PHYSICAL REVIEW A 82, 033827 (2010) Noise figure of amplified dispersive Fourier transformation Keisuke Goda and Bahram Jalali Electrical Engineering Department, University of California, Los Angeles, California 90095, USA (Received 1 June 2010; published 21 September 2010) Amplified dispersive Fourier transformation (ADFT) is a powerful tool for fast real-time spectroscopy as it overcomes the limitations of traditional optical spectrometers. ADFT maps the spectrum of an optical pulse into a temporal waveform using group-velocity dispersion and simultaneously amplifies it in the optical domain. It greatly simplifies spectroscopy by replacing the diffraction grating and detector array in the conventional spectrometer with a dispersive fiber and single-pixel photodetector, enabling ultrafast real-time spectroscopic measurements. Following our earlier work on the theory of ADFT, here we study the effect of noise on ADFT. We derive the noise figure of ADFT and discuss its dependence on various parameters. DOI: 10.1103/PhysRevA.82.033827 PACS number(s): 42.62.Fi, 42.65.Dr, 42.65.Re I. INTRODUCTION Amplified dispersive Fourier transformation (ADFT), also known as time-stretch amplified Fourier transformation [1–6], is a powerful technique for fast real-time spectroscopic mea- surements. ADFT has been used in various applications such as time-stretch analog-to-digital conversion [7], absorption [6] and Raman [3] spectroscopy, and optical frequency- domain reflectometry [4]. Most recently, it has been used to demonstrate an imaging method with a very high frame rate and shutter speed, known as serial time-encoded amplified imaging or microscopy (STEAM) [1,5,8]. ADFT enables fast real-time spectroscopic measurements as it overcomes the limitations of traditional optical spec- trometers. ADFT is based on the exploitation of the analogy between paraxial diffraction and temporal dispersion. The novelty of ADFT is its ability to circumvent the loss inherent in a dispersive medium using internal optical amplification. The amplification is realized using stimulated Raman scattering (SRS) within the dispersive medium. This leads to the lowest- possible noise penalty because, similar to the loss in the disper- sive medium, the SRS gain is also distributed in nature [9,10]. The principle of ADFT is the mapping of the spectrum of an optical pulse into a time-domain waveform using group- velocity dispersion (GVD) and the simultaneous amplification of the waveform in the optical domain. It replaces the spatial disperser (e.g., the diffraction grating or prism) and detector array (e.g., the charge-coupled device, CCD) in conventional spectrometers with a dispersive device (e.g., a dispersive fiber or chirped fiber Bragg grating) and single-pixel photodetector. This greatly simplifies the system and, more importantly, it enables ultrafast real-time spectroscopic measurements at the scan rate equivalent to the pulse repetition rate of the laser. In ADFT, the optical spectrum is measured in the time domain. By measuring the temporal waveform with a single- pixel photodetector, a real-time analog-to-digital converter or digital oscilloscope effectively samples the optical spectrum at ultrahigh scan rates, significantly beyond what is possible with conventional grating-based spectrometers. With distributed optical amplification in the dispersive medium (e.g., the dispersive fiber), ADFT overcomes the fundamental trade-off between loss and dispersion, hence circumventing the loss of sensitivity at high speeds caused by the reduced number of photons that are collected during short integration times—a predicament that affects all conventional spectrometers and imaging systems. Distributed Raman amplification via SRS provides several advantages over discrete optical amplifiers such as rare-earth- doped fiber amplifiers and semiconductor optical amplifiers (SOA’s). First, distributed Raman amplification within the dispersive medium is superior because it maintains a relatively constant signal level throughout the ADFT process. This im- portant property maximizes the signal-to-noise-and-distortion ratio by keeping the signal power away from low-power (noisy) and high-power (nonlinear) regimes. Second, gain is possible at any wavelength as long as a pump field is available at a frequency blue-shifted from the signal by the optical-phonon vibrational frequency [11]. Third, a broad and flexible gain spectrum can be generated by the use of multiple pump fields which may be continuous-wave lasers [1,6] or incoherent light sources [3]. Finally, distributed Raman amplification has a lower noise figure than rare-earth-doped fiber amplifiers and SOA’s. These advantages of Raman amplification over the use of the discrete amplifiers are known in long-haul fiber-optic communication links [11]. Raman-amplified dispersive elements also eliminate the need for high-power optical sources, which can potentially cause damage to the sample under study [1]. The desirable features for the dispersive medium in ADFT are high total dispersion, low loss, large optical bandwidth, and constant dispersion over the bandwidth of interest. At fiber-optic communication wavelengths, the dispersion compensation fiber (DCF) offers an optimum combination of these parameters and has been used in many applications [1,3–6,12–17]. There are also fibers for shorter wavelengths (e.g., 800 nm) that provide adequate dispersion although with much higher loss-to-dispersion ratio than the DCF. Our earlier work on ADFT explained and quantified the spectral resolution of this technique as set by the stationary phase approximation and nonlinear dispersion [2]. In all applications of ADFT, the noise sets the limit on the sensitivity and hence the minimum signal level that can be detected. In this paper, we study the effect of noise on ADFT using quantum noise operators and Langevin noise sources [18–20]. We first derive the noise figure of ADFT and then discuss its depen- dence on various parameters such as the loss and Raman gain coefficients and the pump noise. As numerous applications of ADFT such as spectroscopy [3,6] and imaging [1,4,5] are 1050-2947/2010/82(3)/033827(9) 033827-1 ©2010 The American Physical Society