A Brillouin zone can be thought of as the Fourier Transform of the minimum unit cell of a periodic structure. It's an over-simplified view, but it gives it a fair amount of intuition.
In any periodic structure in which you want to solve a wave equation (say, Schrodinger's equation for an electron in a crystal, or a photon in a Bragg grating), then you want to discover the relationship of frequency to wavenumber. For an electron, frequency and wavenumber multiplied by h-bar are energy and momentum, respectively.
The most common approach to solving it for w-k relationship is to use Bloch waves. Bloch's theorem for waves in a periodic potential states that any wave travelling through a periodic potential can be expressed as the product of a periodic function and a plane wave.
This periodicity makes it so k is not unique, but rather every k is equivalent to k+ 2pi/a, where a is the period of the structure. Because of this, the solution "wraps around" the period boundary.
the same function centered at zero is replicated at k = 2pi/a and -2pi/a and so on. You can think of it instead just wrapping around the boundary.
This is where bands come from (but not necessarily band gaps). For every kvector describing a solution, there are multiple frequencies that can have that kvector because the relationship is periodic (i.e. k = 0 is equivalent to k = 2pi/a, where a is the lattice distance, or the equivalent point one unit cell over).
The Brillouin zone for a fiber Bragg grating is really simple since it is periodic in only one dimension.
Much like the lattice has a unit cell (its minimum unit cell is referred to as theWigner–Seitz cell), the periodic omega-k relationship also has a unit cell since it is periodic. That minimum cell is the Brillouin zone.
Explaining this in full detail is about a 4-week discussion in an undergraduate solid state physics class,
In any periodic structure in which you want to solve a wave equation (say, Schrodinger's equation for an electron in a crystal, or a photon in a Bragg grating), then you want to discover the relationship of frequency to wavenumber. For an electron, frequency and wavenumber multiplied by h-bar are energy and momentum, respectively.
The most common approach to solving it for w-k relationship is to use Bloch waves. Bloch's theorem for waves in a periodic potential states that any wave travelling through a periodic potential can be expressed as the product of a periodic function and a plane wave.
This periodicity makes it so k is not unique, but rather every k is equivalent to k+ 2pi/a, where a is the period of the structure. Because of this, the solution "wraps around" the period boundary.
This is where bands come from (but not necessarily band gaps). For every kvector describing a solution, there are multiple frequencies that can have that kvector because the relationship is periodic (i.e. k = 0 is equivalent to k = 2pi/a, where a is the lattice distance, or the equivalent point one unit cell over).
Much like the lattice has a unit cell (its minimum unit cell is referred to as theWigner–Seitz cell), the periodic omega-k relationship also has a unit cell since it is periodic. That minimum cell is the Brillouin zone.
Explaining this in full detail is about a 4-week discussion in an undergraduate solid state physics class,
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