two locations along the wave path where the impedance changes abruptly; the mouth or aperture of the horn, and the throat where the sides begin to flare out. The amount of reflection at these two sites varies with the flare angle of the horn (the angle the sides make with the axis). In narrow horns with small flare angles most of the reflection occurs at the mouth of the horn. The gain of the antenna is low because the small mouth approximates an open-ended waveguide, with a large impedance step. As the angle is increased, the reflection at the mouth decreases rapidly and the antenna's gain increases. In contrast, in wide horns with flare angles approaching 90° most of the reflection is at the throat. The horn's gain is again low because the throat approximates an open-ended waveguide. As the angle is decreased, the amount of reflection at this site drops, and the horn's gain again increases. This discussion shows that there is some flare angle between 0° and 90° which gives maximum gain and minimum reflection. This is called the optimum horn. Most practical horn antennas are designed as optimum horns. In a pyramidal horn, the dimensions that give an optimum horn are: For a conical horn, the dimensions that give an optimum horn are: where aE is the width of the aperture in the E-field direction aH is the width of the aperture in the H-field direction LE is the slant height of the side in the E-field direction LH is the slant height of the side in the H-field direction d is the diameter of the cylindrical horn aperture L is the slant height of the cone from the apex λ is the wavelength An optimum horn does not yield maximum gain for a given aperture size. That is achieved with a very long horn (an aperture limited horn). The optimum horn yields maximum gain for a given horn length. Tables showing dimensions for optimum horns for various frequencies are given in microwave handbooks. Gain Horns have very little loss, so the directivity of a horn is roughly equal to its gain. The gain G of a pyramidal horn antenna (the ratio of the radiated power intensity along its beam axis to the intensity of an isotropic antenna with the same input power) is: For conical horns, the gain is: where A is the area of the