**2. Leaky wave antennas**

In detail this type of wave radiates continuously along its length, and hence the propagation wavenumber kz is complex, consisting of both a phase and an attenuation constant. Highlydirective beams at an arbitrary specified angle can be achieved with this type of antenna, with a low sidelobe level. The phase constant of the wave controls the beam angle (and this can be varied changing the frequency), while the attenuation constant controls the beamwidth. The aperture distribution can also be easily tapered to control the sidelobe level or beam shape.

All kinds of open planar transmission lines are predisposed to excite leaky waves. There are two kinds of leaky waves. Surface leaky waves radiate power into the substrate. These waves are in most cases undesirable as they increase losses, cause distortion of the transmitted signal and cross-talk to other parts of the circuit. Space leaky waves radiate power into a space and mostly also into the substrate. These waves can be utilized in leaky wave antennas. Leaky-wave antennas can be divided into two important categories, uniform and periodic, depending on the type of guiding structure. A uniform structure has a cross section that is uniform (constant) along the length of the structure, usually in the form of a waveguide that has been partially opened to allow radiation to occur. The guided wave on the uniform structure is a fast wave, and thus radiates as it propagates.

As said previously leaky-wave antennas form part of the general class of travelling-wave antennas which are a class of antennas that use a travelling wave on a guiding structure as the main radiating mechanism [3], as defined by standard IEEE 145-1993: "An antenna that couples power in small increments per unit length, either continuously or discretely, from a travelling wave structure to free space".

Fig. 2. Rectangular metal waveguide with a slit, aperture of the leaky wave antenna.

Leaky-wave antennas are a fast-wave travelling-wave antennas in wich the guided wave is a fast wave, meaning a wave that propagates with a phase velocity that is more than the speed of light in free space.

The slow wave travelling antenna does not fundamentally radiate by its nature, and radiation occurs only at discontinuities (typically the feed and the termination regions). The propagation wavenumber of the travelling wave is therefore a real number (ignoring conductors or other losses). Because the wave radiates only at the discontinuities, the radiation pattern physically arises from two equivalent sources, one at the beginning and one at the end of the structure. This makes it difficult to obtain highly-directive singlebeam radiation patterns. However, moderately directly patterns having a main beam near endfire can be achieved, although with a significant sidelobe level. For these antennas there is an optimum length depending on the desired location of the main beam. An independent control of the beam angle and the beam width is not possible. By contrast, the wave on a leaky-wave antenna (LWA) may be a fast wave, with a phase velocity greater than the speed of light. Leakage is caused by asymmetry, introduced in radiating structure transversal section (e.g.: aperture offset, waveguide shape, etc…), feeding modes or a combination of them. In this type of antennas, the power flux leaking from waveguide to free space ( *Pout* in Fig. 2 and Fig. 3), introduces a loss inside structure, determining a complex propagation wavenumber *<sup>z</sup> k* [4-5]:

$$(\;k\_z = \beta - ja\;)\tag{2}$$

Where is the leakage constant and is the propagation constant . The phase constant of the wave controls the beam angle (and this can be varied changing the frequency), while the attenuation constant controls the beamwidth. Highly-directive beams at an arbitrary specified angle can be achieved with this type of antenna, with a low sidelobe level.

As said previously leaky-wave antennas form part of the general class of travelling-wave antennas which are a class of antennas that use a travelling wave on a guiding structure as the main radiating mechanism [3], as defined by standard IEEE 145-1993: "An antenna that couples power in small increments per unit length, either continuously or discretely, from a

Fig. 2. Rectangular metal waveguide with a slit, aperture of the leaky wave antenna.

Leaky-wave antennas are a fast-wave travelling-wave antennas in wich the guided wave is a fast wave, meaning a wave that propagates with a phase velocity that is more than the

The slow wave travelling antenna does not fundamentally radiate by its nature, and radiation occurs only at discontinuities (typically the feed and the termination regions). The propagation wavenumber of the travelling wave is therefore a real number (ignoring conductors or other losses). Because the wave radiates only at the discontinuities, the radiation pattern physically arises from two equivalent sources, one at the beginning and one at the end of the structure. This makes it difficult to obtain highly-directive singlebeam radiation patterns. However, moderately directly patterns having a main beam near endfire can be achieved, although with a significant sidelobe level. For these antennas there is an optimum length depending on the desired location of the main beam. An independent control of the beam angle and the beam width is not possible. By contrast, the wave on a leaky-wave antenna (LWA) may be a fast wave, with a phase velocity greater than the speed of light. Leakage is caused by asymmetry, introduced in radiating structure transversal section (e.g.: aperture offset, waveguide shape, etc…), feeding modes or a combination of them. In this type of antennas, the power flux leaking from waveguide to free space ( *Pout* in Fig. 2 and Fig. 3), introduces a loss inside structure, determining a complex propagation

> 

of the wave controls the beam angle (and this can be varied changing the frequency), while

specified angle can be achieved with this type of antenna, with a low sidelobe level.

) (2)

is the propagation constant . The phase constant

controls the beamwidth. Highly-directive beams at an arbitrary

travelling wave structure to free space".

speed of light in free space.

wavenumber *<sup>z</sup> k* [4-5]:

the attenuation constant

Where

( *<sup>z</sup> k j*

is the leakage constant and

Moreover the aperture distribution can also be easily tapered to control the sidelobe level or beam shape. Leaky-wave antennas can be divided into two important categories, uniform and periodic, depending on the type of guiding structure.

Fig. 3. Example of periodic leaky wave antenna, using a dielectric substrate upon which are placed rods of other material, even metal, in a periodic layout.

A uniform structure has a cross section that is uniform (constant) along the length of the structure, usually in the form of a waveguide that has been partially opened to allow radiation to occur [6]. The guided wave on the uniform structure is a fast wave, and thus radiates as it propagates. A periodic leaky-wave antenna structure is one that consists of a uniform structure that supports a slow (non radiating) wave that has been periodically modulated in some fashion. Since a slow wave radiates at discontinuities, the periodic modulations (discontinuities) cause the wave to radiate continuously along the length of the structure. From a more sophisticated point of view, the periodic modulation creates a guided wave that consists of an infinite number of space harmonics (Floquet modes) [7]. Although the main (n = 0) space harmonic is a slow wave, one of the space harmonics (usually the n = −1) is designed to be a fast wave, and hence a radiating wave.

### **3. LWA in waveguide**

A typical example of a uniform leaky-wave antenna is a rectangular waveguide with a longitudinal slot. This simple structure illustrates the basic properties common to all uniform leaky-wave antennas. The fundamental *TE*10 waveguide mode is a fast wave, with

2 2 <sup>0</sup>*k* ( ) *a* lower than 0*k* . As mentioned, the radiation causes the wavenumber *<sup>z</sup> k* of the propagating mode within the open waveguide structure to become complex. By means of an application of the stationary-phase principle, it can be found in fact that [5]:

$$\sin \mathcal{G}\_m \equiv \frac{\mathcal{J}}{k\_0} = \frac{c}{\upsilon\_{ph}} \tag{3}$$

where *<sup>m</sup>* is the angle of maximum radiation taken from broadside. As is typical for a uniform LWA, the beam cannot be scanned too close to broadside (*<sup>m</sup>* 0 ), since this corresponds to the cutoff frequency of the waveguide. In addition, the beam cannot be scanned too close to endfire (*<sup>m</sup>* = 90) since this requires operation at frequencies significantly above cutoff, where higher-order modes is in a bound condition or can propagate, at least for an air-filled waveguide. Scanning is limited to the forward quadrant only (0 < *<sup>m</sup>* < ) for a wave travelling in the positive z direction.

Fig. 4. Slotted guide (patented by W. W. Hansen in 1940).

2

This one-dimensional (1D) leaky-wave aperture distribution (see Fig. 4), results in a "fan beam" having a narrow beam in the x-z plane (H plane), and a broad beam in the crossplane. Unlike the slow-wave structure, a very narrow beam can be created at any angle by choosing a sufficiently small value of . From diffraction theory, a simple formula for the beam width, measured between half power points (3dB), is:

$$\mathcal{A}\mathcal{B} \cong \frac{const}{\frac{L}{\lambda\_0} \cos \mathcal{G}\_m} \tag{4}$$

"const" is a parameter which is influenced by the type of aperture and illumination; for example, if at the aperture there's a constant field, const = 0.88 and, if the structure is uniform, const =0.91. As a rule of thumb, supposing:

$$\mathcal{A}\mathcal{B} \cong \frac{1}{\frac{L}{\lambda\_0} \cos \mathcal{G}\_m} \tag{5}$$

a good approximation of beamwidth is yielded, where L is the length of the leaky-wave antenna, and is expressed in radians. For 90% of the power radiated it can be assumed:

$$\frac{L}{k\_0} \equiv \frac{0.18}{\frac{\alpha}{k\_0}} \Rightarrow$$

$$A\mathcal{G} \cong \frac{a}{k\_0}$$

If the antenna has a constant attenuation throughout its length ( ) *z z z* results: <sup>2</sup> ( ) (0) *Pz P e z*

Therefore, being L the length of antenna, if a perfectly matched load is connected at the end of it, it's possible to express antenna efficiency as:

$$\eta\_{rad} = \frac{P(0) - P(L)}{P(0)} = 1 - \frac{P(L)}{P(0)} = 1 - e^{-2\alpha\_z L} \tag{6}$$

Rearranging:

52 Wireless Communications and Networks – Recent Advances

corresponds to the cutoff frequency of the waveguide. In addition, the beam cannot be

significantly above cutoff, where higher-order modes is in a bound condition or can propagate, at least for an air-filled waveguide. Scanning is limited to the forward quadrant

This one-dimensional (1D) leaky-wave aperture distribution (see Fig. 4), results in a "fan beam" having a narrow beam in the x-z plane (H plane), and a broad beam in the crossplane. Unlike the slow-wave structure, a very narrow beam can be created at any angle by

> 0 cos *<sup>m</sup>*

"const" is a parameter which is influenced by the type of aperture and illumination; for example, if at the aperture there's a constant field, const = 0.88 and, if the structure is

0

a good approximation of beamwidth is yielded, where L is the length of the leaky-wave

0

*k*

*L*

*L* 0.18

1

cos *<sup>m</sup>*

is expressed in radians. For 90% of the power radiated it can be assumed:

*const L*

0

) for a wave travelling in the positive z direction.

uniform LWA, the beam cannot be scanned too close to broadside (

Fig. 4. Slotted guide (patented by W. W. Hansen in 1940).

beam width, measured between half power points (3dB), is:

uniform, const =0.91. As a rule of thumb, supposing:

choosing a sufficiently small value of

antenna, and

*<sup>m</sup>* is the angle of maximum radiation taken from broadside. As is typical for a

*<sup>m</sup>* = 90) since this requires operation at frequencies

. From diffraction theory, a simple formula for the

(4)

(5)

*<sup>m</sup>* 0 ), since this

where

only (0 <

*<sup>m</sup>* < 2 

scanned too close to endfire (

$$L = -\frac{\ln(1 - \eta\_{rad})}{2\alpha\_z} \tag{7}$$

For most application, to gain a 90% efficiency, means that the antenna length is within 0 0 10 100 interval.

Fixing the antenna efficiency, using (7), makes possible to express attenuation constant in terms of antenna length, and vice versa. Using antenna efficiencies grater than 90%-95% is not advisable; in fact, supposing constant antenna cross section and, as a consequence, fixed leakage constant, the necessary length L grows exactly as *zL* , which increases asymptotically, as shown in Fig 5. If we want a 100% efficiency ( 1 *rad* ) from (6):

$$P(L) = 0 \implies e^{-2\alpha\_\varepsilon L} = 0 \implies L = \infty$$

we note that is necessary an infinite antenna length.

Substuting (7) in (5), being 0 0 2 / *k* :

$$\mathcal{A}\mathcal{G} \approx \left(\frac{-4\pi}{\ln(1 - \eta\_{rad})\cos\mathcal{G}\_m}\right)\frac{a\_z}{k\_0}$$

Because <sup>2</sup> cos 1 sin , *m m* considering (7)

$$
\Delta \mathcal{G} \approx \left( \frac{-4\pi}{\ln(1 - \eta\_{rad})\sqrt{1 - \left(\frac{\mathcal{J}\_z}{k\_0}\right)^2}} \right) \frac{\alpha\_z}{k\_0} \tag{8}
$$

Since <sup>222</sup> <sup>0</sup> *c z kkk* and having supposed the attenuation constant much smaller than the phase constant, *z z k* , getting:

$$
\Delta \mathcal{G} \approx \left( \frac{-4\pi}{\ln(1 - \eta\_{rad})} \right) \frac{a\_z}{k\_0} \tag{9}
$$

where *<sup>c</sup> k* is the transverse propagation constant. Alternatively, considering (7):

$$A\mathcal{G} \approx \frac{2\pi}{L \cdot k\_c}$$

Using waveguide theory notation, supposing *<sup>c</sup>* the cut-off wavelength:

$$
\Delta \mathcal{G} \approx \frac{\lambda\_c}{L} \tag{10}
$$

(3) and (10), provided the approximations used to be valid, are a valid tool for describing the main parameters of radiated beam.

Fig. 5. Variation of *zL* versus antenna efficiency.

Radiation properties of leaky wave antennas are well described by dispersion diagrams. In fact since leakage occurs over the length of the slit in the waveguiding structure, the whole length constitutes the antenna's effective aperture unless the leakage rate is so great that the power has effectively leaked away before reaching the end of the slit. A large attenuation constant implies a short effective aperture, so that the radiated beam has a large beamwidth. Conversely, a low value of results in a long effective aperture and a narrow beam, provided the physical aperture is sufficiently long.

Moreover since power is radiated continuously along the length, the aperture field of a leakywave antenna with strictly uniform geometry has an exponential decay (usually slow), so that the sidelobe behaviour is poor. The presence of the sidelobes is essentially due to the fact that the structure is finite along z.

When we change the cross-sectional geometry of the guiding structure to modify the value of at some point z, however, it is likely that the value of at that point is also modified slightly. However, since must not be changed, the geometry must be further altered to restore the value of , thereby changing somewhat as well.

In practice, this difficulty may require a two-step process. The practice is then to vary the value of slowly along the length in a specified way while maintaining constant (that is the angle of maximum radiation), so as to adjust the amplitude of the aperture distribution to yield the desired sidelobe performance.

#### **Radiation modes**

54 Wireless Communications and Networks – Recent Advances

4 ln(1 )

 

2 *L kc* 

(3) and (10), provided the approximations used to be valid, are a valid tool for describing the

*c L* 

Using waveguide theory notation, supposing

main parameters of radiated beam.

Fig. 5. Variation of

Conversely, a low value of

*zL* versus antenna efficiency.

provided the physical aperture is sufficiently long.

Radiation properties of leaky wave antennas are well described by dispersion diagrams. In fact since leakage occurs over the length of the slit in the waveguiding structure, the whole length constitutes the antenna's effective aperture unless the leakage rate is so great that the power has effectively leaked away before reaching the end of the slit. A large attenuation constant implies a short effective aperture, so that the radiated beam has a large beamwidth.

results in a long effective aperture and a narrow beam,

where *<sup>c</sup> k* is the transverse propagation constant. Alternatively, considering (7):

0

*<sup>c</sup>* the cut-off wavelength:

(10)

*z*

(9)

0

*<sup>c</sup> rad k k k*

> Let us consider a generic plane wave, whose propagation vector belongs to plane (y-z), directed towards a dielectric film grounded on a perfect electric conductor (PEC) parallel to plane (x-z), as shown in Fig. 6 [8-9].

Fig. 6. Incident wave on a grounded dielectric film, whose thickness is t.

If the incident wave polarization is linear and parallel to the x axis, since both reflection and refraction occur:

$$\begin{cases} E\_{\chi\_0} = A e^{-jk\_{\rm y0} \left( y - t \right)} + C e^{jk\_{\rm y0} \left( y - t \right)} & y \ge t \\ E\_{\chi\_x} = B \cos(k\_{\rm y\_x} y) + D \sin(k\_{\rm y\_x} y) & t \ge y \ge 0 \end{cases}$$

Being the tangent components of electric field null on a PEC surface, B = 0:

$$\begin{cases} E\_{\mathbf{x}\_{0}} = A e^{-jk\_{y0}\{y-t\}} + \mathbf{C} e^{jk\_{y0}\{y-t\}} & y \ge t\\ E\_{\mathbf{x}\_{x}} = D \sin(k\_{y\_{x}}y) & t \ge y \ge 0 \end{cases} \tag{11}$$

One constant can be expressed by the remaining two, as soon as continuity of tangent components of electric field is considered in y = t. First equation of (11) contains an exponential term which, diverging for y → ∞ , violates the radiation condition at infinite distance. Therefore, C ≠ 0 only near plane y = t , at the incidence point. *<sup>z</sup> k* can assume any

value from 0 to 0*k* (i.e.: radiating modes); above it, only discrete values of *kz* exist, identifying the associated guided modes. Since separability condition must be satisfied, in air:

$$k\_0^2 = \alpha^2 \mu\_0 \varepsilon\_0 = k\_{y\_0}^2 + k\_z^2 \text{ where } k\_0 \in \mathfrak{R} \tag{12}$$

For every *<sup>z</sup> k* , it's now possible calculate 0 *<sup>y</sup> k* . In fact, considering only positive solutions:

$$k\_{y\_0} = \sqrt{k\_0^2 - k\_z^2}$$

Obtaining:


Table 1. Wave modes identified by *<sup>z</sup> k* .

Thus, a spectral representation of electromagnetic field near the air-dielectric interface, must contain all values of 0 *<sup>y</sup> k* , from 0 to ∞: the associated integral is complex and slowly convergent. Alternatively, a description, which uses leaky waves and guided modes, both discrete, can well approximate such field.

It's been observed that it's often enough a single leaky wave to obtain a good far field description.

Letting 0 *y y k k* , from (2) and (12), in general:

$$\begin{cases} k\_0 = \beta\_y^2 + \beta\_z^2 - \alpha\_y^2 - \alpha\_z^2 \\ 0 = \beta\_y \alpha\_y + \beta\_z \alpha\_z \end{cases}$$

alternatively

$$\begin{cases} k\_o^2 = \left| \overline{\mathcal{B}} \right|^2 \cdot \left| \overline{\mathcal{a}} \right|^2\\ 0 = \overline{\mathcal{B}} \cdot \overline{\mathcal{a}} \end{cases} \tag{13}$$

Having defined the attenuation and the phase vectors, respectively, as:

$$\begin{aligned} \alpha &= \alpha\_y \overline{y}\_0 + \alpha\_z z\_0 \\\\ \beta &= \beta\_y \overline{y}\_0 + \beta\_z z\_0 \end{aligned}$$

Being 0*k* , from (9) 0 , and 

56 Wireless Communications and Networks – Recent Advances

value from 0 to 0*k* (i.e.: radiating modes); above it, only discrete values of *kz* exist, identifying the associated guided modes. Since separability condition must be satisfied, in

0

For every *<sup>z</sup> k* , it's now possible calculate 0 *<sup>y</sup> k* . In fact, considering only positive solutions:

Thus, a spectral representation of electromagnetic field near the air-dielectric interface, must contain all values of 0 *<sup>y</sup> k* , from 0 to ∞: the associated integral is complex and slowly convergent. Alternatively, a description, which uses leaky waves and guided modes, both

It's been observed that it's often enough a single leaky wave to obtain a good far field

0 0

0 *k*

 

 

*k*

Having defined the attenuation and the phase vectors, respectively, as:

2222

(13)

*y z y z*

*yy zz*

 

<sup>2</sup> <sup>2</sup> <sup>2</sup>

*y* 0 0 *z*

*y* 0 0 *z*

 *y z*

 *y z*

 

 

2 2 *<sup>y</sup>* <sup>0</sup> *<sup>z</sup> k kk*

0

where 0*k* (12)

22 22 0 00 *<sup>y</sup> <sup>z</sup> k kk*

 

**Mode Wave numbers**  Guided *<sup>z</sup>* <sup>0</sup> *k k* <sup>0</sup> *<sup>y</sup> k* Radiating 0 0 *<sup>z</sup> k k* <sup>0</sup> <sup>0</sup> 0 *<sup>y</sup> k k* Evanescent 0 *<sup>z</sup> k j* <sup>0</sup> *<sup>y</sup>* <sup>0</sup> *k k*

air:

Obtaining:

description.

alternatively

Table 1. Wave modes identified by *<sup>z</sup> k* .

discrete, can well approximate such field.

Letting 0 *y y k k* , from (2) and (12), in general:

0

Considering waves propagating in the positive direction of z axis, 0 *<sup>z</sup>* and supposing no losses in z direction, 0 *<sup>z</sup>* , from (13):

$$0 = \beta \cdot a$$

Leaving out , 0 *y y* , two situations can occur: 0 *<sup>y</sup>* and 0 *<sup>y</sup>* . If 0 *<sup>y</sup>* , equations describe a uniform plane wave passing the air-dielectric interface. On the other hand, if 0 *<sup>y</sup>* , two types of superficial waves exist, depending on 0 *<sup>y</sup>* sign:


Fig. 7. Superficial waves at air-dielectric interface when 0 *<sup>y</sup>* .

Because confined superficial waves amplitude decreases exponentially as distance from interface increases, when y is greater than 10 times radiation wavelength, electromagnetic field practically ceases to exist. Improper superficial wave, whose amplitude increases exponentially as distance from interface increases, are not physically possible because they violate the infinite radiation condition.

Removing the hypothesis 0, *<sup>z</sup>* both 0, *<sup>y</sup>* and 0. *<sup>y</sup>*

Fig. 8. General mutual β and α configurations depicting condition 0 .

When losses in dielectric occur, must point towards the inner part of dielectric to compensate such losses (see Fig.8). In the other configuration, when points upwards, even though a non-physical solution is described, the associated wave is useful to describe electromagnetic field near air-dielectric interface.
