In the previous chapter we demonstrated the frequency limitations of direct detection. Phase modulation provides a means of removing the low frequency technical noise from the measurement and overcoming these frequency limitations.
Phase modulation of light in an interferometer shifts both signals and technical optical noise at the same time, so at first sight it would appear to confer little advantage. In this chapter we demonstrate that the internal modulation technique allows shifted technical noise to be suppressed by setting the interferometer to a dark fringe. In addition, we find that for a given amount of electronic noise, there is an optimum level of phase modulation. Fortuitously, the ratio of signal to electronic noise is greatest exactly on a dark fringe. The potential maximum sensitivity in the optimum configuration approaches V(3/2) times that of the direct detection method.
When operating an internal modulation interferometer at the dark fringe, the resulting intensity varies at twice the modulation frequency. The shot noise (proportional to the detected intensity) is therefore a periodic function of time. This periodic shot noise, termed nonstationary61, is shown to be responsible for the V(3/2) decrease in sensitivity, compared to direct detection interferometry.
We begin in section 5.1 by deriving the signal-to-noise ratio for internal phase modulation interferometry under two conditions: pre and post mixer. We then discuss the theoretical and practical implications of our derivation. In section 5.2 we describe the layout of our internal modulation polarimeter experiment and present the results achieved with this device.
The chapter ends w ith an experim ental d em onstration of n o n statio n ary sh o t noise an d its consequences (section 5.3). The use of o p tim al m o d u la tio n /d e m o d u la tio n w aveform s is discussed and an experim ental exam ple using a complex dem odulation function is presented.
5.0 Perspective
A lthough m any internal m o dulation experim ents have been perform ed, few have been published. M an et all62 docum ent an internal m o d u latio n experim ent and derive a basic theory to describe it. This theory how ever does not include the effects of nonstationary shot noise The experiem ent perform ed by M an et al therefore achieves m oderate agreem ent w ith theory.
S tra in 63 has perfo rm ed an internal m odu latio n experim ent w ith Fabry- Perot cavities in the M ichelson interferom eter arms. Strain reports th at the interferom eter approaches shot noise lim ited sensitivity for frequencies above 50kHz.
F ritsc h el12 has also perform ed an internal m odualtion experim ent using Fabry-Perot arm cavities. H ow ever using Fabry-Perot arm s increased the sensitivity of the in terfero m eter to the p o in t w here m echanical noise lim ited perform ance (except in the frequency range around ~ 80kHz w here sensitivity approached the shot noise limit).
Meers et al21 derive an expression for the efficiency of internal m odulation signal extraction. M eers et al show th at in tern al m o d u latio n , u sin g sinusoidal m o d u la tio n /d em o d u latio n , can achieve an efficiency of
V (
2 /3 ) that of direct detection.H ere we p erfo rm a sim ple in tern al m o d u la tio n ex p erim en t u sin g a retardance m odulated polarim eter. Due to the comm on m ode advantages of polarim eters we are able to make shot noise lim ited m easurem ents dow n to ~ 2kHz. We develop a theory th at includes all relevant non-ideal perform ance and achieve good agreem ent w ith experiment.
N iebauer et al61 first identified and analysed nonstationary shot noise and its effects on internal m odualtion interferom eters. Meers et al21 and Mio et al64 have perform ed bench top experim ents dem onstrating nonstationary shot noise and its dependence on dem odulation phase.
O ur ex p erim en t uses a high fringe visibility, large m o d u latio n d ep th interferom eter to investigate non-stationary shot noise. C onsequently we observe a considerably larger effect (~ 4.8dB phase dep en d en t noise) than either Meers or Mio.
N iebauer et al61, Meers et al21 and Schnupp65 have pointed out that the extra signal q u ad ratu re noise caused by non-stationary shot noise can, in principle, be elim inated by using more complex m odulation-dem odulation w aveform s.
We dem onstrate the use of a composite w aveform consisting of bo th a first and th ird harmonic. We show that the third harm onic adds correlated shot noise th at can be used to reduce the resulting shot noise floor in the signal quadrature.
5.1 Derivation of internal modulation theory
As derived in chapter one, a M ichelson interferom eter has a phase response at the detector o utput of
For in tern al m o d u latio n the explicit expression for 0 ab now includes a deliberate m odulation term as well as the signal and DC terms, giving
w here 0 m is the m o d u latio n d ep th in rad ian s, com is the m o d u latio n frequency, and Xs and Xm are arbitary phase offests for the signal and m o dulation frequency com ponents respectively. U sing equation (5.01) we can expand cos 0ab to give
P det =Einc2 R / 2 [ l + V cos 0 ab ] (2.04)
cos e ab = cos e0 cos (0S sin (cos t + Xs)) cos (em sin (com t +
xm))
- cos e0 sin (0S sin (a>s t +xs))
sin (0m sin (com t + xm))- sin 0O sin (0S sin (cos t + Xs)) cos (0m sin (com t + xm))
- sin 0O cos (0S sin (cos t +
xs))
sin (e m sin (com t +xm))
(5.02)S u b stitu tin g eq u atio n (5.02) into eq u atio n (2.04) and u sing the Bessel functions of the first kind to expand the result gives
Equation (5.03) is lim ited to first order in 0S and second order in 0m*. This low order expansion exhibits a DC term and AC com ponents at the signal, m odulation, sum and difference frequencies (interm odulation term s) and at twice the m odulation frequency. Each of the six com ponents varies w ith the interferom eter phase offset 0O.
The inform ation to be recovered is found in the term s involving J i(0S) *
This factor is found in both the direct signal com ponent at frequency cds and in the interm odulation com ponents at sum and difference frequencies com ± cos. The technical noise we seek to avoid is a com bination of intensity and
As 0S is typically less than 1 pradian, all higher order Bessel functions in 0S can be ignored. 0m is typically much larger and we must include the second order Bessel function of 0m for accuracy. It is in fact this second order Besel function that leads to non-stationary shot noise as will be demonstrated in the following analysis.