El reconocimiento del ser de las cosas, no el nihilismo, es nuestra “chance”.

0 0.5 Phase Difference (∆Φ) Error Signal Michelsaon Fringe V o lt a g e ( a rb u n it s ) Phase Difference (∆Φ)

Michelson Fringe - DC offset

0 Lock Point + - V o lt a g e ( a rb u n it s )

Figure 3.6:The offset locking error signal is obtained by taking a DC offset from the voltage at the detector at the output of the interferometer.

3.7 Detection theory

This section introduces the theory of the detection. The techniques used are direct de- tection and standard homodyne detection. This is done using the method of linearized operators. These are discussed in terms of measurements and of a GW signal. Finally, the effect of inefficient detection is reviewed.

3.7.1 Direct detection

The direct detection of light is performed on incidence with a photodiode. The current out of the photodiode, thephotocurrent, is proportional to the incident intensity or photon number. The photocurrent is converted to a voltage and viewed on a CRO for a DC value or a spectrum analyzer for broadband detection. For a light field,

, the photon number using linearized operators, derived in equation 2.23 is given by,

(3.7) then the photocurrent is,

(3.8)

This can also be represented in the frequency domain. Taking the Fourier transform of the linearized operators,

, then calculating the photon number once more, (3.9) to give the photocurrent,

26 Interferometer Control and Detection Theory (3.10)

This equation shows the photocurrent contains a large DC term plus the frequency dependent amplitude quadrature fluctuations, scaled by the amplitude of the field. It is interesting to note the absence of phase quadrature fluctuations in this equation. When detecting a large amplitude field, only the fluctuation in quadrature with the field ampli- tude become important. This is the principle used for thelocal oscillatorexplored in the following section.

3.7.2 Standard Homodyne

The sensitivity of measurement of small optical signal,

can be improved by beating with a large amplitude local oscillator(LO), , using thestandard homodyne configuration. This configuration is shown in the figure 3.7. The fields

and are combined on a beam- splitter, with each output detected with a photodetector. The sum and difference of the photocurrents is taken. The fields at the respective output ports,

and

can be calculated. The phase difference between the fields at interference is . For a balanced beamsplitter

we find3_, (3.11) (3.12)

The phase difference between the fields in this calculation is arbitrary. For mathemat- ical convenience the a

phase shift to is introduced giving,

(3.13) (3.14)

using linearized formalism,

(3.15) (3.16)

3_{The notation used to calculate fields: Introduce an}

on transmission of the beamsplitter. For a bal-

3.7 Detection theory 27 a b c d + − i+ / i- a δavac(t) Loss a)

/

Figure 3.7: a) The standard homodyne configuration. The small signal

is interfered on a bal- anced beamsplitter with the local oscillator,

. The detected photocurrents can be either added or subtracted. b) A model for inefficient detection. A mirror with reflectivity equal to the loss of detection is placed in front of an ideal photodetector.

Taking the photon number at each detector, discarding the 2nd order fluctuation terms, (3.17) (3.18)

here the substitutions for

and

have been made. If

terms without

the amplitude of the LO,

, are discarded. We are left with,

(3.19) (3.20) for the fields at the two photodetectors. Here the fluctuation terms have been simplified using Eulers equation and the substitutions for

and

.

These equations are not particularly enlightening. The largest term is the LO beating with itself,

. There are many cross terms that involve fluctuations which have phases difference dependence. Taking the sum and difference photocurrents we find more in- sight.

28 Interferometer Control and Detection Theory (3.21) (3.22) The sum of photocurrents gives the same fields as if the LO was directly detected. This is because of its large coherent amplitude and our assumption of the signal being negligible. Taking the difference, we find many terms are cancelled out, leaving a cross term between the amplitudes of the LO and signal, with dependence on the phase be- tween them. To measure the amplitude of the signal it is ideal to have the

, that is, in quadrature with the LO. We also find two terms involving the amplitude and phase quadratures fluctuations of the signal with phase dependence. By tuning the phase differ- ence, , measurement of either phase or amplitude quadrature fluctuations of the signal

can be performed. It is important to note the noise of the LO does not enter into this equation. This however, is only the case for perfect subtraction and 50:50 beamsplitter.

This calculation indicates, if one wants to measure a small amplitude signal , it

would it desirable to have the minimum fluctuations on and the phase difference with

the LO,

. If the signal has squeezed fluctuations in the correct quadrature the mea- sured noise can be reduced.

3.7.3 The measurement of squeezing

The measurement of squeezing before it interacts with the interferometer in the exper- iment is performed using the standard homodyne. In this case, the small amplitude signal,

is the squeezing and the LO, is a large amplitude coherent state. The difference current is taken, and the phase difference, is scanned. The fluctuations in each quadra-

ture can be easily measured. A shot noise limit can be easily obtained by blocking the squeezed field and taking the difference photocurrent. In this case the small signal is the vacuum state

3.7.4 The Local oscillator in gravitational wave detectors

A gravitational wave detector uses a LO to improve the measurement sensitivity of the GW signal. The LO for the signal is the field also used for the control of the differential mode of the Michelson. In our interferometer, since we use offset locking, the LO is the DC field coupled to the output of the interferometer when the arms are offset. A vector diagram of fields in the Michelson arms and the resultant, the LO, is shown in figure 3.8. The GW signal, which arises dues to differential motion of the arms, is in phase with the LO. The coherent state fluctuations are shown on the signal. This figure shows that only the noise in phase with the signal is important. This is exploited, by the use of squeezing. A full model of this is found in the following chapter.