# Leaky-Wave Antennas

A diffraction grating spectrally decomposes a pulsed wavefront in space. Another related component common at microwave and mm-wave frequencies is an LWA. LWAs are a class of antennas that use a traveling wave on a guiding structure as the main radiating mechanism [Hessel (1969); Tamir (1969); Oliner and Jackson (2007)]. These antennas are capable of producing narrow beams, with the beam-width limited by the size of the structure. LWAs support a fast wave on the guiding structure, where the phase constant */3(w)* is less than the free-space wavenumber k_{0}. The leaky wave is, therefore, fundamentally a radiating type of wave, which radiates power continuously as it propagates on the guiding structure. Typical implementations of an LWA are in slot-waveguides and printed metamaterial transmission line structures [Caloz et al. (2011); Caloz and Itoh (2006)].

An LWA resembles a diffraction grating in two main respects:

- (1) LWAs are broadband structures and can support pulsed waves with finite signal bandwidths Aw.
- (2) The radiation angle
*в*of an LWA depends on the signal frequency w, as illustrated in Fig. 5.4.

On the other hand, they differ from each other in two respects:

- (1) An equivalent 0
^{th}-order diffraction in an LWA is frequency dependent unlike that in the case of a diffraction grating, where higher-order diffraction orders are used for frequency discrimination. - (2) While a diffracting grating is excited with a 2D wavefront in space, an LWA is fed at a single point in space, as illustrated in Fig. 5.4.

A typical 1D periodic LWA structure may be seen as a uniform structure supporting a slow non-radiating wave, with b_{0}(w) > k_{0} that has been periodically modulated in the longitudinal *(у)-*

direction. The periodic modulation generates an infinite number of space harmonics with propagation constant *p _{n}(a>=* b

_{0}(w) +

*2pn/p*where

*p*is the unit cell period and

*n*is an integer [Hessel (1969)]. Although the main (n = 0) space harmonic is a slow wave, one of the space harmonics (usually

*n*= -1) is designed to be a fast wave, so that

*-k*b

_{0}<_{-1}<

*k*and hence this space harmonic is a radiating wave. With recent developments in the field of LWAs inspired from metamaterial concepts,

_{0},*n*= 0 fundamental space harmonic also radiates inside the fast-wave region enabling a full-space frequency scan including broadside radiation, as will be explained in the next subsection [Caloz and Itoh (2006); Otto et al. (2012); Otto et al. (2014)].

**Figure 5.4 **Spatial-spectral decomposition of a broadband temporal signal using an LWA.

An intuitive derivation of the radiation characteristics of an LWA can be made from a simple wave-propagation argument. Consider the LWA of Fig. 5.4, supporting a guided-wave mode *y{**x**) = y** _{0}e^{-jb(w)y}.* If n

^{th }space harmonic is the radiating leaky-wave mode, the field in the air

region above the aperture (*z >* 0) is given by *y*(x,z) = *y _{0}*

*e*

^{Jb,,x}e

^{jz}*Z*

*,*satisfying

*k*

*0 = 0Z +*

*k*

*Z*

*,*where

*k*is the wavenumber along the z-axis. The angle

_{z}*в(а*

*>*

*)*of the resulting leaky-wave radiation is then given by:

which for small angles around *в* = 0° can be written as:

This small angle approximation is made to establish a link between an LWA and a diffraction grating operated under paraxial conditions. Compared to the frequency-scanning relation of (5.11) of a grating, the above LWA scanning relation has the same form. Therefore, an LWA essentially operates as a diffraction grating, or vice versa.