77 5.12 LA CUENTA DE GASTO
6. LA EDUCACIÓN CULTURAL
Contrary to a homodyne detector where the LO and signal share a common carrier frequency, a heterodyne detector (see fig. 2.3b) employs a frequency detuned LO. For the case where the LO frequency is larger than the signal frequency byΩIF, the LO and signal fields may be
represented by the ansatz (analogous to eq. (2.2.59) for homodyne detection), ˆaLO(sig)(t) =
ˆnLO(sig) +δˆaLO(sig)(t)
e−iωt×
e−i(ΩIFt+θLO)
e−iθsig .
For the reasons detailed above for the case of homodyne detection (formally,ΩIF =0), it is
technically useful to perform balanced detection, i.e. combine the LO and signal on a balanced beam-splitter, using a length-matched interferometer, i.e. the LO and signal arrive at the beam-splitter after propagating for equal times.
Similar to the homodyne case, in the strong LO and length-balanced case, the mean and fluctuating parts of the heterodyne photocurrent (in the strong LO, i.e. ˆnLO
ˆnsig
, and balanced, i.e.ηt = 12, case),
ˆIhet(t) ≈ −2qe ˆnLO ˆnsig sin(θhet+ΩIFt) δ ˆIhet(t) ≈qe 2ˆnLOδ ˆqsigθhet+π/2+ΩIFt(t) (2.2.65) whereθhet:= θsig−θLO. Importantly, the photocurrent is not proportional to a unique signal
quadrature, but in fact, cycles through each quadrature. Despite this fact, the quadrature com- mutation relations eq. (2.2.56) conspire to ensure that the heterodyne photocurrent commutes with itself, viz.2.31
δ ˆIhet(t),δ ˆIhet(t)
=4i q2eˆnLO ·δ(t−t)sin(ΩIF(t−t)) =0,
renderingδ ˆIheta continuous observable.
The spectrum of the heterodyne photocurrent fluctuations is however unlike the homo- dyne spectrum. In fact, the photocurrent fluctuations (eq. (2.2.65)) expressed in terms of the amplitude operators (using eq. (2.2.55)),
δ ˆIhet(t) =qe
ˆnLO
δˆasig(t)e−iθhet−iΩIFt−iπ/2+δˆa†sig(t)eiθhet+iΩIFt+iπ/2
, (2.2.66)
has the two-time correlator (omitting the factor q2
eˆnLO),
δ ˆIhet(t)δ ˆIhet(t)
∝ δˆasig(t)δˆa†sig(t)
e−iΩIF(t−t)+
δˆa†
sig(t)δˆasig(t)
e+iΩIF(t−t)
−δˆasig(t)δˆasig(t)
e−2iθhete−iΩIF(t+t)−
δˆa†
sig(t)δˆasig† (t)
e+2iθhete+iΩIF(t+t),
2.31The second equality, by common abuse of notation, holds in the sense of distribution; i.e. it holds for any arbitrarily close approximation toδ(t).
which is not stationary. The last two terms, being a periodic modulation of a stationary term, give rise to what is called cyclostationary noise [154, 155]. When the signal field carries excitations in a narrow band centred at frequencies much below the intermediate frequency ΩIF, these non-stationary terms may be omitted2.32. The resulting photocurrent correlator,
δ ˆIhet(t)δ ˆIhet(0)
≈ δˆasig(t)δˆasig† (t)
e−iΩIFt+
δˆa†
sig(t)δˆasig(t)
e+iΩIFt = δ ˆq(t)δ ˆq(0) +δ ˆp(t)δ ˆp(0)cosΩIFt
+ δ ˆp(t)δ ˆq(0) +δ ˆq(t)δ ˆp(0)sinΩIFt,
(2.2.67)
is independent of the relative signal-LO phaseθhet. Note that due to simultaneous detection of
conjugate quadratures, any mutual correlations between the two are reflected in the heterodyne photocurrent. Equation (2.2.67) together with the Wiener-Khinchine theorem (eq. (2.1.5)) gives the (single-sided) spectrum of the heterodyne photocurrent:
¯ ShetI [Ω] =q2eˆnLO Ssigaa[Ω+ΩIF] +Ssiga†a†[Ω−ΩIF] ,
expressed in terms of the unsymmetrised power spectral density of the (non-hermitian) flux amplitude operators (as defined in eq. (2.1.7)). The left-hand side, being a single-sided spectrum is defined only forΩ> 0; in particular, fluctuations in the optical field originally about the optical carrier are translated to radio-frequencies,Ω≈ΩIF, about the intermediate frequency.
For a detector with bandwidth much less than 2ΩIF, the spectrum centred aboutΩIF,
¯
ShetI [Ω−ΩIF] ≈qe2ˆnLOSaa[Ω]. (2.2.68) consists of only those components slowly varying with respect toΩIF. In this sense, a hetero-
dyne detector measures the double-sided spectrum of the flux of the optical field, including correlations between its amplitude and phase quadratures.
Explicitly separating out the vacuum fluctuations, from the signal field, i.e. δˆasig → δˆavac+δˆasig, and introducing the efficiencies for the detection, the spectrum of the heterodyne
photocurrent is, ¯ ShetI [Ω−ΩIF] = R 2S¯ !PNE[Ω"] ¯ Shet,detI +4 η·qeR · ! PLO" ¯ Shet,shotI +η2q eR PLOSsigaa[Ω] ! " ¯ Shet,sigI . (2.2.69)
Note that compared to homodyne detection eq. (2.2.62), the shot noise contribution is twice larger, and the signal twice smaller – the former is due to the shot noise from both quadratures being detected, while the latter is due to the signal being spread symmetrically about the intermediate frequency (i.e. double-sidedness). In effect, heterodyne detection is four times less sensitive compared to a homodyne detector. The advantage however is that by detecting both quadratures of the signal field simultaneously, it provides access to correlations between the signal quadratures [162].
2.32In the contrary case, these terms give rise to cyclostationary shot noise [156, 157] – shot noise modulated at ΩIF– in excess of the expectation from a stationary shot noise model. It is generally true that cyclostationary noise may be represented as a sum of correlated stationary noise processes [158] – therefore, it is possible to coherently cancel excess cyclostationary shot noise [156, 159, 160], or use the correlations for benefit [161].
a
b
20 HzBPD
AOM
PZ
T
IS
IS
20 Hz 78 MHz 1 kHz 10 kHz [(photons/s)/H z] 120 100 80 60 40 Frequency [MHz] Detector noise No signalWith signal (unbalanced) With signal (balanced)
1011 1012 1013 1014 1015 1016 1017 1010 109 108
Fig. 2.5 – Design and operation of heterodyne interferometer. (a) Essential design of the balanced heterodyne interferometer used in this thesis. An AOM in the LO path produces the desired frequency shiftΩIF=2π·78 MHz. (b) Heterodyne photocurrent spectrum for the interferometer unbalanced (light red) and balanced (red). Gray shows the electronic noise of the photodetector, and black the shot noise due to the LO. The spectrum is calibrated using the known DC optical power which is reflected as the variance of the carrier beat signal around the intermediate frequencyΩIF=2π·78 MHz.
Design and operation of a realistic heterodyne detector
As illustrated by theoretical considerations, an experimentally practical heterodyne detector inherits all the characteristics of the homodyne detector in fig. 2.4, except for a frequency shifted LO. Figure 2.5a depicts the essential layout of the balanced heterodyne interferometer constructed and employed in this thesis. The substantial difference in the optics is the presence of an acousto-optic modulator (AOM, AA Optoelectronics MT110-B50A1) in the LO arm of the
interferometer. The AOM was operated so as to maximise the diffracted optical power into the first order; at the chosen operation frequencyΩIF =2π·78 MHz, it was possible to attain a
diffraction efficiency>0.8.
Similar to the procedure followed to balance the homodyne detector, the input laser wavelength is modulated to induce interference fringes in the photocurrent. However, in the case of the heterodyne, the mean photocurrent eq. (2.2.65),ˆIhet(t)
∝ sin(θhet+ΩIFt)oscillates
at the offset frequencyΩIF. Therefore, to access the fringes resulting from a modulation of the
phaseθhet, the photocurrent is mixed down using a RF local oscillator at the offset frequency
ΩIF(see schematic in fig. 2.5a). The lengths are balanced by nullifying the fringe frequency.
Unlike the homodyne, the phaseθhetneed not be stabilised, since the photocurrent spectrum
¯
ShetI [Ω](eq. (2.2.65)) is not sensitive to the mean phase.
Figure 2.5b shows the cancellation of input laser noise achieved due to length balance. Shining a LO (PLO≈1 mW) alone gives rise to a shot noise contribution (black trace) ¯Shet,shotI 10·S¯het,detI . For an unbalanced interferometer driven by a (noisy) diode laser, the output photocurrent spectrum gives a direct measure of the laser phase noise transduced by the imbalance of the interferometer (see appendix C). Indeed, the red trace in fig. 2.5b, is consistent with diode laser frequency noise ¯Sω[Ω] =Ω2S¯φ[Ω] ≈2π(35 Hz2/Hz), at Fourier frequencies Ω≈2π·4 MHz from the carrier.