Tailoring Guided Modes in Waveguide Arrays

Die rasante Entwicklung der optischen Datenübertragung erfordert zunehmend optische Bauelemente mit erhöhter Qualität, Zuverlässigkeit und Funktionalität. Allerdings beschränkt die Verwendung herkömmlicher optischer Materialien diese optischen Elemente in ihrer Leistungsfähigkeit. Der Fortschritt in Strukturierungstechnologien ermöglicht die Herstellung so genannter Metamaterialien mit problemangepassten, optimierten Eigenschaften.


Introduction
With the advance of optical data transfer the requirements for quality, reliability and functionality of integrated optics devices have increased.However, the sole use of conventional materials severely constrains the performance of optical elements.More sophisticated fabrication technologies now permit the manufacture of optimized metamaterials.
Photonic crystals as periodic sub-wavelength structures are probably the most prominent example of a metamaterial where, for example, refraction, diffraction and group velocity dispersion can to a large extent be varied.A waveguide array is another prominent example of a metamaterial with controllable optical characteristics /1/.In the array, waves are confined to individual guides (Fig. 1).The evolution of this virtually discretized light is restricted to energy transfer via the evanescent tails of the modes of the corresponding guides.Hence, the dynamics of optical arrays are determined by the characteristics of the guides together with the coupling characteristics between those guides.Here we demonstrate that new types of localized states bound at defects or inhomogeneities in waveguide arrays exist which have no analogon in conventional integrated optics.

Polymer Waveguide Arrays
The investigated arrays consist of 101 waveguides made of an inorganicorganic polymer (n co =1.547) on thermally oxidized silicon wafers (n sub =1.457) with a polymer cladding (n cl =1.544) (see Fig. 1).The samples are fabricated by UV lithography on 4 inch wafers providing propagation lengths of up to 7 cm.All waveguides are 3.5 µm high.Waveguides with a width varying between 2.5 and 4.5 µm provide low-loss single mode waveguiding (< 0.1 dB/cm) for a HeNe-laser light source at λ = 633 nm.A waveguide spacing of between 4 and 5 µm ensures an efficient evanescent coupling of the nearest neighboring guides.Defects and inhomogenities are easily incorporated by locally varying either the width of an individual guide or the spacing around a site.

Defect Modes in Waveguide Arrays
First we checked conventional waveguiding mechanism in an array.To this end a defect is created which consists of a single guide with an increased width of 3.5 µm compared to 3 µm for the rest of the array.As a consequence the propagation constant of the defect guide is increased, whereas the coupling is slightly decreased.Similar to conventional integrated optics, light concentrates around the region of higher effective index and a bound defect mode is born (Fig. 2b).Since the modal fields of such localized states have a flat phase (see Fig. 2a), such localized states are called unstaggered modes.
Next, we investigated deviations from classical waveguiding mechanisms by decreasing the width of the defect waveguide (3 µm compared with 3.5 µm in the rest of the array).Again, the resulting decrease of the propagation constant of the defect is accompanied by only a small increase of the coupling constant.In fact we also observe a guided mode (Fig. 2d) the shape of which differs considerably from that of a unstaggered one.Because fields in adjacent guides are now π out of phase (Fig. 2c) the intensity of this mode becomes zero between the waveguides due to destructive interference from the different modal fields.
Hence in contrast to the unstaggered mode, which is bound by internal reflection, the guiding mechanism of this new, so-called staggered state relies on Bragg-reflection within the periodic structure of the array.
An increase of the coupling constant of the defect can almost exclusively be achieved by decreasing the spacing between the center waveguide and its neighbors (spacing: 4 µm compared with 5 µm in the rest of the array).In this case both unstaggered and staggered modes usually exist simultaneously.An input beam centered on a single waveguide always excites both modes.
At the end facet of the array an interference pattern is observed depending on the current phase difference between the two bound states.If the exciting beam is shifted from the defect guide (n = 0) towards its neighbor (n = ± 1) the phase difference between the staggered and unstaggered modes changes by π.Hence by varying the waveguide of excitation we can switch between destructive and constructive interference in the defect guide or in the neighboring waveguide at the output facet (compare Figs. 3a, b, c).Even if the initial excitation is asymmetric with respect to the defect guide, we never observe an asymmetric guided field at the output.Thus, no asymmetric mode exists, although the defect is multi-mode.
If the coupling constant of the defect waveguide is decreased by increasing the spacing between the center waveguide and its neighbors, no bound states occur.This is somehow counterintuitive.Because the defect tends to be isolated due to the reduced coupling, one might even expect improved guiding properties.Again, the simple classical understanding of mode guiding fails.Light from the defect predominantly couples into Bloch modes with a high transverse velocity.Hence, the excitation will leave a defect with reduced coupling very quickly (Fig. 4a, b).
In contrast to an excitation in the homogenous array (Fig. 4c, d) the defect repels the light causing a dark region around it.

Fig. 3 :
Fig. 3: Interference pattern of a staggered and an unstaggered defect mode for dominant increase of the coupling constant of the defect at a propagation distance of 59.95 mm.dots: experiment, lines: theory, dashed line: position of the excitation (b) excitation of the defect waveguide.(a) and (c) excitation of the left and right nearest neighbor waveguide of the defect.

Fig. 4 :
Fig. 4: Diffraction pattern for an excitation of a repulsive defect ([a] theory, [b] experiment) with reduced coupling and in a homogeneous array ([c] theory, [d] experiment).