## Wavelength-selective directional couplers as ultrafast optical differentiators |

Optics Express, Vol. 19, Issue 8, pp. 7625-7632 (2011)

http://dx.doi.org/10.1364/OE.19.007625

Acrobat PDF (983 KB)

### Abstract

Wavelength-selective directional couplers with dissimilar waveguides are designed for ultrafast optical differentiation within the femtosecond regime (corresponding to processing bandwidths > 10 THz). The theoretically proposed coupler-based differentiators can be produced by wavelength matching of the propagation constants of two different waveguides in the coupler at the center wavelength. A single directional coupler can be designed to achieve either a 2nd-order differentiator or a 1st-order differentiator by properly fixing the product of coupling coefficient and coupling length of the coupler. We evaluated the differentiation errors (~2%) and energetic efficiency (~11% for 1st order differentiation) of the designed optical differentiators through numerical simulations. The proposed design has a strong potential to provide a feasible solution as an integrated differentiation unit device for ultrafast optical signal processing circuits.

© 2011 OSA

## 1. Introduction

1. N. Q. Ngo, S. F. Yu, S. C. Tjin, and C. H. Kam, “A new theoretical basis of higher-derivative optical differentiators,” Opt. Commun. **230**(1-3), 115–129 (2004). [CrossRef]

5. R. Slavík, Y. Park, N. Ayotte, S. Doucet, T.-J. Ahn, S. LaRochelle, and J. Azaña, “Photonic temporal integrator for all-optical computing,” Opt. Express **16**(22), 18202–18214 (2008). [CrossRef] [PubMed]

1. N. Q. Ngo, S. F. Yu, S. C. Tjin, and C. H. Kam, “A new theoretical basis of higher-derivative optical differentiators,” Opt. Commun. **230**(1-3), 115–129 (2004). [CrossRef]

2. M. Kulishov and J. Azaña, “Long-period fiber gratings as ultrafast optical differentiators,” Opt. Lett. **30**(20), 2700–2702 (2005). [CrossRef] [PubMed]

4. M. A. Preciado and M. A. Muriel, “Design of an ultrafast all-optical differentiator based on a fiber Bragg grating in transmission,” Opt. Lett. **33**(21), 2458–2460 (2008). [CrossRef] [PubMed]

6. Y. Park, J. Azaña, and R. Slavík, “Ultrafast all-optical first- and higher-order differentiators based on interferometers,” Opt. Lett. **32**(6), 710–712 (2007). [CrossRef] [PubMed]

1. N. Q. Ngo, S. F. Yu, S. C. Tjin, and C. H. Kam, “A new theoretical basis of higher-derivative optical differentiators,” Opt. Commun. **230**(1-3), 115–129 (2004). [CrossRef]

7. R. Zengerle and O. Leminger, “Wavelength-selective directional coupler made of nonidentical single-mode fibers,” J. Lightwave Technol. **4**(7), 823–827 (1986). [CrossRef]

## 2. Operating principle

**230**(1-3), 115–129 (2004). [CrossRef]

4. M. A. Preciado and M. A. Muriel, “Design of an ultrafast all-optical differentiator based on a fiber Bragg grating in transmission,” Opt. Lett. **33**(21), 2458–2460 (2008). [CrossRef] [PubMed]

_{opt}-ω

_{0}, where ω

_{opt}is the optical frequency and ω

_{0}is the central optical frequency of the signals). Thus first-order optical differentiation can be implemented by use of a (wavelength-selective) optical filter with a transfer function H

_{1}∝ -jω, which depends linearly on frequency (along the whole bandwidth of the signal to be processed). A key feature of a first-order optical differentiator is that it must introduce a π phase shift exactly at the signal’s central frequency, ω

_{0}. Similarly, the transfer function of a second-order differentiator is a parabolic function of frequency, H

_{2}= [H

_{1}]

^{2}∝ -ω

^{2}.

2. M. Kulishov and J. Azaña, “Long-period fiber gratings as ultrafast optical differentiators,” Opt. Lett. **30**(20), 2700–2702 (2005). [CrossRef] [PubMed]

2. M. Kulishov and J. Azaña, “Long-period fiber gratings as ultrafast optical differentiators,” Opt. Lett. **30**(20), 2700–2702 (2005). [CrossRef] [PubMed]

_{1}−β

_{2}where β

_{1}and β

_{2}are the propagation constants for the waveguide 1 and 2 in the coupler, respectively;

*j*is the imaginary unit. We remind the reader that the propagation constants of the waveguides depend on the optical frequency (so-called dispersion characteristics of the waveguides) in a different fashion. Note also that the detuning factor of the coupler does not contain the grating period factor, in contrast to the case of an LPFG-based device. Transmission functions in Eqs. (1) and (2) can be approximated in the vicinity of the resonance frequency ω

_{0}(where β

_{1}≈β

_{2}) by the two first non-zero terms of the respective Taylor expansions:andThe detuning factor in the close vicinity of ω

_{0}can also be expanded in a Taylor series and approximated by Δβ(ω

_{opt}) ≈σ

_{1}(ω

_{opt}- ω), where σ

_{1}= (1/2)[∂β

_{1}(ω

_{0})/∂ω- ∂β

_{2}(ω

_{0})/∂ω]. The maximum coupled power can be obtained at the resonance frequency ω = ω

_{0}, where β

_{1}(ω

_{0}) = β

_{2}(ω

_{0}), for κL = m(π/2), with m = 1, 3, 5,… In this case, it follows from Eq. (3) that the transfer function H

_{1}≈-jω, corresponding to a 1st-order optical differentiator. Similarly, if the coupler parameters are fixed to achieve full energy re-coupling from the waveguide 2 into the waveguide 1, i.e., κL = mπ, with m = 1, 2, 3, …, then it follows from expression (4) that H

_{2}≈-ω

^{2}, corresponding to the spectral response of a 2nd-order optical differentiator. Thus, the wavelength-selective directional coupler can be designed to operate as a 1st-order or as a 2nd-order optical differentiator. We reiterate that as a main advantage, the spectral bandwidth of directional couplers is generally larger than that of the fiber gratings. Moreover, the LPFG provides the 2nd order optical differentiator by cladding-mode full coupling with κL = mπ [2

**30**(20), 2700–2702 (2005). [CrossRef] [PubMed]

## 3. Numerical analysis

_{0}(corresponding to ω

_{0}) can also be approximated by Δβ ≈η(λ−λ

_{0}), where the coupling slope is defined as a function of the wavelength derivatives of the modal propagation constants of each waveguide in the coupler at the center wavelength λ

_{0}, η = ∂β

_{1}(λ

_{0})/∂λ-∂β

_{2}(λ

_{0})/∂λ [9

9. A. K. Das and M. A. Mondal, “Precise control of the center wavelength and bandwidth of wavelength-selective single-mode fiber couplers,” Opt. Lett. **19**(11), 795–797 (1994). [CrossRef] [PubMed]

_{0})

^{Q}], of order Q = 3 and a full-width at half maximum (FWHM) time duration τ

_{0}= 3 ps.

^{−2}in the simulation. The field spectrum of the optical differentiation result was obtained by multiplying the complex spectral transfer function of the optical differentiator under evaluation by the spectrum of the input super-Gaussian input pulse with a center wavelength of 1.55 μm, also shown in Fig. 1(b) (solid curve). Note that the center wavelengths of the input pulse spectrum and devices’ transfer functions (devices resonance wavelengths) should be matched. The temporal differentiated pulse can be retrieved by inverse Fourier transformation of the transmission spectral response of the differentiator.

_{0}(input optical pulse signal is assumed to be centered at ω

_{0}). To be more concrete, the operation bandwidth of the designed optical differentiators is approximately given by the 3dB (i.e. FWHM) spectral bandwidth of the corresponding coupler transmission response. This represents a very reasonable approximation considering that for the 1st-order differentiator, the deviation of the device’s transfer function from a linear fitting curve along this defined bandwidth region is estimated to be less than 1%; similarly, for the 2nd-order differentiator, the relative error of the device’s transfer function, in comparison with the ideal parabolic fitting curve, over the entire defined bandwidth region keeps less than 2.7%.

^{−2}) (a) and coupling slope (L = 8 mm) (b). Practically feasible values for these two parameters have been considered in these evaluations (assuming implementation in a fiber-type waveguide) [7

7. R. Zengerle and O. Leminger, “Wavelength-selective directional coupler made of nonidentical single-mode fibers,” J. Lightwave Technol. **4**(7), 823–827 (1986). [CrossRef]

9. A. K. Das and M. A. Mondal, “Precise control of the center wavelength and bandwidth of wavelength-selective single-mode fiber couplers,” Opt. Lett. **19**(11), 795–797 (1994). [CrossRef] [PubMed]

_{0}is the center wavelength of the coupler’s transfer function. The coupling slope η represents the slope difference (in unit of μm

^{−2}) between the propagation constants of the two waveguides of the coupler.

*p*is a parameter resulting from the curve fitting of the numerically estimated 3dB bandwidth with respect to the coupling length, L and slope η (data shown in Fig. 3). Referring to the report [8

8. K. Morishita, “Wavelength-selective optical-fiber directional couplers using dispersive materials,” Opt. Lett. **13**(2), 158–160 (1988). [CrossRef] [PubMed]

*p*= 3.5 for 1st-order differentiation and

*p*= 6.5 for 2nd-order differentiation, depending on the coupling length and coupling slope, are also presented in Fig. 3 (solid curves). The bandwidth of the 2nd-order differentiator is ~1.86 times larger than that of the 1st-order differentiator, assuming that the couplers have the same parameters in terms of L, η, and center wavelength, λ

_{0}. As evidenced by these results, the coupler-based differentiator can easily provide operation bandwidths in the tens of THz range (corresponding to the femtosecond regime), compared to the few THz operation bandwidths (sub-picosecond regime) that can be offered by LPFG-based differentiators in the best case.

## 4. Discussions on practical design issues

## 5. Conclusion

## Acknowledgments

## References and links

1. | N. Q. Ngo, S. F. Yu, S. C. Tjin, and C. H. Kam, “A new theoretical basis of higher-derivative optical differentiators,” Opt. Commun. |

2. | M. Kulishov and J. Azaña, “Long-period fiber gratings as ultrafast optical differentiators,” Opt. Lett. |

3. | R. Slavík, Y. Park, M. Kulishov, R. Morandotti, and J. Azaña, “Ultrafast all-optical differentiators,” Opt. Express |

4. | M. A. Preciado and M. A. Muriel, “Design of an ultrafast all-optical differentiator based on a fiber Bragg grating in transmission,” Opt. Lett. |

5. | R. Slavík, Y. Park, N. Ayotte, S. Doucet, T.-J. Ahn, S. LaRochelle, and J. Azaña, “Photonic temporal integrator for all-optical computing,” Opt. Express |

6. | Y. Park, J. Azaña, and R. Slavík, “Ultrafast all-optical first- and higher-order differentiators based on interferometers,” Opt. Lett. |

7. | R. Zengerle and O. Leminger, “Wavelength-selective directional coupler made of nonidentical single-mode fibers,” J. Lightwave Technol. |

8. | K. Morishita, “Wavelength-selective optical-fiber directional couplers using dispersive materials,” Opt. Lett. |

9. | A. K. Das and M. A. Mondal, “Precise control of the center wavelength and bandwidth of wavelength-selective single-mode fiber couplers,” Opt. Lett. |

10. | R. Zengerle and O.G. Leminger, “Investigations of fabrication tolerances of narrow bandwidth directional coupler filters in InP,” Integrated Photonics Research, |

11. | D. Marcuse, |

**OCIS Codes**

(320.5540) Ultrafast optics : Pulse shaping

(320.7080) Ultrafast optics : Ultrafast devices

(320.7085) Ultrafast optics : Ultrafast information processing

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: January 21, 2011

Revised Manuscript: February 23, 2011

Manuscript Accepted: March 25, 2011

Published: April 6, 2011

**Citation**

Tae-Jung Ahn and José Azaña, "Wavelength-selective directional couplers as ultrafast optical differentiators," Opt. Express **19**, 7625-7632 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-8-7625

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### References

- N. Q. Ngo, S. F. Yu, S. C. Tjin, and C. H. Kam, “A new theoretical basis of higher-derivative optical differentiators,” Opt. Commun. 230(1-3), 115–129 (2004). [CrossRef]
- M. Kulishov and J. Azaña, “Long-period fiber gratings as ultrafast optical differentiators,” Opt. Lett. 30(20), 2700–2702 (2005). [CrossRef] [PubMed]
- R. Slavík, Y. Park, M. Kulishov, R. Morandotti, and J. Azaña, “Ultrafast all-optical differentiators,” Opt. Express 14(22), 10699–10707 (2006). [CrossRef] [PubMed]
- M. A. Preciado and M. A. Muriel, “Design of an ultrafast all-optical differentiator based on a fiber Bragg grating in transmission,” Opt. Lett. 33(21), 2458–2460 (2008). [CrossRef] [PubMed]
- R. Slavík, Y. Park, N. Ayotte, S. Doucet, T.-J. Ahn, S. LaRochelle, and J. Azaña, “Photonic temporal integrator for all-optical computing,” Opt. Express 16(22), 18202–18214 (2008). [CrossRef] [PubMed]
- Y. Park, J. Azaña, and R. Slavík, “Ultrafast all-optical first- and higher-order differentiators based on interferometers,” Opt. Lett. 32(6), 710–712 (2007). [CrossRef] [PubMed]
- R. Zengerle and O. Leminger, “Wavelength-selective directional coupler made of nonidentical single-mode fibers,” J. Lightwave Technol. 4(7), 823–827 (1986). [CrossRef]
- K. Morishita, “Wavelength-selective optical-fiber directional couplers using dispersive materials,” Opt. Lett. 13(2), 158–160 (1988). [CrossRef] [PubMed]
- A. K. Das and M. A. Mondal, “Precise control of the center wavelength and bandwidth of wavelength-selective single-mode fiber couplers,” Opt. Lett. 19(11), 795–797 (1994). [CrossRef] [PubMed]
- R. Zengerle and O.G. Leminger, “Investigations of fabrication tolerances of narrow bandwidth directional coupler filters in InP,” Integrated Photonics Research, TuB3–1 (1992).
- D. Marcuse, Theory of Dielectric Optical Waveguides (Academic Press Inc., 1991) Chap. 6 and 7.

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