Using Guided Wavelength as a Basis for the Design of Dielectric Rod Antennas
When designing dielectric rod antennas for maximum bandwidth, maximum gain, minimum beamwidth and/or low side lobes, design parameters reported in the literature are provided in terms of freespace wavelength. While this methodology lends itself well to other antenna types, it is not as useful when used to describe dielectric rod antennas. Experimental data of two physically identical antennas measured using identical system parameters, but fabricated from two different dielectric materials, will exhibit very different antenna patterns. Therefore, to state that a dielectric rod antenna of so many freespace wavelengths will produce a pattern with certain beamwidth characteristics does not result in a good rule of thumb that can be applied to another antenna fabricated from a different dielectric material.
This article proposes to establish the use of the material guided wavelength as the basis of describing specifications for dielectric rod antenna fabrication and, in doing so, to provide a basic specification for the design of dielectric rod antennas from a variety of viable dielectric materials that will produce similar antenna pattern characteristics.
The Guided Wavelength
Dielectric rod antennas are typically inserted into circular or rectangular waveguides. As the electromagnetic wave travels down the waveguide, it will encounter the dielectric material. Some of the energy will be reflected back into the waveguide due to the inherent impedance mismatch at the transition between the waveguide and the dielectric material, but most of the energy is transferred into the dielectric material. As the wave exits the waveguide feed, some energy will be refracted from the dielectric surface and radiate into space. Most of the energy will be reflected ("bound") back into the material due to the impedance mismatch between air and the dielectric material (Figure 1 ).
The term "guided wavelength" may be something of a misnomer, as the mechanism of propagation can be characterized as a traveling or surface wave. But Kiely, in his treatise,1 refers to the wave as the "guided wavelength;" Johnson/Jasik,^{2} on the other hand, use the term "trapped wave," so this usage has some precedent.
The freespace wavelength, _{f}, is provided in terms of frequency, f, and the speed of light, c, by the equation
This is the general form used to derive the wavelength from a desired frequency. The freespace wavelength is then related to a physical dimension to arrive at an antenna specification. As an example, an antenna of length 6 _{f} at a frequency of 10 GHz would be 18 cm long physically.
The wavelength of the guided wave, _{g}, can be determined from the permittivity, , and loss tangent, tan , of the dielectric material, and the freespace wavelength:
For loss tangents on the order of 10^{4}, the guided wavelength is approximately
The ratio of the wavelength or phase velocity in a vacuum to that in a dielectric material determines the index of refraction, n, of the dielectric medium and can be expressed in terms of the wavelength in freespace and that of the guided wave:
and from Equation 3, the index of refraction can be expressed in terms of the permittivity as
The guided wavelength traveling within the antenna will be shorter than the freespace wavelength. For cases where the guided wavelength is shorter than the freespace wavelength, n > 1, the phase velocity of the wave will be slower than the speed of light. As an example, for a dielectric material with = 2.5, a frequency of 10 GHz would result in a guided wavelength of 1.9 cm. The index of refraction would then be 1.58. The guided wave is traveling slower than light.
The physical quantity of the material defined in terms of area, length and diameter also affect the length of the guided wave. Circular rod antennas with small constant diameters along the length of the rod will tend to allow more energy to radiate out of the antenna structure as the wave propagates. Because energy is being removed from the material, the guided wavelength will be very close or equal to the freespace wavelength, as opposed to a larger diameter rod in which less energy radiates out and is reflected back into the antenna structure. As stated previously, this causes the guided wavelength to be shorter than the freespace wavelength. The ratio of the freespace wavelength to the guided wavelength defines the efficiency of the antenna
When the area of the rod is small, this ratio tends to become less than unity, and more energy is radiated out of the antenna structure. As the area becomes large, this ratio will be greater than unity, and more energy will propagate through the material. When designing the antenna, it is actually desirable to keep this ratio slightly greater than unity, thus increasing the efficiency. By designing the rod with a k for the feed taper around 1.35 and continuing the taper along the rod so that k approaches 1.2, side lobe levels can be reduced. By maintaining k close to 1.4 with a taper along the rod from feed to termination, the bandwidth of the antenna can be doubled. It is therefore beneficial for dielectric rod antennas that the guided wavelength be less than the freespace wavelength.
When related to an antenna length of a few wavelengths, this can have an impact on antenna design and performance. As an example, consider an antenna with reported specifications of 6 _{f} in length, = 2.2 and a frequency of 10 GHz. Its physical length is 18 cm. The guided wavelength for this antenna would be 2.0 cm, and the length, based on the guided wavelength, would be 9.0 _{g}. If a designer were to review this antenna pattern and determine to fabricate an antenna from a different material with a permittivity of 3.4, then it is probable that he would also use the specification of 6 _{f}, or 18 cm long. Unfortunately, he would be disappointed with the results. This antenna would have a guided wavelength of 1.6 cm, making it 11.25 _{g }long. This antenna would actually be 2.25 guided wavelengths longer. Therefore, basing the design on freespace wavelength can result in confusion and an antenna that will not produce the expected performance. Reported design specifications based on the guided wavelength would highlight that antenna performance is dependent on the dielectric material used and would provide a specification which could be more readily translated to other materials producing comparative results.
Experimental Results
Two dielectric rod antennas were fabricated with the same physical dimensions but of different materials, and their antenna patterns measured. The general antenna design is presented in Figure 2 .
One rod was fabricated from Teflon® and the other from Delrin® blend acetal. Antenna patterns for both rods are presented in Appendix A . As can be observed, the antenna patterns for the two different antennas are not identical with respect to frequency. The antenna fabricated from Delrin provides a favorable antenna pattern at 10 GHz, while the Teflon antenna performs better at 12 GHz. If a center frequency of 10 GHz were chosen for design purposes, then for this case the length of both antennas would be 18 cm. This translates to a design specification of 6 _{f}. Had a designer using the Teflon material based the design on the reported results for the Delrin case, not realizing that performance is dependent on the permittivity of the material, and not being privy to the Teflon data, the performance of the Teflon antenna would not have achieved the desired result.
If the designer had based the design on the guided wave specification, then the results would have been much different. The Delrin specification could have been translated to provide an antenna with different physical dimensions, but similar antenna pattern characteristics.
Table 1 presents the specifications of the freespace and guided wavelengths for the test antennas. Lengths are in centimeters and frequency is in gigahertz. For Teflon, = 2.08, and for Delrin, = 3.1. The length of the rod from feed to termination is 18 cm.
From Figure 3 , it is apparent that the patterns for Teflon at 10 GHz and Delrin at 8 GHz are similar. The same is true for the patterns for Teflon at 12 GHz and Delrin at 10 GHz. From Table 1, the guided wavelength for Teflon at 10 GHz and Delrin at 8 GHz are both close to 2.1 cm. Likewise, for Teflon at 12 GHz and Delrin at 10 GHz, the guided wavelength is close to 1.7 cm. Also, the number of guided wavelength in the physical length for each case is very close. This can also be verified as follows. The guided wavelength for Delrin at 10 GHz would be
The freespace wavelength for Teflon at this guided wavelength would be
This test frequency for this wavelength would be
And indeed, this result is verified from Table 1 and Figure 3: the antenna pattern for Delrin at 10 GHz and 10.6 _{g} would be equivalent to the antenna pattern for Teflon at 12.2 GHz.
Translating Guided Wavelength
A method will now be presented for the determination of the physical length of an antenna based on the guided wavelength for an antenna fabricated from a different dielectric material. Similar methodology could be used to derive other parameters, such as rod diameter or area.
The antenna pattern for Delrin at 10 GHz appears to have favorable characteristics. If a designer wanted to fabricate an antenna from Teflon that would have a similar pattern, then the Delrin 10 GHz results with an antenna length of L_{Del}=10.6 _{g} could be used to determine a comparable design using Teflon.
The guided wavelength for Teflon would be
and the physical length would be
The antenna patterns for a dielectric rod antenna fabricated from Teflon with a physical length of 22 cm and the physical shape shown previously are presented in Appendix B . The actual antennas are shown in Figure 3 .
Table 1  
Material  Frequency  _{f}  No.  _{g}  No. 
Teflon  8  3.75  4.8  2.60  6.90 
Delrin  8  3.75  4.8  2.13  8.50 
Teflon  10  3.00  6.0  2.08  8.65 
Delrin  10  3.00  6.0  1.70  10.60 
Teflon  12  2.50  7.2  1.73  10.40 
Delrin  12  2.50  7.2  1.42  12.70 
Conclusion
The antenna patterns for the 22 cm Teflon antenna and the 18 cm Delrin antenna, when compared with the original Delrin antenna patterns, confirm that an antenna fabricated from a different dielectric material can be specified using the length of a known test sample when given in number of guided wavelengths.
While it is true that the 12 GHz pattern does not match, the antenna was designed for 10 GHz, where the patterns were an excellent match. The 12 GHz result does highlight that the dielectric material directly impacts the antenna pattern and performance, and also indicates that the shape of the antenna does not lend itself well to a wideband design.
It is acknowledged that the same result in Equation 11 could have been derived from test sample data that gives the antenna length in terms of the number of freespace wavelengths, but that would still have required that the number of guided wavelengths be determined prior to solving Equation 11.
Using the number of guided wavelengths to describe the physical length of a dielectric rod antenna therefore highlights an important characteristic. It declares that dielectric rod antenna performance is dependent on the dielectric material used, making guided wavelength the primary specification for defining dielectric rod antennas. This highlights to the designer that dielectric rod antennas are a special class of antenna that cannot be fabricated only from typical antenna specifications; however, when designers see antenna length based on freespace wavelength, they may not realize that this is the case.
Using guided wavelength also clarifies the possible misconception that the number of freespace wavelengths could be used directly in the design of an antenna using a different dielectric material. For example, a designer who was satisfied with the results of the original Delrin data and then rushed off to fabricate an antenna made of Teflon with a physical length of 18 cm would be disappointed with the results and the design process would be delayed.
It is proposed that when presenting dielectric rod antenna data, the length of the antenna be given in terms of the number of guided wavelengths. Then a designer examining antenna pattern results would be able to identify a favorable pattern from the literature, obtain the guided wavelength specification and fabricate an antenna from a desired dielectric material, and achieve similar results in a direct technique. It is further proposed to standardize this term as the "guided wavelength."
References
1. D.G. Kiely, Dielectric Aerials , John Wiley & Sons Inc., New York, NY, 1954, pp. 1580.
2. R.C. Johnson and H. Jasik, Antenna Engineering Handbook , 2nd Edition, McGrawHill, New York, NY, 1984, pp. 12.212.16, 12.21, 12.22.
3. A.R. Von Hippel, Dielectric Materials and Applications , MIT Press, 1954, pp. 913.
Carl L. Everett received his BSEET degree from Texas Technological University, Lubbock, TX, in 1985. Currently, he is an engineer at Bell Helicopter Textron Inc., performing Doppler signature research, antenna design and integration, radar crosssection reduction, and material measurement system development.
Appendix A: Antenna Patterns for Teflon® (Blue) and Delrin® (Red) with LP=18 cm.

Appendix B: Antenna Patterns for Teflon® with LP=22 cm (Blue) and Delrin® with LP=18 cm (Red).
