Time stretch dispersive Fourier transform
Dispersive fourier transform (DFT), otherwise known as time-stretch dispersive Fourier transform, time-stretch transform (TST), temporal Fourier transform or photonic time-stretch (PTS) is a spectroscopy technique that uses optical dispersion instead of a grating or prism to separate the light wavelengths and analyze the optical spectrum in real-time. It employs group-velocity dispersion (GVD) to transform the spectrum of a broadband optical pulse into a time stretched temporal waveform. It is used to perform Fourier transformation on an optical signal on a single shot basis and at high frame rates for real-time analysis of fast dynamic processes. It replaces a diffraction grating and detector array with a dispersive fiber and single-pixel detector, enabling ultrafast real-time spectroscopy and imaging.
DFT is usually used in a two step process. In the first step, the spectrum of an optical broadband pulse is encoded by the information (e.g., temporal, spatial, or chemical information) to be captured. In the next step, the encoded spectrum is mapped by large group-velocity dispersion into a slowed down temporal waveform. At this point the waveform has been sufficiently slowed down so it can be digitized and processed in real-time. Without the time stretch, single shot waveforms will be too fast to be digitized by an analog to digital converters. When needed, the waveform is simultaneously amplified in the dispersive fiber by the process of stimulated Raman scattering. This optical amplification overcomes the thermal noise which would otherwise limit the sensitivity. Subsequent optical pulses perform repetitive measurements at the frame rate of the pulsed laser. Consequently, single shot optical spectra, carrying information from fast dynamic processes, can be digitized and analyzed at high frame rates. The time-stretch dispersive Fourier transformer consists of a low-loss dispersive fiber that is also a Raman amplifier. To create Raman gain, pump lasers are coupled into the fiber by wavelength-division multiplexers, with wavelengths of pump lasers chosen to create a broadband and flat gain profile that covers the spectrum of the broadband optical pulse. Instead of Raman amplification, a discrete amplifier such as an erbium doped optical amplifier or a semiconductor optical amplifier can be placed before the dispersive fiber. However the distributed nature of Raman amplification provides superior signal to noise ratio. Dispersive Fourier Transform has proven to be an enabling technology for wideband A/D conversion (ultra wideband analog to digital converters) and has also been used for high-throughput real-time spectroscopy and imaging (serial time-encoded amplified microscopy (STEAM)).
Recently it has been shown that time-bandwidth product of waveforms can be compressed or expanded by a transformation that includes reshaping the signal in a fashion resembling the graphic art techniques known as Anamorphosis and Surrealism. Dubbed the Anamorphic Stretch Transform (AST), this technique uses a mathematical transformation in which the signal is stretched and warped in a specific fashion prescribed by a newly developed mathematical function called Stretched Modulation Distribution, or Modulation Intensity Distribution (not to be confused with a different function of the same name used in acoustics). This is a 3D plot that describes the dependence of the intensity (power), on the modulation frequency and its time duration, as the signal propagates through a warped dispersive element. The operation spans both near field and far field regimes of dispersive Fourier transform. While making such distinction is unnecessary, in the far field regime AST becomes a generalization of Dispersive Fourier Transform (DFT) where the dispersion causes warped or nonlinear frequency to time mapping. The AST technology makes it possible to not only capture and digitize signals that are faster than the speed of the sensor and the digitizer, but also to minimize the volume of the data generated in the process. The transformation causes the signal to be reshaped is such a way that sharp features are stretched more than coarse features. Upon subsequent sampling, this Self-Adaptive Stretch (SAS) causes more digital samples to be allocated to sharp features where they are needed the most, and fewer to coarse features where they would be redundant. The resulting data compression does not need feature detection or any prior knowledge of the signal. Anamorphic Stretch Transform has shown its capability to compress the Big Data.
Real-time single-shot analysis of spectral noise
Recently PTS has proved invaluable in the study of optical non-linearities in fibers. Correlation properties in both the spectral and temporal domains can be deduced from single-shot PTS data to study the stochastic nature of optical systems. Namely, modulation instability  and supercontiuum generation  in highly non-linear fiber have been studied. It has been suggested that:
"... this technique of real-time spectral noise characterisation should now become the standard approach to analysing noise effects in nonlinear fibre propagation as it provides a powerful means of allowing rapid experimental optimization, and generating large data sets from experiment in this way is orders of magnitude more efficient than stochastic simulations. Indeed, we anticipate wide application of this technique for detailed studies of noise in broadband pulse propagation, the characterization of quantum-optical intensity-correlations in soliton dynamics, and in the study of spectral instabilities in ultrafast nonlinear optics in general."
- K. Goda & B. Jalali, “Dispersive Fourier transformation for fast continuous single-shot measurements," Nature Photonics 7, 102–112 (2013) doi:10.1038/nphoton.2012.359. 
- A. S. Bhushan, F. Coppinger, and B. Jalali, “Time-stretched analogue-to-digital conversion," Electronics Letters vol. 34, no. 9, pp. 839–841, April 1998. 
- Y. Han and B. Jalali, “Photonic Time-Stretched Analog-to-Digital Converter: Fundamental Concepts and Practical Considerations," Journal of Lightwave Technology, Vol. 21, Issue 12, pp. 3085–3103, Dec. 2003. 
- P. Kelkar, F. Coppinger, A. S. Bhushan, and B. Jalali, “Time-domain optical sensing,” Electronics Letters 35, 1661 (1999)
- D. R. Solli, J. Chou, and B. Jalali, "Amplified wavelength–time transformation for real-time spectroscopy," Nature Photonics 2, 48-51, 2008. 
- J. Chou, D. Solli, and B. Jalali, "Real-time spectroscopy with subgigahertz resolution using amplified dispersive Fourier transformation," Applied Physics Letters 92, 111102, 2008. 
- K. Goda, K. K. Tsia, and B. Jalali (2008). "Amplified dispersive Fourier-transform imaging for ultrafast displacement sensing and barcode reading". Applied Physics Letters 93: 131109. arXiv:0807.4967. Bibcode:2008ApPhL..93m1109G. doi:10.1063/1.2992064.
- M. H. Asghari, and B. Jalali, “Anamorphic transformation and its application to time-bandwidth compression,” Physics Arxiv, arXiv:1307.0137, June 2013.
- M. H. Asghari, and B. Jalali, "Anamorphic transformation and its application to time-bandwidth compression," Applied Optics, Vol. 52, pp. 6735-6743 (2013). 
- M. H. Asghari, and B. Jalali, "Demonstration of analog time-bandwidth compression using anamorphic stretch transform," Frontiers in Optics (FIO 2013), Paper: FW6A.2, Orlando, USA. 
- Solli, D. R., Herink, G., Jalali, B. & Ropers, C., “Fluctuations and correlations in modulation instability" 'Nature Photon. 6, 463–468 (2012). 
- B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, & J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation," SCIENTIFIC REPORTS, Volume: 2, Article Number: 882, DOI: 10.1038/srep00882, Published: NOV 28 2012.