## 4 Dispersing the Light
### Fourier Transform Spectrograph  
#### 4.3.1 Concept  
The concept of a Fourier transform spectrograph (FTS) has been around in astronomy for over half a century now [82; 83]. A conventional FTS is an adaptation of a Mach-Zehnder interferometer with a way to precisely control the optical path delay between two identical beams created using a beam splitter. Precise control of the path length necessitates an ultra-stable structure for the FTS. A photonic FTS can easily resolve this issue by having a monolithic and miniaturized on-chip construction which, to a great extent, eliminates mechanical and thermal fluctuations. The schematic of a FTS is shown in Figure 6. The input fiber carrying the astronomical light is coupled with a waveguide which symmetrically splits into two identical waveguides (i.e., a 1 \(\times\) 2 beam splitter). An optical delay modulator introduces a precise and known optical delay in one of the waveguides. The phase modulation is typically achieved using thermal [84; 85] or electro-optical modulation [86; 87; 88]. The beams are combined and the power is measured using a single-element detector. By varying the path delay, an interferogram is constructed. The envelope of this interferogram is the Fourier transform of the incoming spectrum. Thus, the input spectrum is retrieved by inverse Fourier transform of the envelope.  
Figure 6: Schematic of a photonic Fourier transform spectrograph. The input light from a single-mode fiber is coupled to the waveguide. This waveguide splits into two identical arms, with one fixed and the other with a variable path delay introduced (by modulating the refractive index). The two waveguides are combined and the interference pattern (called the interferogram) is measured at the output. The envelope of the interferogram is the Fourier transform of the input signal and is thus extracted in post-processing. A longer path delay is required to get a higher order interference fringe and thus a higher spectral resolution.  
Photonic FTSs, like their conventional analogues, have several advantages [83]. The FTSs are unique in terms of providing arbitrarily large spectral resolution (\(\sim\)10-100 m/s) over a broad band of wavelengths with excellent sensitivity. The downside is the integration time required for obtaining the spectrum. Unlike dispersive spectrometers, FTSs do not experience a degradation in the signal-to-noise ratio at a high spectral resolution, since the power is not distributed across small spectral channels [89]. A convenient feature of FTSs is that they can offer flexible resolving power (governed by the extent of optical depths covered) depending on the scientific requirement, making them versatile. Moreover, for photonic FTSs, the detector can be a single-element detector directly butt-coupled to the photonic chip or channeled via a SMF. There is renewed interest in FTSs for exoplanet characterization, thanks to the plentiful discoveries of exoplanets from the _Kepler_ mission [90; 91]. FTSs are also vital for measuring the radial velocities in binary star systems [92]. Photonic single-mode FTSs are poised to provide a diffraction-limited, stable platform for performing sensitive high spectral resolution exoplanet studies. By using distinguishing spectral features (such as certain molecular lines in the mid-IR), an exoplanet spectral feature can be extracted from the interferogram without physically separating the star and planet light (such as in a nulling interferometer) [93].