[Slide 1] Hey everyone, welcome to lecture 1.7.
So continuing on the topic of 2D photonic crystals, now we're going to talk
about the symmetries that allow us to go from a Brillouin zone to a reducible
[Slide 2] Brillouin zone and thus simplify the kind of calculations that you have to perform.
And so, we can understand bandstructure symmetries in a very formal way.
In fact, as like a broad set of operators
that would commute with the eigenproblem operator.
So if you recall the eigenproblem itself that we're using,
the so-called master equation.
Curl of epsilon inverse, curl of h.
Then that equation basically can be associated with an operator times h.
[Slide 3] Okay, so then the curl of epsilon inverse curl is written as an operator, right?
Which of course we then subsequently turn into matrix in our calculation.
But before we do that actually as a peer operator,
we can actually try to commute it with various symmetry operators.
And so there are several different kinds of symmetry operators that would commute
with it.
And then of course,
we already know that because of periodicity we have a Brillouin zone and
so on, but then where are the types of symmetries that we're interested in?
So the first symmetry, may be the most obvious, maybe it's so
obvious that you didn't even think of it, would be time-reversal invariance.
And so the reason why this is so common is because any
operator that has real observables, like say in this case like a frequency,
that's real, would be called Hermitian, okay?
And then Hermitian operators will always commute with a time-reversal operator,
so that means that if you run everything in forward or
then run in reverse then the laws are the same.
So in other words, if you refract 96% of light as it goes through glass
coming from this direction, then you go back from the angle of reflection and
then it will reflect 96% again.
Okay, so that's like simple example.
But, it's very abstract concept,
which just tells us that we can always reverse time and
thus actually simplify the bandstructure itself because reversing
time is actually equivalent to reversing the k vector in this case.
And so then that means that omega n(k) would be equal to omega n (-k).
And so that like one-to-one mapping actually allows us to
cut the irreducible Brillouin zone in half.
And so that alone saves us a lot of time in our calculation.
But that's not the only thing that we can do.
[Slide 4] There are other types of symmetries.
So another class of symmetries is called a mirror-plane symmetry.
And so basically the mirror-plane operator is defined
such that when you apply it to the H field then you recover H but
then plus or minus 1 times the H field.
Okay, so then you can think of the eigen values of
the mirror operator as either being +1 or -1.
Obviously, each field would have to respect one or the other of these.
And it kind of makes sense if you think about like, in a bandstructure
calculation context that you could have like, these two classes of solutions.
And in fact we have seen them already but I didn't say it explicitly.
Basically we have the transverse electric and transverse magnetic fields.
And so in the case of transverse electric,
because it's all in the plane, then that means that it would respect
a mirror plane symmetry reflection, and then transfers magnetic, would as well.
But the difference is that transfers magnetic when you flip it,
you would get a +1, and transfers electric, you would get a -1.
And so that's giving rise with those polarizations.
But you can also do mirror planes in the plane, if you will.
So you could have, some structure that's like a plus sign.
And so you can have reflection like both in this direction, along the plus, and
then this direction along the plus sign, right.
And so if that's the whole symmetry of the problem,
then you actually have two mirror plane symmetries.
But then there's a third type of symmetry that's also very important
called rotational symmetry.
And so when we think about this very easily.
It's just like, if you can rotate the whole lattice
by a certain amount of degrees, okay?
So say like 90 degrees, like in the case of a square lattice and
it's exactly the same, then we could say that has a rotational symmetry.
And so of course, it depends on the periodicity,
also formally known as the crystalgraphic point group.
Okay, so in 2D, then it's the difference between a square or a rectangular or
a triangular lattice, right.
And so of course the square lattice has this 4-fold rotational symmetry.
You actually have 6-fold symmetry for the triangular lattice.
You can have 3-fold symmetry for
a modified triangular lattice with two basis elements.
If you don't have 4-fold, 3-fold, or 6-fold symmetry,
then you don't actually have rotational symmetry for a truly periodic system.
Instead would be called a quasi-periodic system.
And so, if you think about a soccer ball, soccer balls usually have 5-fold
pattern which kind of looks like a pentagon, which tiles the whole surface.
But if you try to project that out into real space,
you see that you can take one unit and then see it repeated
at a uniform distance away over and over and over again.
Instead, you have kind of a very well ordered pattern, but
it's not perfectly periodic.
And so this gives rise to some interesting phenomenon, as well.
Although unfortunately, because of time,
we can't cover a lot of detail in this class.
But you can certainly read up on it.
There have been a number of papers about how to calculate quasicrystal
bandstructures in the last maybe, decade or so.
So with that, thank you.
In next lecture we'll be discussing how do you apply all these
symmetries to more complex bandstructures.