Download Band structure of cavity-type hypersonic phononic

Band structure of cavity-type hypersonic phononic crystals fabricated by
femtosecond laser-induced two-photon polymerization
Supplementary material
FIG. S1. Dispersion curves obtained along the ΓM direction for two values of the transverse sound velocity (a,b) cT = 1400 m/s, (c,d) cT =
1700 m/s The upper panels (a) and (c) give the relative longitudinal (L) vs. transverse (T) character (color scale bar) of the dispersion
curves. The lower panels (b) and (d) give the BLS intensities obtained for P66/(P11+P12) = -0.2.
In Fig. S1, the band diagrams of the air-filled hPnC are computed for two different transverse sound velocity cT of the matrix.
In the two upper panels, the color scale indicates the polarization of the different modes, whereas in the same dispersion
curves of the two lower panels, the contribution of these modes to the BLS spectra is illustrated by a color scale bar. For the
latter, the experimental BLS intensities were best represented when the photoelastic coefficients (Eq.1) assuming P 66/
(P11+P12) is negative (~ -0.2), P11/ (P11+P12) = 0.3 and P12/ (P11+P12) = 0.7 (Fig. S1b,d). For example, mode 2 in Fig. 3b (close
to the M point of the Brillouin zone) exhibits significant contribution to the BLS spectrum and is strongly longitudinal (L)
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FIG. S2. Theoretical and experimental dispersion curves along ΓX and ΓM directions of the square patterned polymer filled with Cargille
liquid (cf = 1670 m/s). cL = 2900 m/s and cT = 1400 m/s (a-d) and cT = 1700 m/s (e-h). The color scale gives the relative longitudinal (L) vs.
transverse (T) polarizations of the modes (a,c,e,g) and the BLS intensity (b,d,f,h).
FIG. S3. Theoretical and experimental BLS spectra along the ΓX and ΓM directions, for two values of q. The ratio P/(P11+P12) is fixed in
each row (P/(P11+P12) = 4 is the optimal value).
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Figure S2 is a replica of Fig, S1 for the filled-hPnC using liquid Cargille with longitudinal cf = 1670 m/s. Two different
transverse sound velocity cT of the matrix are assumed in Fig. S2a,c,e,f along the two symmetry directions, ΓX and ΓM with
the color scale bar indicating the polarization character of the observed modes. The same dispersion curves of the four lower
panels indicate (color scale bar) the relative contribution of these modes to the BLS spectra. For the latter, the experimental
BLS intensities were best represented (Eq. 4 along ΓX and ΓM) when the photoelastic coefficient P in the fluid is higher
compared to (P11+P12) in the ZPO solid with P11, P12 used in Fig. S1. According to Fig. S3, P/(P 11+P12) = 4 leads to a good
agreement with the BLS intensities.
FIG. S4. Effective sound velocity (ceff) normalized to the longitudinal sound velocity of the matrix (cL) along ΓX and ΓM for two different
infiltrated liquids (Cargille ,cf = 1670 m/s and polydimethylsiloxane (PDMS), cf = 1000m/s)
The normalized ceff/cL, effective sound velocity to the matrix longitudinal acoustic cL, decreases with filling fraction
(porosity) for both air- and liquid –filled hPnC but the drop of ceff is direction dependent. Moreover, infiltration does not
necessarily leads to higher ceff compared to the air-filled hPnC as clearly revealed in the case of PDMS with lower sound
velocity cf (1000 m/s) than Cargille (cf = 1670 m/s) but similar densities (ρ). Hence, the longitudinal modulus M = ρcceff2 can
be tuned for the same hPnC by liquid infiltration and is direction dependent. The trend in Fig.S4 cannot be captured by
effective medium expressions, e.g., Wood’s law.
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