What is a Photonic Crystal?
 The concept of photonic crystals was introduced by Eli Yablonovitch
and Sajeev John in 1987.
 Periodic Structure: A photonic crystal is a material with a periodic
arrangement of dielectric materials in which the refractive
index changes periodically..
 Light Manipulation: It can control and manipulate the flow of light.
 Photonic Band Gap: Photonic crystals create a frequency range
(band gap) where light of specific wavelengths cannot propagate.
Goal and Application of the Study
 Goal: The study aims to analyze the band gap in a photonic crystal made of GaAs
pillars.
 Extract Band Gaps: It focuses on extracting the band gaps for the lowest bands of the
crystal.
 Application: This type of analysis helps in designing optical devices that rely on
controlling the propagation of light.
 Engineering Band Gaps: By designing specific structures, you can block or allow certain
wavelengths of light, like an optical filter.
Complexity of the Band-Gap Analysis
 Frequency-Dependent Refractive Index: The refractive index of GaAs changes with
frequency, making the analysis complex.
 Wave Vector Ramping: The wave vector k must be gradually varied (or ramped) to
construct the band diagram.
 Nonlinear Eigenvalue Problem: Combining the frequency dependency and wave vector
ramping requires reformulating the problem as a nonlinear eigenvalue problem.
 Solver Challenges: A stationary solver is used with the eigenvalue as an unknown.
 Field Normalization: The electric field needs to be normalized across the domain to
simplify the calculations.
How to Solve the Problem
 Nonlinear Eigenvalue Problem: Reformulate the problem as a nonlinear eigenvalue
problem to solve for the frequencies.
 Stationary Solver: Use a stationary solver to treat the eigenvalue as an unknown.
 Normalization: Normalize the electric field to ensure that the average field is unity
across the domain.
 Update Refractive Index: The nonlinear solver updates the refractive index with each
found eigenvalue to handle frequency dependency.
 Parametric Sweep: A parametric solver is used to sweep the wave vector k and
compute the band diagram.
Wave Vector and Floquet Periodicity
 Wave Vector (k): Defines the propagation of waves in simulations. The wave vector for
the propagating wave enters the simulation as Floquet periodicity
_
= 𝐸𝑧 1 𝑒 𝑖β where β=kd (phase factor determined by ‘k’ and
distance ‘d’ between periodic boundaries).
 Floquet Condition: 𝐸𝑧 2
 Reciprocal lattice vectors derived from primitive vectors define the range of k in
photonic crystals, simplifying under specific orientations.
 Reciprocal Lattice Vector
2𝜋(𝑎2 ∗ 𝑎3)
2𝜋(𝑎3 ∗ 𝑎1)
,
𝑏2 =
𝑎1. (𝑎2 ∗ 𝑎3)
𝑎1. (𝑎2 ∗ 𝑎3)
• a1 and a2 are perpendicular to each other and to a3
𝑎1
𝑎2
𝑏1 = 2𝜋
,
𝑏2 = 2𝜋
|𝑎1|. |𝑎1|
|𝑎2|. |𝑎2|
𝑏1 =
Defining refractive index
• In the expression section
Definition ->expression
C_const is speed of light , f is the frequency which will be provided later
Floquet Periodic Boundary Conditions
• Electric Field Condition:
• 𝐸𝑑𝑠𝑡 = 𝐸𝑠𝑟𝑐 𝑒 −𝑖𝑘𝑓(𝑟𝑑𝑠𝑡−𝑟𝑠𝑟𝑐) , H𝑑𝑠𝑡 = 𝐻𝑠𝑟𝑐 𝑒 −𝑖𝑘𝑓𝑡(𝑟𝑑𝑠𝑡−𝑟𝑠𝑟𝑐)
• Dst stands for destination
• Src stands for source
Mesh Optimization
Field distribution for K=0
• Solved linear eigenvalue problem
Field distribution for non zero K values
• Need to solve nonlinear eigenvalue problem
• Normalize the integral in unit cell
• Here we use eigenfrequency from sol1 as an initial values
Electric field distribution at k=0.5
• Z-component of electric field having eigenfrequency of band 5
Dispersion plot or Band diagram
Analysis of
the Electric
Field Plots
Figure 1
 Figure 1 shows the z-component of the electric field for k = 0,
depicting the field distribution in the crystal.
 Figure 2 shows the same field for k = 0.5 for the fifth band,
highlighting how the field evolves as k increases.
Figure 2
Analysis of the Band Diagram
 Degeneracy at k = 0: At k = 0, bands 2 and 3 are degenerate, meaning they have the
same frequency at this point.
 Degeneracy at k = 0.5: Bands 1 and 2, and bands 4 and 5, become degenerate again
at k = 0.5.
 Band Gap Between Bands 3 and 4: There is a frequency range between bands 3 and
4 where there are no states, corresponding to a band gap.
 Wave Vector Sweep: The band diagram shows the results of sweeping the wave
vector k from 0 to 0.5.
 Practical Insight: The band diagram is key to understanding which frequencies can
propagate and which are blocked.
Conclusion of the Study
 Band Gap Identification: The study successfully identifies band gaps in the photonic
crystal, especially between bands 3 and 4.
 Importance of k-Ramping: Sweeping the wave vector k is crucial for understanding how
different frequencies propagate in the crystal.
 Complex Solver Setup: Handling the frequency-dependent refractive index and wave
vector ramping required a nonlinear eigenvalue approach.
 Applications: This analysis is useful for designing photonic devices like optical filters,
waveguides.