4.1 Detection of optical beats

What is usually called an optical beat or simply a beat is the sinusoidal behaviour of the intensity of two overlapping and coherent fields. For example, if we superpose two fields of slightly different frequency, we obtain
E = E cos(ω t) + E cos(ω t) 0 1 0 2 2 2 2 2 (73 ) P = E = E0 (cos(ω1t) + cos (ω2t) + 2cos(ω1t)cos(ω2t)) = E20 (cos2(ω1t) + cos2(ω2t ) + cos(ω+t ) + cos(ω − t)),
with ω+ = ω1 + ω2 and ω− = ω1 − ω2. In this equation the frequency ω− can be very small and can then be detected with the photodiode as illustrated in Figure 22View Image.
P = E2 (1 + cos(ω t)) (74 ) diode 0 −
Using the same example photodiode as before: in order to be able to detect an optical beat ω− would need to be smaller than 100 MHz. If we take two, sightly detuned Nd:YAG lasers with f = 282 THz, this means that the relative detuning of these lasers must be smaller than 10–7.

In general, for a field with several frequency components, the photodiode signal can be written as

|E |2 = E ⋅ E∗ = ∑N N∑ aa ∗ei(ωi−ωj)t. i=0j=0 i j (75 )
For example, if the photodiode signal is filtered with a low-pass filter, such that only the DC part remains, we can compute the resulting signal by looking for all components without frequency dependence. The frequency dependence vanishes when the frequency becomes zero, i.e., in all parts of Equation (75View Equation) with ωi = ωj. The output is a real number, calculated like this:
∑ ∑ ∗ x = aiaj with {i,j | i,j ∈ {0, ...,N } ∧ ωi = ωj}. (76 ) i j

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