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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 4 — Feb. 19, 2007
  • pp: 1579–1587
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Control of directional evanescent coupling in fs laser written waveguides

Alexander Szameit, Felix Dreisow, Thomas Pertsch, Stefan Nolte, and Andreas Tünnermann  »View Author Affiliations


Optics Express, Vol. 15, Issue 4, pp. 1579-1587 (2007)
http://dx.doi.org/10.1364/OE.15.001579


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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

Systems of evanescently coupled optical waveguides are of great interest over the last several years since such systems exhibit unique imaging [1

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

] and propagation [2

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]

] properties caused by the anisotropy of the medium which is due to the discreteness of the physical system. A peculiar feature of waveguide lattices is the possibility of exciting so-called discrete spatial solitons [3

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

]. While one-dimensional discrete solitons were shown in waveguide arrays fabricated by a variety of techniques [4

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

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

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]

], two-dimensional discrete solitons are only possible in fibers [7

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]

], optical arrays induced in photorefractives [8

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]

] and fs laser written waveguide arrays [9

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]

]. Despite of such nonlinear effects also a variety of interesting two-dimensional linear effects in waveguide arrays have been discovered and studied during the last years [10

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

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]

]. A further step in the investigation of discrete propagation is the directional specific tuning of the two-dimensional evanescent coupling between adjacent waveguides. However, this is not possible in optically induced arrays in photorefractives and fiber waveguide arrays due to the circular shape of the waveguides and the strong difficulties to precisely adjust the waveguide separations in different directions. Only the fs writing technique provides the feasibility of a precise adjustment of the evanescent coupling.

By focusing ultrashort laser pulses into bulk material, in the focal region nonlinear absorption is taking place leading to optical breakdown and the formation of a microplasma, inducing a permanent refractive index change in the material [12

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]

]. The dimensions of these changes are approximately the same as the size of the focal region. By moving the sample transversely with respect to the beam a continuous modification is obtained and a waveguide is created. These waveguides can be written along arbitrary paths since the only limiting factor in the placement of the focus is the focal length of the writing objective. Furthermore, all structural changes are permanent and therefore non-sensitive to external conditions. This allows to fabricate two-dimensional waveguide arrays with strong boundary effects [7

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]

]. In addition the linear and nonlinear properties of every single waveguide can be controlled precisely by choosing appropriate writing parameters [13

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]

]. Up to now only the use of fs laser structuring allows the fabrication of three-dimensional nonlinear devices with sharp boundaries and controlled coupling properties.

In this paper we present detailed investigations on the properties of two-dimensional evanescent coupling in femtosecond laser written waveguide arrays, for the first time. The coupling constants in different directions for elliptical and circular waveguides are directly measured. It will be shown that the anisotropy of the coupling of elliptical waveguides can be used for the specific enhancement of the diagonal coupling in square waveguide arrays by simply changing the array orientation. Furthermore, using circular waveguides, a circular array with isotropic coupling properties is fabricated.

For the determination of the coupling constant in waveguide arrays, usually calculated output patterns are fitted to the experimental data with the coupling constant as a free parameter. Therefore, especially in two-dimensional arrays where the diagonal coupling cannot be neglected, the parameter space is multi-dimensional, which makes a correct interpretation of the output patterns difficult. In order to precisely tune the coupling in the array, we produced pairs of waveguides with different parameters by fs direct writing and determine the directional dependant coupling constant. This provides the basis for creating the desired waveguide arrays.

Fig. 1. Scheme of the writing process in a transparent bulk material by use of fs laser pulses. Inset: Measured profile of refractive index change.

2. Experimental results

For the fabrication of the waveguides we used a Ti:Sapphire laser system (RegA/Mira, Coherent Inc.) with a repetition rate of 100 kHz, a pulse duration of about 150 fs and 0.3 μJ pulse energy at a laser wavelength of 800 nm. The beam was focused into a polished fused-silica sample by a 20x microscope objective with a numerical aperture of 0.45 (Fig. 1). The writing velocity was 1250 μm/s, performed by a high precision positioning system (ALS 130, Aerotech). The resulting index changes are approximately the size of the focal area of the writing objective, which can be described by

w0=M2λπNA
b=M2λπNA2.
(1)

Here 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]

]

n2(x,y)=neff2λ24π2ΔA(x,y)A(x,y),
(2)

The propagation of light in a dispersion-free and lossless array of identical equally spaced homogeneous waveguides can be modeled by using a coupled mode approach in which only the amplitudes an(z) in the nth 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

15. A. Yariv, Optical Electronics, 4th ed. (Saunders College Publ., 1991).

].

iddzan(z)+c(an+1(z)+an1(z))=0
(3)

with c as the coupling constant. It can be formally shown, that the coupling constant from waveguide n to waveguide n+1 reads as [15

15. A. Yariv, Optical Electronics, 4th ed. (Saunders College Publ., 1991).

]

c~ωΔεn+1En(x,y)En+1(x,y)dxdy,
(4)

where Δεn+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.

In the case of only two waveguides, Eq. (3) reduces to

iddzA(z)+cB(z)=0
iddzB(z)+cA(z)=0.
(5)

When A(z) is the excited waveguide (A(0)=1, B(0)=0) , then the solution is simply

A(z)=coscz
B(z)=isincz.
(6)

The output intensities as measurable quantities give

c=1zarctanIBIA
(7)

with IA = |A(z)|2 and IB = |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 = lc 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(clc)=0 which yields for the coupling length

lc=π2c.
(8)

For the experiments we wrote waveguide pairs with a length of 10 mm from 12 μm to 36 μm separation (measured from center to center), with orientation angles 0 °, 15°, 30°, … and 90° (Fig. 2). We irradiated the waveguides with laser light from a tunable He-Ne laser at 633 nm, 612 nm, 594 nm and 532 nm and laser light from a Ti:Sa laser at 800 nm. The resulting coupling constants for λ = 800 nm are shown in Fig. 3(a). The decay of the coupling is exponential with increasing waveguide separation. This is intuitive since for gaussian mode profiles the overlap in Eq. (4), describing the coupling strength, also decreases exponentially. As a consequence of the elliptical shape of the waveguides, the coupling is highly non-isotropic with a maximum in vertical and a minimum in horizontal direction. This can be understood by the different mode overlap in the different directions since the propagating mode profiles are elliptical. Furthermore, the coupling for different wavelengths differs due to the presence of ω in Eq. (4). In Fig. 3(b), the coupling constants for an angle of 45 ° are shown for different wavelengths, exhibiting the highly dispersive character of the evanescent coupling. The decrease of the coupling with increasing separation is exponential for all wavelengths, but shows a different slope, relying on the extension of the propagating mode beyond the waveguide. With these data it is possible to fabricate waveguide arrays with predetermined coupling constants in different directions.

Fig. 2. a) Microscope image of waveguide pairs with a separation of 20 μm written in different orientations for the measurement of the coupling constant. b) Resulting output patterns at 633 nm for a propagation length of 10 mm. The excited waveguide is marked by a white circle.
Fig. 3. (a) Coupling constant for different waveguide separations and different coupling angles at λ = 800 nm. (b) Coupling constant for different wavelengths and different waveguide separations at a orientation angle of 45°.
Fig. 4. (a) Microscope images of square waveguide arrays with a waveguide separation of 18 μm exhibiting orientation angles 0°, 15°, 30° and 45°. (b-e) Corresponding (rotated) output patterns for excitation of the center waveguide at λ = 614 nm. (f-i) Corresponding calculated output patterns. The excited waveguide is marked by a white circle.

However, various applications rely on an isotropic coupling of the single waveguides. A possible solution is to reduce the ellipticty of the waveguides. For this purpose it is necessary to shape the focus of the writing objective. This can be achieved by using a slit, which reduces the numerical aperture of the beam transversely to the writing direction. This results in a broadening of the beam while the effective Rayleigh length does not change as strong (note that the slit reduces the numerical aperture only transversal to the writing direction) [16

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]

]. Therefore, the resulting waveguides exhibit a reduced ellipticity. To minimize the losses due to the slit, the cross-section of the laser beam is adjusted by a telescope consisting of two cylindrical lenses increasing the laser beam power passing the slit (Fig. 5 left).

Fig. 5. Left: Setup for the fabrication of cylindrical waveguides. Right: Comparison of elliptical and circular waveguides.

With this setup it was possible to fabricate waveguides with a ratio of the transverse cross-sections of 1:1.5 in contrast to the waveguides written without shaping the focus which exhibit a ratio of 1:4.6 (Fig. 5 right). Due to the strongly reduced ellipticity of the resulting refractive index profile (Fig. 6, bottom) also the mode profiles for the investigated wavelengths are highly circular (Fig. 6a-e). Hence the mode overlap between adjacent waveguides is independent of the coupling direction. This allows for the fabrication of waveguide arrays with isotropic coupling. To analyze this behavior we fabricated again waveguide pairs from 12 μm separation to 36 μm, with orientation angles 0°, 15°, 30°, … and 90… and illuminated the waveguides with laser light from a tunable He-Ne laser at 633 nm, 612 nm, 594 nm and 532 nm and laser light from a Ti:Sa laser at 800 nm. The resulting coupling constants for λ = 800 nm are shown in Fig. 7(a). The decay of the coupling constant is exponential, however, the anisotropy of the coupling is strongly reduced. In Fig. 7(b) the coupling constant is shown as a function of the orientation angle at a waveguide separation of 18 μm. The characteristics of the measurements were fitted with an expression for the radius R of an ellipse depending of the angle φ

R=p1εcosφ
(9)

with 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 εell=0.6 and for the circular waveguides we found εcirc=0.15 which is only 25 % of the former value. Therefore, the evanescent coupling of circular waveguides are almost isotropic.

This is of specific interest for the fabrication of circular arrays with circular waveguides since there the coupling angle changes considerably. We fabricated an array consisting of 32 waveguides with a waveguide separation of 28 μm and a length of 12 cm. The overall diameter of the array is 286 μm (Fig. (8 a). In Fig. 8(b) the array has been excited with red laser light at λ = 633 nm. In comparison, Fig. 8(c) shows the calculated output pattern for a coupling length of lc≈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

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]

] for which an isotropic coupling is mandatory.

Fig. 6. (a-e) Mode profile of a circular waveguide for different input wavelengths. Also the measured profile of the refractive index change at λ = 633 nm is shown.
Fig. 7. (a) Coupling constant for different waveguide separations and different coupling angles for cylindrical waveguides at λ = 800 nm. (b) Comparison of the dependence of the coupling on the direction at 18 μm waveguide separation.

3. Conclusion

In conclusion we systematically investigated the evanescent coupling of elliptical and circular fs laser written waveguides by analyzing the coupling in waveguide pairs. Elliptical waveguides exhibiting a strong anisotropy in coupling were used to specifically increase diagonal coupling in square waveguide arrays. This is comparable to the coupling in three directions as observed in hexagonal arrays and thus allows to investigate the transition from a cubic array to a hexagonal arrangement. Using circular waveguides we fabricated a circular waveguide array with isotropic coupling, which is the base for a variety of optical switching and routing devices.

Fig. 8. (a) Microscope image of a circular waveguide array with a diameter of 286 μm. (b) Measured intensity output pattern for λ = 633 nm and (c) corresponding theoretical calculation assuming isotropic coupling. The excited waveguide is marked by a white circle.

The authors wish to thank Stefan Fahr and Holger Hartung for fruitful discussions. A. Szameit was supported by a grant from the Jenoptik AG.We further acknowledge support by the Deutsche Forschungsgemeinschaft (Research Unit 532 “Nonlinear spatial-temporal dynamics in dissipative and discrete optical systems”).

References and links

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]

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]

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]

15.

A. Yariv, Optical Electronics, 4th ed. (Saunders College Publ., 1991).

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]

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]

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


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References

  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]
  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]
  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]
  15. A. Yariv, Optical Electronics, 4th ed. (Saunders College Publ., 1991).
  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]
  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]

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