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- [Voiceover] Let's talk about
Single Slit Interference.

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Now if I were you, I'd already be upset

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and a little mad.

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Single Slit Interference?

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Interference?

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Wave Interference is by definition

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multiple waves overlapping
at a single point.

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So how could a Single Slit ever produce

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multiple waves that could overlap?

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I mean, when we had a double slit--

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if I put a barrier in here--

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and we have a double slit.

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At least then--

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okay, I send in my wave.

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It gets over to here.

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There's a small hole.

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We know what waves do at a
small hole, they diffract.

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Which is to say, they spread out.

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At least with a double slit,

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you would have two waves spreading out.

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Now they can overlap.

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Interference.

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But for a single slit,

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how are we ever going to get this?

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Well, I never really told you,

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why do the waves spread out at a hole?

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Why does diffraction happen at all?

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Why, when waves encounter a hole,

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do they spread out?

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And the answer to this question is the key

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to Single Slit Interference.

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And the answer to why
they spread out at a hole

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is something called Huygen's Principle.

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And I can't say it.

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This is a Dutch physicist, scientist,

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who figured this out.

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Huygen's Principle.

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And I apologize right now to
all the Dutch people out there,

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I'm butchering this name.

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Huygen's Principle, easier
to spell than to say.

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What he said, he figured
out something ingenious.

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He figured out this.

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If you've got a wave coming in,

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these wave fronts.

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Remember these wave fronts are like peaks.

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And in between them are
the troughs or the valleys.

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If you've got a wave front coming in,

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propagating this way.

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You can say, "Yeah, that wave front

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moves from here to there."

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That's what it does.

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Or, he realized, with a wave,

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you can treat every point on this wave

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as a source of another wave
that spreads out spherically.

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If, in the forwards direction,

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this wave spreads out spherically.

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This point here.

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He said that, a wave front you can
think of as an infinite source of waves.

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Each point is the source of another wave.

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And you're thinking, this
is horribly complicated.

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What kind of mess is
this going to give you?

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Well, if you add this up, these are going

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to interfere with each other,

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constructive, destructive,

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in a way that just gives you

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this same wave front right back.

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This is crazy, but true.

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If you let every point on this
wave be another wave source,

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it will just add up to
another wave front here.

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You will just get the same thing back.

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And this is the key to understanding

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why diffraction happens.

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It's because the wave was
already diffracting, so to speak.

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It was already doing diffraction.

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Every point on here was doing diffraction.

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It's just, it always added up
with the other waves around it

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and every other point and
gave you the same wave back.

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But when there's a barrier,

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when there's something in the way,

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these here can't re-join
up with their buddies.

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You just get this one here spreading out.

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And then this one down here spreads out.

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All the rest of these get blocked.

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Now that these are blocked,

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they're not going to get to interfere

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constructively and destructively
with these points here.

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And so what do you see
when it hits the hole?

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You just see this thing spreading out.

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So, it was always
diffracting, so to speak.

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We just didn't notice it
because it always added up.

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When you've got a hole or a barrier,

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that's when we actually notice it.

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And this is the key to Single
Slit Interference, because

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if I get rid of all of that,

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if we imagine our wave
coming in here like this.

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Well, this wave's going to hit here.

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Every point's the source of another wave.

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So this point's going
to start spreading out.

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this point's going to start spreading out.

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When we have a Single Slit,

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we really have infinitely
many sources of waves here.

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And since some of them are blocked,

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we could see an interference pattern

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over here on the wall,

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because these can interact
and interfere with each other.

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What interference pattern
are we going to see?

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Well, on the wall over here
we see a big ol' bright spot,

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right in the middle.

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And if I were guessing, I
would've thought that was it.

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Big ol' bright spot,

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because you're shining a
light through a small hole.

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Single hole, you would get
a big bright spot there.

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The weird thing is, this jumps back up

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goes to a minimum.

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A zero point.

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And then jumps back up,

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and then it comes back up again.

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And you get this.

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These are going to be not very pronounced.

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These aren't very pronounced.

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You get a big bright spot in the middle.

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These are relatively weak compared to

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other interference patterns
that we've looked at.

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And down here, it jumps
up a little bit again,

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over and over here.

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So this is the pattern you see.

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How can we get this?

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How do we analyze it?

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That's what we're going
to try to figure out.

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Figure that out?

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Okay, well this is a--

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I said there's infinitely
many sources here.

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With when this wave gets to here.

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That would take a long time to draw.

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I'm going to draw eight.

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So, let's say there's
one, two, eight sources.

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Let's just imagine there's eight here.

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To make this a little big
easier to think about.

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and the weird part is that
this jumps back up here.

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So let's look at this minimum right here.

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Let's look at this point
where it goes to zero.

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This destructive point.

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So the wave from this top most point,

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this wave from the top
most, upper most point,

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has to travel a certain
distance to get there.

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I'm going to also look
at the fifth one down.

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This one that's basically half way.

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How about these two?

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If these two interfere destructively,

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the argument I'm going to make is

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if these two interfere destructively,

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all the rest of them are going

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to have to interfere destructively.

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Why?

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Well, we know how to play this game.

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Let's draw out right angle line here.

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There we go.

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And so we know that,
okay, if these are going

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to interfere destructively
this is the extra path length.

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This extra path length
of this second wave,

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this lower middle wave has to travel.

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Has to be, what?

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If I want destructive over here,

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it's got to be a half wave length,

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three halves wave length,

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five halves wavelength.

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That's much it has be in
order to be destructive.

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So if this is the first point,

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let's just say that's one
half of a wave length.

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And what's the relationship
between the angle

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that this is at on the wall,
compared to the center line?

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Well, we already figured that out.

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Remember, that relationship was

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d sin theta equals the path
length difference between these.

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That we derived.

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This screen had to be very far away

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compared to the width of the hole.

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But that relationship still applies.

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What would d be in this case?

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Now we have to be pretty careful.

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We have to be careful because

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this hole has a certain width.

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We'll call that width w.

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So if this hole has a certain width w,

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how far apart are these?

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These are not w apart.

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These are w over two apart.

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And so what's the relationship here

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for the path length
distance between these two?

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Well if they're w over two apart,

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I have d sign theta as the
path length difference,

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so d would be w over two.

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Times sin of the angle that
this makes to this point

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on the wall.

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And if their path length
difference is lambda over two,

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then that would be destructive.

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So equals lambda over two.

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And this is a little weird already,

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because look, I can cancel off the two's.

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And what do I get?

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I get that w, the width of
the entire width of the slit,

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times sin of theta equals lambda.

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This is giving me destructive.

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Remember before, all of the points

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that were integer wavelengths
were giving me constructive.

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This time it's giving me a
destructive point over here.

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And the reason is we played this game

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where w is the hole width.

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These are only w over two apart.

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That two cancels with that two.

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Okay, but I didn't really
prove that this hole,

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that they should all be destructive yet.

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This is just for these two.

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We've got infinitely many more in here.

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How are we ever going to show
that if these two cancel,

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the rest of them cancel?

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Well, we'll just pair them off.

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Look at this.

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Now imagine you come down one.

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I go to this one,

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I consider this wave that
makes it over to here.

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And the next wave, down
from this other one here.

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Okay, so I move this one down a smidgen,

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I move this one down a smidgen.

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I imagine these two waves
traveling a certain distance

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to get over to this point.

227
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What relationship holds between these two?

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I can do the same thing.

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These are also w over two apart.

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So this here, is also w over two.

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So I'd get the same relationship.

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I'd get w over two.

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Sin of, is that going
to be the same angle?

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Yeah, it's the same angle.

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Same point on the wall.

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This is really far away

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so these approximations are whole,

238
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where these line are
supposed to be approximately

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parallel because the screen or

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the wall's very far away
in comparison to the width.

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That equals...

242
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well, that's going to be the same thing.

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I've got a w over two,
times sin of the same angle.

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Shoot, that's got to equal the same thing

245
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that it did up here.

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If the angle's the same,

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my w over two is the same.

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That's also going to
equal half a wavelength.

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That's also going to be destructive.

250
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These two will also
interfere destructively.

251
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And I can keep playing this game.

252
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I can pick this point here, over to here.

253
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And the next one down.

254
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These two would have to be destructive.

255
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I can pair them off and
keep pairing them off.

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I get destructive for all of them.

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I could annihilate all of
them by pairing them off

258
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and finding a partner that's
destructive to that one.

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And so, this really is
a destructive point.

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This point over here,

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all the light is gone.

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Completely annihilated.

263
00:09:51,638 --> 00:09:53,166
Gives you destructive.

264
00:09:53,166 --> 00:09:57,114
So the short of it, is that
this relationship here,

265
00:09:57,114 --> 00:10:00,544
this relationship that w, this slit width,

266
00:10:00,544 --> 00:10:04,360
times sin of theta, the angle, same angle

267
00:10:04,360 --> 00:10:05,409
we've always been defining it as,

268
00:10:05,409 --> 00:10:09,463
equals integer wavelengths.

269
00:10:09,463 --> 00:10:11,434
This time got to be careful though,

270
00:10:11,434 --> 00:10:14,290
this time this gives
you destructive points.

271
00:10:14,290 --> 00:10:15,644
Not the constructive points.

272
00:10:15,644 --> 00:10:17,239
It was always constructive before.

273
00:10:17,239 --> 00:10:19,780
This gives you destructive points now.

274
00:10:19,780 --> 00:10:21,200
And you might be upset.

275
00:10:21,200 --> 00:10:22,839
You might say, "Hold on a minute,

276
00:10:22,839 --> 00:10:24,530
"we only proved this for,

277
00:10:24,530 --> 00:10:26,378
"this was just for n equals one.

278
00:10:26,383 --> 00:10:27,590
"Or m equals one.

279
00:10:27,590 --> 00:10:29,028
"One wavelength.

280
00:10:29,028 --> 00:10:31,763
"You didn't prove this for
anything besides n equals one.

281
00:10:31,763 --> 00:10:34,412
Well, you can just as easily show

282
00:10:34,412 --> 00:10:37,743
that three lambda over two
would also give destructive.

283
00:10:37,743 --> 00:10:39,742
Or five lambda over two.

284
00:10:39,742 --> 00:10:41,941
That would give us all
the odd integers here.

285
00:10:41,941 --> 00:10:46,079
So m, m here can be...

286
00:10:46,079 --> 00:10:47,138
it can't be zero.

287
00:10:47,138 --> 00:10:48,539
We'll talk about that in a minute.

288
00:10:48,539 --> 00:10:52,798
It could be one, two, three,
four, five, and so on.

289
00:10:52,798 --> 00:10:54,545
One we already showed.

290
00:10:54,545 --> 00:10:56,479
Three you get,

291
00:10:56,479 --> 00:10:58,633
well, if you made this
three halves wavelength,

292
00:10:58,633 --> 00:10:59,943
that's also destructive.

293
00:10:59,943 --> 00:11:00,914
That'd be three.

294
00:11:00,914 --> 00:11:02,064
Five halves wavelength,

295
00:11:02,064 --> 00:11:03,672
the two's are always cancelling.

296
00:11:03,672 --> 00:11:06,108
So five halves wavelength would work.

297
00:11:06,108 --> 00:11:08,371
What about the even integers?

298
00:11:08,371 --> 00:11:09,602
How do we get these?

299
00:11:09,602 --> 00:11:12,596
Well, those come from
the fact that I didn't

300
00:11:12,596 --> 00:11:17,596
have to pair these off with
the top one and the middle one.

301
00:11:17,806 --> 00:11:20,445
That's dividing this into w over two.

302
00:11:20,445 --> 00:11:24,375
So pairing them off, by
lengths of w over two.

303
00:11:24,375 --> 00:11:25,501
I can pair them off.

304
00:11:25,501 --> 00:11:27,979
I can divide this by any even integer.

305
00:11:27,979 --> 00:11:30,803
I can imagine pairing off instead of doing

306
00:11:30,803 --> 00:11:33,080
the top most one and the middle one.

307
00:11:33,080 --> 00:11:38,080
I can do the top most one
and skip one down here.

308
00:11:38,105 --> 00:11:39,917
And so I can pair these off,

309
00:11:39,917 --> 00:11:44,463
if I divide this into
this distance right here.

310
00:11:46,094 --> 00:11:47,908
That distance would be, what?

311
00:11:47,908 --> 00:11:49,269
That'd be w over four.

312
00:11:49,269 --> 00:11:51,279
And so I can imagine pairing off,

313
00:11:51,279 --> 00:11:53,455
okay if these two cancel,

314
00:11:54,331 --> 00:11:56,111
if those two points cancel,

315
00:11:56,111 --> 00:11:59,262
then the next one down,

316
00:11:59,262 --> 00:12:01,642
so this one here...

317
00:12:02,934 --> 00:12:06,400
And this one here would also
cancel by the same reasoning.

318
00:12:06,400 --> 00:12:08,911
And so, I can play the same game now,

319
00:12:08,911 --> 00:12:11,143
but w over four would be how I divide it.

320
00:12:11,143 --> 00:12:12,280
I can't divide it by anything.

321
00:12:12,280 --> 00:12:13,575
I can't divide it by three.

322
00:12:13,575 --> 00:12:17,836
Like 2.5, because I always want
to pair these off in two's.

323
00:12:17,836 --> 00:12:19,942
Always two's, that's my whole plan.

324
00:12:19,942 --> 00:12:21,269
That's my whole strategy here,

325
00:12:21,269 --> 00:12:23,233
to cancel these in two's.

326
00:12:23,233 --> 00:12:26,274
And I can do that by dividing
this by any even integer.

327
00:12:26,274 --> 00:12:28,006
So w over four would work.

328
00:12:28,006 --> 00:12:29,100
What would that give us?

329
00:12:29,100 --> 00:12:31,815
Okay, w over four with the
distance between these,

330
00:12:31,815 --> 00:12:36,509
times sin theta, equals, let's
just say it's the first one,

331
00:12:36,509 --> 00:12:37,680
half of a wavelength.

332
00:12:37,680 --> 00:12:40,983
Well if I solve this,
if I move the four over,

333
00:12:42,120 --> 00:12:46,011
I get w sin theta equals two lambda.

334
00:12:46,011 --> 00:12:49,397
So the two's also give us
destructive interference.

335
00:12:49,397 --> 00:12:50,676
I can divide by eight.

336
00:12:50,676 --> 00:12:52,673
That would give us four,
once I move it over.

337
00:12:52,673 --> 00:12:54,638
I can divide by any even integer,

338
00:12:54,638 --> 00:12:56,670
any integer here is going to give us

339
00:12:56,670 --> 00:12:58,477
a destructive point on the wall.

340
00:12:58,477 --> 00:13:01,411
So this would be m equals one.

341
00:13:01,411 --> 00:13:04,002
This would be m equals two.

342
00:13:04,002 --> 00:13:05,993
And so on, upwards.

343
00:13:05,993 --> 00:13:07,960
So this relationship right here gives you

344
00:13:07,960 --> 00:13:08,837
all the destructive points.

345
00:13:08,837 --> 00:13:11,569
How come m equals zero is
not a destructive point?

346
00:13:11,569 --> 00:13:13,377
Well, m equals zero is
right in the middle.

347
00:13:13,377 --> 00:13:15,310
That's the most constructive point.

348
00:13:15,310 --> 00:13:16,545
That's the brightest spot.

349
00:13:16,545 --> 00:13:20,538
So m equals zero is not
a destructive point.

350
00:13:20,538 --> 00:13:24,080
But any other integer does
give you a destructive point.

351
00:13:24,080 --> 00:13:26,933
So this is the formula for
the destructive points,

352
00:13:26,933 --> 00:13:30,879
w is the entire width of the Single Slit.

353
00:13:30,879 --> 00:13:34,904
Theta is the angle, the way we
normally measure angle here,

354
00:13:34,904 --> 00:13:36,777
you imagine a center line like that.

355
00:13:36,777 --> 00:13:39,575
Imagine a line up to
your point on the wall.

356
00:13:39,575 --> 00:13:42,645
This angle here would be theta.

357
00:13:42,645 --> 00:13:46,033
And m is any integer that is not zero.

358
00:13:46,033 --> 00:13:49,603
Lambda is the wavelength
of the actual light

359
00:13:49,603 --> 00:13:51,834
that you're sending in here.

360
00:13:51,834 --> 00:13:54,081
Now this just gives you
the destructive points.

361
00:13:54,081 --> 00:13:56,173
You might wonder, "Hey, I'm clever.

362
00:13:56,173 --> 00:13:59,340
"If the integers are giving
us destructive points,

363
00:13:59,340 --> 00:14:03,643
"then the half integers should
give us the constructive points?"

364
00:14:03,643 --> 00:14:07,437
If w sin theta equals,
you know, lambda over two,

365
00:14:07,437 --> 00:14:10,041
or three lambda over two,

366
00:14:10,041 --> 00:14:13,645
is this going to give
us constructive points?

367
00:14:13,645 --> 00:14:16,546
And eh, not really.

368
00:14:16,546 --> 00:14:19,247
So, there's some complications here.

369
00:14:19,247 --> 00:14:21,137
And if you're interested in why

370
00:14:21,137 --> 00:14:24,008
this does not give the
constructive points,

371
00:14:24,008 --> 00:14:25,677
I'm going to make another video.

372
00:14:25,677 --> 00:14:26,938
Watch that one.

373
00:14:26,938 --> 00:14:28,733
Because if you've been
paying close attention,

374
00:14:28,733 --> 00:14:31,306
you should be upset
about something else too.

375
00:14:31,306 --> 00:14:33,771
You should be upset about
something earlier I've said,

376
00:14:33,771 --> 00:14:35,705
that might make it seem like we can prove

377
00:14:35,705 --> 00:14:37,405
this does not happen.

378
00:14:37,405 --> 00:14:39,138
With the diffraction grading,

379
00:14:39,138 --> 00:14:40,980
if you were paying close attention,

380
00:14:40,980 --> 00:14:45,112
we "proved," quote unquote,
that these do not occur.

381
00:14:45,112 --> 00:14:47,546
And if you're upset by any of that,

382
00:14:47,546 --> 00:14:49,542
or you want to know why
the constructive formula

383
00:14:49,542 --> 00:14:51,929
does not exactly give
you constructive points,

384
00:14:51,929 --> 00:14:53,239
watch that video.

385
00:14:53,239 --> 00:14:54,971
If you're happy with what we do know.

386
00:14:54,971 --> 00:14:57,906
That this gives you the
destructive points on the wall,

387
00:14:57,906 --> 00:00:00,000
then you're good.