# Calculating Electromagnetic Responses in Periodic Structures With FDTD Periodic Boundary Conditions

### Key Takeaways

• According to Bloch’s theorem, the fields inside a periodic structure are periodic up to a particular phase difference and attenuation.

• Periodic boundary conditions can be classified into two types: direct field methods and field transformation methods.

• FDTD periodic boundary conditions reduce the computational cost of simulation, as only one unit cell is simulated to obtain a complete electromagnetic response. There are several periodic structures, such as metamaterials, waveguides, phased array antennas, and diffraction gratings, in modern electromagnetics. When these periodic devices are modeled for computational electromagnetic analysis using numerical methods, for instance, finite-difference time-domain (FDTD), we can observe that the fields inside the structure also take on the same symmetry and periodicity. The existence of periodicity in the physical structure and fields can be used to simulate the entire system, especially in FDTD

In FDTD modeling, periodic boundary conditions are applied to simulate the response of the entire system. The simulation of one unit cell with  FDTD periodic boundary conditions can determine the solution to electromagnetic problems in periodic electromagnetic devices or structures.

## The FDTD Method

The concept of the periodic electromagnetic structure can be applied to electromagnetic bandgap materials, metamaterials, phased array antennas, and frequency selective surfaces. Most of these applications use computational electromagnetics to calculate electromagnetic responses, scattering, and the radiation phenomenon.

The most frequently used numerical method in computational electromagnetics is FDTD. Its simplicity and ease of implementation make it popular compared to other numerical methods. The  FDTD method is efficient in modeling both periodic and non-periodic structures.

### FDTD Periodic Boundary Conditions

When modeling periodic structures using FDTD, we can see the repetition of the smallest unit cell or grid to infinity.  In certain periodic structures, the electric and magnetic fields present in the grid also share periodicity. The advantage of the periodicity in structure and fields is utilized in FDTD simulation as FDTD periodic boundary conditions. The FDTD periodic boundary condition reduces the computational cost of simulation, as only one unit cell is simulated to obtain the complete electromagnetic response.

### Bloch’s Theorem

The mathematical background of FDTD periodic boundary conditions is Floquet’s theorem or Bloch’s theorem. Bloch’s theorem describes the fields inside periodic structures. According to Bloch’s theorem, the fields inside a periodic structure are periodic up to a particular phase difference and attenuation. Applying Bloch’s theorem to FDTD modeling, the fields in each grid share periodicity and this is conceptualized as FDTD periodic boundary conditions.

## FDTD Periodic and Absorbing Boundary Conditions

In FDTD modeling and simulation, periodic as well as absorbing boundary conditions are used whenever required.

In electromagnetic problems such as scattering and radiation, the electromagnetic fields propagate to infinity. The exact numerical modeling of this electromagnetic problem requires infinite computational memory. However, we convert the unlimited physical domain into the limited computational domain with FDTD absorbing boundary conditions.

Consider an array antenna that is periodic in structure. When modeling periodic structures such as array antennas, one-dimensional or two-dimensional periodicity exists. FDTD periodic boundary conditions can be implemented for the computation of such periodic structures. The FDTD simulation of periodic structures simulates a single cell terminated by periodic boundary conditions. The FDTD periodic boundary conditions shrink the infinite periodic structure into a single grid that undergoes numerical calculation. The FDTD simulation in periodic structures shows increased computational efficiency with the implementation of periodic boundary conditions.

## The Classifications of FDTD Periodic Boundary Conditions

FDTD periodic boundary conditions can be classified into two types:

1. Direct field method - The direct field method solves Maxwell’s equations directly with periodic boundary conditions. There is no need for any field transformation in this method. Some direct field methods include:

1. Normal incidence method

2. Sine-cosine method

3. Multiple unit cell method

4. Angled update method

2. Field transformation method - The field transformation technique in this type of periodic boundary condition introduces auxiliary fields to eliminate the need for time-advanced data. The field transformation method is preferred over the direct field method in most periodic electromagnetic problems. Examples of this method include:

1. Multi-spatial grid method

2. Spectral FDTD method

3. Split-field method

FDTD periodic boundary conditions replicate the unit cell to achieve electromagnetic solutions in periodic structures. Cadence offers FDTD simulation tools to simplify the computation of electromagnetic problems in both periodic and non-periodic structures.