## Photonic fractional Fourier transformer with a single dispersive device |

Optics Express, Vol. 21, Issue 7, pp. 8558-8563 (2013)

http://dx.doi.org/10.1364/OE.21.008558

Acrobat PDF (1608 KB)

### 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.

© 2013 OSA

## 1. Introduction

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]

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]

## 2. Theory

*p*th fractional Fourier transform of a temporal signal

*f*(

*t*) will be denoted by

*F*[

_{p}*f*(

*t*)](

*t*), whose mathematical definition, is given by [8,9

_{p}9. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A **10**(10), 2181–2186 (1993). [CrossRef]

*c*(

*p*) =

*ε*exp(−

*jπ*/4)[csc(

*pπ*/2)]

^{1/2}; where

*p*is the fractional-Fourier order,

*t*is the variable in the fractional domain, and

_{p}*ε*is an arbitrary fixed parameter with time dimension. We emphasize that the FrFT belongs to a two-parameter family of transformations, i.e. the fractional order

*p*and the scaling parameter

*ε*. In a FrFT, a change of the scaling parameter thoroughly changes the functional form of the output; as opposed to the ordinary Fourier transform, where only a scaling in the reciprocal variable is accomplished [10

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]

*H*(

*ω*) have flat amplitude and quadratic phase response (i.e., linear group delay) over a certain spectral bandwidth Δ

*ω*:where

_{H}*ω*is the baseband angular frequency, and Φ

_{20}is the first-order dispersion coefficient. Additionally, and only for simplicity, the average time delay has been ignored. Its impulse response −for complex envelopes of pulses with bandwidths narrower than Δ

*ω*− will be given by:The propagation of a single pulse in a linear dispersive regime can be expressed by convolving

_{H}*f*(

*t*) with

*h*(

*t*). Therefore, the output pulse

*f*(

_{o}*t*) will be given by:Intensities in Eqs. (1) and (4) can be made equal, provided:Thus, the output of a linear dispersive system inherently provides a time-enlarged version in intensity of the FrFT. This result could be considered the time-domain counterpart of the existing equivalence between Fresnel diffraction and FrFT, which has been shown before for the spatial case [11

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]

*p*th order can be interpreted as a rotation in the time-frequency plane by

*p*π/2, the details of this transformation can be found in Ref [13

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]

*ω*and time width Δ

*t*. When there is a negligible dispersion within the spectral bandwidth of the input signal, i.e.:Equation (2) reduces to the unity. Therefore, the spectrum of the input signal remains unchanged, which is equivalent to no transformation at all, i.e., a FrFT of degree zero. On the contrary, in the high dispersion limit, i.e.:the quadratic term in

*t*can be neglected, see Eq. (4), reducing to an ordinary Fourier transform. Equation (7) is the time-domain equivalence of the Fraunhofer approximation. Hence, under the conditions given by Eq. (7), the output signal envelope is, within a phase factor, proportional to the Fourier transform of the input signal envelope evaluated at the angular frequency

*ω*=

*t*/Φ

_{20}. Between the limits given by Eqs. (6) and (7), a continuum of FrFTs ranging from

*p*= 0 to

*p*= 1 can be achieved.

*z*is the fiber length,

*β*

_{20}represents the second-order derivative of the propagation constant respect to the angular frequency

*ω*, and

*D*is the dispersion parameter. Another solution widely used to provide enough dispersion is by using LCFBGs, because they can be designed with flat reflectivity and linear group delay within the operative bandwidth. Equation (3) also describes the impulse response of LCFBGs; so the reflection of a pulse from a LCFBG is mathematically identical to Fresnel diffraction.

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]

*ϕ*

_{20}and first-order dispersion factor Φ´´

_{20}are given by [15

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]

## 3. Numerical results

*F*(

*ω*) = ℑ[

*f*(

*t*)] by the complex grating spectral response

*H*(

*ω*). The output temporal waveform is then recovered by taking the inverse Fourier transform, i.e.,

*f*(

_{o}*t*) = ℑ

^{−1}[

*F*(

*ω*)

*H*(

*ω*)]. Finally, since optical detection follows a square law, the detected waveform represents the FrFT of

*p*th order of the input signal, i.e. |

*f*(

_{o}*t*)|

_{p}^{2}= |

*F*[

_{p}*f*(

*t*)](

*t*)|

_{p}^{2}. In the following, we will design the LCFBG to work as a fractional Fourier transformer. Later, this design will be numerically tested by using a typical input signal.

*OptiGrating*. Regarding the design constraints of the LCFBG, the grating bandwidth determines the minimum time-width of the input pulses to be processed. Whereas, the LCFBG dispersion is determined by both the fractional order parameter

*p*and desired scale factor

*ε*, see Eq. (5). In this work, the LCFBG is designed to provide a dispersion coefficient of 1218 ps/nm over a 200 GHz bandwidth centered at the optical wavelength of 1552.52 nm. The device with this characteristic is 200 mm long and has a grating-period variation from 0.53496

*μ*m to 0.53441

*μ*m (unapodized uniform profile). By using transfer matrix theory and multilayer theory

*H*(

*ω*) = |

*H*(

*ω*)|exp[

*jΨ*(

*ω*)] is computed. Figures 1(a) and 1(b) show both the LCFBG reflectivity |

*H*(

*ω*)|

^{2}, and group delay

*τ*= ∂Ψ/∂

_{d}*ω*, respectively. The first-order dispersion coefficient of the LCFBG Φ´

_{20}can be obtained from Φ´

_{20}= ∂

*τ*/∂

_{d}*ω*= (λ

_{0}

^{2}/2

*πc*)∂

*τ*/∂

_{d}*λ*, giving Φ´

_{20}= 1.55 × 10

^{−21}s

^{2}. It can be observed that the LCFBG provides the required features, and both the linearity of the group delay and the flatness of the reflectivity can be improved by an appropriate apodization of the grating`s refractive index perturbation [16

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]

*L*= (

*c*/2

*n*)Φ

_{m}_{20}Δ

*ω*, where

_{G}*c*is the speed of light in vacuum,

*n*is the effective refractive index, Δ

_{m}*ω*is the grating operative optical bandwidth, and Φ

_{G}_{20}is given by Eq. (5). Of course, the spectral bandwidth of the input signal Δ

*ω*should be lower than Δ

*ω*.

_{G}*p*and scale factors

*ε*are available when the dispersion is fixed to a given value −in our case Φ´

_{20}= 1.55 × 10

^{−21}s

^{2}−; from these, we select

*p*= 0.35 and

*ε*

^{2}= (1/2

*π*) × 10

^{−19}s

^{2}. Figure 2(b) shows the mathematically obtained 0.35th order FrFT, of the input signal shown in Fig. 2(a), which consists of a twin optical pulse of 6 ps (FWHM) each separated by 50 ps. Figure 2(c) shows the simulated reflected pulse in the LCFBG. By comparing Figs. 2(b) and 2(c) we conclude that they are identical, except by the scaling in the time domain, which can be easily corrected according to the time scaling of Eq. (5). Figure 2(d) shows both signals, i.e. the mathematically obtained FrFT and the reflected pulse in the LCFBG after applying Eq. (5) in the later for time-scale correction. It can be observed that both signals superimpose upon the other, indicating the close resemblance between them. The average time delay that the reflected signal in the LCFBG undergoes because of the group delay effect has been corrected in Fig. 2.

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]

*ϕ*

_{20}= 1.11 × 10

^{20}rad × Hz

^{2}; plus a LCFBG with a first-order dispersion coefficient of Φ´´

_{20}= 1.32 × 10

^{−21}s

^{2}, see Eq. (9). Usually quadratic-phase modulation is achieved by driving an electro-optic modulator with an electrical sinusoidal signal at the modulation frequency

*ω*−being

_{m}*ϕ*

_{20}∝

*ω*− ; which is only approximately quadratic at a cusp of the sinusoid. Therefore, the optical input signal must be restricted to a usable time aperture

_{m}^{2}*T*centered on a cusp. Thus, by limiting the influence of higher order (fourth) phase terms to a 2%, results in

_{a}*T*≈

_{a}*ϕ*

_{20}

^{−1/2}[17

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

## 4. Experimental results

*C*= −11. In this way the input light pulse were modeled by

*f*(

*t*) = sech(

*t*/

*T*

_{0})exp(−

*jCt*

^{2}/2

*T*

_{0}

^{2}), with

*T*

_{0}= 12 ps, which corresponds to a FWHM of 21 ps. The light pulses were detected with a 50 GHz bandwidth sampling oscilloscope. The dispersion line was made with a commercially available optical fiber (SM980, low numerical aperture Fibercore), with a measured first order dispersion

*D*= −44 ps/nm × Km at the lasing wavelength. Three different fiber lengths were used: 101 m, 214 m, and 315 m; by using Eq. (8), this results in the following first-order dispersion coefficients Φ

_{20}= 2.54 ps

^{2}, 5.4 ps

^{2}, and 7.94 ps

^{2}, respectively. Finally, by using Eq. (5) and by selecting

*ε*

^{2}= 500 ps

^{2}, we obtain the following fractional orders

*p*= 0.0203, 0.043, and 0.063, for each fiber length, respectively.

*ε*

^{2}= 500 ps

^{2}) of the hyperbolic secant profile shown in Fig. 3(b), for

*p*= 0.0203 [Fig. 3(c)],

*p*= 0.043 [Fig. 3(d)], and

*p*= 0.063 [Fig. 3(e)]. For each theoretically obtained FrFT, it is also shown the experimentally detected output light pulse after propagating by a fiber length of 101 m, 214 m, and 315 m, respectively. The time scale of the propagated pulse was modified according to Eq. (5), with the corresponding fractional order. There is a good degree of resemblance between analytically obtained and experimentally detected FrFT. There is also a deviation that increases with the fractional order in the trailing edge, but this could be motivated by the fact that the proposed input profile does not fit the experimental data perfectly.

## 5. Conclusions

## Acknowledgments

## References and links

1. | C. Cuadrado-Laborde and M. V. Andrés, “In-fiber all-optical fractional differentiator,” Opt. Lett. |

2. | C. Cuadrado-Laborde and M. V. Andrés, “Proposal and design of an all-optical in-fiber fractional integrator,” Opt. Commun. |

3. | C. Cuadrado-Laborde, “Proposal and design of a photonic in-fiber fractional Hilbert transformer,” IEEE Photon. Technol. Lett. |

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. |

5. | T. Alieva, M. J. Bastiaans, and M. L. Calvo, “Fractional transforms in optical information processing,” EURASIP J. Appl. Sig. P. |

6. | M. A. Muriel, J. Azaña, and A. Carballar, “Real-time Fourier transformer based on fiber gratings,” Opt. Lett. |

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. |

8. | H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, |

9. | A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A |

10. | W. Lohmann, Z. Zalevsky, R. G. Dorsch, and D. Mendlovic, “Experimental considerations and scaling property of the fractional Fourier transform,” Opt. Commun. |

11. | J. Hua, L. Liu, and G. Li, “Observing the fractional Fourier transform by free-space Fresnel diffraction,” Appl. Opt. |

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. |

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. |

14. | A. W. Lohmann and D. Mendlovic, “Fractional Fourier transform: photonic implementation,” Appl. Opt. |

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. |

16. | K. Ennser, M. N. Zervas, and R. Laming, “Optimization of apodized linearly chirped fiber gratings for optical communications,” IEEE J. Quantum Electron. |

17. | B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. |

**OCIS Codes**

(060.2310) Fiber optics and optical communications : Fiber optics

(070.2575) Fourier optics and signal processing : Fractional Fourier transforms

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: November 13, 2012

Revised Manuscript: January 28, 2013

Manuscript Accepted: February 7, 2013

Published: April 1, 2013

**Citation**

C. Cuadrado-Laborde, A. Carrascosa, A. Díez, J. L. Cruz, and M. V. Andres, "Photonic fractional Fourier transformer with a single dispersive device," Opt. Express **21**, 8558-8563 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-7-8558

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### References

- 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 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]
- 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]
- T. Alieva, M. J. Bastiaans, and M. L. Calvo, “Fractional transforms in optical information processing,” EURASIP J. Appl. Sig. P.10, 1498–1519 (2005).
- 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]
- H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001).
- A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A10(10), 2181–2186 (1993). [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]
- 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]
- 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]
- 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]
- A. W. Lohmann and D. Mendlovic, “Fractional Fourier transform: photonic implementation,” Appl. Opt.33(32), 7661–7664 (1994). [CrossRef] [PubMed]
- 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]
- 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]

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