Resonant light transport through Mie modes in photonic glasses

We present an optical characterization of photonic glasses, composed of randomly arranged, monodisperse dielectric spheres packed at high filling fractions. We provide a detailed optical study of the resonant behavior of diffuse light transport through such systems. By means of independent static and dynamic measurements, we show resonances in the transport mean free path, diffusion constant, and also energy velocity of light. We also show that the main transport parameters can be controlled by varying the sphere diameter.

Figure 1

(Color online) Modulus of the scattered field for a sphere diameter $d=1100\mathrm{nm}$ and refractive index $n=3.4$ in the case of low $(d∕\lambda =0.33)$ and high energy $(d∕\lambda =1.65)$. The confinement of the mode is proportional to the $Q$ factor of the spheres, which increases with the field energy.

Figure 2

(Color online) Top: (Left) Scanning electron microscopy (SEM) image from the photonic glass surface composed of polystyrene (PS) spheres with a diameter of $1200\mathrm{nm}$. (Right) SEM image from $\mathrm{Ti}{\mathrm{O}}_2$ powder surface with an average particle diameter $850\mathrm{nm}$. Bottom: Corresponding histogram of particle sizes from the photonic glass (left) and $\mathrm{Ti}{\mathrm{O}}_2$ powder (right).

Figure 3

(Color online) Top: picture shows an opal-based photonic crystal (left) which shows visible iridescences due to Bragg reflections and a photonic glass (right) white and without any trace of Bragg features. Bottom: (Left) SEM image from a photonic crystal cleaved edge (scale bar is $10\mu \mathrm{m}$). (Right) SEM image from a photonic glass surface (scale bar is $10\mu \mathrm{m}$). Both samples are made of PS spheres (2% of polidispersity measured by TEM) and have centimeters squared areas and millimeters thickness.

Figure 4

Plot of two-body sphere-sphere potential $U\left(r\right)$ vs reduced distance $r∕d$ for different electrolyte concentrations. The total potential is relative to the thermal energy.

Figure 5

Plot of the average ff of a photonic glass as a function of electrolyte (salt) concentration. Spheres that composed the glass are $1220\mathrm{nm}$ diam. Filling fraction can be estimated by weighting the samples when their geometry is known. A concentration $0\mathrm{M}$ gives rise to a well known opal-based photonic crystal where the total volume occupied by the spheres is 74% of the total volume of the unit cell in an fcc lattice.

Figure 6

Scheme of the light intensity vs distance into a slab of a photonic glass. Extrapolation length $z_e$ and penetration length $z_p$ are shown.

Figure 7

(Color online) Cartoon of the experimental setup to perform static measurements. A slab of photonic glass with thickness $L$ is placed in the entrance of an integrating sphere and illuminated with white light. Diffuse light is measured in the detector.

Figure 8

Normalized total transmission of white light through photonic glasses as a function of the reduced parameter $dn∕\lambda$. Samples are composed by spheres with four different diameters, and thickness is about $100\mu \mathrm{m}$ in all cases.

Figure 9

Total transmission of white light through two different reference samples as a function of the reduced parameter $dn∕\lambda$. The dashed line represents total transmission through a photonic glass made of PS spheres of $d=200\mathrm{nm}$. Solid line represents total transmission through polydisperse $\mathrm{Ti}{\mathrm{O}}_2$ of averaged $d=850\mathrm{nm}$ (SEM inset of the sample where the scale bar represents $200\mathrm{nm}$). Both present nonresonant light transmission.

Figure 10

Inverse of total light transmission of white light through photonic glasses made of PS spheres with $d=1220\mathrm{nm}$ as a function of wavelength. (a) For low values of $L$ $({\mathcal{l}}_a>L)$. In this case, thicknesses vary from $50\text{to}250\mu \mathrm{m}$. (b) For high values of $L$ $({\mathcal{l}}_a<L)$, up to $1600\mu \mathrm{m}$, transmission shows a linear dependence on thickness except for those wavelengths where absorption is not negligible. Three particular wavelengths are marked with a triangle $({\lambda }_1=744\mathrm{nm})$, a square $({\lambda }_2=828\mathrm{nm})$, and a circle $({\lambda }_3=875\mathrm{nm})$.

Figure 11

Plot of $T^{-1}$ as a function of the thickness for three different wavelengths. (a) For low $L({\mathcal{l}}_a⪢L)$, linear dependence is clearly shown for $\lambda =744$, 828 and $875\mathrm{nm}$ as an indication of the validity of the Ohm’s law for this wavelengths which present, however, different slopes. (b) For high $L$ $({\mathcal{l}}_a<L)$, exponential behavior is revealed for $\lambda =875\mathrm{nm}$ related to a water absorption.

Figure 12

(Color online) Experimental transport mean free path in a photonic glass made of PS spheres of $d=1220\mathrm{nm}$. The data are obtained by fitting experimental $T^{-1}$ curves. Two particular wavelengths are marked with a triangle $({\lambda }_1=744\mathrm{nm})$, a square $({\lambda }_2=828\mathrm{nm})$.

Figure 13

Total white light transmission through photonic glasses composed by PS spheres with $200\mathrm{nm}$ (straight line), $1000\mathrm{nm}$ (dashed-dot line) and $1200\mathrm{nm}$ (dashed line) as a function of wavelength. All the spectra present a peak absorption at $\lambda =875\mathrm{nm}$ (marked with a circle) related to water.

Figure 14

Plot of the absorption length, ${\mathcal{l}}_a$, from a photonic glass made of PS with $d=1220\mathrm{nm}$. It shows clearly two absorption peaks at $\lambda =875\mathrm{nm}$, $\lambda =730\mathrm{nm}$ related to liquid water absorptions.

Figure 15

Experimental set up to perform dynamical measurements. A ultrashort laser pulse is sent to the sample. A spectrometer collects the pulse temporally spreaded by the sample and sends it to a streak camera.

Figure 16

(Color online) Time-resolved transmission measurements at $\lambda =744$ and $828\mathrm{nm}$ where the different slope of $T\left(t\right)$ gives rise to a minimum $[D\left(744\right)=166{\mathrm{m}}^2∕\mathrm{s}]$ and a maximum $[D\left(828\right)=205{\mathrm{m}}^2∕\mathrm{s}]$ in the diffusion constant. In this case, $L=1120\mu \mathrm{m}$. (Straight lines represent the fit of the experimental measurements with the diffusion equation.)

Figure 17

(Color online) Experimental diffusion constant in a photonic glass made of PS spheres with $d=1220\mathrm{nm}$ obtained by fitting experimental $T\left(t\right)$ curves. Two particular wavelengths are marked with a triangle $({\lambda }_1=744\mathrm{nm})$ and a square $({\lambda }_2=828\mathrm{nm})$.

Figure 18

Experimental values of the energy velocity as a function of the wavelength for a photonic glass made of PS sphere with diameter $d=1220\mathrm{nm}$ and for $\mathrm{Ti}{\mathrm{O}}_2$ powder. For the photonic glass, a full oscillation of around 5% amplitude is shown. The averaged medium group velocity is $0.77c$. For the $\mathrm{Ti}{\mathrm{O}}_2$ powder, flat velocity dispersion is shown, obtained with the same procedure (note the different scales on the $y$ axis).

Figure 19

Plot of the transport mean free path: experimental (stars), theoretical prediction of single-sphere Mie theory [9] for ideal spheres (straight line).

Figure 20

Experimental (stars) and theoretical (straight line) transmission from a PG with spheres $d=550\mathrm{nm}$. The dotted line shows the calculated structure factor $S\left(q\right)$.

Figure 21

Analytical calculations of the transport mean free path for four values of sphere polidispersity, p, and for two different values of sphere refractive index. Sphere diameters are $1\mu \mathrm{m}$ and $\mathrm{ff}=0.5$.