It is important that the Michelson be operated on a dark fringe in order to reject laser noise, but we here confront the fact that we cannot actually directly detect the electric field at the asymmetric (output) port, and certainly not at the frequencies of oscillation used in current interferometers (where the laser field has angular frequency ω0 ≈1015rad/s). What we can actually detect is the average power, using (for IR and visible light) a photodetector. The power is given generally by
P =E∗E, (3.17)
whereE∗ is the complex conjugate ofE. We will also lazily consider the power at the photodetector to be our final output, although this is not strictly true. The photodetector converts the power hitting it (with some efficiency) into an electrical current or voltage; it is this electrical signal which is actually measured, but the distinction is not terribly important for this discussion.
The power transmitted to the asymmetric port of a Michelson interferometer with perfect end mirrors is
PAS = 4|EIN2 |(tbsrbs)2sin2φ−, (3.18) where the differential phaseφ− isk(2∆lx−2∆ly) =k∆l−.
The signal gain we expect is proportional to the derivative dPdφAS
− , dPAS
dφ− ∝2 sinφ−cosφ−, (3.19)
which is zero when φ− = 0. Thus, for small deviations around the dark fringe, we have no signal.
This is a common situation, and much of interferometer design is concerned with how to solve this problem. However, there is a straightforward solution: we can use a different field, co-incident on the photodetector, as a local oscillator ELO. In this case the field and detected power at the
photodetector become
EP D =ELO+ESIG
PP D =|EP D∗ EP D| (3.20)
=|(ELO+ESIG)∗(ELO+ESIG)|,
where the cross terms in the product,ESIG∗ ELO+ESIGE∗LO, can allow us to recover a linear signal aroundφ−= 0 even when|ESIG|= 0 atφ−= 0, for the proper choice of local oscillator field. The two sections which follow describe two of the possible choices for such a local oscillator.
Classically this signal will be limited by shot noise, the fluctuations in laser power due to photon counting statistics (for a quantum mechanical discussion, refer to [44]). The shot noise level is given by the power incident on the photodetector:
δPshot= r2hc
λ Pinc. (3.21)
If we set up our interferometer such that our local oscillator field is proportional to the input field ELO ∝Ein, then (cf. equation (3.16)) the shot noise limited signal-to-noise ratio will scale as:
SN Rshot∝p
Pin. (3.22)
This is a fairly general result, and explains why increasing the power illuminating the interferometer can improve the sensitivity.
3.5.1 Optical Heterodyne Detection
One possible choice for a local oscillator is a frontal phase modulation sideband. The usual setup includes an electro-optic phase modulator installed with the input beam (cf. figure 3.5). These devices (also called Pockels cells) can be used to apply a sine wave modulation of depth Γ and angular frequency Ωm on the phase of the input light E0eiω0t yielding the field incident on the interferometer,
E(t) =E0ei[ω0t+Γcos(Ωm)t]. (3.23) We can use the Jacobi-Anger expansion which relates trigonometric exponentials in terms of Bessel functions [45],
eiΓcos(Ω)t=
∞
X
n=−∞
inJn(Γ)einΩt, (3.24)
yielding (keeping only terms to first order):
E(t) =E0eiω0t[J0(Γ)−iJ−1(Γ)e−iΩmt+iJ1(Γ)eiΩmt+...], (3.25) which for Γ.0.5 is approximately
E(t)'E0eiω0t
(1−Γ2 4 ) +iΓ
2 e−iΩmt+iΓ 2 eiΩmt
. (3.26)
Then the first-order modulation sidebands can function as local oscillators. This scheme is known asheterodyne detection, because the local oscillator is at a different frequency from the carrier field.
After photodetection, there is a beat signal in the photocurrent at the difference frequency (in this case, Ωm). The desired readout signal (in this case, the Michelson length) appears as a modulation of this beat signal. The modulation sidebands are typically at radio frequency (RF), and this technique is thus sometimes known as RF readout. We will refer to the RF local oscillator fields as
‘RF sidebands.’
We can consider the case where we have audio sidebands on the carrier field and on the RF sideband fields we are using as a local oscillator. If we let, e.g.,Fωindicate the complex amplitude of a field oscillating at frequencyω0+ω, the field at the photodetector can be written
EP D=eiω0t(F−Ωme−iΩmt+FΩmeiΩmt
+F−ωae−iωat+Fωaeiωat (3.27)
+F−(Ωm±ωa)e−i(Ωm±ωa)t+F(Ωm±ωa)ei(Ωm±ωa)t (3.28) +F0),
where we have again only kept first-order terms at each modulation, and we have lumped together the audio sidebands on the RF sidebands. Applying equation (3.17) yields 49 terms. We will ignore terms at 2Ωmand 2ωa, and collect terms aroundωa,Ωm,Ωm±ωa, and at DC:
P(ω= 0) =F−ΩmF−Ω∗ m+FΩmFΩ∗m+F−ωaF−ω∗ a+FωaFω∗a+F0F0∗ (3.29a) P(ω≈Ωm) = 2×Ren
F−ωae−iωat+Fωaeiωat+F0
×
F−Ωme−iΩmt+FΩmeiΩmt∗o
(3.29b) + 2×Ren
F0×h
F−(Ωm±ωa)e−i(Ωm±ωa)t+F(Ωm±ωa)ei(Ωm±ωa)ti∗o P(ω≈ωa) = 2×Ren
F0×
F−ωae−iωat+Fωaeiωat∗o
, (3.29c)
where Re indicates taking the real part of a complex number. Equation (3.29a) is the total light power on the photodetector. Equation (3.29c) is essentially homodyne detection (discussed more in section 3.5.2). Equation (3.29b) is the heterodyne signal. We can now demodulate the signal
by using an electronic mixer to multiply the photodetector output by a cos(Ωmt) (the mixer really mulitplies by a square wave but the distinction is not critical) and low-pass filtering the mixer output to eliminate components remaining at Ωmand 2Ωm. The in-phase mixer outputMI is
MI =Re
F0∗FΩm+F0F−Ω∗ m (3.30a)
+ Re
(F−∗ωaFΩ∗m+F−∗ωaF−∗Ωm+FωaFΩm+FωaF−Ωm)eiωat (3.30b) + Ren
(F0F−(Ω∗ m+ωa)+F0∗F−(Ωm−ωa)+F0∗F(Ωm+ωa)+F0F(Ω∗m−ωa))eiωato
. (3.30c)
Multiplying by a sin(Ωmt) for the quadrature phase output MQ would give us the negative of the imaginary part instead:
MQ =−Im
F0∗FΩm+F0F−∗Ωm (3.31a)
− Im
(F−∗ωaFΩ∗m+F−∗ωaF−∗Ωm+FωaFΩm+FωaF−Ωm)eiωat (3.31b)
− Imn
(F0F−∗(Ωm+ωa)+F0∗F−(Ωm−ωa)+F0∗F(Ωm+ωa)+F0F(Ω∗ m−ωa))eiωato
. (3.31c) One convenient improvement is to use a so-calledI&Qdemodulator which simultaneously mul- tiplies the photodetector output by both a cos Ωmt (the in-phase) and a sin Ωmt (the quadrature- phase), yielding theI0andQ0phased outputs. Then these outputs can be combined using a rotation matrix to achieve an arbitrary demodulation phase φD, such thatIφD is equivalent to mixing the PD output with cos(Ωmt+φD),
IφD
QφD
=
cosφD −sinφD
sinφD cosφD
I0
Q0
. (3.32)
Generally, the I-phase signal will contain information about the relative optical phase of the F0
and F±Ωm fields; the Q-phase signal will contain information about the imbalances in the F±Ωm fields (see [43] for a derivation).
3.5.1.1 Schnupp Asymmetry for Michelson Length Sensing
We can transmit the RF sidebands to the asymmetric port of a Michelson interferometer using a Schnupp asymmetry [46], which is a macroscopic length difference between the arms of the Michelson.
We recall the Michelson length asymmetry (macroscopic) as
l−=lx−ly. (3.33)
We will as usual denote microscopic deviations from the conditionl−=nλby ∆l−. If the arms of the Michelson are of equal length (l−= 0), and moreover ∆l−= 0, then all frequencies of light will
undergo the same destructive interference at the asymmetric port. If the arms are of unequal length (l− 6= 0), however, only at those wavelengths for whichφ− =nπ will the asymmetric port actually be dark.
We can apply a frontal phase modulation to the input beam at Ωm. The transmission for the modulation sidebands to the asymmetric port is
Tmich= sin2
Ωml− c
. (3.34)
We can recover a linear sensing function for the Michelson displacement by using this transmitted pair of RF sideband field as a local oscillator in an RF heterodyne detection scheme. This technique allows us to operate a Michelson on a carrier dark fringe.
3.5.2 Optical Homodyne Detection
A technique complementary to heterodyne detection (cf. section3.5.1) ishomodyne detection. For homodyne detection, the local oscillator employed to downconvert the audio frequency sidebands is a field oscillating at the same frequency as the carrier field. In this case, the audio sidebands are directly downconverted to DC by the photodetection process (cf. equation (3.29c)), and can simply be read out from the photocurrent. This technique is generally simpler than heterodyne detection, because it requires less in the way of electronics and altogether avoids the modulation/demodulation process, but in most cases it is more vulnerable to several types of noise—especially laser noise and the so-called 1/f or flicker noise that plagues basically all electronics at frequencies near DC. We will revisit homodyne detection in the context of DC readout in chapter4.