# Periodic Structure

## Periodic Structure

in microwave technology, a structure, or system, that can be superimposed on itself by a parallel displacement over a certain finite distance. The minimum value *d* of this distance is called the period. Strictly speaking, periodic structures are infinite and serve as idealized models for the theoretical study of real objects. In practice, limited portions of periodic structures are used and are by convention also called periodic structures. Depending on the number of independent directions of displacement, we speak of one-dimensional, two-dimensional, or three-dimensional periodic structures (Figures 1 and 2). One-dimensional and two-dimensional periodic structures are employed as slow-wave systems, antennas, and diffraction gratings. Two-dimensional and three-dimensional periodic structures are used to construct lenses, prisms, and other devices that fix the direction of propagation of electromagnetic waves.

Any component *A* of the electric and magnetic fields at a point of a periodic structure with the coordinate *z* (the direction of the structure’s periodicity coincides with the z-axis) can be represented as a series:

Each term of the series is called a space harmonic. Here, *a _{m}* is the amplitude of the mth space harmonic and depends on the form of the periodic structure,

*ω*is the angular frequency of the electromagnetic oscillations,

*t*is the time,

*β*=

_{m}*β*+ (2

*πm*/

*d*) is the wave number of the

*m*th space harmonic, and

*i*is the unit imaginary number. The basic characteristics of a periodic structure are (1) the space harmonics’ retardation coefficients

*n*=

_{m}*β*/

_{m}c*ω*, which by definition agree with the refractive index in optics and are numerically equal to the ratios of the phase velocity

*c*of the wave in free space to the phase velocities

*ω*/

*β*of the harmonics in the periodic structure; (2) the group velocity

_{m}*dω*/

*dβ*, which coincides in direction with the direction of energy transfer for the electromagnetic waves; and (3) the dispersion, which characterizes the dependence of the retardation coefficient

_{m}*n*on the wavelength λ in free space. The phase velocity of the wave is determined from the retardation coefficient, and the group velocity can be obtained from the dispersion. The phase velocities and retardation coefficients of the space harmonics differ, but the harmonics’ group velocities are identical.

In microwave electronic devices that use periodic structures as interaction structures, the velocity of the electrons is usually

close to the phase velocity of the wave but may differ from the group velocity in both magnitude and direction. Agreement between the directions of the phase and group velocities of a wave (positive dispersion) is typical for an oscillation-amplifying mode, and opposing directions for these velocities (negative dispersion) is characteristic for an oscillation-generating mode.

### REFERENCES

Aizenberg, G. Z.*Antenny ul’trakorotkikh voln*. Moscow, 1957.

Taranenko, Z. I., and Ia. K. Trokhimenko.

*Zamedliaiushchie sistemy*. Kiev, 1965.

Silin, R. A., and V. P. Sazonov.

*Zamedliaiushchie sistemy*. Moscow, 1966.

R. A. SILIN