 ### Introduction

Coherence is a phenomenon which appears in many situations. In this course we will concentrate on electromagnetic signals, waves, and fields. We'll use examples from simple radios to millimetre waves and optics. This wide variety has been chosen deliberately so as to emphasise the general/universal nature of coherent behaviour. A signal or wave is said to be coherent if its behaviour at various times/places is linked in a deterministic way. Most textbooks and discussions tend to concentrate on simple examples like sinewaves. This is mathematically convenient, but it's useful to remember that many other kinds of field or wave pattern can show coherence. For example, the coherent properties of a ‘white’ light can be exploited in an interferometer to obtain a spectrum of the light. We'll be looking at interferometry later. This first section will concentrate on the use of mixers and heterodyne receivers to detect and process coherent signals.

Note that we'll sometimes be talking about fields & waves, at other times about signals. The term ‘signal’ is generally used in information theory and communications to indicate that the pattern of variations of some quantity is being used to communicate some information. Here we don't really need to worry much about the precise distinction between ‘waves’, ‘fields’, and ‘signals’. In each case we're interested in the behaviour of something which links together what we see in two (or more) locations in space-time.

Some of the most common requirements when dealing with coherent signals and waves are that we should measure the size (amplitude or power) and the frequency of the signal/wave. This requirement arises in many situations. For example, we may need to do this to be able to control the output from a laser or to recover information (e.g. music from a distant radio station). These measurements can be done in various ways. In this section we will begin by looking at how Mixers can be used to make Heterodyne measurements of the properties of a signal. Mixers appear in various forms throughout physics & engineering. (We'll ignore those which appear in the kitchen!) Here we will consider two examples. The first is a conventional electronic diode of the kind used in radios, TV's, and other electronic systems like radar & millimeter-wave receivers. The second is an optical photodetector.

### Mixer Diodes.

Conventional electronic diodes are made by joining together two dissimilar materials. The two most common types are PN-Junctions and Schottky diodes. The PN-Junction is formed by joining two pieces of semiconductor — one piece doped N-type, the other P-type. The Schottky diode is made by joining together a suitable piece of metal and a piece of semiconductor. The general properties of these diodes are illustrated in figure 1.1. The first property to note is that the diode only passes electric current when the applied voltage between its terminals/wires has the polarity referred to as forward biassed. When reverse biassed, almost no current can get through the diode. The second property is that — unlike a resistor — the size of the diode current when forward biassed is not simply proportional to the applied voltage between the wires. The diode is therefore said to be a Nonlinear device. Most physics textbooks will tell you that the I-V dependence of a diode has an exponential form, typically quoting an equation like  where and h are constants whose values depend upon the details of the diode, T is the diode's temperature in Kelvins and k is Boltsmann's constant. In practice, the shape of a diode's I-V curve depends on the details of how it was made, hence many real diodes only approximate to exponential behaviour. In general, the I-V behaviour of a real diode can be represented in terms of a suitable polynomial series,  where the values are chosen to suit the diode being considered. For our purposes it's convenient to simplify this and assume that we have a Square Law diode whose behaviour obeys  This square law assumption makes the following explanation much easier to follow. It also makes the diode work very well! As a result, many real diodes are deliberately manufactured to behave in a square law manner, so the assumption turns out to be a good description of reality. Consider the situation illustrated in figure 1.2. Here we are applying a combination of three input voltages to a square law diode; an input signal, , an steady (d.c.) level, , and a sinewave, . The total voltage across the diode at any instant will therefore be  where we can say that  For simplicity we can begin by assuming that the input signal is also a sinewave   We can also arrange that — i.e. the diode is always forward biassed. For a square law device this means that the diode current at any instant will be  This can be rearranged into the form     Looking at expression 1.8 we can see that, having applied a pair of input voltage fluctuations at the frequencies & , we set up current fluctuations at the frequencies, , , , and , as well as at the frequencies , and . There is also a steady current component (i.e. a ‘zero Hertz’). This pattern of current fluctuations can be treated as the ‘output’ from the diode when it is driven by the input voltages we've chosen. We can now guide a portion of this output current out via a suitable filter which only passes frequencies similar to the Difference Frequency, . As a result we can get an output voltage  We'll be considering the uses of this effect later on. For now, it is enough to note three significant points
• The amplitude of is proportional to the amplitude, A, of the input signal.
• The frequency, , of the output depends on the signal frequency, .
• The phase of the output is the same as that of the signal.

As a result, we can expect a measurement of the amplitude, frequency, and phase of the output to provide information about the amplitude, frequency, and phase of the input signal. In effect the heterodyne system lets us ‘shift’ the frequency of a signal whilst preserving any information we might need to know about its amplitude, etc.

Consider an example where we have an input signal at a frequency of, say 50 GHz (50×109 Hz). This comes from a ‘distant’ signal source — i.e. one which isn't a part of our measurement system. The signal is applied to the diode along with an output at, say, GHz, from a Local Oscillator (LO)— i.e. one which is part of our measurement system. The diode's non-linearity is said to ‘mix’ these two signals to produce an output at the frequency, = 1 MHz. This frequency is much lower than that of the original signal, hence it is much easier to amplify, filter, and measure its properties. The diode, local oscillator, and filter act together as a Heterodyne Receiver. Heterodyning is the process of ‘beating together’ or Mixing two different frequencies to obtain an output at some other, related frequency. When used in this way the diode is called a Mixer Diode.

The heterodyne receiver is an example of a system which uses the technique called Frequency Conversion. An input at the frequency is converted to an output at . For obvious reasons this is also called Down Conversion. In this case the output filter selects an output frequency which is lower than either the signal or local oscillator frequency. If we'd wished we could have used a filter which only passed frequencies around the Sum Frequency, . The heterodyne system would then perform Up Conversion on the input signal. In either case it is normal to refer to the filtered output from the mixer as the Intermediate Frequency (or ‘IF’) output. This is because the output from the mixer and filter is often processed by other circuits before being reconverted into the output finally required.   Content and pages maintained by: Jim Lesurf (jcgl@st-and.ac.uk)
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University of St. Andrews, St Andrews, Fife KY16 9SS, Scotland.