We study the localisation and control of high frequency sound in a dual-core square-lattice photonic crystal fibre preform. The coupled states of two neighboring acoustic resonances are probed using an interferometric set up, and experimental evidence is obtained for odd and even symmetry trapped states. Full numerical solutions of the acoustic wave equation show the existence of a two-dimensional sonic band gap, and numerical modelling of the strain field at the defects gives results that agree well with the experimental observations. The results suggest that sonic band gaps can be used to manipulate sound with great precision and enhance its interaction with light.
©2003 Optical Society of America
There is currently keen interest in materials that are periodically microstructured so as to exhibit photonic band gaps - frequency ranges where all optical vibrations are switched off [1–3]. Sonic vibrations can be controlled in a similar way [4–8]. Here we report the observation of high frequency (~25 MHz) sonic band gaps (SBGs) in rods of silica glass with square-lattice arrays of hollow micro-channels running along their length. By incorporating point defects in the form of filled-in channels, localised resonances appear [9,10]. Driving a dual-defect structure with a piezoelectric transducer, we monitor the vibrations using laser interferometry. Numerical solutions of the full acoustic wave equation, for the sonic band structure and the fields at the defects, are in good agreement with the measurements. The results point to a new class of ultra-efficient photo-sonic device in which both sound and light are controlled with great precision and their interactions enhanced.
The glass samples were prepared following a procedure commonly used to fabricate photonic crystal fibres [11,12]. First of all, precision-made tubes and rods of silica glass (~1 mm in diameter) were stacked into a large-scale “preform” of the desired structure. This was then fused together and reduced in size, roughly by a factor of 20 in transverse linear dimension, using an optical fibre drawing tower. These intermediate preforms were used directly in the work reported here (normally they are drawn down once more to form a fibre). A scanning electron micrograph of the cross-section of the sample used in the experiments is given in Fig. 1 - note the small interstitial holes. Two structural defects were placed in next-nearest-neighbour sites. Their effect is to “clear a space” for localised resonances to form at frequencies within the SBG. These resonances can be excited by placing a transducer directly inside the defects themselves . In this work we excite them by means of evanescent wave tunnelling through the SBG region. When the frequency of the transducer (attached to the side of the preform) coincides with the frequency of a defect resonance, energy is efficiently transferred. At other intra-SBG frequencies this transfer is suppressed, while at frequencies outside the SBG there will be no localisation of energy at the defect - the entire preform will vibrate.
A longitudinal-mode piezoelectric disk (motion normal to the plane, 8 mm in diameter, high frequency cut-off at 35 MHz) was used. In this configuration (Fig. 2), only acoustic waves with displacements perpendicular to the axis of the air holes, i.e., waves with in-plane motion, are excited. Both shear and longitudinal waves will appear in the structure, since they are coupled together at the air-glass interfaces. It is well known that vibrations cause modifications in the refractive index through the strain-optical effect . As a result, a laser beam travelling through a vibrating medium will suffer a phase change related to the strength of the vibrations; this can be easily measured with an interferometer. Using this technique we studied the acoustic vibrations in the solid glass “cores” of the preform defects.
The preform was placed in one arm of a Michelson interferometer. Two objective lenses were used to launch the light into the selected core. The detected signal from the interferometer contained harmonics of the frequency of excitation, the amplitudes of which can be related to the amplitude of the phase change in the core region. A second Michelson interferometer, in which the transducer was used as one of the mirrors, was set up to monitor the amplitude of the transducer vibration. Since the efficiency of the transducer was strongly dependent on frequency, this second interferometer turned out to be essential to normalise the transducer amplitude over all drive frequencies.
Figure 3 shows the change in phase as a function of frequency when the laser beam was launched into the upper and the lower cores. In both cases, two sharp peaks were observed at 23.00 MHz and 23.25 MHz. These peaks stand out clearly against a background of low-level vibrations. Their widths are ~90 kHz, indicating a quality factor Q~300 for the related resonances. The fact that both cores resonate simultaneously at the two peaks shows that there is strong coupling between them. Furthermore, the almost-equal amplitudes in each core at resonance suggest that the cores are well-matched in resonant frequency, and that the lower/higher frequency peaks are associated with even/odd modes of the coupled cores (see discussion below).
To our knowledge, specific calculations for the case of an array of air holes in fused silica have not been reported although similar structures have been studied theoretically . Using a technique based on a development of the Rayleigh method for elastodynamic materials , we obtained the sonic band structure of the square lattice including interstitial holes. The density of fused silica is ρ=2200 kg/m3 and the Lamé coefficients are µ=31.15 GPa & λ=16.05 GPa . For the particular experimental case considered here, the wave motion is in the transverse plane and is described by the Navier vector wave equations:
where u(r)=(u(r),ν(r)) is the displacement vector at position r=(x,y) and ω is the angular frequency of vibration. The boundary conditions assume traction-free surfaces at each hole, i.e., the components of the stress tensor vanish at the boundaries. As the cylinders are considered to be infinitely long, the problem is inherently two-dimensional. Finally, the periodicity implies that u must satisfy Bloch’s theorem:
where the Bloch vector k B lies entirely in the (x,y) plane and R p points to the center of the pth cavity. Using the multi-pole method described in , we find the set of normal frequencies ω for a given k B, corresponding to propagating modes. The resulting sonic band structure, for a defect-free lattice with the same hole size and spacing as the preform, is presented in Fig. 4. A full SBG appears in the frequency range 21.8–25.0 MHz. The resonances observed in the experiments sit well within this range, suggesting that the sound is indeed trapped in the cores by a SBG.
Next we modelled the preform complete with defects. This was done by tiling a supercell containing N×N periods with the two defects at its centre. Although a band diagram can be calculated for such a structure, the number of dispersion curves will dramatically increase compared to Fig. 4. We therefore concentrated on analysing the localised modes at the defects, solving the equations for a 9×9 super-cell with Bloch conditions imposed at the exterior boundary. A finite element approach was used with Neumann boundary conditions at the air-glass interfaces. The results are presented in Fig. 5 as density plots of the dilatation (∂u/∂x+∂v/∂y) and the shear strain (∂u/∂y+∂v/∂x)/2. Five localised modes in total were found, with resonant frequencies 23.04, 23.47, 23.55, 24.15 and 24.32 MHz (Fig. 5). Of these, only the four higher frequency modes show significant dilatation - required if the resonances are to be detectable in the interferometer (for glass, shear strain produces only very small - if any - changes in refractive index, whereas dilatation has a large effect ). In mode a the dilatation is symmetrical about the horizontal axis, corresponding to an “even” mode of the dual-core system, i.e., the two cores resonate in phase. Mode b, on the other hand, has odd symmetry, i.e., the two cores resonate out of phase. (It is interesting that the shear strain has the opposite symmetry in each case.) Mode c is once again even in dilatational strain, and mode d exhibits very small (even) dilatational strain. The frequencies predicted by the theory are ~1 MHz higher than in the experiments. We attribute this to the limitations of the numerical modelling - since it can only treat circular inclusions, it is unable to take account of the narrow points of fusion between glass tubes and rods in the real structure. This is likely to make the modelled structure stiffer and thus increase its resonant frequencies. The next question is: which two of the predicted resonant modes correspond to the two peaks seen in the experiment (Fig. 3)? It is known that the frequency splitting between different spatial modes of a dual-core system increases with the strength of coupling between the cores. Inspection of the real structure (Fig. 1) shows that the central hole in the real structure is somewhat squashed, permitting the core fields to interact more strongly than in the ideal structure. This suggests that the theoretical frequency splitting should be somewhat smaller than the experimental one. This makes it likely that modes a & b (frequency splitting 80 kHz in the model are the ones seen in the experiment (splitting 250 kHz). It is also likely that resonances c & d (and also the resonance predicted at 23.04 MHz) contribute to some of the smaller peaks in the phase response curve in Fig. 3.
In conclusion, the ability to precisely manipulate high frequency acoustic fields suggests many possibilities for new devices. For example, consider a laser beam launched equally into the two cores of a dual-core preform resonating in its odd mode, and then re-combined at the output. As the relative phase between the cores changes, the wave amplitude at the output will oscillate. The result is an efficient optical modulator. If the structures are scaled down in size to the point where the cores become single mode waveguides (as in a photonic crystal fibre), the acoustic resonant frequencies scale up to into the GHz range (although the acoustic losses increase with frequency, they only become significant at multi-GHz frequencies). The almost perfect spatial overlap of acoustic and optical fields, in a very small area, points to a new family of highly efficient acousto-optic devices based on photo-sonic effects, such as optically pumped acoustic oscillators [17,18]. It may even prove possible to suppress acoustic vibrations in the fibre core, suggesting that high power single-frequency laser light could be transmitted without the onset of stimulated Brillouin scattering.
A. Dièz acknowledges the financial support of the Comisión Interministerial de Ciencia y Tecnologia (CICYT) of Spain. His current address is Departamento de Física Aplicada, Universidad de Valencia, Dr. Moliner 50, E-46100, Spain. E. Marin’s current address is Laboratoire TSI - UMR CNRS 5516, Bâtiment F-10, rue Barrouin, 42000 Saint-Etienne, France. The authors would like to thank M. Franczyk and B.J. Mangan for preform fabrication. The work was supported by the UK Engineering and Physical Sciences Research Council.
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