Drastically increase the antenna bandwidth. Double Resonant Slot Antennas with Enhanced Bandwidth If the slot antenna is fed near one edge by a narrow microstrip line at a proper location and for a proper slot width, at a frequency above that of the first resonance, a fictitious short circuit near the microstrip feed may be created. Since the magnetic slot is a balanced antenna, a 1:1 balun should be used if it is fed. Indicating that the resonant frequency is too low and the wire too long.

A quarter wave antenna is sometimes used because it's shorter than a full wave antenna.
A quarter wave works better than a random wave antenna because of harmonics, I think that's the right word. You get less SWR (google it) using a quarter wave than you would with a 3/16 or a 5/16 wave antenna.
THe feed line does not enter into the calculation if it is shielded wire or a balance line. If you just feed the antenna with a hunk of wire then all of the wire not becomes an antenna and all of your design and math goes out the window, but less of your signal goes out of your antenna.
An antenna is able to receive a signal of any frequency. But frequencies that are multibles of dividibles of the antenna length are recieved much more stronger than those that are at odds with the math.

Slot Antenna

Figure 1: The length of a slot determines the resonant frequency, the width of the slit determines the broad bandwidth of the slot radiator.

Resonant Frequency Of Slot Antenna Signal

Figure 1: The length of a slot determines the resonant frequency, the width of the slit determines the broad bandwidth of the slot radiator.

Slot Antenna

Slot radiators orslot antennas are antennas that are used in the frequency range from about 300 MHz to 25 GHz. They are often used in navigation radar usually as an array fed by a waveguide. But also older large phased array antennas used the principle because the slot radiators are a very inexpensive way for frequency scanning arrays. Slot antennas are an about λ/2 elongated slot, cut in a conductive plate (Consider an infinite conducting sheet), and excited in the center. This slot behaves according to Babinet's principle as resonant radiator. Jacques Babinet (1794 - 1872) was a French physicist and mathematician, formulated the theorem that similar diffraction patterns are produced by two complementary screens (Babinet's principle). This principle relates the radiated fields and impedance of an aperture or slot antenna to that of the field of a dipole antenna. The polarization of a slot antenna is linear. The fields of the slot antenna are almost the same as the dipole antenna, but the field’s components are interchanged: a vertical slot has got an horizontal electric field; and the vertical dipole has got a vertical electrical field.

The impedance of the slot antenna (Z_s) is related to the impedance of its complementary dipole antenna (Z_d) by the relation:

Z_d · Z_s = η²/4

where

Z_s = impedance of the slot antenna
Z_d = impedance of its dual antenna
η = intrinsic impedance of free space.

(1)

It follows for Z_s = 485 Ω.

The band width of a narrow rectangular slot is equal to that of the related dipole, and is equal to half the bandwidth of a cylindrical dipole with a diameter equal to the slot width. Figure 2 shows slot antennas different from the rectangular shape that increasing the bandwidth of the slot antenna.

Figure 2: Various broadband slot antenna.

Although the theory requires an infinite spread conductive surface, the deviation from the theoretical value is small when the surface is greater than the square of the wavelength. The feeding of the slot antenna can be done with ordinary two-wire line. The impedance is dependent on the feeding point, as in a dipole. The value of 485 Ω applies only to a feeding point at the center. A shift of the feed point from the center to the edge steadily decreases the impedance.

The application of slot antennas can be versatile. They can replace dipoles e.g. if it is required a polarization perpendicular to the longitudinal extension of the radiator. If a dipole is used for feeding of a parabolic antenna to generate a vertically orientated but horizontally polarized fan beam, then this dipole must be orientated horizontally. This would mean that the edge surfaces of the parabolic reflector will not be sufficiently illuminated, but a lot of energy above and below the reflector would be lost. In addition, the length of the dipole is extended in a plane, in which is demanding a point like source of radiation for the focus of the parabolic reflector. If this dipole is replaced by a slot antenna, in this case don't appear these disadvantages.

Slots in waveguides

Figure 3: Various slot arrangements in a waveguide.

Slot antennas in waveguides provide an economical way of the design of antenna arrays. The position, shape and orientation of the slots will determine how (or if) they radiate. Figure 3 shows a rectangular waveguide with a drawn with red lines snapshot of the schematic current distribution in the waveguide walls. If slots are cut into the walls, so the current flow is affected more or less depending on the location of the slot. If the slots are sufficiently narrow so the slots B and C (Fig. 3) have little influence on the current distribution. These two slots radiate not (or very little). The slots A and D represent barriers to the current flow. Thus, this current flow acts as an excitation system for the slot, this one acts as radiator. Since the wave in the waveguide moves forward, these drawn lines migrate in the direction of propagation. The slot gets one always alternating voltage potential at its slot edges (depending on the frequency in the waveguide). The power that the slot radiates can be altered by moving the slots closer or farther from the edge. The slots A and D (as drawn in Figure 3) have the strongest coupling to the RF energy transported in the waveguide. In order to reduce this coupling, for example the slot A could be moved closer to one of the shorter waveguide walls. Rotating of the slots would have a the same effect (an angle between the orientations of A and B or C and D). The coupling of this rotated slot ist a factor of about sin² of the rotating angle θ.

Slotted Waveguide Antennas

Figure 4: Basic geometry of a slotted waveguide antenna (The slot radiators are on the wider wall of the rectangular waveguide.)

Several slot radiators in a waveguide form a group antenna. The waveguide is used as the transmission line to feed the elements. In order for radiate in the correct phase, all single slots must be cutted in the distance of the wavelength, that is valid for the interior of the waveguide. This wavelength differs from the wavelength in free space and is a function of the wider side a of a rectangular waveguide. Usually this wavelength is calculated for the TE₁₀ mode by:

a = length of the wider side of the rectangular waveguides
λ_h = “guided” wavelength (within the waveguide)
λ = wavelength in free space(2)

Resonant Frequency Of Slot Antenna Mounts

Figure 5: Basic geometry of a slotted waveguide antenna with rotated slot antennas on the narrower wall.

The wavelength within the waveguide is longer than in free space. The distance of the slot radiators in the group is set at this wavelength to a value that is slightly larger than the wavelength λ in the free space. The number and the size of the sidelobes is affected so unfavorably. The slots are often attached to the left and right eccentrically (with reduced coupling). If mounted on the narrow side of the waveguide, it may happen t