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Optical real-time Fourier transformation with kilohertz resolutions
Hugues Guillet de Chatellus, Luis Romero Cortés, and José Azaña
Hugues Guillet de Chatellus,1,2,3,* Luis Romero Cortés,1 and José Azaña1
1Institut National de la Recherche Scientifique—Energie, Matériaux et Télécommunications (INRS-EMT), 1650 Boul. Lionel-Boulet, Varennes, Québec J3X 1S2, Canada
2CNRS, LIPHY, F-38000 Grenoble, France
3Université Grenoble Alpes, LIPHY, F-38000 Grenoble, France
*Corresponding author: hugues.guilletdechatellus@ujf-grenoble.fr
H Guillet de Chatellus
L Cortés
J Azaña
Hugues Guillet de Chatellus, Luis Romero Cortés, and José Azaña, "Optical real-time Fourier transformation with kilohertz resolutions," Optica 3, 1-8 (2016)
Optical amplifiers
Radio frequency photonics (060.5625)
Analog optical signal processing (070.1170)
Spectrum analysis (070.4790)
Continuous optical signal processing (070.6020)
Revised Manuscript: November 26, 2015
Manuscript Accepted: November 26, 2015
Real-time Fourier transformation (RTFT) of signals is a fundamental concept that enables Fourier analysis at speeds beyond the limitations of conventional digital signal processing engines. In the optical domain, RTFT is commonly performed by inducing large amounts of group-velocity dispersion on the signal (time-varying light wave) of interest to map the signal's frequency spectrum along the time domain. However, the optical frequency resolution of this method is typically restricted above the gigahertz range, which represents a critical limitation for applications in real-time spectroscopy, ultrafast detection, imaging and sensing, and, more particularly, for photonic-assisted generation and processing of radio-frequency (RF) signals. Here we propose a new concept for realization of RTFT that involves superposition of multiple signal replicas that are shifted simultaneously along the temporal and frequency domains, leading to an output temporal waveform that effectively maps the input optical spectrum. This configuration overcomes the frequency-resolution limitations of dispersion-based RTFT schemes, while providing the desired signal's spectrum with minimal latency, equal to the inverse of the frequency resolution. We experimentally demonstrate a practical implementation of the concept on optical signals using a frequency shifted feedback (FSF) laser, achieving a frequency resolution of ≃30 kHz and a time–bandwidth product exceeding 400, while the predicted linearity of the frequency-to-time mapping process is shown over a 20 GHz bandwidth. The introduced concept should be of general interest for high-speed, real-time Fourier analysis, beyond the optical-domain implementation reported here; moreover, this work also paves the way for novel applications of FSF lasers in several areas, including high-precision metrology and optical or RF waveform synthesis and processing.
Optical frequency domain reflectometry based on real-time Fourier transformation
Yongwoo Park, Tae-Jung Ahn, Jean-Claude Kieffer, and José Azaña
Real-time Fourier transformation of lightwave spectra and application in optical reflectometry
Antonio Malacarne, Yongwoo Park, Ming Li, Sophie LaRochelle, and José Azaña
Real-time Fourier transformation based on the bandwidth magnification of RF signals
Yan Zheng, Jilong Li, Yitang Dai, Feifei Yin, and Kun Xu
Real-time broadband radio frequency spectrum analyzer based on parametric spectro-temporal analyzer (PASTA)
Liao Chen, Yuhua Duan, Haidong Zhou, Xi Zhou, Chi Zhang, and Xinliang Zhang
Ultrahigh dispersion of broadband microwave signals by incoherent photonic processing
Yongwoo Park and José Azaña
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Lacot, E.
Levit, B.
Li, I.
Marhic, M. E.
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Pinckney, J. T.
Quiring, V.
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Sohler, W.
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Tong, Y. C.
Tsang, H. K.
Tsia, K. K.
van Howe, J.
Wei, X.
Weiner, A. M.
Wong, K. K. Y.
Xi, L.
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Zhang, C.
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IEEE Microw. Wireless Compon. Lett. (1)
Fig. 1. (a) Sketch of a generic FSF cavity. The AOFS is driven at frequency f s = ω s / 2 π , resulting in a frequency shift per round trip equal to f s . The cavity round-trip time is τ c = 1 / f c = 2 π / ω c . (b) Optical spectrum of the output of the FSF laser seeded by a monochromatic wave at angular frequency ω 0 . The envelope of the spectrum of the FSF frequency comb is the function H (see text).
Fig. 2. (a) Representation of the response of the FSF cavity to a single input frequency. A single seed frequency generates a train of pulses at a repetition rate ω s / 2 π . (b) Principle of FTM in a FSF cavity: the temporal shape of the output pulses maps the spectrum of the input light field.
Fig. 3. Numerical example of the latency of the FSF-RTFT technique. Left: time-limited input waveform of duration T at the input of the FSF loop. The spectral content consists of five frequency components. Right: intensity at the output of the FSF loop. After a transient time τ L (or latency), the output signal maps repeatedly the input spectrum.
Fig. 4. Sketch of the fiber FSF laser. The loop is seeded with a phase-modulated CW laser. The AOFS is driven by an arbitrary function generator (AFG). An optical bandpass filter (OBPF) enables control of the spectral bandwidth of the laser and limits the ASE, and an optical isolator (OI) prevents backward laser operation in the loop. A heterodyne interferometer (bottom) enables measurement of the spectrum of the phase-modulated input (see text).
Fig. 5. (a) Output time traces recorded for an input laser phase modulated by a sine wave at different frequencies (50, 450, and 1050 kHz). No averaging is performed. (b) Linearity of the frequency-to-time mapping: the time delay between the "carrier" and the "sideband" pulses is plotted versus the PM frequency offset from DC, 1, 10, 15, and 20 GHz. A linear frequency to time mapping curve has been numerically extrapolated over the whole PM frequency range, according to Eq. (4).
Fig. 6. (a) Experimental output time traces recorded with different values of the PM frequency f m around 1 GHz. No averaging is performed. (b) The time traces are overlapped to synchronize the "carrier" pulses. (c) Zoom on the "sideband" pulses, showing the capability of the technique to distinguish frequency shifts with the predicted frequency resolution of ∼ 30 kHz .
Fig. 7. Top: optical spectra of the input laser after two different PM formats, reconstructed from heterodyne beatings with the seed laser (linear scale). The zero-frequency lines are added manually (see text). The PM functions are sine waves at frequencies of 200 (left) and 100 kHz (right). Bottom: time traces recorded at the output of the FSF loop seeded by the corresponding signals (16 traces averaged). The time axis in the bottom plots has been scaled in relation to the frequency axis of the upper plots, according to the FTM scaling law given in Eq. (4).
(1) E ( t ) = E 0 e − i ω 0 t ∑ n H ( n ω s ) e − i n ω s t e i n ω 0 τ c e i n ( n + 1 ) 2 ϕ ,
(2) ∑ n H ( n ω s ) e − i n ω s t ∝ ∑ n h ( t − n 2 π ω s ) ,
(3) E ( t ) = E 0 e − i ω 0 t ∑ n h ( t − ω 0 ω s τ c − n 2 π ω s ) .
(4) τ = τ c ω 0 ω s ,
(5) δ f = ω s ω c 2 π Δ ω .
(6) E out ( t ) = ∫ E ˜ in ( ω ) e − i ω t ∑ n h ( t − ω ω s τ c − n 2 π ω s ) d ω .