2.3 Basics of Detection: Heterodyne and homodyne readout techniques

Let us now address the question of how one can detect a GW signal imprinted onto the parameters of the light wave passing through the interferometer. The simple case of a Michelson interferometer considered in Section 2.1.2 where the GW signal was encoded in the phase quadrature of the light leaking out of the signal(dark) port, does not exhaust all the possibilities. In more sophisticated interferometer setups that are covered in Sections 5 and ??, a signal component might be present in both quadratures of the outgoing light and, actually, to different extent at different frequencies; therefore, a detection method that allows measurement of an arbitrary combination of amplitude and phase quadrature is required. The two main methods are in use in contemporary GW detectors: these are homodyne and heterodyne detection [36Jump To The Next Citation Point, 138Jump To The Next Citation Point, 164Jump To The Next Citation Point, 79Jump To The Next Citation Point]. Both are common in radio-frequency technology as methods of detection of phase- and frequency-modulated signals. The basic idea is to mix a faint signal wave with a strong local oscillator wave, e.g., by means of a beamsplitter, and then send it to a detector with a quadratic non-linearity that shifts the spectrum of the signal to lower frequencies together with amplification by an amplitude of the local oscillator. This topic is also discussed in Section 4 of the Living Review by Freise and Strain [59] with more details relevant to experimental implementation.

2.3.1 Homodyne and DC readout

Homodyne readout.
Homodyne detection uses local oscillator light with the same carrier frequency as the signal. Write down the signal wave as:

S(t) = Sc(t)cosω0t + Ss(t)sinω0t (37 )
and the local oscillator wave as:
L (t) = Lc (t)cos ω0t + Ls(t)sinω0t. (38 )
Signal light quadrature amplitudes Sc,s(t) might contain GW signal Gc,s(t) as well as quantum noise nc,s(t) in both quadratures:
S (t) = G (t) + n (t), c,s c,s c,s
while the local oscillator is a laser light with classical amplitudes:
(L (0),L (0)) = (L0cos ϕLO, L0sinϕLO ), c s
where we introduced a homodyne angle ϕLO, and laser noise lc,s(t):
(0) Lc,s(t) = L c,s + lc,s(t).
Note that the local oscillator classical amplitude L 0 is much larger than all other signals:
L ≫ max [l ,G ,n ]. 0 c,s c,s c,s
Let mix these two lights at the beamsplitter as drawn in the left panel of Figure 11View Image and detect the two resulting outgoing waves with two photodetectors. The two photocurrents i1,2 will be proportional to the intensities I 1,2 of these two lights:
--------- (L + S )2 L2 [ ] i1 ∝ I1 ∝ ---------= -0-+ L0 (Gc + lc + nc)cos ϕLO + L0 (Gs + ls + ns )sin ϕLO + 𝒪 G2c,s,l2c,s,n2c,s , ----2---- 2 (L-−-S)2- L20- [ 2 2 2 ] i2 ∝ I2 ∝ 2 = 2 − L0 (Gc − lc + nc)cos ϕLO − L0 (Gs − ls + ns )sin ϕLO + 𝒪 Gc,s,lc,s,nc,s ,
where -- A stands for time averaging of A (t) over many optical oscillation periods, which reflects the inability of photodetectors to respond at optical frequencies and thus providing natural low-pass filtering for our signal. The last terms in both expressions gather all the terms quadratic in GW signal and both noise sources that are of the second order of smallness compared to the local oscillator amplitude L0 and thus are omitted in further consideration.
View Image

Figure 11: Schematic view of homodyne readout (left panel) and DC readout (right panel) principle implemented by a simple Michelson interferometer.

In a classic homodyne balanced scheme, the difference current is read out that contains only a GW signal and quantum noise of the dark port:

i = i − i ∝ 2L [(G + n )cosϕ + (G + n ) sin ϕ ]. (39 ) hom 1 2 0 c c LO s s LO
Whatever quadrature the GW signal is in, by proper choice of the homodyne angle ϕLO one can recover it with minimum additional noise. That is how homodyne detection works in theory.

However, in real interferometers, the implementation of a homodyne readout appears to be fraught with serious technical difficulties. In particular, the local oscillator frequency has to be kept extremely stable, which means its optical path length and alignment need to be actively stabilized by a low-noise control system [79Jump To The Next Citation Point]. This inflicts a significant increase in the cost of the detector, not to mention the difficulties in taming the noise of stabilising control loops, as the experience of the implementation of such stabilization in a Garching prototype interferometer has shown [60, 75, 74Jump To The Next Citation Point].

DC readout.
These factors provide a strong motivation to look for another way to implement homodyning. Fortunately, the search was not too long, since the suitable technique has already been used by Michelson and Morley in their seminal experiment [109]. The technique is known as DC-readout and implies an introduction of a constant arms length difference, thus pulling the interferometer out of the dark fringe condition as was mentioned in Section 2.1.2. The advantage of this method is that the local oscillator is furnished by a part of the pumping carrier light that leaks into the signal port due to arms imbalance and thus shares the optical path with the signal sidebands. It automatically solves the problem of phase-locking the local oscillator and signal lights, yet is not completely free of drawbacks. The first suggestion to use DC readout in GW interferometers belongs to Fritschel [63] and then got comprehensive study by the GW community [138Jump To The Next Citation Point, 164Jump To The Next Citation Point, 79Jump To The Next Citation Point].

Let us discuss how it works in a bit more detail. The schematic view of a Michelson interferometer with DC readout is drawn in the right panel of Figure 11View Image. As already mentioned, the local oscillator light is produced by a deliberately-introduced constant difference δL of the lengths of the interferometer arms. It is also worth noting that the component of this local oscillator created by the asymmetry in the reflectivity of the arms that is always the case in a real interferometer and attributable mostly to the difference in the absorption of the ‘northern’ and ‘eastern’ end mirrors as well as asymmetry of the beamsplitter. All these factors can be taken into account if one writes the carrier fields at the beamsplitter after reflection off the arms in the following symmetric form:

out E0 ¯ℰ - En (t) = − √--(1 − 𝜖n)cosω0 (t − 2Ln ∕c) = − √--(1 − Δ 𝜖)cosω0(t + ΔL ∕c), 2 2 out E0-- -¯ℰ-- - E e (t) = − √2-(1 − 𝜖e)cosω0 (t − 2Le ∕c) = − √2-(1 + Δ 𝜖) cosω0(t − ΔL ∕c),
where 𝜖n,e and Ln,e stand for optical loss and arm lengths of the corresponding interferometer arms, Δ 𝜖 = 𝜖2n(−1𝜖−e¯𝜖) is the optical loss relative asymmetry with ¯𝜖 = (𝜖n + 𝜖e)∕2, -- ℰ = E0(1 − ¯𝜖) is the mean pumping carrier amplitude at the beamsplitter, ΔL = Ln − Le and ¯ Ln+Le- t = t + 2c. Then the classical part of the local oscillator light in the signal (dark) port will be given by the following expression:
(0) Eount(t) −-Eoeut(t)- ¯ ω0ΔL-- ¯ ¯ ω0-ΔL- ¯ L DC(t) = √2-- = ℰΔ 𝜖cos c cosω0 t + ℰ sin c sin ω0t, (40 ) ◟---√-◝◜(0)----◞ ◟--√-◝◜(0)-◞ 2Lc 2Ls
where one can define the local oscillator phase and amplitude through the apparent relations:
∘ --------(-------)-- 1 ω0ΔL (0) ω0ΔL 2 ω0E0 ΔL tan ϕDC = ---tan ------ L DC ≃ ℰ¯ (Δ 𝜖)2 + ------ ≃ --------, (41 ) Δ 𝜖 c c c
where we have taken into account that ω0ΔL ∕c ≪ 1 and the rather small absolute value of the optical loss coefficient max [𝜖 ,𝜖 ] ∼ 10 −4 ≪ 1 n e available in contemporary interferometers. One sees that were there no asymmetry in the arms optical loss, there would be no opportunity to change the local oscillator phase. At the same time, the GW signal in the considered scheme is confined to the phase quadrature since it comprises the time-dependent part of ΔL and thus the resulting photocurrent will be proportional to:
--------- ( )2 iDC ∝ (L + S )2 ≃ L (0) + 2L (0)(locut+ nc)cos ϕDC + 2L (0)(Gs + lsout+ ns)sinϕDC, (42 ) DC DC DC
where locu,ts denote the part of the input laser noise that leaked into the output port:
out in ω0ΔL-- in ω0ΔL-- in inω0-ΔL- lc (t) ≃ lc Δ 𝜖cos c − ls sin c ≃ lc Δ 𝜖 − ls c , (43 ) ω ΔL ω ΔL ω ΔL loust(t) ≃ licnsin -0----+ lins Δ 𝜖 cos-0--- ≃ linc -0----+ lins Δ 𝜖, (44 ) c c c
and nc,s stand for the quantum noise associated with the signal sidebands and entering the interferometer from the signal port.

In the case of a small offset of the interferometer from the dark fringe condition, i.e., for ω0 ΔL ∕c = 2πΔL ∕λ0 ≪ 1, the readout signal scales as local oscillator classical amplitude, which is directly proportional to the offset itself: L(D0C)≃ 2πE0 ΔλL0-. The laser noise associated with the pumping carrier also leaks to the signal port in the same proportion, which might be considered as the main disadvantage of the DC readout as it sets rather tough requirements on the stability of the laser source, which is not necessary for the homodyne readout. However, this problem, is partly solved in more sophisticated detectors by implementing power recycling and/or Fabry–Pérot cavities in the arms. These additional elements turn the Michelson interferometer into a resonant narrow-band cavity for a pumping carrier with effective bandwidth determined by transmissivities of the power recycling mirror (PRM) and/or input test masses (ITMs) of the arm cavities divided by the corresponding cavity length, which yields the values of bandwidths as low as ∼ 10 Hz. Since the target GW signal occupies higher frequencies, the laser noise of the local oscillator around signal frequencies turns out to be passively filtered out by the interferometer itself.

DC readout has already been successfully tested at the LIGO 40-meter interferometer in Caltech [164] and implemented in GEO 600 [77, 79Jump To The Next Citation Point, 55] and in Enhanced LIGO [61, 5]. It has proven a rather promising substitution for the previously ubiquitous heterodyne readout (to be considered below) and has become a baseline readout technique for future GW detectors [79].

2.3.2 Heterodyne readout

Up until recently, the only readout method used in terrestrial GW detectors has been the heterodyne readout. Yet with more and more stable lasers being available for the GW community, this technique gradually gives ground to a more promising DC readout method considered above. However, it is instructive to consider briefly how heterodyne readout works and learn some of the reasons, that it has finally given way to its homodyne adversary.

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Figure 12: Schematic view of heterodyne readout principle implemented by a simple Michelson interferometer. Green lines represent modulation sidebands at radio frequency ΩRF and blue dotted lines feature signal sidebands

The idea behind the heterodyne readout principle is the generalization of the homodyne readout, i.e., again, the use of strong local oscillator light to be mixed up with the faint signal light leaking out the dark port of the GW interferometer save the fact that local oscillator light frequency is shifted from the signal light carrier frequency by Ω ∼ RF several megahertz. In GW interferometers with heterodyne readout, local oscillator light of different than ω0 frequency is produced via phase-modulation of the pumping carrier light by means of electro-optical modulator (EOM) before it enters the interferometer as drawn in Figure 12View Image. The interferometer is tuned so that the readout port is dark for the pumping carrier. At the same time, by introducing a macroscopic (several centimeters) offset ΔL of the two arms, which is known as Schnupp asymmetry [134Jump To The Next Citation Point], the modulation sidebands at radio frequency ΩRF appear to be optimally transferred from the pumping port to the readout one. Therefore, the local oscillator at the readout port comprises two modulation sidebands, Lhet(t) = L+ (t) + L− (t), at frequencies ω0 + ΩRF and ω0 − ΩRF, respectively. These two are detected together with the signal sidebands at the photodetector, and then the resulting photocurrent is demodulated with the RF-frequency reference signal yielding an output current proportional to GW-signal.

This method was proposed and studied in great detail in the following works [65, 134, 60, 75, 74, 104, 116] where the heterodyne technique for GW interferometers tuned in resonance with pumping carrier field was considered and, therefore, the focus was made on the detection of only phase quadrature of the outgoing GW signal light. This analysis was further generalized to detuned interferometer configurations in [36Jump To The Next Citation Point, 138] where the full analysis of quantum noise in GW dual-recycled interferometers with heterodyne readout was done.

Let us see in a bit more detail how the heterodyne readout works as exemplified by a simple Michelson interferometer drawn in Figure 12View Image. The equation of motion at the input port of the interferometer creates two phase-modulation sideband fields (L+ (t) and L− (t)) at frequencies ω0 ± ΩRF:

[ ] [ ] L (t) = L (0) + l cos(ω + Ω )t + L (0) + l sin(ω + Ω )t, + [ c+ c+] 0 RF [ s+ s+ ] 0 RF (0) (0) L− (t) = L c− + lc− cos(ω0 − ΩRF )t + L s− + ls− sin (ω0 − ΩRF )t,
where (0) L(c,s)± stand for classical quadrature amplitudes of the modulation (upper and lower) sidebands4 and l(c,s)± (t) represent laser noise around the corresponding modulation frequency.

Unlike the homodyne readout schemes, in the heterodyne ones, not only the quantum noise components ω0 nc,s falling into the GW frequency band around the carrier frequency ω0 has to be accounted for but also those rallying around twice the RF modulation frequencies ω0 ± 2ΩRF:

Shet(t) = (Gc + n ωc0)cosω0t + (Gs + n ωs0)sin ω0t +n ω0+2ΩRF cos(ω + 2Ω )t + n ω0+2 ΩRFsin(ω + 2Ω )t (45 ) cω −2Ω 0 RF sω− 2Ω 0 RF +n c0 RF cos(ω0 − 2ΩRF )t + n s0 RFsin(ω0 − 2ΩRF )t. (46 )
The analysis of the expression for the heterodyne photocurrent
----------- --- --- --- ------------ ------ ihet ∝ (Lhet + S )2 = S2 + L2 + L2 + 2(L+ + L− )S + 2L+L − (47 ) + −
gives a clue to why these additional noise components emerge in the outgoing signal. It is easier to perform this kind of analysis if we represent the above trigonometric expressions in terms of scalar products of the vectors of the corresponding quadrature amplitudes and a special unit-length vector T H [ϕ ] = {cos ϕ, sin ϕ}, e.g.:
Shet ≡ (G + nω0)T ⋅ H [ω0t] + nTω +2Ω ⋅ H [(ω0 + ΩRF )t] + nTω− 2Ω ⋅ H [(ω0 − 2 ΩRF )t] 0 RF 0 RF
where T G = {Gc, Gs} and ω ω T n ωα = {nc,n s}. Another useful observation, provided that ω0 ≫ max [Ω1,Ω2 ], gives us the following relation:
----------------------------- [ ] H [(ω0 + Ω1 )t]HT [(ω0 + Ω2 )t] = 1 cos(Ω1 − Ω2)t − sin(Ω1 − Ω2 )t = 1ℙ [(Ω1 − Ω2 )t]. 2 sin(Ω1 − Ω2 )t cos(Ω1 − Ω2)t 2
Using this relation it is straightforward to see that the first three terms in Eq. (47View Equation) give DC components of the photocurrent, while the fifth term oscillates at double modulation frequency 2ΩRF. It is only the term ------------ 2(L+ + L − )S that is linear in GW signal and thus contains useful information:
------------ 2(L+ + L− )S ≃ Ic(t)cosΩRFt + Is(t)sin ΩRFt + {oscillations at frequency 3ΩRF }

where

Ic(t) = (G + n ω + nω +2Ω )T ⋅ L (0)+ (G + n ω + nω −2Ω )T ⋅ L (0), 0 0 RF + (0) 0 0 RF − (0) Is(t) = − i(G + nω0 + n ω0+2ΩRF)T ⋅ σy ⋅ L + + i(G + nω0 + nω0−2ΩRF )T ⋅ σy ⋅ L − ,
and σy is the 2nd Pauli matrix:
[ ] σy = 0 − i . i 0
In order to extract the desired signal quadrature the photodetector readout current ihet is mixed with (multiplied by) a demodulation function D (t) = D0 cos(ΩRFt + ϕD) with the resulting signal filtered by a low-pass filter with upper cut-off frequency Λ ≪ ΩRF so that only components oscillating at GW frequencies ΩGW ≪ ΩRF appear in the output signal (see Figure 12View Image).

It is instructive to see what the above procedure yields in the simple case of the Michelson interferometer tuned in resonance with RF-sidebands produced by pure phase modulation: L (0c+) = L(c0−)= 0 and L (0s+)= L(s0−)= L0. The foregoing expressions simplify significantly to the following:

( ) ω n ωs0− 2ΩRF nωs0+2ΩRF ( ω −2Ω ω +2Ω ) Ic(t) = 2L0 Gs + ns0 + ---------+ --------- and Is(t) = − L0 ns0 RF − ns0 RF . 2 2
Apparently, in this simple case of equal sideband amplitudes (balanced heterodyne detection), only single phase quadrature of the GW signal can be extracted from the output photocurrent, which is fine, because the Michelson interferometer, being equivalent to a simple movable mirror with respect to a GW tidal force as shown in Section 2.1.1 and 2.1.2, is sensitive to a GW signal only in phase quadrature. Another important feature of heterodyne detection conspicuous in the above equations is the presence of additional noise from the frequency bands that are twice the RF-modulation frequency away from the carrier. As shown in [36Jump To The Next Citation Point] this noise contributes to the total quantum shot noise of the interferometer and makes the high frequency sensitivity of the GW detectors with heterodyne readout 1.5 times worse compared to the ones with homodyne or DC readout.

For more realistic and thus more sophisticated optical configurations, including Fabry–Pérot cavities in the arms and additional recycling mirrors in the pumping and readout ports, the analysis of the interferometer sensitivity becomes rather complicated. Nevertheless, it is worthwhile to note that with proper optimization of the modulation sidebands and demodulation function shapes the same sensitivity as for frequency-independent homodyne readout schemes can be obtained [36]. However, inherent additional frequency-independent quantum shot noise brought by the heterodyning process into the detection band hampers the simultaneous use of advanced quantum non-demolition (QND) techniques and heterodyne readout schemes significantly.


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