# Real-Time Measurement of Photodissociation with a Static Modulated Fourier Transform Spectrometer
## 2 Methods  
### A Static Modulated Fourier Transform Spectrometer composed of a Modified Sagnac Interferometer  
In this paper, a static modulation Fourier transform spectrometer was implemented, based on a modified Sagnac interferometer to perform a real-time measurement. The interferometer consisted of two plane mirrors and one beam splitter, as shown in Figure 1. The beamsplitter was tilted 45\({}^{\circ}\) from the optical axis, and the two mirrors were each tilted 22.5\({}^{\circ}\) from the optical axis. Unlike a conventional dynamic modulated Fourier transform spectrometer that measures the interferogram as a temporal function, the static modulated Fourier transform spectrometer, as used in this study, measured the interferogram as a spatial function. The interferogram, composed of spatially distributed optical signal along the optical path difference, was measured by a one-dimensional detector array.  
Figure 1: A static modulated Fourier transform spectrometer composed of a modified Sagnac interferometer.  
In order to obtain an interferogram, the pumping light was irradiated to the sample and then the fluorescence light from the sample was measured by the interferometer. The fluorescence light incident to the interferometer was split into two beams at the beam splitter. Those two beams travelled different optical paths, returning to the beam splitter. If the two mirrors were located the equal distances from the beam splitter, the optical path lengths of the two beams should be same, and no interferogram was observed at the focal plane. When the mirror M2 was displaced by \(a\) from the symmetrical position, IP was indicated in a dotted line and the optical path difference between two split beams generated an interferogram at the focal plane in the overlapped sector of two beams, where a one-dimensional detector was placed. The distance between the two beams \(l\) was determined by the position of M2 in Equation (1).  
\[l=\sqrt{2a} \tag{1}\]  
where \(l\) is the distance between two beams and \(a\) is the mirror displacement from the initial symmetrical position IP.  
At the focal plane, the interferogram appeared in the form of the interference between two beams. In the modified Sagnac interferometer, the optical path difference between two beams was equal to that at the focal plane in the case of the double slit. The interferogram occurred in the overlapped sector and was measured by a one-dimensional detector array. Since the spectral resolution of the interferometer was defined as the reciprocal of the maximum optical path difference, it can be expressed as Equation (2).  
\[\delta_{\text{Sagnac}}=\frac{1}{\Delta}=\frac{f}{l\cdot N\cdot y_{\text{min}}}, \tag{2}\]  
where \(\delta_{\text{Sagnac}}\) is the spectral resolution, \(\Delta\) is an optical path difference,\(f\) is the focal length of the lens, \(y_{\text{min}}\) is the width of the single pixel, and \(N\) is the number of pixels of the detector.  
In order to obtain a better spectral resolution in the static modulated Fourier transform spectroscopy, a longer maximum optical path difference is required. Since the sampling frequency of the spectrometer is inversely proportional to the optical path difference, the sampling frequency decreases as the maximum optical path difference increases. Since the incorrect data sampling of the optical signal appears as aliasing in the spectrum, the spectral resolution of the spectrometer without aliasing is defined as Equation (3).  
\[N\cdot\delta_{\text{Sagnac}}\geq 2\overline{v}_{\text{max}}, \tag{3}\]  
where \(\overline{v}_{\text{max}}\) is the maximum wavenumber of the measured optical signal.  
According to Nyquist sampling theory in Equation (3), the sampling frequency in the static modulated Fourier transform spectrometer is at least twice the wavenumber of the measured optical signal. When the wavenumber band of the optical signal to be measured is wide, only the sampling frequency for the largest wavenumber needs to be considered.