#### Quasi-optical line of sight

Figure 1: Very low-flying targets can approach unnoticed below the horizon

#### Quasi-optical line of sight

Electromagnetic waves behave almost like light rays and can therefore be calculated according to optical laws. This is not surprising, because light is also considered an electromagnetic wave. The only difference is the frequency, which is much higher for light than for electromagnetic waves in the radar frequency range.

The line of the quasi-optical view is somewhat longer than the optical view because of the more pronounced diffraction at lower frequencies. Thus, the radar horizon is farther away than the optical horizon. Obstacles along the way also have a different effect. A few trees have a fatal influence on the optical line of view, while electromagnetic waves may be able to penetrate this obstacle.

An elevated observer location is just as effective in optics as an elevated antenna setup is for electromagnetic waves since the curvature of the earth’s surface causes flat objects to disappear very quickly behind the horizon.

The diagram in Figure 1 illustrates the visibility of low-flying aircraft, which is limited by the curvature of the earth.

Figure 2: Two right-angled triangles help to calculate the quasi-optical visibility.

The derivation of the maximum range calculation depending on the antenna height and target height may made according to simple trigonometric rules, which are shown in figure 2. The overall range results from the parts R1 and R2, which may be calculated according to the law of Pythagoras:

(1)

The term for the square of the antenna height in this expression is small compared to the other term and can be ignored, leaving the approximation:

(2)

The resulting error is usually less than 1%. The same procedure can be made in the second triangle for the distance R2 concerning the height of the target:

(3)

From both partial distances, the maximum range can be determined:

(4)

The equivalent earth radius req takes into account the standard refraction and is equal to 4/3 of the mean earth radius of 6 371 km. This root expression in front of the parenthesis can be summed to a constant factor, also taking into account the different units of measurement (on the left side of the equation Rmax is in kilometers, on the right side the heights and req in meters).

(5)

Of particular importance is the line of quasi-optical sight for maritime radars since their mounting height is limited by the height of the mast. This effect is especially true for smaller yachts, where the mandatory navigation radar should be mounted as high up as possible.

Assume that the navigation radar on a warship is at a height of 20 m above sea level. A small fishing boat reaches a height of 5 m with its superstructure. At what distance would the fishing boat be visible on the radar?

(6)

Now calculate again using a complete equation (without this approximation of omitting a term):

(7)

In comparison with the above result, the difference here is only 50 m, which may be ignored since both results depend on the current refraction, which was assumed to be only a standard quantity.