4.2 Signal demodulation

A typical application of light modulation, is its use in a modulation-demodulation scheme, which applies an electronic demodulation to a photodiode signal. A ‘demodulation’ of a photodiode signal at a user-defined frequency ωx, performed by an electronic mixer and a low-pass filter, produces a signal, which is proportional to the amplitude of the photo current at DC and at the frequency ω ± ω 0 x. Interestingly, by using two mixers with different phase offsets one can also reconstruct the phase of the signal, or to be precise the phase difference of the light at ω0 ± ωx with respect to the carrier light. This feature can be very powerful for generating interferometer control signals.

Mathematically, the demodulation process can be described by a multiplication of the output with a cosine: cos(ωx + φx ) (φx is the demodulation phase), which is also called the ‘local oscillator’. After the multiplication was performed only the DC part of the result is taken into account. The signal is

N N S = |E |2 = E ⋅ E∗ = ∑ ∑ a a∗ei(ωi−ωj)t. (77 ) 0 i j i=0 j=0
Multiplied with the local oscillator it becomes
( ) S1 = S0 ⋅ cos(ωxt + φx) = S0 1 ei(ωxt+φx) + e− i(ωxt+ φx) N∑ N∑ (2 ) = 12 aia∗j ei(ωi− ωj)t ⋅ ei(ωxt+φx) + e−i(ωxt+ φx) . (78 ) i=0 j=0
With A = a a∗ ij i j and eiωijt = ei(ωi−ωj)t we can write
( ∑N ∑N ∑N ) ( ) S1 = 1- Aii + (Aij eiωijt + A ∗ e−iωijt) ⋅ ei(ωxt+φx) + e− i(ωxt+ φx) . (79 ) 2 i=0 i=0 j=i+1 ij
When looking for the DC components of S1 we get the following [20]:
∑ 1 − iφ ∗ iφ S1,DC = 2(Aij e x + A ij e x) with {i,j | i,j ∈ {0,...,N } ∧ ωij = ωx} i∑j { −iφx} (80 ) = ℜ Aij e . ij
This would be the output of a mixer and a subsequent low-pass filter. The results for φx = 0 and φx = π∕2 are called in-phase and in-quadrature, respectively (or also first and second quadrature). They are given by
S = ∑ ℜ {A }, 1,DC,phase ij ij S1,DC,quad = ∑ ℑ {Aij}. (81 ) ij
If only one mixer is used, the output is always real and is determined by the demodulation phase. However, with two mixers generating the in-phase and in-quadrature signals, it is possible to construct a complex number representing the signal amplitude and phase:
∑ z = aia∗j with {i,j | i,j ∈ {0,...,N } ∧ ωij = ωx }. (82 ) ij

Often several sequential demodulations are applied in order to measure very specific phase information. For example, a double demodulation can be described as two sequential multiplications of the signal with two local oscillators and taking the DC component of the result. First looking at the whole signal, we can write:

S2 = S0 ⋅ cos(ωxt + φx) cos(ωyt + φy ). (83 )
This can be written as
1 S2 = S0 2(cos(ωyt + ωxt + φy + φx) + cos(ωyt − ωxt + φy − φx)) (84 ) = S0 12(cos(ω+t + φ+ ) + cos(ω − t + φ− )),
and thus reduced to two single demodulations. Since we now only care for the DC component we can use the expression from above (Equation (82View Equation)). These two demodulations give two complex numbers:
∑ z1 = ij Aij with {i,j | i,j ∈ {0,...,N } ∧ ωi − ωj = ω+}, z2 = ∑ A with {k, l | k,l ∈ {0, ...,N } ∧ ω − ω = ω }. (85 ) ij kl k l −
The demodulation phases are applied as follows to get a real output (two sequential mixers)
x = ℜ { (z e− iφx + z eiφx) e−iφy}. (86 ) 1 2
In a typical setup, a user-defined demodulation phase for the first frequency (here φx) is given. If two mixers are used for the second demodulation, we can reconstruct the complex number
− iφx iφx z = z1 e + z2 e . (87 )
More demodulations can also be reduced to single demodulations as above.
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