further than the wave
from the 2nd hole but it's going to travel
1.1 plus 1.1 wavelengths further than the 1st hole.
So it's going to be 2.2.
This wave from the 3rd hole is going to travel
2.2 wavelengths further than the wave
from the 1st hole.
Where would that be?
So, one, two wavelengths .2 would be
even further down the line here.
Not completely at the bottom but
further down the line.
So it would be, at this point over here
and you can keep doing this and let's
just see what happens.
Let's just do a couple more.
This wavelength has to travel this far.
Now you can probably see the pattern.
This wave through the 4th hole has to travel
1.1 wavelengths further than the wave from the 3rd.
It travels 2.2 wavelengths further
than the wave from the 2nd.
It travels 3.3 wavelengths further
than the wave from the 1st.
So if I compare this wavelength
where this wave is on it's cycle
compared to the 1st, then I'm going
to be 3.3 wavelengths.
So if I go one, two, three and then
.3 would be down here somewhere.
I'd get one that's at .4, I'd get one
that at .5, I'd get one that's at .6,
one at .7, one at .8, one at .9,
one at, well .10, which is back to
a whole wavelength again,
a whole wavelengths difference.
What is this going to be?
If all of these waves are overlapping
like this at this point that's only
slightly deviated from this other point.
What am I going to see there?
Well, look at you could pair these off.
This point and this point are going
to interfere completely destructively.
One's at a maximum, one's at a minimum,
one's at the peak, one's at the trough,
so you get zero and you can keep pairing these off.
This one here and that one there
completely annihilate each other, completely destructive.
You keep doing this, this one here and this one there
completely destructive and you can
keep finding points that completely
destruct each other and that means
that you're going to get no intensity
at all, even slightly off from
this magical point, this magical
integer wavelength point.
What that means is instead of getting a blur,
instead of getting this, instead of
getting this smudgy pattern.
You're going to get, at the bright spots,
a dot and then another dot at
the bright spot and in between
these bright spots, you will get darkness.
Which is great, it's great because
it's easier to measure so that's one good thing
and it relies on this fact that
if you've got multiple holes,
100's even 1,000's of holes,
if you go off even slightly
because this isn't going to always
match up perfectly, you keep going down the line
one of these are going to interfere
constructively with another and you can keep
pairing these off and then the 2nd
would interfere destructively with
a different one and the 3rd with
a different one and the 4th with
a different one and they're always
going to match up so you can destroy them all
and you get destructive point in between.
That's why it's dark.
Now at first I didn't like that argument.
I had to go back over that and make sure
that made sense to me.
If it doesn't make sense to you
go through, you've got to hammer it out,
try it out, draw it out.
I've tried my best to explain it here
but I have to admit, it's a difficult
one to comprehend but that's the idea.
So this is great actually.
These multiple holes here are giving us
points that are points that are
spaced out and clearly delineated.
Now I can clearly see, okay if I wanted
to measure this distance I don't have to guess.
There's no guesswork, one dot's here
one dot's there, bright, bright, no smudginess.
That's one great thing about this.
The other great thing is because I've got more holes,
this brightness lasts longer.
These bright spots will keep going further,
I can see this travel further
down the line then I would with the smudginess
because I've got multiple holes
interacting and I'll have more
intense dots, it's brighter.
The dots are more delineated
and they're typically brighter,
they're more intense.
So we give this a special name
because this is so useful, these multiple holes
instead of a double slit,
these multiple holes are more useful.
We call this a diffraction grading.
So this is a diffraction grading
and it's more useful than a double slit
in many ways because it gives you
clearly delineated dots and it let's
you see them more clearly.
How many holes are there in a diffraction grading?
Well typically, these are rated
in lines per centimeter.
In other words, if you took a diffraction grading
and you asked, how many holes are there?
The lines or the holes.
How many holes are there per centimeter?
Well, the lines could be the blocked parts
and the holes could be the parts
where there's no blocked part
but regardless, there will be as many
lines as there are holes.
How many holes are there per centimeter?
There's typically 1,000's of lines
per centimeter in a diffraction grading.
That's how small these are spaced together.
You might worry though, isn't the math
going to be complicated here?
We've got all these holes here.
It turns out no.
This is the best part.
By far the best part.
This relationship still holds.
All that math we did, still fine
and because these all line up
for the good points, the magic points.
These points where they all overlap
in such a way that they're one wavelength further.
Those line up perfectly so that
no matter how many holes you go down the line,
they're all perfectly constructively interfering
and so you get this equation.
It's the same equation that we had before
where d sin theta equals m lambda
gives you the constructive points
for a diffraction grading interference
pattern on the wall.