W e analyse a sim ple M ichelson interferom eter as depicted in Fig. (2.2). Light enters a 50:50 beam splitter and propagates along tw o different arm s w hich im p art different phase delays, and is recom bined at the original beam splitter. The accum ulated phase difference 0ab(t) betw een the tw o fields at the beam combiner is the sum of the intrinsic DC phase difference 0O due to the different optical paths travelled by the two beam s, and the externally induced phase difference 0(t), due to a transducer, m odulator or therm ally induced drift for instance.
A llow ing for non-ideal optical characteristics in the interferom eter such as different optical losses in the two arm s or an asym metric beam splitter, the optical pow er available at the detector situated at the antisym m etric port of Fig. (2.2) is given by (see equations (2.02), (2.03) and (2.04) for details):
P det = E inc2 R /2 [ 1 + V COS 0ab ] (4.01)
w here Einc2 is the in p u t optical pow er, R is the m ean m irror reflectivity and V is the fringe visibility. The o u tp u t intensity reaches a m inim um at this p o rt w hen 0ab = n.
If the tim e d ep en d en t p art of 0ab is a sim ple sinusoidal m o d u latio n (the signal), we m ay write
w here 0O is the optical phase offset in the absence of signal, 0S is the optical phase m odulation am plitude due to the signal at frequency cos, and %s is the signal phase offset at t = 0. Putting equation (4.02) into (4.01) yields the time dependence of the optical power at the detector
p det = Eine2 R / 2
{
1 +v
cos0o cos [6 S sin (cos t +xs)]
- V
sin0o sin [0S sin (cos t +xs)] }
= Einc2 R /2 { [ l + V JO(0S) COS0Q + 2 V J2(0s)cos0ocos[2 (cost + x s)l + ...]
- 2 V sin0o [ji(0s)sin (o)s t + %s) + J3(0s)sin[3(cos t +
Xs) ] + - ] }
(4.03)
w h ere Jn(0 s) are Bessel functions of the first kind. For sm all phase m o d u la tio n s 0 S << 1, harm onics above the fu n d am e n ta l m o d u la tio n frequency can be neglected as the higher order Bessel functions are small.
W hen this m o d u lated light field is incident on a photo detector w ith responsivity p(>.) am p eres/w att, the induced photocurrent is p(X)Pdet. This resp o n siv ity can be ex p ressed in term s of the p h o to d io d e q u a n tu m efficiency r\ using equation (2.29). The photocurrent of equation (4.03) is well approxim ated by a DC com ponent and an oscillating com ponent at signal frequency cos. The DC photocurrent is
idc = PEinc2R /2 ( 1 + V Jo(0s) COS0O) (4.04)
w hile the desired AC com ponent in the photocurrent is:
he = [p Einc2 R V Ji(0s) sin0o] sin (o)s t +
xs)
(4.05)The instantaneous signal pow er at cos delivered to a tuned AC receiver or spectrum analyser w ith in p u t im pedance r is iac2 r, tim e-averaged over one signal cycle gives
As expected, the absolute received signal power is strongest when the interferometer is operated midway between adjacent bright and dark fringes, where the derivative of transmission with respect to phase is largest (0O = 7t/2), while no signal power is obtained when operating about the turning points ( 0O = 0,7T, ...) in the transmission function. This can be clearly seen in Fig. (2.3).
Shot noise appears superimposed on the photocurrent as a quadrature- independent flat spectrum fluctuation. The shot noise current RMS amplitude is given by the square root of equation (2.28):
<isho.>RMS= (2eidcB)1/2 (4.07)
where idc is the DC photocurrent and B is the detection bandwidth. At a phase insensitive receiver, these current fluctuations produce a noise power (<iqn>RMs)2r in addition to any signal power in the modulation frequency band being observed. From equation (4.07) and the expression for DC photocurrent in equation (4.04), we obtain the average shot noise power at the photodetector output as a function of the phase offset 0o:
P sh o t = 2 e idc B r
= P s h o tm a x ( l + V Jo(Q s ) COS0o) / 2 (4.08)
where P sh o t max = [2ep Ejnc2 RB r] is the shot noise which would be observed
at a bright fringe in an ideal interferometer (V=l, R=l). This shot noise is equally distributed over both quadratures in the RF domain, at all frequencies, provided the DC light level is significantly larger than the optical modulation due to the signal (ie stationary statistics apply).
In a shot noise limited sensor, the signal-to-noise ratio is obtained by dividing the signal power in equation (4.06) by the shot noise power in equation (4.08). In general, however, technical optical noise, and unrelated electronic noise contribute to the total noise in the measurement. The total noise then becomes ( P sh o t + P te c h + P e ie c tro n ic )- In the limit where P tech is
Ptotal — Pshot + £ Pshot max
= Pshot max [(f+26) + VJq(0s ) COS0o] / 2 (4.09)
where e is the ratio P eiectronic / Pshot max • This leads to the following signal-
to-noise power ratio as a function of the DC phase offset 0o in the interferometer:
S /N = (p Ej^2 R J12(6s)/2eB)
{v
2sin
2e
0 / [(l+2e) - VJ0(9S) cos60]}(4.10)
For very small signals (0S « 1), equation (4.10) approximates to :
S /N = (p Einc2 R6S2 / 8eB)
{v
2sin2e0 / [(l+2e) + V (l-0s2/4) cos60]} (4.11)Equation (4.11) is valid for all sensitive interferometer applications. If 0S is comparable to unity then the detected signal will be extremely large and the signal-to-noise ratio will not be a practical issue.