DC Signal

Signal Extraction

3.1 Optically Heterodyned Signals

3.1.1 DC Signal

The process of mixing and low pass filtering can be mathematically described as a product with a cosine at the demodulation frequency n with phase

/3,

which is then averaged over period T, equal to the period of the modulation.[32]

Vout(t)

=~it ^IEPDI

^{2 cos (r!t'}

⁺

^f3)dt'

t-T

(3.6)

=~

{

E;E2e-it3} (3.7)

This is assuming the detected power of Eq. (3.2), from which it can be seen that a factor of 2 in the signal has been lost. This is because the multiplication at n down- converts the half signal to DC, but also up-converts half the signal to 2n, where it's lost in the low pass filter.

Some care must be taken in extending this to the phase modulated case, where there are two RF sidebands. Eq. (3. 7) assumes that E1 is at a lower frequency than E2 which has consequences on the sign of the demodulation phase. If E1 were at a higher frequency, the demodulation phase would have the opposite sign in the exponential.

The demodulated signal for phase modulated input light of Eq. (3.4) is

Vout(t) =~ {E:.Eoe-i/3

+

E~E+e-i/3}

=~

{

E₀(E_ei.8

+

E+e-i13)*}

(3.8) (3.9)

The demodulation phase f3

=

0 is usually referred to as the "inphase" demodulation, and f3 = 1r /2 is the "quadrature" demodulation. The signal at a specific photodiode at a specific demodulation phase will be referred to as a "signal port."

Signal Sensitivity

Eq. (3. 7) is proportional to the measured voltage as the fields evolve due to the motion of the mirrors in the interferometer. When the interferometer is locked, the deviation away from a nominal DC value (typically 0), due to the fluctuation in a degree of freedom, will be small. A Taylor series expansion around the lock point indicates that the first derivative gives the "gain" of the signal relative to the degree of freedom with which the derivative is taken. A matrix, known as the "matrix of discriminants," can be formed which gives the derivatives at each signal port with respect to every degree of freedom. The photodiode will be referred to by the letter of the transmission function which propagates the light from the input to the photodiode. The matrix element at the photodiode which has a transmission function "t" for the jth degree of freedom is given by

(3.10)

It should be noted at this time that there will be three photodiodes used, and the transmission functions will be indicated by r, for the reflected photodiode, p for the pickoff photodiode, and d for the dark port photodiode. The subscript t in this instance should not be confused with the transmissivity of a mirror. The subscript of f3t likewise indicates the photodiode output to which the demodulation is applied.

The subscripts of M will indicate the photodiode and degree of freedom, respectively.

Two things have been assumed here. First, the amplitudes E₁ and E₂ incident on the photodiode are taken to be E1a1t1 and E1a₂t_{2 ,} respectively, where the t/s are the transmission functions, and the ai 's are the input field amplitudes relative to the amplitude of the laser light, E1• Second, it's assumed that the relative phase of a1

and a2 is ^1r /2. Casting the matrix element in this form has the utility of relating the

signal sensitivity to the interferometer transmission functions. The actual values of the incident field amplitudes are then constant scaling factors for all matrix elements of a single signal port.

The phase modulated input field has a matrix described by

(3.11)

A row in the matrix of discriminants shows the relative DC sensitivity of a signal port to the degrees of freedom of the interferometer. This can be used to decide which signal ports should be used to control each of the degrees of freedom.

Excunples

A couple of examples, using the phase modulated input fields, are good pedagog- ical tools for developing some intuition about what these demodulated signals are measuring. It's first assumed that Eo is real. The RF sidebands are assumed equal, E+

=

E_, and imaginary. The derivatives of the carrier and RF sidebands are as- sumed to be imaginary and real, respectively. This is true for fields which are incident on an interferometer which has strictly resonant or anti-resonant conditions. ¹

The measured inphase (/3

=

0) signal is

(3.12) The matrix element associated with this is found to be

M _{t,j ex} 2^('>: _~ ^{ _{o¢j t+} ^{oto *}

₊

_{to o¢j} ^ot+

*}

(3.13)

(

1 oto 1 ot+)

ex 2t₀t+ - - - -

-ito o¢j - it+ o¢j (3.14)

The quantity

-itk

^{~ is} easily recognized as the derivative of the phase of ^tk with respect to the parameter B. This indicates that the inphase demodulation is sensitive

1 In general, there may also be an overall phase factor which can be removed, maintaining the assumption of real carrier and imaginary RF sidebands.

to the difference in the rates of change of phase between the carrier and the RF sidebands. Another way to say this is that the inphase demodulation measures the differential phase modulation of the carrier and RF sidebands relative to each other.

The quadrature demodulation is orthogonal to the inphase demodulation, (3 =

1r /2. The measured signal is

(3.15)

and its associated matrix element is

(3.16)

Here, it's seen that the quadrature demodulation measures the differential amplitude modulation of the two RF sidebands.

It will be seen later that the assumptions about the relative phases of the carrier and RF sidebands incident on the photodiode typically do not hold for detuned inter- ferometers. However, the concepts embodied in these simple examples are generally useful.