Strong enhancement of effective refractive index in structured colloids (TiO₂@Silica): Localization of light

Abstract

We use non-resonant Raman scattering to demonstrate a large enhancement of the effective refractive index experienced by Raman photons in a scattering medium comprising spatially-correlated photonic structures of core–shell TiO₂@Silica scatterers mixed with silica nanoparticles and suspended in ethanol. We show that the high refractive index extends outside the physical boundary of the medium, which is attributed to the evanescent contributions of electromagnetic modes that are strongly localized within the medium. Notably, the effective enhancement can be observed even at very low intensities of Raman emission. This anomalous non-linear phenomenon could be explained by the successive polarization of valence electrons to virtual states induced by the strong photon correlations in the strongly localized electromagnetic modes. The enhancement of refractive index and its extension in the vicinity of the medium's interface provide new opportunities for controlling the electromagnetic fields in advanced photonic devices.

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Introduction

The enhancement of the light–matter coupling in dielectric-scattering media, in addition to delving into fundamental concepts of physics, can also present advanced applications.^1–3 In recent years, dramatic progress in the photonics field in engineered-disordered media has been seen.^1,4–7 Engineered or correlated disorder in photonic assemblies can yield anomalous and impactful optical phenomena such as localization of light, enhanced light–matter interaction, unidirectional invisibility and single mode lasing. Several works on these topics have been published elsewhere.^1,8–11 In particular, localization of light in a three-dimensional (3D) dielectric system shows prospects for completely new optical phenomena.^2,3 However, it has proven elusive,^12–16 which has led to extensive debate.¹⁷ Absorption, non-linear phenomena and the near field dipole–dipole coupling were claimed to hamper localization of light. In fact, renormalization of the absorption coefficient (enhanced absorption) in a disordered scattering medium at the mobility edge was first predicted theoretically¹⁸ and later experimentally shown.¹⁹ In our recent works,^1,4,5 we showed that the correlation in the scatterers’ position, due to the Coulomb interaction between scatterers, might give rise to strongly localized electromagnetic modes (localization of light) and associated phenomena, such as strong enhancement of the Raman scattering cross section and very low random lasing threshold. The latter have been observed in correlated colloidal structures composed of core–shell TiO₂@Silica scatterers suspended in ethanol with or without addition of silica nanoparticles.^1,4,5 The silica NPs (20 nm diameter) enhances the particle–particle interaction between the TiO₂@Silica NPs (scatterers). These silica NPs should be localized at the intermediate regions between TiO₂@Silica NPs, due to their higher mobility and the strong Coulomb repulsion exerted on them by the larger TiO₂@Silica NPs (500 nm diameter). In addition, due to the high dielectric permittivity of silica NPs,²⁰ an increase in the solvation layer, effective viscosity and Coulomb potential is expected in the intermediate region between TiO₂@Silica NPs,²¹ favoring the scatterer–scatterer interaction and, therefore, the correlation in their positions. Consequently, silica NPs would act like pivots or bridges for the Coulomb interaction between TiO₂@Silica NPs.

In this work, through non-resonant Raman scattering, strong enhancement of the effective refractive index experienced by Raman photons at the sample input interface is shown in a TiO₂@Silica ethanol suspension with addition of silica NPs, which was suggested^1,4,19,22,23 and inferred^24,25 in our previous works. This strong enhancement of the effective refractive index experienced by Raman photons represents a new kind of non-linear phenomenon, especially considering the low intensities involved in the Raman emission. In previous works, we showed strong enhancement in the Raman cross section (non-resonant)^1,4 and absorption coefficient^19,24,25 near the sample input interface. The enhancement of the non-resonant Raman scattering together with an enhanced absorption should also imply an increase in the effective refractive index (n_S). It should be noted that according to the Kramers–Kronig relationship, an enhanced absorption should be linked to an enhancement of the refractive index. Additionally, an enhancement of the non-resonant Raman scattering should be a consequence of the increase in the polarization of valence electrons to virtual states, which, in turn, leads to an increase in the refractive index. A parallel of this phenomenon in electronic systems was first addressed by Campagnano and Nazarov, who proposed a dynamic barrier at the border of a disordered electronic medium at localization.²⁶

Materials and methods

Core–shell TiO₂@Silica NPs, synthesized by an improved Stöber method,¹⁹ were dispersed in ethanol (HPLC) solution with the TiO₂ core (rutile) with filling fractions (FF_TiO2) of 0.26%, 0.54%, 1.35%, 4.8%, 8.8% and 12.1%. Additionally, silica NPs (20 nm diameter) with filling fractions (FF_SiO2) of 0.75% and 1.5% were added to these TiO₂@Silica NP suspensions. The TiO₂@Silica NPs used in the current experiments are the same as those used in our previous reports.^{1,4,5,19,22–25} TEM, EELS (electron energy loss spectroscopy) mapping (Si) and energy dispersive X-ray fluorescence (ED-XRF) spectroscopy of the core–shell TiO₂@Silica NPs were performed and reported in our previous work.^4,19 The mass percentage ratio (Ti/Si), determined by ED-XRF, was Ti₇₂/Si₂₈, which allows estimating a silica shell thickness of ∼40–45 nm. Diffraction patterns from the TiO₂@Silica suspensions showing the correlation of the scatterers’ position can be found in our previous works.^5,27 Details of the synthesis of TiO₂@Silica NPs can be found in the ESI.†

A Micro-Raman LabRAMAN HR Horiba Scientific with a CW 532 nm laser as an excitation source (50 mW with power fluctuation ± 1%) was used for Raman scattering measurements. Micro-Raman measurements were done using a Leica 50× objective with the numerical aperture NA = 0.55. A schematic diagram of the Raman measurement can be found in our previous works.^1,4 Three arrangements were performed, one without using slides to cover the sample (air) and the other two using either glass (refractive index for BK7 is n_Glass = 1.52) or sapphire (n_Al2O3 = 1.78) slides to cover the sample. For these air, glass and sapphire (slide) arrangements, the pumping spot sizes were 0.8 μm, 1.2 μm and 1.4 μm, respectively. The latter represents pumping power densities of 10 × 10¹⁰ W m⁻², 4.4 × 10¹⁰ W m⁻² and 3.2 × 10¹⁰ W m⁻² for the air, glass and sapphire (slide) arrangements, respectively. For the above three arrangements, the Raman signature of the TiO₂ core (rutile, 410 nm mean diameter) was monitored as a function of FF_TiO2.

Fig. 1 shows illustrations of Raman pumping and collection corresponding to the three setups used for collecting the Raman signal and studying the enhancement of the effective refractive index at the sample–slide interface with air, glass and sapphire as cover slides (from left to right). For pumping and collection of the Raman signal, θ_CAir, θ_CGlass, θ_CAl2O3 cone angles and the depth of focus (d_B(M)) are only determined by the classical refractive indexes of slides, n_CM (n_CAir, n_CGlass, n_CAl2O3) and the numerical aperture (NA) of the pumping-collection objective, NA = 0.55. The pumping-collection depth (d_B(M)) is calculated through , where λ = 532 nm is the pumping wavelength, n_CM is the refractive index of the slide and NA = 0.55 is the numerical aperture of the objective. More details about the Raman measurements can be found in the ESI.†

Fig. 1 Illustrations of Raman pumping and collection using the 50× objective and air, glass and sapphire (Al₂O₃) slides (from left to right). The green and orange arrows represent the pumping and Raman emission, respectively. An enhancement in the effective refractive index of the sample leads to a decrease in the Raman cone angle (θ_SAir, θ_SGlass, θ_SAl2O3) collected by each detection system (air, glass and sapphire). θ_CSAir, θ_CSGlass and θ_CSAl2O3 are the classical Raman cone angles of pumping by each setup and would also be the collected angle if the effective refractive index of the sample was not enhanced. θ_CAir, θ_CGlass and θ_CAl2O3 are the angles of the entry and collection cone (Raman) at the air–sample or air–slide interfaces, which determine the pump and collection depth into the sample (d_B). These angles are insensitive to the dynamics of the refractive indexes of the sample and the slide at the sample–slide interface.

Results

An enhancement in the effective refractive index of the sample (n_S) at the sample–slide interface, experienced by Raman photons, should induce a reduction of the cone angle of the Raman signal collected from the sample (θ_SAir, θ_SGlass, θ_SAl2O3; Fig. 1), due to light refraction at the sample–slide or sample–air interfaces. In turn, an increase of the internal reflection is also expected at these interfaces, which would reduce the Raman signal collected. This n_S (sample) enhancement at the sample–slide interface could be extended to the refractive index of slides (sapphire and glass), due to the interaction with the evanescent wave coming from the electromagnetic modes strongly localized in the sample near the input interface (sample–slide). For the measurement without a slide (air), the enhancement of refractive index of air at the sample–air interface through the interaction with the evanescent wave would not be noticeable, since the electric susceptibility of air (χ_eAir) is close to zero, so the enhanced χ_eAir value would also be close to zero . Therefore, an n_S enhancement for the measurement without a slide (air) should result in a lower collected Raman signal than expected without n_S enhancement, since an enhanced n_S would lead to an increase in the n_S/n_Air contrast at the sample–air interface, inducing a reduction in the Raman collection angle by refraction and an increase in the internal reflection. Following the above arguments, we could demonstrate an increase of the effective refractive index of the sample (n_S) experienced by Raman photons at the sample–slide interface for FF_TiO2 > 0.26%.

The Raman signal, measured experimentally (I_Exp) for each arrangement (air and slides), is plotted as a function of FF_TiO2 for both FF_SiO2 (Fig. 2a and b). As shown for the three arrangements, the Raman signal increases quicker than linearly as the FF_TiO2 is increased. The solid red, black and blue lines represent the expected linear behavior of the Raman signal in FF_TiO2 for the air, glass and sapphire arrangements, respectively.

Fig. 2 Comparison of the Raman signal and Raman enhancements measured with the three setups (sapphire slide, glass slide and air). For the three arrangements, Raman signals for the samples with FF_SiO2 = 0.75% (a) are stronger than those with FF_SiO2 = 1.5% (b). Measured Raman enhancements for the samples with FF_SiO2 = 0.75% (c) and FF_SiO2 = 1.5% (d). The corrected Raman enhancements at FF_SiO2 = 0.75% (e) and FF_SiO2 = 1.5% (f) are higher than two orders for arrangements using glass and sapphire slides. The inset shows a zoomed-in view of the first three FF_TiO2 (0.26, 0.54 and 1.06%).

The strongest Raman signal is collected for the air arrangement. This can be associated with the small spot size and depth of focus (higher numerical aperture) compared to the other arrangements (sapphire and glass slides), leading to an increase in the pumping power density per unit volume. The faster than linear increase in the Raman signal in FF_TiO2 represents a per-particle enhanced Raman scattering cross section.

Conventionally, the Raman signal must be proportional to both the emitter concentration (FF_TiO2) and excitation intensity. The enhancement of the Raman signal or Raman scattering cross section per particle as a function of FF_TiO2, determined by normalizing the Raman signal by FF_TiO2 and by the Raman signal at FF_TiO2 = 0.26%, is illustrated in Fig. 2c and d. Even though the Raman signal is higher for the air arrangement (no slide), the enhancement in the Raman signal is stronger for the glass slide arrangement. This is the consequence of the index matching at the sample–glass interface, which would reduce the collection losses by internal reflection caused by an n_S enhancement at the sample–glass interface. We also note that the experimental setup (50× objective with NA = 0.55) allows a collection depth (d_B(M)) that is strongly sensitive to the pumping arrangement (sapphire and glass slides, and air). Therefore, for a better comparison of the enhancement as a function of FF_TiO2, one needs a correction of the Raman intensity considering the pumping and collection depths, particularly for shorter scattering lengths (<d_B(M)). Thus, the measured Raman signal is corrected for the effective depth of pumping and collection and for the reflection losses at the air–slide or air–sample interfaces as:

In eqn (1), I_{Exp(Al2O3,Glass,Air)} is the corrected Raman signal. d_B(M) and Ref_CM are the depth of pumping and collection for each slide and the reflection coefficient averaged over the pumping and collection cone angle at the air–slide or air–sample interfaces, respectively (see the ESI†).

For the above three setups (air, glass slide and sapphire slide), presented in Fig. 1, the Raman signal for each FF_TiO2 > 0.26%, corrected by the effective depth of pumping and collection as well as by the reflected light at the air–slide or air–sample interfaces (I_{Exp(Al2O3,Glass,Air)}), was normalized by FF_TiO2 and the Raman signal at FF_TiO2 = 0.26% which allows the determination of the corrected Raman enhancement at each FF_TiO2. For the above three setups, the determined-corrected Raman enhancement was plotted as a function of FF_TiO2 as presented in Fig. 2e and f. For the three arrangements and FF_TiO2 < 4.8%, the corrected Raman enhancement at FF_SiO2 = 1.5% is stronger than that at FF_SiO2 = 0.75%. This behavior is reverted for FF_TiO2 ≥ 4.8%, which has been explained in our recent previous work⁴ through an increase of the residual absorption effect associated with the silica NPs (FF_TiO2 ≥ 4.8%). For all FF_TiO2 with both 0.75% and 1.5% FF_SiO2, the lowest Raman enhancement was observed for the measurements without a slide (air), which can be explained through the collection losses due to internal reflection and light refraction at the sample–air interface caused by the enhanced refractive index of the sample near the sample–air interface. Additionally, for FF_TiO2 < 1.35%, the strongest Raman enhancement was observed for the measurements with the sapphire slide. However, for FF_TiO2 ≥ 4.8% with both 0.75% and 1.5% FF_SiO2, the strongest Raman enhancement was achieved by using the glass slide. For FF_TiO2 = 1.35%, Raman enhancements are similar for the measurements with both sapphire and glass slides. This result can be explained by the refractive index matching at the sample–slide (air, glass or sapphire) interfaces, which could be evidence of the dynamics of the effective refractive index in the sample interface. An index matching at the sample–slide interface reduces the collection losses by both the internal reflection and the decrease of the collection cone angle by light refraction at the sample–slide interface.

Next, in order to determine the losses by a possible n_S enhancement, for each FF_TiO2 > 0.26%, a scaling or normalization factor was calculated for the glass slide (I_0.26%(Glass)/I_Exp(Glass)), where I_0.26%(Glass) and I_Exp(Glass) are the corrected Raman signals obtained with the glass slide for FF_TiO2 = 0.26% and FF_TiO2 > 0.26%, respectively. The Raman signals measured at the lowest sample concentration, FF_TiO2 = 0.26%, was taken as a reference because there was no Raman signal enhancement⁴ and presumably there should be no n_S enhancement either. We have taken as a reference the ratio of the Raman signals between FF_TiO2 = 0.26% and FF_TiO2 > 0.26% measured with the glass slide because the classical refractive indexes of the sample (n_CS) and glass slide (n_CGlass) are very close. Therefore, if we preliminarily assume that a similar n_Glass enhancement would also be induced at the sample–glass interface due to the interaction with the evanescent wave, the enhanced n_S and n_Glass at the sample–glass interface would also be very close. Thereby, the Raman signal losses due to the enhancement of the effective refractive indexes at the sample–glass interface (n_S and n_Glass) would be minimal. Clearly, the intensity of the evanescent wave decreases as it penetrates in the glass slide from the sample–slide interface. But as a first approximation, we will consider a similar enhancement of the electrical susceptibility of the glass (χ_eGlass) and the sample (χ_eS). With this normalization factor, we can obtain the expected values of the Raman signal enhancement if there were no losses due to the refractive index enhancement using the equation:

where (I_{RS(Al2O3,Glass,Air)}) values will be used later. For each slide or air, the cone angle of the emitted-collected Raman signal, coming from the sample before refraction at the sample–slide or sample–air interfaces (θ_SM; θ_SAir, θ_SGlass, θ_SAl2O3), must be linked to θ_CM (θ_CAir, θ_CGlass, θ_CAl2O3), as shown in Fig. 1, using the following Snell expression:where n_S and n_M are the effective refractive indexes of the sample and slide at the sample–slide interface, respectively. θ_SM without n_S enhancement (θ_CS) can be determined by evaluating the classical values of refractive indexes of the sample (n_CS) and slide (n_CM) in the above equation (eqn (3)). θ_CS can be determined as:

Thereby, we can estimate a classical value of the cone angle of the collected Raman signal from the sample without n_S enhancement (θ_CS). For FF_TiO2 = 4.8% and FF_SiO2 = 0.75%, if we consider n_CS = 1.44 and θ_CAir ≈ 33.4° (air, without slides), θ_CS can be determined as follows:

This classical θ_CS ≈ 22.5° (without enhancement) is the same for all microscope slides and air at FF_TiO2 = 4.8% and FF_SiO2 = 0.75% and only changes as a function of FF_TiO2.

n _CAir = 1 remains constant even if the effective refractive index of the sample is enhanced (bigger than its classical index, n_S > n_CS). This is because the electric susceptibility of air is very close to zero, so its enhanced electric susceptibility must also be close to zero. However, θ_SAir would be lower than θ_CS, so a decrease in the collected Raman signal is expected. For FF_TiO2 > 0.26%, the I_Exp(Air) values are scaled as described by eqn (2), obtaining I_RS(Air) values. The ratio (I_0.26%(Air)/I_RS(Air)) = F_Air is defined as the deviation factor for the measurement without a slide (air). Note that F_Air is the ratio between the expected enhancement (if there were no losses in the collection) and the measured enhancement of the Raman signal, i.e., F_Air represents the losses of collected Raman intensity (eqn (5)).

Note that (I_Exp(Glass)/I_0.26%(Glass)) represents the expected enhancement of the Raman signal for each FF_TiO2 > 0.26%, as it was preliminarily assumed that there are no Raman collection losses using a glass slide. On the other hand, the collected Raman intensity (I_CR) is proportional to the total emitted Raman signal (I_Te) in the θ_SM cone angle, i.e., I_CR ∝ I_Te(θ_SM) in the θ_SM cone angle. I_Te(θ_SM) can be estimated by considering that the emitted Raman signal (I_eR) is a Lambertian function in the emission angle θ (I_eR(θ) ∝ I_e0cos θ), where I_e0 is a constant that is proportional to the effective pumping intensity.

Integrating it, we have:

In this way, θ_SM for the measurement without a slide (θ_SAir) can be estimated using the following expression:

Applying the Snell equation for this maximum angle of collection coming from the sample (θ_SAir) and assuming n_Air = 1, we obtain:

where θ_CAir = 33.4° is the half collection cone angle (see Fig. 1). In the above consideration, we did not introduce the losses by internal reflection at the sample–air interface (R_SAir), which can be estimated using Fresnel equations considering this enhanced refractive index, n_S, which has been preliminarily determined. 〈R_SAir〉 is the average loss by internal reflection on the collection cone (between 0° and θ_SAir) from the sample. Thereby, the more accurate n_S value can be estimated as follows:

From this n_S value, 〈R_SAir〉 is again determined, and a more accurate n_S value is again calculated by iteration, until n_S tends to reach the same value between iterations. We remark that the n_S value has been determined by considering that, for the measurement with a glass slide, there is no deviation between the expected and measured Raman enhancements, i.e., I_0.26%(Glass)/I_RS(Glass) value is equal to unity, which was justified by assuming a similar enhancement for n_S and n_Glass at the sample–slide interface. But the latter approximation was used in order to estimate a preliminary n_S value.

From this enhanced n_S value, the enhanced value of n_Al2O3, induced by the interaction with the evanescent wave at the sample–sapphire interface, can be estimated through the deviation factor with a sapphire cover slide, F_Al2O3 = (I_0.26%(Al2O3)/I_RS(Al2O3)). θ_SAl2O3 can be expressed by applying a similar equation to eqn (8), substituting F_Air by F_Al2O3:

Applying the Snell equation for this maximum angle of collection from the sample (θ_SAl2O3), we obtain:

where θ_CAl2O3 = 18° is the cone angle of pumping and collection on the sapphire cover slide (see Fig. 1). From this preliminarily calculated n_Al2O3 value, the average losses by internal reflection at the sample–sapphire interface (〈R_SAl2O3〉) can be estimated using Fresnel's equations. Thereby, the more accurate n_Al2O3 value can be estimated as follows:

From this n_Al2O3 value, 〈R_SAl2O3〉 was again determined, and a more accurate n_Al2O3 value was again calculated by iteration, until n_Al2O3 tends to reach the same value between iterations. It is important to remark again that in the latter considerations we assumed as a first approximation that n_Glass at the sample–glass interface undergoes an enhancement similar to that of the sample. However, the effective refractive index and susceptibility of the slide should decrease progressively from the sample–slide interface since the interaction of the evanescent wave decreases as it penetrates in the glass slide. This implies that the effective enhancement of susceptibility of the slide at the sample–glass interface should be less than that of the sample. The effective n_Glass value (enhanced) at the sample–glass interface can be estimated theoretically from the enhanced n_Al2O3 value determined above. Assuming that the classical susceptibilities of glass (χ_eCGlass) and sapphire (χ_eCAl2O3) at the sample–slide interface should be enhanced approximately equally, the enhanced n_Glass can be estimated as:

where χ_eAl2O3 and χ_eGlass are the enhanced susceptibilities of sapphire and glass at the sample–slide interface, respectively. From the enhanced n_Glass value, above calculated, we can estimate the glass deviation factor (F_Glass) for each FF_TiO2 and FF_SiO2, and then apply the Snell equation for this maximum angle of collection from the sample using the glass slide (θ_SGlass) and apply eqn (7) for F_Glass:where θ_CGlass = 21.2° is the collection cone angle on the glass cover slide (see Fig. 1). Note that θ_CS is only determined by FF_TiO2. From the F_Glass values and eqn (2) and (5), new corrected values of and were obtained. Consecutively, n_S, n_Al2O3 and n_Glass values were again calculated (more accurately) from and values. Fig. S3 (ESI†) shows a schematic diagram of the flow of operations to determine the effective refractive indexes of the sample and slides (n_S, n_Al2O3 and n_Glass) at the sample–slide interface.

Fig. 3a and b show the effective refractive indexes (n_S) and electric susceptibilities (χ_eS) of the samples, experienced by emitted Raman photons, at the input sample–slide interface, extracted using the above method, as a function of FF_TiO2, respectively.

Fig. 3 Determination of the effective refractive index and the corresponding electric susceptibilities extracted by the losses of the Raman signals. (a) Strong enhancement of the effective refractive index (n_S) and (b) electric susceptibility (χ_eS) of the sample is observed as FF_TiO2 is increased. Black inverted triangles in (a) represent the classical refractive index (n_CS) and in (b) represent the classical electric susceptibility (χ_CeS) of the sample calculated by Maxwell–Garnett approximation; (c and d) electric susceptibility enhancements of the samples (blue triangles), sapphire (black stars) and glass (red triangles) extracted from measurements using samples with FF_SiO2 = 0.75% (c) and FF_SiO2 = 1.5% (d).

A monotonic n_S and χ_eS increase is observed as FF_TiO2 is increased. For all FF_TiO2, the effective refractive index and electric susceptibility of the samples with FF_SiO2 = 1.5% are higher than those of the samples with FF_SiO2 = 0.75%. This behavior is expected for FF_TiO2 < 4.8%, since Raman enhancement for FF_TiO2 < 4.8% is stronger in the samples with FF_SiO2 = 1.5%, (stronger enhanced Raman signal) in comparison with the samples with FF_SiO2 = 0.75% (Fig. 2). However, for FF_TiO2 ≥ 4.8%, the above behavior is not intuitive since the enhanced Raman signal is larger for the samples with FF_SiO2 = 0.75%. This may be the result of a higher residual absorption in the samples with FF_SiO2 = 1.5% (larger total surface area of SiO₂ NPs),⁴ which for FF_TiO2 ≥ 4.8% would begin to contribute appreciably to the value of the deviation factors used to calculate the effective refractive index. This effect would be particularly pronounced for the F_Air values, due to the higher internal reflection (higher refractive index contrast) at this sample–air interface. For FF_TiO2 ≥ 4.8%, the collected Raman signal can be reduced (mainly in air) by two main causes: (i) the decrease of the collection cone angle due to the refraction of light at the sample–slide interface (approximation used for the refractive index determination) and (ii) a more strongly enhanced residual absorption (particularly, for the samples with FF_SiO2 = 1.5%). The latter hypothesis would imply that the F_Air value includes implicitly additional losses by residual absorption (FF_TiO2 ≥ 4.8%), which means that the F_Air value, extracted from the measurements and used for refractive index determination, would be effectively increased by the residual absorption (mainly for FF_SiO2 = 1.5% and FF_TiO2 ≥ 4.8%). The latter would limit the accuracy of the above method to only a negligible residual absorption.

Fig. 3c and d show the enhancement of the electric susceptibilities (sample, sapphire and glass) experienced by Raman photons and extracted from the measurements in the samples with 0.75% and 1.5% FF_SiO2, respectively. For all FF_TiO2, the enhancements of the effective susceptibilities (sample, sapphire and glass) for FF_SiO2 = 1.5% are stronger than for FF_SiO2 = 0.75%. Again, for the samples with FF_TiO2 ≥ 4.8% and FF_SiO2 = 1.5%, this may be due to a higher loss of the Raman signal given its higher residual absorption. Additionally, for both FF_SiO2, the enhancement of the slide's susceptibilities (sapphire and glass slides) is less than that of the sample's, which can be explained by the decrease of the evanescent wave interaction coming from the sample entering the slides.

The extended enhancement of the effective refractive index to the immediate vicinity of the sample edge through interaction with the evanescent wave coming from the sample represents a phenomenon to highlight that makes it a powerful sensing tool with great potential applications. It should be noted that different from the SERS phenomenon, where strong enhancement of light–matter coupling occurs very close to the nanometallic structure and at a specific wavelength and bandwidth (plasmon resonance),²⁸ the strong enhancement of the effective refractive index, described in this work, is extended to longer distances from the sample surface (extension of the evanescent wave) and possibly to a wider range of wavelengths (see our previous work).⁴ This means that the evanescent wave extension range in the near vicinity of the medium is considerably larger (hundreds of nanometers) than that of SERS (several nanometers).

For a negligible residual absorption, the enhancement factor of the sample's electric susceptibility, G_χe, experienced by the Raman photons is:

where E_Loc is the locally enhanced electromagnetic field resulting from recurrent scattering in effective cavities at the Raman emission frequency (ω_R) and E₀ represents the electromagnetic field of the Raman emission. This is a linear dependence of the susceptibility enhancement, experienced by the Raman photons, with the enhancement of the Raman electromagnetic field trapped in strongly localized electromagnetic modes (effective cavities).

On the other hand, the enhancement factor of the Raman signal (G_R) due to strongly localized electromagnetic field is defined as:

where ω_P and ω_R are the pumping and emission Raman frequencies, respectively. E_Loc is the locally enhanced electromagnetic field resulting from recurrent scattering in effective cavities, at the pumping (ω_P) and emission Raman (ω_R) frequencies. E₀(ω_P) and E₀(ω_R) represent the electromagnetic field of the incoming laser wave (pumping) and the Raman emission, respectively. If ω_P ≈ ω_R, the scattering strength could be considered to be approximately the same for both electromagnetic waves and a fourth power dependence of the Raman signal enhancement is expected. In this way, if similar enhancements of locally confined electromagnetic fields are assumed for pumping (ω_P) and Raman emission (ω_R) frequencies, then (G_χe)⁴ ≈ G_R. Clearly, the coherence of Raman emission is lower than that of the Raman pump laser, which could reduce the confinement for the Raman emission. Consequently, the electromagnetic field enhancement of the Raman emission would be less than that experienced by the Raman pump laser.

For the glass slide arrangement, Fig. 4a and b show a comparison of the Raman enhancement (G_R) and the fourth power of the electrical susceptibility enhancement (G_χe)⁴ for the samples with FF_SiO2 = 0.75% and FF_SiO2 = 1.5%, respectively. For FF_SiO2 = 0.75% (Fig. 4a), where the residual absorption effect should show less influence, a similar monotonic increase of G_R and (G_χe)⁴ in FF_TiO2 (seemingly a power law) is observed. Nevertheless, (G_χe)⁴ increase is slightly lower (lower slope) than G_R increase for FF_TiO2 ≥ 4.8%. The latter could be a consequence of a decrease in light confinement for Raman emission in comparison with the Raman pumping source, due to the low coherence of the Raman emission.

Fig. 4 Comparison of the Raman enhancement measured using a glass slide to cover the samples, G_R (black open squares), and fourth power of the electric susceptibility enhancement, (G_χe)⁴ (red open circles), for the samples with (a) FF_SiO2 = 0.75% and (b) FF_SiO2 = 1.5%.

For FF_SiO2 = 1.5% (Fig. 4b), a similar monotonic increase of G_R and (G_χe)⁴ in FF_TiO2 is also observed. Nevertheless, (G_χe)⁴ values are slightly higher than G_R values, which could be attributed to the effect of residual absorption. Note that the residual absorption should contribute for both a decrease of the Raman enhancement and an increase of the electrical susceptibility enhancement.

It is important to highlight that this enhancement of the effective refractive index at the sample–slide interface, experienced by Raman photons, is experienced for the Raman emission intensity that is much lower than the Raman pumping intensity. Taking into account the nonlinear refractive index coefficient recently reported for rutile films (n₂ = −1.36 × 10⁻¹⁶ m² W⁻¹)²⁹ and the power densities used for pumping (∼10¹¹ W m⁻²), both for a wavelength of 532 nm, the variation of refractive index (Δn rutile) must be around Δn ∼ −10⁻⁵. This further suggests that the observed enhancement of the effective refractive index experienced by the Raman emission (power density several orders lower) represents an anomalous non-linear phenomenon that cannot be explained in the context of the classical nonlinear optical framework, i.e., it would be essentially and conceptually different from a classical non-linear phenomenon.

Contrary to a classical non-linear phenomenon, which is due to the increased probability of simultaneous multi-photon interaction (very strong electromagnetic field), we propose that this anomalous non-linear phenomenon would be induced by the increased probability of sequential photon interactions, due to the strong correlation of the photons in the strongly localized electromagnetic modes. The latter can be understood as successive polarization of valence electrons to virtual states. The considerable benefit of sequential photon interaction is that it emerges even at very low intensity of the incoming electromagnetic field, i.e., there is no threshold for this phenomenon.

Further studies on the enhancement of the Raman signal and refractive index in this correlated-structured scattering system are called for in order to deepen the understanding on these strongly localized electromagnetic modes (localization of light) and associated phenomena.

Conclusion

By studying the Raman signal in this structured scattering medium with three configurations (two different cover slides and air), the effective refractive indexes of the structured medium (sample) and slides (sapphire and glass) experienced by Raman photons at the sample–slide interface were extracted. A strong enhancement of the effective refractive index experienced by the Raman photons is determined. This finding is attributed to the electromagnetic modes strongly localized in this structured scattering medium and the evanescent wave interaction coming from them, whose origin is induced by the correlation in the scatterers’ position. The latter is caused by a strong Coulomb interaction between TiO₂@Silica NPs (scatterers), which is favored by the presence of silica NPs of 20 nm size. The silica NPs act like pivots or bridges for the interaction between TiO₂@Silica NPs, leading to a stronger correlation in the scatterers’ positions which, in turn, favors interferential processes (higher density and Q-factors of strongly localized electromagnetic modes).

The enhancement of the effective refractive index of both sapphire and glass slides, due to the evanescent wave interaction, is weaker than that of the sample, which is explained through the decrease of the evanescent wave interaction coming from the sample entering the slides. This strong enhancement of the effective refractive index, extended to the vicinity of loss-less materials, opens up a number of interesting sensing applications that require extended volumes of interaction and spectral sensitivities, which are way outside the SERS capabilities. A study of the detection sensitivity of molecules immobilized on a solid interface in contact with this structured medium is called for in order to demonstrate possible applications. The findings presented here open an avenue for designing and manufacturing advanced photonics devices.

Author contributions

F.C.M. & E.J.V. synthesized the nanoparticles; J.D., N.U.W. & E.J.V. performed the Raman experiments; J.D., N.U.W., F.C.M., A.D. & E.J.V. analyzed the results and prepared the manuscript; E.J.V. wrote the manuscript and guided the research.

Data availability

Data for this article, including collected Raman signals for each experimental arrangement and concentrations of TiO₂@Silica and silica NPs, are available at the following link: https://drive.google.com/drive/folders/1VqnzGWPZPumwiIMeqFd-SvNBbO1U5WQv?usp=sharing.

Conflicts of interest

The authors declare no conflicts of interest.

Acknowledgements

We gratefully acknowledge financial support from FAPESP (grants 2022/02525-2; 2019/06334-4; 2021/04334-7; 2017/11986-5 and 2017/10765-5), CNPq (grants 313703/2021-3, 435260/2018-9, 465423/2014-0) and IPEN/CNEN 2020.06.IPEN.33.PD. This work was partially funded by the Office of Naval Research under N00014-20-1-2789.

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Footnote

† Electronic supplementary information (ESI) available: Theoretical description of the TiO₂@Silica Coulomb interaction in the colloidal suspension; description of the Raman measurement setups and Raman emission spectra; mathematical description for the Raman signal correction by pumping and collection depth; the effective refractive indexes of the sapphire and glass slides at the sample–slide interface extracted from measurements; schematic description of the flow of operations executed to determine the effective refractive indexes. See DOI: https://doi.org/10.1039/d4nr03626c

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