Recasting equation 2.15 in terms of ∆ℓeffect yields [1]
|∆ℓeffect|= λ∆Ieffect
π(Imax−Imin). (2.16) From the above equation, it is apparent that, provided we know the wave- length of the light and the maximum and minimum intensities of the interfer- ence pattern, measurement of the change in light intensity of the interference pattern will allow for calculation of the induced optical path-length difference
∆ℓeffect.
2.4 Electronic Stabilization of the Interferom-
mechanical vibrations, thermal drift and field-induced strains, etc. Since the sensitivity of the interferometer is limited by these influences, it is essential that they be minimized, allowing the sensitivity limits of the instrument to approach the theoretical limit as closely as practically feasible.
Passive stabilization of the interferometer from mechanical vibrations can be achieved to an extent through resting it on a relatively massive table which is in turn supported by, though pneumatically isolated from, a base table. Such means of passive stabilization have been described, for example, by [32]. Min- imization of thermal effects can be attempted through temperature-control of the environment. Such passive stabilization, which can be extremely costly to implement, is never entirely effective in isolating the interferometer from all noise, and this has led to the development of relatively cheap and effective active stabilization techniques.
Two distinct methods for the active stabilization of interferometers are uti- lized in most studies reported in the literature. Optical stabilization is em- ployed in the so-called phase-quadrature interferometer originally proposed by Peck and Obetz [33]. This method proves to be inappropriate for our investigation, since it requires the introduction of a shift in phase between the two perpendicular components of the light in one arm of the interferom- eter, whereas we are passing linearly-polarized light through the samples to determine separately the refractive indices parallel and perpendicular to the applied electric field. We turn our attention instead to the electronic means of stabilization, which employs electronic feedback to mitigate against the
effects of low-frequency environmental noise.
The principles of the electronic stabilization of an interferometer have been described elsewhere [7], and are worth reviewing briefly. Consider a material which introduces an optical path-length change (much smaller in magnitude than the wavelength of the probing light) in one arm of an interferometer, via, for example, an induced effect such as the electric-field-induced electro-optic effect. It is possible to express the optical path-length difference between the two interfering waves as [32]
∆ℓ= (ℓ1−ℓ2) +ℓeffect+ℓnoise (2.17)
where ℓ1 and ℓ2 are the respective static distances along the two arms of the interferometer, ℓeffect is the small (and typically modulated) optical path- length change introduced by the effect under investigation, and ℓnoise is the nett (dynamic) optical path-length variation due to all low-frequency noise contributions. By adjusting the static optical path-length difference such that the interferometer is being operated at a point of maximum sensitivity according to equation 2.10, we have
(ℓ1−ℓ2) +ℓnoise= λ4(2n+ 1), where n= 0,1,2,· · · . (2.18)
Here the phase difference between the two interfering waves is given by
2π π (2.19)
The cumulative effect of this time-varying noise ℓnoise will be to disturb the interferometer such that it cannot maintain balance at the point of maxi- mum sensitivity, and will consequently compromise the accurate and precise measurement of ℓeffect.
Electronic stabilization of the interferometer is concerned with cancelling, or at least minimizing, the noise contributions to the phase difference ∆φ aris- ing through the termℓnoise. Using a photodiode detector to monitor the light intensity at a point on the interference pattern yields a voltage signal which is proportional to the light intensity at the point. By measuring the photodiode output voltage corresponding to the light intensity at the point of maximum interferometer sensitivity, given by equation 2.11, it becomes possible to de- termine, through the drift in photodiode output voltage, the extent of the noise-induced drift from this point of maximum sensitivity. An active ele- ment in the interferometer can be continuously driven to compensate for this low-frequency drift, thereby actively maintaining the interferometer at its point of maximum sensitivity. The active element is typically a piezoelectric transducer or a condenser microphone which drives the mirror in the reference arm of the interferometer, thereby varying the reference arm’s path length to compensate for the noise in equation 2.19 [7, 10, 14, 29, 30, 32, 34, 35]. In- terferometeric measurement sensitivities of up to 10−3Å have been achieved through this technique of electronic stabilization of the interferometer [7, 10].
For the interferometer used in this study, a piezoelectric transducer coupled
to a feedback circuit has been employed as the means of electronic stabiliza- tion. The reference-arm mirror was attached to this piezoelectric transducer.
Further details are provided in Chapter 3.
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