Abstract

The last 20 years have seen the spectacular emergence of optically assisted solutions for the analog generation and processing of radio-frequency (RF) signals [J. Lightwave Technol. 27, 314 (2009) [CrossRef]  ]. Among these, real-time Fourier transformation (FT) is of particular importance for signal processing and filtering, but is of limited use for analyzing signals with time-dependent spectrum, especially chirped RF waveforms. On the contrary, fractional Fourier transformation (FrFT), which decomposes a waveform onto a continuous basis of linearly chirped functions, provides a generalization of FT and constitutes a valuable tool for analyzing signals with time-dependent spectrum [Signal Process. 91, 1351 (2011) [CrossRef]  ]. Here we prove a new and simple concept, enabling agile computation of the FrFT of both optical and RF signals in real time, with minimum latency time and a frequency resolution in the tens of kHz range. We demonstrate two practical applications of the technique: the absolute measurement of RF/optical chirp rates and the detection of weak chirped RF signals buried under noise. The introduced concept should be of practical interest for RF signal filtering and radar signal processing.

© 2017 Optical Society of America

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References

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  1. V. Namias, IMA J. Appl. Math. 25, 241 (1980).
    [Crossref]
  2. E. Sejdić, I. Djurović, and L. Stanković, Signal Process. 91, 1351 (2011).
    [Crossref]
  3. C. McBride and F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
    [Crossref]
  4. H. M. Ozaktas, M. A. Kutay, and D. Mendlovic, Adv. Imaging Electron Phys. 106, 239 (1999).
    [Crossref]
  5. H. M. Ozaktas, D. Mendlovic, L. Onural, and B. Barshan, J. Opt. Soc. Am. A 11, 547 (1994)
    [Crossref]
  6. P. Pellat-Finet, Opt. Lett. 19, 1388 (1994).
    [Crossref]
  7. H. Kolner, IEEE J. Quantum Electron. 30, 1951 (1994).
    [Crossref]
  8. S. Coëtmellec, M. Brunel, D. Lebrun, and C. Özkul, J. Opt. Soc. Am. A 18, 2754 (2001).
    [Crossref]
  9. Cuadrado-Laborde, A. Carrascosa, A. Díez, J. L. Cruz, and M. V. Andres, Opt. Express 21, 8558 (2013).
    [Crossref]
  10. W. Lohmann and D. Mendlovic, Appl. Opt. 33, 7661 (1994).
    [Crossref]
  11. V. Torres-Company, J. Lancis, and P. Andres, Prog. Opt. 56, 1 (2011).
    [Crossref]
  12. H. Guillet de Chatellus, E. Lacot, W. Glastre, O. Jacquin, and O. Hugon, Phys. Rev. A 88, 033828 (2013).
    [Crossref]
  13. H. Guillet de Chatellus, L. Romero Cortés, and J. Azaña, Optica, 3, 1 (2016).
    [Crossref]
  14. N. Levanon and E. Mozeson, Radar Signals (Wiley, 2004)
  15. Schnebelin and H. Guillet de Chatellus, Appl. Opt. 56, A62 (2017).
    [Crossref]
  16. J. Capmany, G. Li, C. Lim, and J. Yao, Opt. Express 21, 22862 (2013).
    [Crossref]
  17. J. Yao, J. Lightwave Technol. 27, 314 (2009).
    [Crossref]

2017 (1)

2016 (1)

2013 (3)

2011 (2)

V. Torres-Company, J. Lancis, and P. Andres, Prog. Opt. 56, 1 (2011).
[Crossref]

E. Sejdić, I. Djurović, and L. Stanković, Signal Process. 91, 1351 (2011).
[Crossref]

2009 (1)

2001 (1)

1999 (1)

H. M. Ozaktas, M. A. Kutay, and D. Mendlovic, Adv. Imaging Electron Phys. 106, 239 (1999).
[Crossref]

1994 (4)

1987 (1)

C. McBride and F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[Crossref]

1980 (1)

V. Namias, IMA J. Appl. Math. 25, 241 (1980).
[Crossref]

Andres, M. V.

Andres, P.

V. Torres-Company, J. Lancis, and P. Andres, Prog. Opt. 56, 1 (2011).
[Crossref]

Azaña, J.

Barshan, B.

Brunel, M.

Capmany, J.

Carrascosa, A.

Coëtmellec, S.

Cruz, J. L.

Cuadrado-Laborde,

Díez, A.

Djurovic, I.

E. Sejdić, I. Djurović, and L. Stanković, Signal Process. 91, 1351 (2011).
[Crossref]

Glastre, W.

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

Guillet de Chatellus, H.

Hugon, O.

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

Jacquin, O.

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

Kerr, F. H.

C. McBride and F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[Crossref]

Kolner, H.

H. Kolner, IEEE J. Quantum Electron. 30, 1951 (1994).
[Crossref]

Kutay, M. A.

H. M. Ozaktas, M. A. Kutay, and D. Mendlovic, Adv. Imaging Electron Phys. 106, 239 (1999).
[Crossref]

Lacot, E.

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

Lancis, J.

V. Torres-Company, J. Lancis, and P. Andres, Prog. Opt. 56, 1 (2011).
[Crossref]

Lebrun, D.

Levanon, N.

N. Levanon and E. Mozeson, Radar Signals (Wiley, 2004)

Li, G.

Lim, C.

Lohmann, W.

McBride, C.

C. McBride and F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[Crossref]

Mendlovic, D.

Mozeson, E.

N. Levanon and E. Mozeson, Radar Signals (Wiley, 2004)

Namias, V.

V. Namias, IMA J. Appl. Math. 25, 241 (1980).
[Crossref]

Onural, L.

Ozaktas, H. M.

H. M. Ozaktas, M. A. Kutay, and D. Mendlovic, Adv. Imaging Electron Phys. 106, 239 (1999).
[Crossref]

H. M. Ozaktas, D. Mendlovic, L. Onural, and B. Barshan, J. Opt. Soc. Am. A 11, 547 (1994)
[Crossref]

Özkul, C.

Pellat-Finet, P.

Romero Cortés, L.

Schnebelin,

Sejdic, E.

E. Sejdić, I. Djurović, and L. Stanković, Signal Process. 91, 1351 (2011).
[Crossref]

Stankovic, L.

E. Sejdić, I. Djurović, and L. Stanković, Signal Process. 91, 1351 (2011).
[Crossref]

Torres-Company, V.

V. Torres-Company, J. Lancis, and P. Andres, Prog. Opt. 56, 1 (2011).
[Crossref]

Yao, J.

Adv. Imaging Electron Phys. (1)

H. M. Ozaktas, M. A. Kutay, and D. Mendlovic, Adv. Imaging Electron Phys. 106, 239 (1999).
[Crossref]

Appl. Opt. (2)

IEEE J. Quantum Electron. (1)

H. Kolner, IEEE J. Quantum Electron. 30, 1951 (1994).
[Crossref]

IMA J. Appl. Math. (2)

V. Namias, IMA J. Appl. Math. 25, 241 (1980).
[Crossref]

C. McBride and F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. A (2)

Opt. Express (2)

Opt. Lett. (1)

Optica (1)

Phys. Rev. A (1)

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

Prog. Opt. (1)

V. Torres-Company, J. Lancis, and P. Andres, Prog. Opt. 56, 1 (2011).
[Crossref]

Signal Process. (1)

E. Sejdić, I. Djurović, and L. Stanković, Signal Process. 91, 1351 (2011).
[Crossref]

Other (1)

N. Levanon and E. Mozeson, Radar Signals (Wiley, 2004)

Supplementary Material (1)

NameDescription
» Supplement 1       additional calculations and results

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Figures (6)

Fig. 1.
Fig. 1.

Fractional Fourier transform (FrFT) in the time-frequency plane. (a) Square modulus of the FrFT of order α of a square function. (b) Illustration of the capability of the FrFT to separate a signal from a noise background [5]. When the signal and the noise overlap in both the time and frequency domains, they cannot be separated by temporal or spectral filtering. On the contrary, for a proper choice of α, the FrFT of the total time–frequency distribution leads to two separable components along the intermediate direction u.

Fig. 2.
Fig. 2.

Optical implementations of the FrFT in the space and time domains. (a) FrFT by propagation through the free space (resp. through a GVD medium). The propagated wavefront (resp. signal) maps the FrFT of the input one, multiplied by a quadratic phase term [6]. (b) Free space–lens–free space (resp. GVD–time lens (TL)–GVD) implementation of the FrFT. The output wavefront (resp. signal) maps the FrFT of the input one with no additional chirp term [10]. (c) Implementation of the FrFT by mapping the input wavefront (resp. signal) into the spectral domain by means of a spatial (resp. temporal) ff setup, and application of a subsequent quadratic spectral phase by free-space (resp. GVD) propagation. This last scheme is equivalent to the operation mode of the proposed FSL-based implementation.

Fig. 3.
Fig. 3.

Real-time FrFT of RF signals in the FSL. (a) Sketch of the experimental system. The FSL is seeded with an optical signal (in blue) obtained by modulating a CW laser (frequency: f0) with the RF input signal (in red). A photodiode detects the intensity of the output optical signal (in blue) and generates an RF waveform (in red) mapping the intensity of the optical field, i.e., the square modulus of the FrFT of the input RF signal. (b) The optical signal at the output of the FSL consists of the superposition of replicas of the input one, shifted along both the temporal and frequency domains, and multiplied by a quadratic phase factor, in green (see Supplement 1). (c) This process is equivalent to the mapping of the input RF signal into the optical frequency domain, combined with the multiplication by a quadratic phase (dashed line). The corresponding time trace (intensity) maps periodically the square modulus of the FrFT of the RF input signal.

Fig. 4.
Fig. 4.

Experimental demonstration of analog FrFT of a square input signal. A 5 µs duration square signal modulates the intensity of the input CW laser. Top: temporal traces recorded at the output of the FSL for different values of fs (with τc=116  ns). No averaging is performed on the traces. Notice that when fs=1/τc=8.6  MHz, Δfs=0: the output trace maps the square modulus of the FT of the input square signal (sinc2 function). Bottom traces: numerical computations of the corresponding FrFT of the square function (the scaling factor is τ=0.5 µs, see Supplement 1, sections 1 and 4).

Fig. 5.
Fig. 5.

Application of FrFT for arbitrary RF chirp measurement. Top left: The FrFT consists of projecting the time–frequency distribution onto a direction intermediate between time and frequencies (symbolized by the orange, red, and green arrows). When the projection is perpendicular to that of the chirp spectrogram, the signal becomes FT-limited and compressed (green arrow) as compared to non-perpendicular projections (orange and red directions). The chirp rate is deduced from the corresponding value of fs (see text).

Fig. 6.
Fig. 6.

Filtering of a chirp waveform from a noisy background. Top left: simplified representation of the time–frequency distribution of the signal (central line) surrounded by two noise components. As said, the FrFT corresponds to projecting the time–frequency distribution onto specific directions (orange, red, and green arrows). The orange trace corresponds to Δfs=0 (flat spectral phase): the output waveform maps the square modulus of the input FT. When the projection is perpendicular to the direction of the chirp, the signal and the noise components are temporally well separated (green arrow).

Equations (2)

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FrFTα[f](x)=Aαei2tanαx2f(y)ei2cotα(xcosαy)2dy
=Aαei2cotαx2f(y)ei2cotαy2eixysinαdy,