1. 1st Brillouin zone and designation of its high symmetry lines & points
A. Both finite and infinite arrays
Photonic array type
1st Brillouin zone and its high symmetry lines
Square / rectangular lattice (array of identical waveguides)
Hexagonal lattice (array of identical waveguides)
Centered rectangular lattice (array of identical waveguides) x y z
1st Brillouin zone:
High symmetry lines:
𝑘 𝑥 is in units of 𝜋 Λ 𝑥 (i.e. 1 along 𝑘 𝑥 axis means 𝑘 𝑥 = 𝜋 Λ 𝑥 )
𝑘 𝑦 is in units of 𝜋 Λ 𝑦 (i.e. 1 along 𝑘 𝑦 axis means 𝑘 𝑦 = 𝜋 Λ 𝑦 )
for a square array x y z
1st Brillouin zone:
High symmetry lines:
𝑘 𝑥 is in units of 𝜋 Λ 𝑥 (i.e. 1 along 𝑘 𝑥 axis means 𝑘 𝑥 = 𝜋 Λ 𝑥 )
𝑘 𝑦 is in units of 𝜋 Λ 𝑦 (i.e. 1 along 𝑘 𝑦 axis means 𝑘 𝑦 = 𝜋 Λ 𝑦 ) x y z
1st Brillouin zone:
High symmetry lines:
𝑘 𝑥 is in units of 𝜋 Λ 𝑥 (i.e. 1 along 𝑘 𝑥 axis means 𝑘 𝑥 = 𝜋 Λ 𝑥 )
𝑘 𝑦 is in units of 𝜋 Λ 𝑦 (i.e. 1 along 𝑘 𝑦 axis means 𝑘 𝑦 = 𝜋 Λ 𝑦 )

2. 1st Brillouin zone and its high symmetry lines
Photonic array type
1st Brillouin zone and its high symmetry lines
Square / rectangular superlattice
Hexagonal superlattice with hexagonal supercells
Centered rectangular superlattice x y z
1st Brillouin zone:
High symmetry lines:
supercell
for a square superlattice
𝑘 𝑥 is in units of 𝜋 Λ 𝑥 (i.e. 1 along 𝑘 𝑥 axis means 𝑘 𝑥 = 𝜋 Λ 𝑥 )
𝑘 𝑦 is in units of 𝜋 Λ 𝑦 (i.e. 1 along 𝑘 𝑦 axis means 𝑘 𝑦 = 𝜋 Λ 𝑦 ) x y z
1st Brillouin zone:
High symmetry lines:
supercell
𝑘 𝑥 is in units of 𝜋 Λ 𝑥 (i.e. 1 along 𝑘 𝑥 axis means 𝑘 𝑥 = 𝜋 Λ 𝑥 )
𝑘 𝑦 is in units of 𝜋 Λ 𝑦 (i.e. 1 along 𝑘 𝑦 axis means 𝑘 𝑦 = 𝜋 Λ 𝑦 ) x y z
1st Brillouin zone:
High symmetry lines:
supercell
𝑘 𝑥 is in units of 𝜋 Λ 𝑥 (i.e. 1 along 𝑘 𝑥 axis means 𝑘 𝑥 = 𝜋 Λ 𝑥 )
𝑘 𝑦 is in units of 𝜋 Λ 𝑦 (i.e. 1 along 𝑘 𝑦 axis means 𝑘 𝑦 = 𝜋 Λ 𝑦 )

3. 1st Brillouin zone and its high symmetry lines
Photonic array type
1st Brillouin zone and its high symmetry lines
Square lattice of quadrangular spots (↔ square lattice of holes)
Kagome superlattice of triangular spots (↔ honeycomb lattice holes) x y z
1st Brillouin zone:
High symmetry lines:
𝑘 𝑥 is in units of 𝜋 Λ 𝑥 (i.e. 1 along 𝑘 𝑥 axis means 𝑘 𝑥 = 𝜋 Λ 𝑥 )
𝑘 𝑦 is in units of 𝜋 Λ 𝑦 (i.e. 1 along 𝑘 𝑦 axis means 𝑘 𝑦 = 𝜋 Λ 𝑦 ) x y z
1st Brillouin zone:
High symmetry lines:
supercell
𝑘 𝑥 is in units of 𝜋 Λ 𝑥 (i.e. 1 along 𝑘 𝑥 axis means 𝑘 𝑥 = 𝜋 Λ 𝑥 )
𝑘 𝑦 is in units of 𝜋 Λ 𝑦 (i.e. 1 along 𝑘 𝑦 axis means 𝑘 𝑦 = 𝜋 Λ 𝑦 )
see next 2 pages for1D arrays & 2D finite arrays

4. 1st Brillouin zone and its high symmetry line
Photonic array type
1st Brillouin zone and its high symmetry line
1D square array of identi-cal waveguides (conside-red as 1D superlattice)
1D hexagonal array of iden- tical waveguides (considered as 1D superlattice)
1D square superlattice
1D hexagonal superlattice x y z
1st Brillouin zone:
High symmetry line:
𝑘 𝑥 is in units of 𝜋 Λ 𝑥 (i.e. 1 along 𝑘 𝑥 axis means 𝑘 𝑥 = 𝜋 Λ 𝑥 )
𝑘 𝑦 is in units of 𝜋 Λ 𝑦 (i.e. 1 along 𝑘 𝑦 axis means 𝑘 𝑦 = 𝜋 Λ 𝑦 )
supercell x y z
1st Brillouin zone:
High symmetry line:
supercell
𝑘 𝑥 is in units of 𝜋 Λ 𝑥 (i.e. 1 along 𝑘 𝑥 axis means 𝑘 𝑥 = 𝜋 Λ 𝑥 )
𝑘 𝑦 is in units of 𝜋 Λ 𝑦 (i.e. 1 along 𝑘 𝑦 axis means 𝑘 𝑦 = 𝜋 Λ 𝑦 ) x y z
1st Brillouin zone:
High symmetry line:
supercell
𝑘 𝑥 is in units of 𝜋 Λ 𝑥 (i.e. 1 along 𝑘 𝑥 axis means 𝑘 𝑥 = 𝜋 Λ 𝑥 )
𝑘 𝑦 is in units of 𝜋 Λ 𝑦 (i.e. 1 along 𝑘 𝑦 axis means 𝑘 𝑦 = 𝜋 Λ 𝑦 ) x y z
1st Brillouin zone:
High symmetry line:
supercell
𝑘 𝑥 is in units of 𝜋 Λ 𝑥 (i.e. 1 along 𝑘 𝑥 axis means 𝑘 𝑥 = 𝜋 Λ 𝑥 )
𝑘 𝑦 is in units of 𝜋 Λ 𝑦 (i.e. 1 along 𝑘 𝑦 axis means 𝑘 𝑦 = 𝜋 Λ 𝑦 )

5. B. Finite arrays only: Photonic array type
1st Brillouin zone and its high symmetry line
Arbitrary 2D array, based on square or rectangular lattice
Arbitrary 2D array, based on hexagonal lattice
Arbitrary 2D array, based on centered rectangular lattice x y z
1st Brillouin zone:
High symmetry lines:
𝑘 𝑥 is in units of 𝜋 Λ 𝑥 (i.e. 1 along 𝑘 𝑥 axis means 𝑘 𝑥 = 𝜋 Λ 𝑥 )
𝑘 𝑦 is in units of 𝜋 Λ 𝑦 (i.e. 1 along 𝑘 𝑦 axis means 𝑘 𝑦 = 𝜋 Λ 𝑦 )
for a square array x y z
1st Brillouin zone:
High symmetry lines:
𝑘 𝑥 is in units of 𝜋 Λ 𝑥 (i.e. 1 along 𝑘 𝑥 axis means 𝑘 𝑥 = 𝜋 Λ 𝑥 )
𝑘 𝑦 is in units of 𝜋 Λ 𝑦 (i.e. 1 along 𝑘 𝑦 axis means 𝑘 𝑦 = 𝜋 Λ 𝑦 ) x y z
1st Brillouin zone:
High symmetry lines:
𝑘 𝑥 is in units of 𝜋 Λ 𝑥 (i.e. 1 along 𝑘 𝑥 axis means 𝑘 𝑥 = 𝜋 Λ 𝑥 )
𝑘 𝑦 is in units of 𝜋 Λ 𝑦 (i.e. 1 along 𝑘 𝑦 axis means 𝑘 𝑦 = 𝜋 Λ 𝑦 )