Abstract

In this work we used the temporal analog of spatial Fresnel diffraction to design a temporal fractional Fourier transformer with a single dispersive device, in this way avoiding the use of quadratic phase modulators. We demonstrate that a single dispersive passive device inherently provides the fractional Fourier transform of an incident optical pulse. The relationships linking the fractional Fourier transform order and scaling factor with the dispersion parameters are derived. We first provide some numerical results in order to prove the validity of our proposal, using a fiber Bragg grating as the dispersive device. Next, we experimentally demonstrate the feasibility of this proposal by using a spool of a standard optical fiber as the dispersive device.

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  1. C. Cuadrado-Laborde and M. V. Andrés, “In-fiber all-optical fractional differentiator,” Opt. Lett.34(6), 833–835 (2009).
    [CrossRef] [PubMed]
  2. C. Cuadrado-Laborde and M. V. Andrés, “Proposal and design of an all-optical in-fiber fractional integrator,” Opt. Commun.283(24), 5012–5015 (2010).
    [CrossRef]
  3. C. Cuadrado-Laborde, “Proposal and design of a photonic in-fiber fractional Hilbert transformer,” IEEE Photon. Technol. Lett.22(1), 33–35 (2010).
    [CrossRef]
  4. C. Cuadrado-Laborde, M. V. Andrés, and J. Lancis, “Self-referenced phase reconstruction proposal of GHz-bandwidth non-periodical optical pulses by in-fiber semi-differintegration,” Opt. Commun.284(24), 5636–5640 (2011).
    [CrossRef]
  5. T. Alieva, M. J. Bastiaans, and M. L. Calvo, “Fractional transforms in optical information processing,” EURASIP J. Appl. Sig. P.10, 1498–1519 (2005).
  6. M. A. Muriel, J. Azaña, and A. Carballar, “Real-time Fourier transformer based on fiber gratings,” Opt. Lett.24(1), 1–3 (1999).
    [CrossRef] [PubMed]
  7. J. Azaña, L. R. Chen, M. A. Muriel, and P. W. E. Smith, “Experimental demonstration of real-time Fourier transformation using linearly chirped fibre Bragg gratings,” Electron. Lett.35(25), 2223–2224 (1999).
    [CrossRef]
  8. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001).
  9. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A10(10), 2181–2186 (1993).
    [CrossRef]
  10. W. Lohmann, Z. Zalevsky, R. G. Dorsch, and D. Mendlovic, “Experimental considerations and scaling property of the fractional Fourier transform,” Opt. Commun.146(1-6), 55–61 (1998).
    [CrossRef]
  11. J. Hua, L. Liu, and G. Li, “Observing the fractional Fourier transform by free-space Fresnel diffraction,” Appl. Opt.36(2), 512–513 (1997).
    [CrossRef] [PubMed]
  12. H. M. Ozaktas, S. Ö. Arık, and T. Coşkun, “Fundamental structure of Fresnel diffraction: natural sampling grid and the fractional Fourier transform,” Opt. Lett.36(13), 2524–2526 (2011).
    [CrossRef] [PubMed]
  13. C. Cuadrado-Laborde, R. Duchowicz, R. Torroba, and E. E. Sicre, “Fractional Fourier transform dual random phase encoding of time-varying signals,” Opt. Commun.281(17), 4321–4328 (2008).
    [CrossRef]
  14. A. W. Lohmann and D. Mendlovic, “Fractional Fourier transform: photonic implementation,” Appl. Opt.33(32), 7661–7664 (1994).
    [CrossRef] [PubMed]
  15. C. Cuadrado-Laborde, P. Costanzo-Caso, R. Duchowicz, and E. E. Sicre, “Periodic pulse train conformation based on the temporal Radon-Wigner transform,” Opt. Commun.275(1), 94–103 (2007).
    [CrossRef]
  16. K. Ennser, M. N. Zervas, and R. Laming, “Optimization of apodized linearly chirped fiber gratings for optical communications,” IEEE J. Quantum Electron.34(5), 770–778 (1998).
    [CrossRef]
  17. B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron.30(8), 1951–1963 (1994).
    [CrossRef]

2011 (2)

C. Cuadrado-Laborde, M. V. Andrés, and J. Lancis, “Self-referenced phase reconstruction proposal of GHz-bandwidth non-periodical optical pulses by in-fiber semi-differintegration,” Opt. Commun.284(24), 5636–5640 (2011).
[CrossRef]

H. M. Ozaktas, S. Ö. Arık, and T. Coşkun, “Fundamental structure of Fresnel diffraction: natural sampling grid and the fractional Fourier transform,” Opt. Lett.36(13), 2524–2526 (2011).
[CrossRef] [PubMed]

2010 (2)

C. Cuadrado-Laborde and M. V. Andrés, “Proposal and design of an all-optical in-fiber fractional integrator,” Opt. Commun.283(24), 5012–5015 (2010).
[CrossRef]

C. Cuadrado-Laborde, “Proposal and design of a photonic in-fiber fractional Hilbert transformer,” IEEE Photon. Technol. Lett.22(1), 33–35 (2010).
[CrossRef]

2009 (1)

2008 (1)

C. Cuadrado-Laborde, R. Duchowicz, R. Torroba, and E. E. Sicre, “Fractional Fourier transform dual random phase encoding of time-varying signals,” Opt. Commun.281(17), 4321–4328 (2008).
[CrossRef]

2007 (1)

C. Cuadrado-Laborde, P. Costanzo-Caso, R. Duchowicz, and E. E. Sicre, “Periodic pulse train conformation based on the temporal Radon-Wigner transform,” Opt. Commun.275(1), 94–103 (2007).
[CrossRef]

2005 (1)

T. Alieva, M. J. Bastiaans, and M. L. Calvo, “Fractional transforms in optical information processing,” EURASIP J. Appl. Sig. P.10, 1498–1519 (2005).

1999 (2)

J. Azaña, L. R. Chen, M. A. Muriel, and P. W. E. Smith, “Experimental demonstration of real-time Fourier transformation using linearly chirped fibre Bragg gratings,” Electron. Lett.35(25), 2223–2224 (1999).
[CrossRef]

M. A. Muriel, J. Azaña, and A. Carballar, “Real-time Fourier transformer based on fiber gratings,” Opt. Lett.24(1), 1–3 (1999).
[CrossRef] [PubMed]

1998 (2)

W. Lohmann, Z. Zalevsky, R. G. Dorsch, and D. Mendlovic, “Experimental considerations and scaling property of the fractional Fourier transform,” Opt. Commun.146(1-6), 55–61 (1998).
[CrossRef]

K. Ennser, M. N. Zervas, and R. Laming, “Optimization of apodized linearly chirped fiber gratings for optical communications,” IEEE J. Quantum Electron.34(5), 770–778 (1998).
[CrossRef]

1997 (1)

1994 (2)

A. W. Lohmann and D. Mendlovic, “Fractional Fourier transform: photonic implementation,” Appl. Opt.33(32), 7661–7664 (1994).
[CrossRef] [PubMed]

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron.30(8), 1951–1963 (1994).
[CrossRef]

1993 (1)

Alieva, T.

T. Alieva, M. J. Bastiaans, and M. L. Calvo, “Fractional transforms in optical information processing,” EURASIP J. Appl. Sig. P.10, 1498–1519 (2005).

Andrés, M. V.

C. Cuadrado-Laborde, M. V. Andrés, and J. Lancis, “Self-referenced phase reconstruction proposal of GHz-bandwidth non-periodical optical pulses by in-fiber semi-differintegration,” Opt. Commun.284(24), 5636–5640 (2011).
[CrossRef]

C. Cuadrado-Laborde and M. V. Andrés, “Proposal and design of an all-optical in-fiber fractional integrator,” Opt. Commun.283(24), 5012–5015 (2010).
[CrossRef]

C. Cuadrado-Laborde and M. V. Andrés, “In-fiber all-optical fractional differentiator,” Opt. Lett.34(6), 833–835 (2009).
[CrossRef] [PubMed]

Arik, S. Ö.

Azaña, J.

M. A. Muriel, J. Azaña, and A. Carballar, “Real-time Fourier transformer based on fiber gratings,” Opt. Lett.24(1), 1–3 (1999).
[CrossRef] [PubMed]

J. Azaña, L. R. Chen, M. A. Muriel, and P. W. E. Smith, “Experimental demonstration of real-time Fourier transformation using linearly chirped fibre Bragg gratings,” Electron. Lett.35(25), 2223–2224 (1999).
[CrossRef]

Bastiaans, M. J.

T. Alieva, M. J. Bastiaans, and M. L. Calvo, “Fractional transforms in optical information processing,” EURASIP J. Appl. Sig. P.10, 1498–1519 (2005).

Calvo, M. L.

T. Alieva, M. J. Bastiaans, and M. L. Calvo, “Fractional transforms in optical information processing,” EURASIP J. Appl. Sig. P.10, 1498–1519 (2005).

Carballar, A.

Chen, L. R.

J. Azaña, L. R. Chen, M. A. Muriel, and P. W. E. Smith, “Experimental demonstration of real-time Fourier transformation using linearly chirped fibre Bragg gratings,” Electron. Lett.35(25), 2223–2224 (1999).
[CrossRef]

Coskun, T.

Costanzo-Caso, P.

C. Cuadrado-Laborde, P. Costanzo-Caso, R. Duchowicz, and E. E. Sicre, “Periodic pulse train conformation based on the temporal Radon-Wigner transform,” Opt. Commun.275(1), 94–103 (2007).
[CrossRef]

Cuadrado-Laborde, C.

C. Cuadrado-Laborde, M. V. Andrés, and J. Lancis, “Self-referenced phase reconstruction proposal of GHz-bandwidth non-periodical optical pulses by in-fiber semi-differintegration,” Opt. Commun.284(24), 5636–5640 (2011).
[CrossRef]

C. Cuadrado-Laborde, “Proposal and design of a photonic in-fiber fractional Hilbert transformer,” IEEE Photon. Technol. Lett.22(1), 33–35 (2010).
[CrossRef]

C. Cuadrado-Laborde and M. V. Andrés, “Proposal and design of an all-optical in-fiber fractional integrator,” Opt. Commun.283(24), 5012–5015 (2010).
[CrossRef]

C. Cuadrado-Laborde and M. V. Andrés, “In-fiber all-optical fractional differentiator,” Opt. Lett.34(6), 833–835 (2009).
[CrossRef] [PubMed]

C. Cuadrado-Laborde, R. Duchowicz, R. Torroba, and E. E. Sicre, “Fractional Fourier transform dual random phase encoding of time-varying signals,” Opt. Commun.281(17), 4321–4328 (2008).
[CrossRef]

C. Cuadrado-Laborde, P. Costanzo-Caso, R. Duchowicz, and E. E. Sicre, “Periodic pulse train conformation based on the temporal Radon-Wigner transform,” Opt. Commun.275(1), 94–103 (2007).
[CrossRef]

Dorsch, R. G.

W. Lohmann, Z. Zalevsky, R. G. Dorsch, and D. Mendlovic, “Experimental considerations and scaling property of the fractional Fourier transform,” Opt. Commun.146(1-6), 55–61 (1998).
[CrossRef]

Duchowicz, R.

C. Cuadrado-Laborde, R. Duchowicz, R. Torroba, and E. E. Sicre, “Fractional Fourier transform dual random phase encoding of time-varying signals,” Opt. Commun.281(17), 4321–4328 (2008).
[CrossRef]

C. Cuadrado-Laborde, P. Costanzo-Caso, R. Duchowicz, and E. E. Sicre, “Periodic pulse train conformation based on the temporal Radon-Wigner transform,” Opt. Commun.275(1), 94–103 (2007).
[CrossRef]

Ennser, K.

K. Ennser, M. N. Zervas, and R. Laming, “Optimization of apodized linearly chirped fiber gratings for optical communications,” IEEE J. Quantum Electron.34(5), 770–778 (1998).
[CrossRef]

Hua, J.

Kolner, B. H.

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron.30(8), 1951–1963 (1994).
[CrossRef]

Laming, R.

K. Ennser, M. N. Zervas, and R. Laming, “Optimization of apodized linearly chirped fiber gratings for optical communications,” IEEE J. Quantum Electron.34(5), 770–778 (1998).
[CrossRef]

Lancis, J.

C. Cuadrado-Laborde, M. V. Andrés, and J. Lancis, “Self-referenced phase reconstruction proposal of GHz-bandwidth non-periodical optical pulses by in-fiber semi-differintegration,” Opt. Commun.284(24), 5636–5640 (2011).
[CrossRef]

Li, G.

Liu, L.

Lohmann, A. W.

Lohmann, W.

W. Lohmann, Z. Zalevsky, R. G. Dorsch, and D. Mendlovic, “Experimental considerations and scaling property of the fractional Fourier transform,” Opt. Commun.146(1-6), 55–61 (1998).
[CrossRef]

Mendlovic, D.

W. Lohmann, Z. Zalevsky, R. G. Dorsch, and D. Mendlovic, “Experimental considerations and scaling property of the fractional Fourier transform,” Opt. Commun.146(1-6), 55–61 (1998).
[CrossRef]

A. W. Lohmann and D. Mendlovic, “Fractional Fourier transform: photonic implementation,” Appl. Opt.33(32), 7661–7664 (1994).
[CrossRef] [PubMed]

Muriel, M. A.

M. A. Muriel, J. Azaña, and A. Carballar, “Real-time Fourier transformer based on fiber gratings,” Opt. Lett.24(1), 1–3 (1999).
[CrossRef] [PubMed]

J. Azaña, L. R. Chen, M. A. Muriel, and P. W. E. Smith, “Experimental demonstration of real-time Fourier transformation using linearly chirped fibre Bragg gratings,” Electron. Lett.35(25), 2223–2224 (1999).
[CrossRef]

Ozaktas, H. M.

Sicre, E. E.

C. Cuadrado-Laborde, R. Duchowicz, R. Torroba, and E. E. Sicre, “Fractional Fourier transform dual random phase encoding of time-varying signals,” Opt. Commun.281(17), 4321–4328 (2008).
[CrossRef]

C. Cuadrado-Laborde, P. Costanzo-Caso, R. Duchowicz, and E. E. Sicre, “Periodic pulse train conformation based on the temporal Radon-Wigner transform,” Opt. Commun.275(1), 94–103 (2007).
[CrossRef]

Smith, P. W. E.

J. Azaña, L. R. Chen, M. A. Muriel, and P. W. E. Smith, “Experimental demonstration of real-time Fourier transformation using linearly chirped fibre Bragg gratings,” Electron. Lett.35(25), 2223–2224 (1999).
[CrossRef]

Torroba, R.

C. Cuadrado-Laborde, R. Duchowicz, R. Torroba, and E. E. Sicre, “Fractional Fourier transform dual random phase encoding of time-varying signals,” Opt. Commun.281(17), 4321–4328 (2008).
[CrossRef]

Zalevsky, Z.

W. Lohmann, Z. Zalevsky, R. G. Dorsch, and D. Mendlovic, “Experimental considerations and scaling property of the fractional Fourier transform,” Opt. Commun.146(1-6), 55–61 (1998).
[CrossRef]

Zervas, M. N.

K. Ennser, M. N. Zervas, and R. Laming, “Optimization of apodized linearly chirped fiber gratings for optical communications,” IEEE J. Quantum Electron.34(5), 770–778 (1998).
[CrossRef]

Appl. Opt. (2)

Electron. Lett. (1)

J. Azaña, L. R. Chen, M. A. Muriel, and P. W. E. Smith, “Experimental demonstration of real-time Fourier transformation using linearly chirped fibre Bragg gratings,” Electron. Lett.35(25), 2223–2224 (1999).
[CrossRef]

EURASIP J. Appl. Sig. P. (1)

T. Alieva, M. J. Bastiaans, and M. L. Calvo, “Fractional transforms in optical information processing,” EURASIP J. Appl. Sig. P.10, 1498–1519 (2005).

IEEE J. Quantum Electron. (2)

K. Ennser, M. N. Zervas, and R. Laming, “Optimization of apodized linearly chirped fiber gratings for optical communications,” IEEE J. Quantum Electron.34(5), 770–778 (1998).
[CrossRef]

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron.30(8), 1951–1963 (1994).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

C. Cuadrado-Laborde, “Proposal and design of a photonic in-fiber fractional Hilbert transformer,” IEEE Photon. Technol. Lett.22(1), 33–35 (2010).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Commun. (5)

C. Cuadrado-Laborde, M. V. Andrés, and J. Lancis, “Self-referenced phase reconstruction proposal of GHz-bandwidth non-periodical optical pulses by in-fiber semi-differintegration,” Opt. Commun.284(24), 5636–5640 (2011).
[CrossRef]

C. Cuadrado-Laborde and M. V. Andrés, “Proposal and design of an all-optical in-fiber fractional integrator,” Opt. Commun.283(24), 5012–5015 (2010).
[CrossRef]

W. Lohmann, Z. Zalevsky, R. G. Dorsch, and D. Mendlovic, “Experimental considerations and scaling property of the fractional Fourier transform,” Opt. Commun.146(1-6), 55–61 (1998).
[CrossRef]

C. Cuadrado-Laborde, R. Duchowicz, R. Torroba, and E. E. Sicre, “Fractional Fourier transform dual random phase encoding of time-varying signals,” Opt. Commun.281(17), 4321–4328 (2008).
[CrossRef]

C. Cuadrado-Laborde, P. Costanzo-Caso, R. Duchowicz, and E. E. Sicre, “Periodic pulse train conformation based on the temporal Radon-Wigner transform,” Opt. Commun.275(1), 94–103 (2007).
[CrossRef]

Opt. Lett. (3)

Other (1)

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001).

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Figures (3)

Fig. 1
Fig. 1

(a) LCFBG reflectivity versus optical wavelength, the grating has a 200 GHz bandwidth centered at 1552.52 nm. (b) LCFBG reflection group delay; the grating provides a linear group delay with a slope (dispersion coefficient) of 1218 ps/nm.

Fig. 2
Fig. 2

(a) Input signal consisting in a twin Gaussian optical pulse of 6 ps each (FWHM) separated by 50 ps. (b) Mathematically obtained 0.35th order FrFT with scale factor ε 2 = (1/2π) × 10−19 s2. (c) Simulated response after reflection in the LCFBG shown in Fig. 1 of the input pulse shown in (a). (d) The same signals shown in (b) and (c) are represented together for comparison purposes, except that now the signal shown in (c) has a time-scaling correction.

Fig. 3
Fig. 3

(a) Scheme of the experimental setup. (b) Input pulse provided by the modelocked laser (solid curve), and its corresponding fitting by a sech profile (scatter points). (c) Mathematically obtained 0.0203th order FrFT of the sech profile shown in (b), and experimentally detected output light pulse after propagation in a fiber length of 101 m. (d)-(e) Same as in (c) but for 0.043th and 0.063th order FrFTs and propagations of 214 m and 315 m, respectively.

Equations (9)

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

F p [ f( t ) ]( t p )=c( p ) f( t )exp[ jπ( t 2 + t p 2 ) ε 2 cot( pπ 2 ) ] exp[ j2π t p t ε 2 csc( pπ 2 ) ]dt,
H( ω )=exp[ j Φ 20 ω 2 /2 ],
h( t )=exp[ j t 2 / 2 Φ 20 ].
f o ( t )= d t f( t )exp[ j ( t t ) 2 / 2 Φ 20 ] .
Φ 20 =( ε 2 / 2π )tan( pπ /2 ), t p t cos( pπ /2 ).
Φ 20 Δ ω 2 2π;
Δ t 2 / Φ 20 2π,
Φ 20 =z β 20 , D=2πc β 20 / λ 2 ,
ϕ 20 =( 2π / ε 2 )tan( pπ /4 ), Φ 20 =( ε 2 / 2π )sin( pπ /2 ).

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