Phase Modulation/Demodulation Characteristics. In the Vetra-Piezo step-scan interferometer, phase modulation (ΦM) is produced by dithering the fixed mirror along the retartdation direction at a contant frequency. The DSP-controlled piezoelectric transducers actuate this dithering. The term "phase modulation" comes from the fact that the optical path difference modulation, or dithering, actually modulates the derivative, or phase, of the IR intensity. There are two important parameters for ΦM: modulation frequency fΦM and amplitude ε. The ΦMfrequency (Hz) refers to the number of mirror oscillations per second and the ΦM amplitude measures the distance of the mirror oscillation and is usually expressed in terms of the He-Ne laser wavelength (λHe-Ne). the raw step-scan ΦM interferogram can be expressed by modifying Equation (1-1), i.e.
The integrand in the above equation can be expanded and further expressed in terms of Bessel functions Jn(2πσε), where n=0,1,2,3,……， and sin(2mπfΦMt), where m=1,2,3…… When a DSP board or a phase sensitive lock-in amplifier is used to demodulate the detector signal at the fundamental ΦM frequency, only the J1（2πσε）term is significant, and therefore, the demodulated ΦM interferogram can be expressed as
Thus, the ΦM interferogram (in the absence of phase errors) is antisymmetric with respect to the zero path difference (ZPD) and is the derivative of the amplitude modulation interferogram. In addition, the integrand is convolved with the first order Bessel function, J1（2πσε）, and thus the demodulated ΦM interferogram and single-beam profiles depend on ΦM amplitude, ΦM frequency and energy of the light.
The design of the Vectra-Piezo step-scan interferometer allows ΦM frequencies and amplitudes to vary over the wide ranges typically needer for near-IR and mid-IR experiments. It offers better stability and flexibility than systems that rely on dithering a much heavier moving assembly for ΦM, because the DSP easily controls the relatively lighter "fixer" mirror assembly. Figure 1.7 shows the representative laser signals for the Vectra-Piezo interferometer at a ΦM frequency of 400Hz and amplitudes from 0.5-3.5 λHe-Ne, as recorded on a digital oscilloscope. The nearly perfect repeating laser fringes show that the mirror position and the ΦM amplitude are very stable and reproducible. The ability to independently control the ΦM frequency and amplitude enables users to optimize the experiment for the spectral region of interest. The modulation effciency varies as a function of infrared wavenumber and the modulation amplitude. When the ΦM amplitude is fixed, the wavenumber-dependent profile (normally referred as the phase modulation characteristics) is given by the absolute values of the first order Bessel function. The theoretical phase modulation characteristics at amplitudes of 0.5, 1.5, 2.5, and 3.5 λHe-Ne are shown in Figure 1.8.
The Bessel function characteristics contribute to the throughput profile of ΦM experiment. In genreal, the first lobe at the lowest wavenumber region is the largest, the lobes at higher energies become smaller in magnitude. All lobes becomes narrower and shift to lower energies as the ΦM amplitude increase. Figure 1.9 illustrates single-beam spectra measured with a TGS detector at a ΦM frequency of 400Hz and a ΦM amplitudes varying from 0.5 to 5.5 λHe-Ne. The results are plotted on the same scale. These throughput curves are results of the combined effects of the spectrometer's normal throughput and the ΦM modulation characteristics.
The throughput for the mid-IR region, especially the fingerprint region, is improved significantly from ΦM amplitude of 0.5 to 3.5 λHe-Ne. However, the throughput in the near-IR region is reduced at ΦM amplitudes of 2.5 and 3.5 λHe-Ne because of the presence of nodes. Thus ΦM amplitude should be selected according to the spectra range of interest. ΦM amplitude of 0.5 λHe-Ne or 1.5 λHe-Ne are appropriate for near IR experiments. A ΦM amplitude of 3.5λHe-Ne provides excellent coverage over the entire mid-IR region. Larger ΦM amplitudes shift the largest lobe to lower energies and bring nodes (zero effciency) into mid-IR region. Therefore, a ΦM amplitude of 3.5 λHe-Ne is often chosen to optimize mid-IR throughput.
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