Problem: Fourier Transform Infrared Spectroscopy

  
Figure 8.1: Schematic sketch of the essential features of a Fourier transform infrared (FTIR) spectrometer.

      The Michelson interferometer is the basis of the most widely used spectrometers for infrared (IR) spectroscopy. The spectrometer consists of a ``white" light source, a beam splitter sending the light to a fixed mirror and a moving mirror, a sample to be measured, and the detector (see Figure 8.1). A spectrum is obtained by recording the intensity of the radiation at the detector, as a function of the position of the moving mirror. Usually, two spectra are recorded: one with the sample placed in the path of the light, and another one without the sample. The two spectra are ratioed point by point, and the result is Fourier-transformed to obtain the frequency-dependent transmission of the sample. The reflectivity of the sample can be measured similarly.

(a) Assume that the detector is pointlike and its sensitivity does not depend on the frequency of incident radiation, the source provides uniform intensity at all frequencies, and the displacement x of the moving mirror is measured from the point where the optical path length in the two arms is equal. Show that at finite the quantity

 

is indeed proportional to the frequency-dependent transmission of the sample, and determine the constant . (Note: In optical spectroscopy the units of wavenumbers, , are often used instead of frequency, .)   

(b) What determines the frequency resolution of the instrument?  

(c) The detector signal is amplified by a a device of upper cutoff frequency kHz. The mirror moves with a velocity v=2.5 cm/sec. What is the high frequency cutoff of the measured spectrum?