1. Introduction

Systems of evanescently coupled optical waveguides are of great interest over the last several years since such systems exhibit unique imaging [1] and propagation [2] 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]. While one-dimensional discrete solitons were shown in waveguide arrays fabricated by a variety of techniques [4, 5, 6], two-dimensional discrete solitons are only possible in fibers [7], optical arrays induced in photorefractives [8] and fs laser written waveguide arrays [9]. 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, 11]. 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]. 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]. In addition the linear and nonlinear properties of every single waveguide can be controlled precisely by choosing appropriate writing parameters [13]. 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.

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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

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]

where 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.

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 a_n(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].

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]

where Δε_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.

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

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

The output intensities as measurable quantities give

with 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_c 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(cl_c)=0 which yields for the coupling length

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.

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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°.

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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.

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The anisotropic coupling in elliptic fs laser written waveguides is a fertile degree of freedom in the design of two-dimensional waveguide arrays allowing the precise tuning of the evanescent coupling between adjacent waveguides by choosing a specific angle for the coupling direction without changing the waveguide separation. For example, it is possible to study the influence of diagonal coupling in square waveguide arrays only by tilting the array geometry (Fig. 4a). To show that behavior we designed a 5×5 square waveguide array exhibiting a length of 1 cm and a desired coupling length of l^h_c = 0.3 cm in horizontal and l^v_c = 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 l^h_c = l^v_c = 0.45 cm. The diagonal coupling from the upper left corner to the lower right corner is increased to $l_c^d=\frac{1}{3}{l}_c^h=0.15\mathrm{cm}$. Since now evanescent coupling in three directions influences the propagation, the tilted array at 45 ° is comparable in principle to a hexagonal array [10]. Hence, it is possible to analyze the transition from a pure square to a hexagonal lattice by investigating the intensity output at different tilting angles. Therefore, using the anisotropy of elliptical waveguides is an efficient way of designing and fabricating waveguide arrays with specific tuned coupling in horizontal, vertical and diagonal direction. Such systems have not been realized before and will give additional insight in discrete diffraction in two-dimensional waveguide arrays.

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]. 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.

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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

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 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.

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 l_c≈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] 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.

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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.

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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.

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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”).