Abstract

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 30kHz 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.

© 2015 Optical Society of America

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References

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2014 (2)

J. Zhang and J. Yao, “Time-stretched sampling of a fast microwave waveform based on the repetitive use of a linearly chirped fiber Bragg grating in a dispersive loop,” Optica 1, 64–69 (2014).
[Crossref]

C. Zhang, X. Wei, M. E. Marhic, and K. K. Y. Wong, “Ultrafast and versatile spectroscopy by temporal Fourier transform,” Sci. Rep. 4, 5351 (2014).

2013 (5)

2011 (1)

2010 (1)

2009 (1)

K. Goda, K. K. Tsia, and B. Jalali, “Serial time-encoded amplified imaging for real-time observation of fast dynamic phenomena,” Nature 458, 1145–1149 (2009).
[Crossref]

2008 (1)

D. R. Solli, J. Chou, and B. Jalali, “Amplified wavelength-time transformation for real-time spectroscopy,” Nat. Photonics 2, 48–51 (2008).
[Crossref]

2007 (1)

2006 (2)

2005 (1)

I. Li, J. D. McKinney, and A. M. Weiner, “Photonic synthesis of broadband microwave arbitrary waveforms applicable to ultra-wideband communication,” IEEE Microw. Wireless Compon. Lett. 15, 226–228 (2005).
[Crossref]

2004 (1)

L. P. Yatsenko, B. W. Shore, and K. Bergmann, “Theory of a frequency-shifted feedback laser,” Opt. Commun. 236, 183–202 (2004).
[Crossref]

2000 (1)

J. Azaña and M. A. Muriel, “Real-time optical spectrum analysis based on the time-space duality in chirped fiber gratings,” IEEE J. Quantum Electron. 36, 517–526 (2000).
[Crossref]

1997 (1)

Y. C. Tong, L. Y. Chan, and H. K. Tsang, “Fiber dispersion or pulse spectrum measurement using a sampling oscilloscope,” Electron. Lett. 33, 983–985 (1997).
[Crossref]

1990 (1)

P. D. Hale and F. V. Kowalski, “Output characterization of a frequency shifted feedback laser: theory and experiment,” IEEE J. Quantum Electron. 26, 1845–1851 (1990).
[Crossref]

1987 (1)

F. V. Kowalski, J. A. Squier, and J. T. Pinckney, “Pulse generation with an acousto-optic frequency shifter in a passive cavity,” Appl. Phys. Lett. 50, 711–713 (1987).
[Crossref]

1983 (1)

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Elsevier, 2007).

Ahn, T.-J.

Azaña, J.

Berger, N. K.

Bergmann, K.

L. P. Yatsenko, B. W. Shore, and K. Bergmann, “Theory of a frequency-shifted feedback laser,” Opt. Commun. 236, 183–202 (2004).
[Crossref]

Chan, L. Y.

Y. C. Tong, L. Y. Chan, and H. K. Tsang, “Fiber dispersion or pulse spectrum measurement using a sampling oscilloscope,” Electron. Lett. 33, 983–985 (1997).
[Crossref]

Chou, J.

D. R. Solli, J. Chou, and B. Jalali, “Amplified wavelength-time transformation for real-time spectroscopy,” Nat. Photonics 2, 48–51 (2008).
[Crossref]

Dezfooliyan, R.

Fisher, B.

Foster, M. A.

Gaeta, A. L.

Glastre, W.

Goda, K.

K. Goda and B. Jalali, “Dispersive Fourier transformation for fast continuous single-shot measurements,” Nat. Photonics 7, 102–112 (2013).
[Crossref]

K. Goda, K. K. Tsia, and B. Jalali, “Serial time-encoded amplified imaging for real-time observation of fast dynamic phenomena,” Nature 458, 1145–1149 (2009).
[Crossref]

Guillet de Chatellus, H.

Hale, P. D.

P. D. Hale and F. V. Kowalski, “Output characterization of a frequency shifted feedback laser: theory and experiment,” IEEE J. Quantum Electron. 26, 1845–1851 (1990).
[Crossref]

Hugon, O.

Jacquin, O.

Jalali, B.

K. Goda and B. Jalali, “Dispersive Fourier transformation for fast continuous single-shot measurements,” Nat. Photonics 7, 102–112 (2013).
[Crossref]

K. Goda, K. K. Tsia, and B. Jalali, “Serial time-encoded amplified imaging for real-time observation of fast dynamic phenomena,” Nature 458, 1145–1149 (2009).
[Crossref]

D. R. Solli, J. Chou, and B. Jalali, “Amplified wavelength-time transformation for real-time spectroscopy,” Nat. Photonics 2, 48–51 (2008).
[Crossref]

Jannson, T.

Kieffer, J.-C.

Kowalski, F. V.

P. D. Hale and F. V. Kowalski, “Output characterization of a frequency shifted feedback laser: theory and experiment,” IEEE J. Quantum Electron. 26, 1845–1851 (1990).
[Crossref]

F. V. Kowalski, J. A. Squier, and J. T. Pinckney, “Pulse generation with an acousto-optic frequency shifter in a passive cavity,” Appl. Phys. Lett. 50, 711–713 (1987).
[Crossref]

Lacot, E.

Levit, B.

Li, I.

I. Li, J. D. McKinney, and A. M. Weiner, “Photonic synthesis of broadband microwave arbitrary waveforms applicable to ultra-wideband communication,” IEEE Microw. Wireless Compon. Lett. 15, 226–228 (2005).
[Crossref]

Li, J.

Marhic, M. E.

C. Zhang, X. Wei, M. E. Marhic, and K. K. Y. Wong, “Ultrafast and versatile spectroscopy by temporal Fourier transform,” Sci. Rep. 4, 5351 (2014).

Marklof, J.

McKinney, J. D.

I. Li, J. D. McKinney, and A. M. Weiner, “Photonic synthesis of broadband microwave arbitrary waveforms applicable to ultra-wideband communication,” IEEE Microw. Wireless Compon. Lett. 15, 226–228 (2005).
[Crossref]

Muriel, M. A.

J. Azaña and M. A. Muriel, “Real-time optical spectrum analysis based on the time-space duality in chirped fiber gratings,” IEEE J. Quantum Electron. 36, 517–526 (2000).
[Crossref]

Park, Y.

Pinckney, J. T.

F. V. Kowalski, J. A. Squier, and J. T. Pinckney, “Pulse generation with an acousto-optic frequency shifter in a passive cavity,” Appl. Phys. Lett. 50, 711–713 (1987).
[Crossref]

Quiring, V.

S. Reza, R. Ricken, V. Quiring, and W. Sohler, “High resolution optical frequency domain ranging with an integrated frequency shifted feedback (FSF) laser,” in European Conference on Lasers and Electro-Optics 2007, Conference Digest (Optical Society of America, 2007), paper CJ3-5.

Reza, S.

S. Reza, R. Ricken, V. Quiring, and W. Sohler, “High resolution optical frequency domain ranging with an integrated frequency shifted feedback (FSF) laser,” in European Conference on Lasers and Electro-Optics 2007, Conference Digest (Optical Society of America, 2007), paper CJ3-5.

Ricken, R.

S. Reza, R. Ricken, V. Quiring, and W. Sohler, “High resolution optical frequency domain ranging with an integrated frequency shifted feedback (FSF) laser,” in European Conference on Lasers and Electro-Optics 2007, Conference Digest (Optical Society of America, 2007), paper CJ3-5.

Salem, R.

Shore, B. W.

L. P. Yatsenko, B. W. Shore, and K. Bergmann, “Theory of a frequency-shifted feedback laser,” Opt. Commun. 236, 183–202 (2004).
[Crossref]

Sohler, W.

S. Reza, R. Ricken, V. Quiring, and W. Sohler, “High resolution optical frequency domain ranging with an integrated frequency shifted feedback (FSF) laser,” in European Conference on Lasers and Electro-Optics 2007, Conference Digest (Optical Society of America, 2007), paper CJ3-5.

Solli, D. R.

D. R. Solli, J. Chou, and B. Jalali, “Amplified wavelength-time transformation for real-time spectroscopy,” Nat. Photonics 2, 48–51 (2008).
[Crossref]

Squier, J. A.

F. V. Kowalski, J. A. Squier, and J. T. Pinckney, “Pulse generation with an acousto-optic frequency shifter in a passive cavity,” Appl. Phys. Lett. 50, 711–713 (1987).
[Crossref]

Tian, F.

Tong, Y. C.

Y. C. Tong, L. Y. Chan, and H. K. Tsang, “Fiber dispersion or pulse spectrum measurement using a sampling oscilloscope,” Electron. Lett. 33, 983–985 (1997).
[Crossref]

Tsang, H. K.

Y. C. Tong, L. Y. Chan, and H. K. Tsang, “Fiber dispersion or pulse spectrum measurement using a sampling oscilloscope,” Electron. Lett. 33, 983–985 (1997).
[Crossref]

Tsia, K. K.

K. Goda, K. K. Tsia, and B. Jalali, “Serial time-encoded amplified imaging for real-time observation of fast dynamic phenomena,” Nature 458, 1145–1149 (2009).
[Crossref]

van Howe, J.

Wei, X.

C. Zhang, X. Wei, M. E. Marhic, and K. K. Y. Wong, “Ultrafast and versatile spectroscopy by temporal Fourier transform,” Sci. Rep. 4, 5351 (2014).

Weiner, A. M.

R. Dezfooliyan and A. M. Weiner, “Photonic synthesis of high fidelity microwave arbitrary waveforms using near field frequency to time mapping,” Opt. Express 21, 22974–22987 (2013).
[Crossref]

I. Li, J. D. McKinney, and A. M. Weiner, “Photonic synthesis of broadband microwave arbitrary waveforms applicable to ultra-wideband communication,” IEEE Microw. Wireless Compon. Lett. 15, 226–228 (2005).
[Crossref]

Wong, K. K. Y.

C. Zhang, X. Wei, M. E. Marhic, and K. K. Y. Wong, “Ultrafast and versatile spectroscopy by temporal Fourier transform,” Sci. Rep. 4, 5351 (2014).

Xi, L.

Xu, C.

Yao, J.

Yatsenko, L. P.

L. P. Yatsenko, B. W. Shore, and K. Bergmann, “Theory of a frequency-shifted feedback laser,” Opt. Commun. 236, 183–202 (2004).
[Crossref]

Zhang, C.

C. Zhang, X. Wei, M. E. Marhic, and K. K. Y. Wong, “Ultrafast and versatile spectroscopy by temporal Fourier transform,” Sci. Rep. 4, 5351 (2014).

Zhang, J.

Zhang, X.

Adv. Opt. Photon. (1)

Appl. Phys. Lett. (1)

F. V. Kowalski, J. A. Squier, and J. T. Pinckney, “Pulse generation with an acousto-optic frequency shifter in a passive cavity,” Appl. Phys. Lett. 50, 711–713 (1987).
[Crossref]

Electron. Lett. (1)

Y. C. Tong, L. Y. Chan, and H. K. Tsang, “Fiber dispersion or pulse spectrum measurement using a sampling oscilloscope,” Electron. Lett. 33, 983–985 (1997).
[Crossref]

IEEE J. Quantum Electron. (2)

J. Azaña and M. A. Muriel, “Real-time optical spectrum analysis based on the time-space duality in chirped fiber gratings,” IEEE J. Quantum Electron. 36, 517–526 (2000).
[Crossref]

P. D. Hale and F. V. Kowalski, “Output characterization of a frequency shifted feedback laser: theory and experiment,” IEEE J. Quantum Electron. 26, 1845–1851 (1990).
[Crossref]

IEEE Microw. Wireless Compon. Lett. (1)

I. Li, J. D. McKinney, and A. M. Weiner, “Photonic synthesis of broadband microwave arbitrary waveforms applicable to ultra-wideband communication,” IEEE Microw. Wireless Compon. Lett. 15, 226–228 (2005).
[Crossref]

J. Lightwave Technol. (3)

Nat. Photonics (2)

K. Goda and B. Jalali, “Dispersive Fourier transformation for fast continuous single-shot measurements,” Nat. Photonics 7, 102–112 (2013).
[Crossref]

D. R. Solli, J. Chou, and B. Jalali, “Amplified wavelength-time transformation for real-time spectroscopy,” Nat. Photonics 2, 48–51 (2008).
[Crossref]

Nature (1)

K. Goda, K. K. Tsia, and B. Jalali, “Serial time-encoded amplified imaging for real-time observation of fast dynamic phenomena,” Nature 458, 1145–1149 (2009).
[Crossref]

Opt. Commun. (1)

L. P. Yatsenko, B. W. Shore, and K. Bergmann, “Theory of a frequency-shifted feedback laser,” Opt. Commun. 236, 183–202 (2004).
[Crossref]

Opt. Express (4)

Opt. Lett. (1)

Optica (1)

Phys. Rev. A (1)

H. Guillet de Chatellus, E. Lacot, W. Glastre, O. Jacquin, and O. Hugon, “Theory of Talbot lasers,” Phys. Rev. A 88, 033828 (2013).
[Crossref]

Sci. Rep. (1)

C. Zhang, X. Wei, M. E. Marhic, and K. K. Y. Wong, “Ultrafast and versatile spectroscopy by temporal Fourier transform,” Sci. Rep. 4, 5351 (2014).

Other (2)

G. P. Agrawal, Nonlinear Fiber Optics (Elsevier, 2007).

S. Reza, R. Ricken, V. Quiring, and W. Sohler, “High resolution optical frequency domain ranging with an integrated frequency shifted feedback (FSF) laser,” in European Conference on Lasers and Electro-Optics 2007, Conference Digest (Optical Society of America, 2007), paper CJ3-5.

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Figures (7)

Fig. 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.
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.
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.
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.
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.
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.
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).

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

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 ,
n H ( n ω s ) e i n ω s t n h ( t n 2 π ω s ) ,
E ( t ) = E 0 e i ω 0 t n h ( t ω 0 ω s τ c n 2 π ω s ) .
τ = τ c ω 0 ω s ,
δ f = ω s ω c 2 π Δ ω .
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 .

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