30 S. Yoshida et al. Fig. 5.3 Carrier fringe configuration When the above signal is taken with a digital camera whose exposure time is , the output signal S( ) is proportional to the time integration of I(t). Here if the exposure time is significantly longer than the period of the oscillation!, i.e., > >2 /!, the oscillatory term on the right-hand side of Eq. (5.4) is much less than the first two non-oscillatory terms. By integrating the signal in the form of Eq. (5.4) for the exposure time, we obtain the following approximated expression of the signal. S. / D Z 0 I.t/dt Š2I0 f1CcosıoJ0 .ı/g (5.5) Our purpose here is to extract ıDkd and thereby estimate the oscillation amplitude ı. It is apparent that the sensitivity of the signal is proportional to the factor cosıo. Apparently, unlike the one-dimensional analysis with a fast photo-diode, the constructive interference condition ıo Dn provides the highest sensitivity. This is naively understood as follows. In this case, the cosine function is integrated. The integration of a cosine function is essentially a sine function. Therefore, this time the phase providing the highest sensitivity is where the sensitivity is the lowest for the other case. 5.2.3 Phase Fluctuation due to Environmental Disturbance In the normal incident configuration, the phase noise infiltrates into the cosıo term as a function of time. If the frequency of the phase noise is low, there is no way to distinguish the change in the output signal S( ) is due to J0(ı) or deviation from the initial value of ıo due to the phase noise unless the noise is characterized and the signal is high-pass filtered. Before discussing the carrier fringe configuration, it is worth while estimating the magnitude of the low frequency phase noise under a realistic condition. When the interferometer is placed in air, temperature fluctuation plays a significant role. The optical phase change due to the temperature dependence of the refractive index of air can be expressed as follows. d' D2 l @n @T dT (5.6) Here is the wavelength, l is the path length, n is the refractive index of air and dT is the temperature change. The temperature coefficient @n/@T of air is 0.87 10 6 (1/ıC) [5]. The arm length of the interferometer used in this experiment is 10 (cm). The wavelength of the laser used in this study is 632.8 nm. So, the phase change due to a temperature change of ˙0.1 ıC over the round trip in the interferometric arm is 20 (cm)/632.8 (nm) 0.87 10–6 0.1 D2.75% (of the period 2 ). Our measurement of air temperature in the interferometer arms indicate as much as 0.4 ıC within a typical time of the interferometric experiment. The phase error due to this air temperature change is 2.75 4 D11.0% of the period.
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