## Control of directional evanescent coupling in fs laser written waveguides

Optics Express, Vol. 15, Issue 4, pp. 1579-1587 (2007)

http://dx.doi.org/10.1364/OE.15.001579

Acrobat PDF (512 KB)

### Abstract

We investigate the evanescent coupling of femtosecond laser written waveguides with elliptical and circular shape. A directional tuning of the coupling properties is realized in a cubic array by tilting the elliptical waveguides. This allows to specifically pronounce diagonal coupling. In contrast, directional insensitive coupling is demonstrated in a circular waveguide array based on circular waveguides.

© 2007 Optical Society of America

## 1. Introduction

1. H. Haus and L. Molter-Orr, “Coupled multiple waveguide systems,” IEEE J. Quantum Electron. **19**,840–844 (1983). [CrossRef]

2. T. Pertsch, T. Zentgraf, U. Peschel, A. Braeuer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett. **88**,0939,>011–4 (2002). [CrossRef]

3. D. Christodoulides and R. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. **13**,794–796 (1988). [CrossRef] [PubMed]

4. H. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. **81**,3383–3386 (1998). [CrossRef]

5. A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tuennermann, and F. Lederer, “Discrete Nonlinear Localization in Femtosecond Laser Written Waveguides in Fused Silica,” Opt. Exp. **13**,10,552–10,557 (2005). [CrossRef]

6. A. Fratalocchi, G. Assanto, K. Brzdakiewicz, and M. Karpierz, “Discrete propagation and spatial solitons in nematic liquid crystals,” Opt. Lett. **29**,1530–1532 (2004). [CrossRef] [PubMed]

7. T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tuennermann, “Discrete diffraction in two-dimensional arrays of coupled waveguides in silica,” Opt. Lett. **29**,468–470 (2004). [CrossRef] [PubMed]

8. J. Fleischer, M. Segev, N. Efremidis, and D. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature **422**,147–150 (2003). [CrossRef] [PubMed]

9. A. Szameit, D. Bloemer, J. Burghoff, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, “Two-dimensional soliton in cubic fs laser written waveguide arrays in fused silica,” Opt. Exp. **14**,6055–6062 (2006). [CrossRef]

10. A. Szameit, D. Bloemer, J. Burghoff, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, “Hexagonal waveguide arrays written with fs-laser pulses,” Appl. Phys. B. **82**,507–512 (2006). [CrossRef]

11. H. Trompeter, W. Krolikowski, D. Neshev, A. Desyatnikov, A. Sukhorukov, Y. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch Oscillations and Zener Tunneling in Two-Dimensional Photonic Lattices,” Phys. Rev. Lett. **96**,0539,031–4 (2006). [CrossRef]

12. K. Itoh, W. Watanabe, S. Nolte, and C. Schaffer, “Ultrafast Processes for Bulk Modification of Transparent Materials,” MRS Bulletin **31**,620–625 (2006). [CrossRef]

7. T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tuennermann, “Discrete diffraction in two-dimensional arrays of coupled waveguides in silica,” Opt. Lett. **29**,468–470 (2004). [CrossRef] [PubMed]

13. D. Bloemer, A. Szameit, F. Dreisow, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, “Measurement of the nonlinear refractive index of fs-laser-written waveguides in fused silica,” Opt. Exp. **14**,2151–2157 (2006). [CrossRef]

## 2. Experimental results

*w*

_{0}is the radius of the focal spot and

*b*is the Rayleigh length.

*M*

^{2}characterizes the difference between a real laser beam and a diffraction-limited Gaussian beam. By measuring the near-field profile at a wavelength of 800 nm and solving the Helmholtz-Equation [14

14. I. Mansour and F. Caccavale, “An improved procedure to calculate the refractive index profile from the measured near-field intensity,” J. Lightwave Technol. **14**,423–428 (1996). [CrossRef]

*A*(

*x*,

*y*) is the modal field and

*n*

_{eff}is the effective refractive index of the propagating mode, the refractive index modifications were evaluated. Resulting from the shape of the focal area, the waveguides exhibit a high ellipticity with diameters of approximately 4×13 μm

^{2}. We further measured the losses by a cut-back method and obtained a value of < 0.4 dB/cm.

*a*(

_{n}*z*) in the

*n*th waveguide evolve during propagation while the field shapes remain constant. Assuming evanescent coupling only between adjacent waveguides this induces transverse dynamics of the propagating field. Energy exchange is caused by the overlap of the evanescent tails of the guided modes. In a planar array this behavior can be adequately modeled by a set of coupled differential equations [15].

*c*as the coupling constant. It can be formally shown, that the coupling constant from waveguide

*n*to waveguide

*n*+1 reads as [15]

_{n+1}is the refractive index change and

*E*

_{n,n+1}(

*x*,

*y*) denotes the field shapes of the guided modes in the waveguides

*n*and

*n*+1, respectively. Therefore, besides the increase of the refractive index in the waveguides the coupling is dispersive and highly dependant on mode overlap and mode shape.

*A*(

*z*) is the excited waveguide (

*A*(0)=1,

*B*(0)=0) , then the solution is simply

*I*= |

_{A}*A*(

*z*)|

^{2}and

*I*= |

_{B}*B*(

*z*)|

^{2}. This is the base for an exact measurement of the coupling constant

*c*between two adjacent waveguides, since the intensities can be determined very accurately. A very useful parameter is the coupling length, defining the propagation distance

*z*=

*l*after which in a two-waveguide system all of the guided power in the excited waveguide has been coupled in the adjacent one. In that case, it follows from Eq. (6) cos(

_{c}*cl*)=0 which yields for the coupling length

_{c}*l*= 0.3 cm in horizontal and

^{h}_{c}*l*= 0.6 cm in vertical direction at 614 nm. From our previous analysis we found a required waveguide separation of 22 μm. For a tilting angle of 0° the diagonal coupling can be completely neglected. This is due to the increased waveguide separation in these directions which are √2 times the separation in the horizontal or vertical direction, so that the coupling constant is decreased accordingly. Therefore, after the short propagation length, there is almost no energy flow in the diagonal directions. However, for an increasing tilting of the array the influence of the diagonal coupling from the upper left corner to the lower right corner also increases and reaches a maximum at 45° (Fig. 4b-e). In comparison calculated output patterns are shown in Fig. 4(f-i). This behavior can be understood by considering the orientation of the waveguides. For an increasing tilting the ellipse of the waveguides is more and more oriented in one diagonal direction yielding a larger overlap of the propagating modes. At 45 μ the waveguide ellipses directly point in diagonal direction, providing a stronger coupling although the distance is still larger by a factor of √2. The second diagonal direction can still be neglected since the mode overlap remains small due to the elliptical waveguide shape. Due to the tilting angle of 45 ° both, horizontal and vertical coupling are equal, exhibiting a value of

^{v}_{c}*l*=

^{h}_{c}*l*= 0.45 cm. The diagonal coupling from the upper left corner to the lower right corner is increased to

^{v}_{c}10. A. Szameit, D. Bloemer, J. Burghoff, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, “Hexagonal waveguide arrays written with fs-laser pulses,” Appl. Phys. B. **82**,507–512 (2006). [CrossRef]

16. M. Ams, G. Marshall, D. Spence, and M. Withford, “Slit-beam shaping method for femtosecond laser direct-write fabrication of symmetric waveguides in bulk glasses,” Opt. Exp. **13**,5676–5681 (2005). [CrossRef]

*R*of an ellipse depending of the angle φ

*p*as the half-parameter and ε as the numerical eccentricity, whereas the latter describes the deviation of the ellipse from a circle. Circular shape is obtained for ε = 0. For the elliptical waveguides we obtained a value of ε

_{ell}=0.6 and for the circular waveguides we found ε

_{circ}=0.15 which is only 25 % of the former value. Therefore, the evanescent coupling of circular waveguides are almost isotropic.

*l*≈2 cm, which is in excellent agreement to the experimental data proving not only the high isotropy of the coupling but also the high precision of the fabricated array. Due to the isotropic coupling the resulting output patterns are highly symmetric and independent of the excited waveguide. Therefore, such devices are highly appropriate to the use as circular switching and routing devices [17

_{c}17. W. Krolikowski, U. Trutschel, M. Cronin-Golomb, and C. Schmidt-Hattenberger, “Solitonlike optical switching in a circular fiber array,” Opt. Lett. **19**,320–322 (1994). [CrossRef] [PubMed]

## 3. Conclusion

## References and links

1. | H. Haus and L. Molter-Orr, “Coupled multiple waveguide systems,” IEEE J. Quantum Electron. |

2. | T. Pertsch, T. Zentgraf, U. Peschel, A. Braeuer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett. |

3. | D. Christodoulides and R. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. |

4. | H. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. |

5. | A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tuennermann, and F. Lederer, “Discrete Nonlinear Localization in Femtosecond Laser Written Waveguides in Fused Silica,” Opt. Exp. |

6. | A. Fratalocchi, G. Assanto, K. Brzdakiewicz, and M. Karpierz, “Discrete propagation and spatial solitons in nematic liquid crystals,” Opt. Lett. |

7. | T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tuennermann, “Discrete diffraction in two-dimensional arrays of coupled waveguides in silica,” Opt. Lett. |

8. | J. Fleischer, M. Segev, N. Efremidis, and D. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature |

9. | A. Szameit, D. Bloemer, J. Burghoff, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, “Two-dimensional soliton in cubic fs laser written waveguide arrays in fused silica,” Opt. Exp. |

10. | A. Szameit, D. Bloemer, J. Burghoff, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, “Hexagonal waveguide arrays written with fs-laser pulses,” Appl. Phys. B. |

11. | H. Trompeter, W. Krolikowski, D. Neshev, A. Desyatnikov, A. Sukhorukov, Y. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch Oscillations and Zener Tunneling in Two-Dimensional Photonic Lattices,” Phys. Rev. Lett. |

12. | K. Itoh, W. Watanabe, S. Nolte, and C. Schaffer, “Ultrafast Processes for Bulk Modification of Transparent Materials,” MRS Bulletin |

13. | D. Bloemer, A. Szameit, F. Dreisow, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, “Measurement of the nonlinear refractive index of fs-laser-written waveguides in fused silica,” Opt. Exp. |

14. | I. Mansour and F. Caccavale, “An improved procedure to calculate the refractive index profile from the measured near-field intensity,” J. Lightwave Technol. |

15. | A. Yariv, |

16. | M. Ams, G. Marshall, D. Spence, and M. Withford, “Slit-beam shaping method for femtosecond laser direct-write fabrication of symmetric waveguides in bulk glasses,” Opt. Exp. |

17. | W. Krolikowski, U. Trutschel, M. Cronin-Golomb, and C. Schmidt-Hattenberger, “Solitonlike optical switching in a circular fiber array,” Opt. Lett. |

**OCIS Codes**

(130.3120) Integrated optics : Integrated optics devices

(140.7090) Lasers and laser optics : Ultrafast lasers

(230.7370) Optical devices : Waveguides

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: October 13, 2006

Revised Manuscript: December 19, 2006

Manuscript Accepted: December 19, 2006

Published: February 19, 2007

**Citation**

Alexander Szameit, Felix Dreisow, Thomas Pertsch, Stefan Nolte, and Andreas Tünnermann, "Control of directional evanescent coupling in fs laser written waveguides," Opt. Express **15**, 1579-1587 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-4-1579

Sort: Year | Journal | Reset

### References

- H. Haus and L. Molter-Orr, "Coupled multiple waveguide systems," IEEE J. Quantum Electron. 19, 840-844 (1983). [CrossRef]
- T. Pertsch, T. Zentgraf, U. Peschel, A. Braeuer, and F. Lederer, "Anomalous refraction and diffraction in discrete optical systems," Phys. Rev. Lett. 88, 0939,011-4 (2002). [CrossRef]
- D. Christodoulides and R. Joseph, "Discrete self-focusing in nonlinear arrays of coupled waveguides," Opt. Lett. 13, 794-796 (1988). [CrossRef] [PubMed]
- H. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998). [CrossRef]
- A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tuennermann, and F. Lederer, "Discrete Nonlinear Localization in Femtosecond Laser Written Waveguides in Fused Silica," Opt. Exp. 13, 10,552-10,557 (2005). [CrossRef]
- A. Fratalocchi, G. Assanto, K. Brzdakiewicz, and M. Karpierz, "Discrete propagation and spatial solitons in nematic liquid crystals," Opt. Lett. 29, 1530-1532 (2004). [CrossRef] [PubMed]
- T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tuennermann, "Discrete diffraction in two-dimensional arrays of coupled waveguides in silica," Opt. Lett. 29, 468-470 (2004). [CrossRef] [PubMed]
- J. Fleischer, M. Segev, N. Efremidis, and D. Christodoulides, "Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices," Nature 422, 147-150 (2003). [CrossRef] [PubMed]
- A. Szameit, D. Bloemer, J. Burghoff, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, "Two-dimensional soliton in cubic fs laser written waveguide arrays in fused silica," Opt. Exp. 14, 6055-6062 (2006). [CrossRef]
- A. Szameit, D. Bloemer, J. Burghoff, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, "Hexagonal waveguide arrays written with fs-laser pulses," Appl. Phys. B. 82, 507-512 (2006). [CrossRef]
- H. Trompeter, W. Krolikowski, D. Neshev, A. Desyatnikov, A. Sukhorukov, Y. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch Oscillations and Zener Tunneling in Two-Dimensional Photonic Lattices," Phys. Rev. Lett. 96, 0539,031-4 (2006). [CrossRef]
- K. Itoh, W. Watanabe, S. Nolte, and C. Schaffer, "Ultrafast Processes for Bulk Modification of Transparent Materials," MRS Bulletin 31, 620-625 (2006). [CrossRef]
- D. Bloemer, A. Szameit, F. Dreisow, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, "Measurement of the nonlinear refractive index of fs-laser-written waveguides in fused silica," Opt. Exp. 14, 2151-2157 (2006). [CrossRef]
- I. Mansour and F. Caccavale, "An improved procedure to calculate the refractive index profile from the measured near-field intensity," J. Lightwave Technol. 14, 423-428 (1996). [CrossRef]
- A. Yariv, Optical Electronics, 4th ed. (Saunders College Publ., 1991).
- M. Ams, G. Marshall, D. Spence, and M. Withford, "Slit-beam shaping method for femtosecond laser direct-write fabrication of symmetric waveguides in bulk glasses," Opt. Exp. 13, 5676-5681 (2005). [CrossRef]
- W. Krolikowski, U. Trutschel, M. Cronin-Golomb, and C. Schmidt-Hattenberger, "Solitonlike optical switching in a circular fiber array," Opt. Lett. 19, 320-322 (1994). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.