rus
Issue #4/2025
A. R. Gainutdinov
The Influence of the Dielectric Particle’s Porosity on the Position of the Mie Resonances
The Influence of the Dielectric Particle’s Porosity on the Position of the Mie Resonances
DOI: 10.22184/1993-7296.FRos.2025.19.4.296.303
The dielectric particles with a high refractive index demonstrate Mie resonances in the
infrared region of spectrum, where the typical absorption bands of many hydrocarbons are
observed. The article provides a numerical determination of the dependence of the
refractive index of a dielectric particle located in a target substance matrix on the particle
porosity value. It is shown that an increased particle porosity leads to a linear decrease in
the particle refractive index and to a linear shift of the Mie resonance to the region of short
wavelengths. An estimate is given for the shift in the spectral range of the Mie resonance
manifestation when the particle porosity value is changed by a few percent.
The dielectric particles with a high refractive index demonstrate Mie resonances in the
infrared region of spectrum, where the typical absorption bands of many hydrocarbons are
observed. The article provides a numerical determination of the dependence of the
refractive index of a dielectric particle located in a target substance matrix on the particle
porosity value. It is shown that an increased particle porosity leads to a linear decrease in
the particle refractive index and to a linear shift of the Mie resonance to the region of short
wavelengths. An estimate is given for the shift in the spectral range of the Mie resonance
manifestation when the particle porosity value is changed by a few percent.
Influence of Dielectric Particle Porosity on the Position of Mie Resonances
A. R. Gainutdinov
Kazan (Volga Region) Federal University, Kazan, Russia
The dielectric particles with a high refractive index demonstrate Mie resonances in the infrared region of spectrum, where the typical absorption bands of many hydrocarbons are observed. The article provides a numerical determination of the dependence of the refractive index of a dielectric particle located in a target substance matrix on the particle porosity value. It is shown that an increased particle porosity leads to a linear decrease in the particle refractive index and to a linear shift of the Mie resonance to the region of short wavelengths. An estimate is given for the shift in the spectral range of the Mie resonance manifestation when the particle porosity value is changed by a few percent.
Keywords: Mie resonances, express detection of low hydrocarbon concentrations, dielectric particles, refractive index, particle porosity, silicon particles
Article received: 22.04.2025
Article accepted: 19.05.2025
INTRODUCTION
One of the urgent problems of up-to-date photonics is the registration of various substances with low concentrations. Each substance has specific absorption bands by which it can be identified. However, at the low substance concentrations, it is necessary to enhance their intensity, but not in the entire spectrum of wavelengths, but only at certain wavelengths relevant to the absorption bands of a specific target substance. Such selective enhancement can be achieved by using dielectric particles with a high refractive index added to the matrix of the material under study. This type of enhancement is explained by the excitation of magnetic and electric Mie-type resonances in the dielectric particles [1–3]. This type of selective enhancement of the optical response is necessary in many applications, for example, in the development of IR spectroscopic sensors for express detection of low hydrocarbon concentrations. The Mie resonances are actively observed and studied in the photonic crystals and metamaterials, as well as in the quantum technologies [4–8].
In the context of hydrocarbon absorption line studies, it is important to understand that the Mie resonances can localize and enhance the electromagnetic field near the particle surface. This enhanced field can significantly increase the interaction of light with the hydrocarbon molecules located near or on the particle surface, thereby enhancing their light absorption. Enhancement of the hydrocarbon absorption by the Mie resonances occurs due to the several mechanisms: an increase in the local light intensity, an increased interaction time of light with the substance, and the coincidence of resonance frequencies with the absorption frequencies of hydrocarbons [9–12]. Enhancement of the hydrocarbon absorption by the Mie resonances offers the challenges for various applications: hydrocarbon sensorics, photocatalysis, improved hydrocarbon adsorption, infrared spectroscopy. However, despite the strong potential, the application of Mie resonances to enhance hydrocarbon absorption is limited due to the fact that the enhancement efficiency significantly depends on the exact match between the Mie resonance frequency of the particles and absorption frequency of the hydrocarbons [13–15]. In this regard, precise control of the size, shape, and refractive index of the particles is required. Thus, the aim of this paper is to study the influence of particle porosity on the position of Mie resonances.
At present, there are many various methods for synthesizing particles. The process of particle generation is comprehensive and multistage [16–22]. The particles obtained during such synthesis have pores that affect the physicochemical properties of both the particle and the material generated on its basis [23–26]. If the particle porosity is not considered, while mistakenly assuming that it is dense (without pores), this fact will affect the refractive index value selected for calculations. Therefore, the Mie resonances will occur at other frequencies that do not correspond to the absorption frequencies of the target substance. Thus, the particles with a certain porosity are required, since the particle porosity affects the physicochemical properties of both the particle and the material generated on its basis. Currently, there are several methods for determining the particle porosity [27–29]. Each of them has its own advantages and disadvantages. For practical applications, an express method for determining the total porosity of particles is required. It shall be simple, non-destructive, and not requiring comprehensive sample preparation. Therefore, an important task is the experimental determination of porosity that was proposed by us in the paper [30].
THEORETICAL ANALYSIS
There are many models describing the relations between the effective refractive index of a medium and the volume fractions of its components [31]. Each of these models corresponds to the certain properties of the components: physical, geometric, aggregate, etc. To describe the effective refractive index of a microparticle, the effective medium approximation (EMA) method was used. This approximation is applied for a model of a statistical isotropic medium where there are no distinguished directions and takes into account the mutual influence of pores [32]. The silicon particles with air-filled pores were considered. Silicon is one of the most accessible substances with a high refractive index. This fact makes it possible to obtain the Mie resonances for the IR region of spectrum. This region of spectrum demonstrates most of the typical absorption bands of many substances, including hydrocarbons. Thus, determination of low hydrocarbon concentrations is an urgent task. This effective medium approximation can be also applied for other substances. In the case of a porous particle, the Bruggeman model (self-consistent model) is the most accurate one. This model is suitable for situations where there is no clearly determined matrix and inclusions, but there is a mixture of two or more phases. This model is described by the Bruggeman formula [32]:
(1 − P) × + P × = 0,
where np is the refractive index of the particle, nsi is the refractive index of silicon and nair is the refractive index of air. This equation is solved numerically in relation to np. Obviously, when P is increased, the contribution of the second term is raised, and in the extreme case at P = 1, the refractive index of the particle np becomes equal to the refractive index of the pores nair. Thus, the particle refractive index depends on its porosity, and determination of dependence of the Mie resonance position on the particle porosity is an extremely important task.
The objects of the theoretical study were silicon particles with a radius of 0.5 μm, illuminated by a broadband light source in the wavelength range of 1–5 μm. The refractive indices of the particle were calculated on the basis of the Bruggeman formula. The particle environment in the simulation was air. The Mie resonance spectra of the dielectric particle were calculated. The Mie resonances were determined for various components of the electromagnetic field: magnetic dipole, electric dipole, magnetic quadrupole, and electric quadrupole. The scattering maxima in the Mie theory are observed under the resonance conditions. The resonances occur when the incident electromagnetic radiation efficiently excites any natural oscillations of the electromagnetic field inside the particle. These oscillations, in turn, lead to the enhanced energy scattering. The scattering cross section is described by the following formula [33]:
Csca = ∑∞n = 1 2n + 1|an|2 + |bn|2, (1)
where k is the wave-number vector, an and bn are the Mie coefficients that depend on the particle radius, the wave-number vector and are expressed through the spherical Bessel and Hankel functions of the first kind. In order for the scattering maximum to be observed, at least one or more of the Mie coefficients (an or bn) shall be maximal. Moreover, the Mie coefficients are functions of the dimensional parameter x = k ∙ r, where r is the particle radius, and the relative refractive index m = where np is the particle refractive index and nenv is the refractive index of the surrounding medium. Near the resonance values x, one or more Mie coefficients can be increased sharply. This leads to a maximum in the scattering cross section.
RESULTS AND DISCUSSION
The dependences of the normalized scattering cross-section on the wavelength for various refractive indices were determined and their graphical illustrations were prepared (Fig. 1). In the Mie scattering theory, the normalized scattering cross-section is a dimensionless value specifying the efficiency of light scattering by a spherical particle in relation to its geometric size. Fig. 1 shows that with an increased refractive index of the particle at a constant refractive index of the medium, the Mie resonances relevant to various harmonics are shifted to the region of longer waves. The air with a refractive index of nenv = 1 was taken as the surrounding medium. The peaks designated by MD correspond to the magnetic-dipole harmonics, and the peaks designated by ED correspond to the electric-dipole harmonics. The weaker peaks in the region of 1.5–2 μm are associated with the resonances of higher harmonics: quadrupole and octupole.
Next, the relations between the particle refractive index and the volume fraction of available pores were calculated. The particles were made of silicon, and the pores were filled with air. The refractive index values of the particle np were found numerically using the formula 1. The porosity was varied in the range from 0 to 50%. The dependency graph of the refractive index np and the particle porosity is shown in Fig. 2.
The graph shows that with the increasing porosity (volume fraction of pores), the effective refractive index of the particle is decreased linearly. Then, the wavelength values relevant to the maxima of light scattering were calculated depending on the particle porosity.
Figure 3 shows the process of linear decrease in the Mie resonance wavelength for various components (magnetic dipole, electric dipole, magnetic quadrupole, and electric quadrupole) when the particle porosity is increasing.
This conclusion is very important for the practical use of particles in the enhanced light scattering method. The availability of pores will deviate the Mie resonances from the target values. When the particle porosity is changed by only 5%, the magnetic dipole peak shifts by 50 nm. For example, for alkanes, in particular, asymmetric and symmetric vibrations of C−H bonds, the absorption peak is at 3 360–3 390 nm and 3 470–3 500 nm, respectively, i. e. the interval for absorption waves is 30 nm [34]. Thus, any change in porosity by only 5% in a particle placed in a matrix of the target substance of interest leads to the fact that the enhanced light scattering will be observed at the wavelengths that do not correspond to the typical absorption bands of the target substances. That is, the absorption region does not fall into the spectral range to identify concentration of the substance of interest.
CONCLUSION
Detection of low substance concentrations is one of the urgent tasks in photonics. Each substance has some typical absorption bands. It is necessary to enhance absorption not in the entire spectrum of wavelengths, but at the certain wavelengths relevant to the absorption bands of a specific substance. The method is required that allows selective light amplification at the wavelengths of interest. This type of selected enhancement of the optical response is needed in many applications, for example, in the development of IR spectroscopic sensors for rapid detection of low hydrocarbon concentrations. One solution to this problem is the use of dielectric particles with a high refractive index. These particles enhance the light scattering at certain wavelengths. The position of Mie resonance depends on many properties of both the particle and its environment. Most particles have pores that affect the overall refractive index of the particle. In this paper, the dependences of refractive index of a dielectric particle and the wavelength of Mie resonances on the dielectric particle porosity are calculated and plotted. It is shown that an increase in the particle porosity leads to a linear decrease in the particle refractive index and to a linear shift of the Mie resonance to the shorter wavelengths. It is indicated that even small changes in the particle porosity lead the Mie resonance shift. This shift results in the enhanced light scattering being observed at the wavelengths that do not correspond to the typical absorption bands of the target substances.
ACKNOWLEDGEMENTS
The author expresses his gratitude to M. Kh. Salakhov, Doctor of Physical and Mathematical Sciences, A. R. Gazizov, Ph.D. in Physical and Mathematical Sciences, and A. I. Garifullin, Ph.D. in Physical and Mathematical Sciences for consultations during the work.
Author
Gainutdinov Azat Radikovich, postgraduate student, Kazan (Volga Region) Federal University; e-mail: azat794@mail.ru; Kazan, Russia.
ORCID: 0009-0000-8711-8711
CONFLICT OF INTEREST
The author confirms that there is no conflict of interest.
A. R. Gainutdinov
Kazan (Volga Region) Federal University, Kazan, Russia
The dielectric particles with a high refractive index demonstrate Mie resonances in the infrared region of spectrum, where the typical absorption bands of many hydrocarbons are observed. The article provides a numerical determination of the dependence of the refractive index of a dielectric particle located in a target substance matrix on the particle porosity value. It is shown that an increased particle porosity leads to a linear decrease in the particle refractive index and to a linear shift of the Mie resonance to the region of short wavelengths. An estimate is given for the shift in the spectral range of the Mie resonance manifestation when the particle porosity value is changed by a few percent.
Keywords: Mie resonances, express detection of low hydrocarbon concentrations, dielectric particles, refractive index, particle porosity, silicon particles
Article received: 22.04.2025
Article accepted: 19.05.2025
INTRODUCTION
One of the urgent problems of up-to-date photonics is the registration of various substances with low concentrations. Each substance has specific absorption bands by which it can be identified. However, at the low substance concentrations, it is necessary to enhance their intensity, but not in the entire spectrum of wavelengths, but only at certain wavelengths relevant to the absorption bands of a specific target substance. Such selective enhancement can be achieved by using dielectric particles with a high refractive index added to the matrix of the material under study. This type of enhancement is explained by the excitation of magnetic and electric Mie-type resonances in the dielectric particles [1–3]. This type of selective enhancement of the optical response is necessary in many applications, for example, in the development of IR spectroscopic sensors for express detection of low hydrocarbon concentrations. The Mie resonances are actively observed and studied in the photonic crystals and metamaterials, as well as in the quantum technologies [4–8].
In the context of hydrocarbon absorption line studies, it is important to understand that the Mie resonances can localize and enhance the electromagnetic field near the particle surface. This enhanced field can significantly increase the interaction of light with the hydrocarbon molecules located near or on the particle surface, thereby enhancing their light absorption. Enhancement of the hydrocarbon absorption by the Mie resonances occurs due to the several mechanisms: an increase in the local light intensity, an increased interaction time of light with the substance, and the coincidence of resonance frequencies with the absorption frequencies of hydrocarbons [9–12]. Enhancement of the hydrocarbon absorption by the Mie resonances offers the challenges for various applications: hydrocarbon sensorics, photocatalysis, improved hydrocarbon adsorption, infrared spectroscopy. However, despite the strong potential, the application of Mie resonances to enhance hydrocarbon absorption is limited due to the fact that the enhancement efficiency significantly depends on the exact match between the Mie resonance frequency of the particles and absorption frequency of the hydrocarbons [13–15]. In this regard, precise control of the size, shape, and refractive index of the particles is required. Thus, the aim of this paper is to study the influence of particle porosity on the position of Mie resonances.
At present, there are many various methods for synthesizing particles. The process of particle generation is comprehensive and multistage [16–22]. The particles obtained during such synthesis have pores that affect the physicochemical properties of both the particle and the material generated on its basis [23–26]. If the particle porosity is not considered, while mistakenly assuming that it is dense (without pores), this fact will affect the refractive index value selected for calculations. Therefore, the Mie resonances will occur at other frequencies that do not correspond to the absorption frequencies of the target substance. Thus, the particles with a certain porosity are required, since the particle porosity affects the physicochemical properties of both the particle and the material generated on its basis. Currently, there are several methods for determining the particle porosity [27–29]. Each of them has its own advantages and disadvantages. For practical applications, an express method for determining the total porosity of particles is required. It shall be simple, non-destructive, and not requiring comprehensive sample preparation. Therefore, an important task is the experimental determination of porosity that was proposed by us in the paper [30].
THEORETICAL ANALYSIS
There are many models describing the relations between the effective refractive index of a medium and the volume fractions of its components [31]. Each of these models corresponds to the certain properties of the components: physical, geometric, aggregate, etc. To describe the effective refractive index of a microparticle, the effective medium approximation (EMA) method was used. This approximation is applied for a model of a statistical isotropic medium where there are no distinguished directions and takes into account the mutual influence of pores [32]. The silicon particles with air-filled pores were considered. Silicon is one of the most accessible substances with a high refractive index. This fact makes it possible to obtain the Mie resonances for the IR region of spectrum. This region of spectrum demonstrates most of the typical absorption bands of many substances, including hydrocarbons. Thus, determination of low hydrocarbon concentrations is an urgent task. This effective medium approximation can be also applied for other substances. In the case of a porous particle, the Bruggeman model (self-consistent model) is the most accurate one. This model is suitable for situations where there is no clearly determined matrix and inclusions, but there is a mixture of two or more phases. This model is described by the Bruggeman formula [32]:
(1 − P) × + P × = 0,
where np is the refractive index of the particle, nsi is the refractive index of silicon and nair is the refractive index of air. This equation is solved numerically in relation to np. Obviously, when P is increased, the contribution of the second term is raised, and in the extreme case at P = 1, the refractive index of the particle np becomes equal to the refractive index of the pores nair. Thus, the particle refractive index depends on its porosity, and determination of dependence of the Mie resonance position on the particle porosity is an extremely important task.
The objects of the theoretical study were silicon particles with a radius of 0.5 μm, illuminated by a broadband light source in the wavelength range of 1–5 μm. The refractive indices of the particle were calculated on the basis of the Bruggeman formula. The particle environment in the simulation was air. The Mie resonance spectra of the dielectric particle were calculated. The Mie resonances were determined for various components of the electromagnetic field: magnetic dipole, electric dipole, magnetic quadrupole, and electric quadrupole. The scattering maxima in the Mie theory are observed under the resonance conditions. The resonances occur when the incident electromagnetic radiation efficiently excites any natural oscillations of the electromagnetic field inside the particle. These oscillations, in turn, lead to the enhanced energy scattering. The scattering cross section is described by the following formula [33]:
Csca = ∑∞n = 1 2n + 1|an|2 + |bn|2, (1)
where k is the wave-number vector, an and bn are the Mie coefficients that depend on the particle radius, the wave-number vector and are expressed through the spherical Bessel and Hankel functions of the first kind. In order for the scattering maximum to be observed, at least one or more of the Mie coefficients (an or bn) shall be maximal. Moreover, the Mie coefficients are functions of the dimensional parameter x = k ∙ r, where r is the particle radius, and the relative refractive index m = where np is the particle refractive index and nenv is the refractive index of the surrounding medium. Near the resonance values x, one or more Mie coefficients can be increased sharply. This leads to a maximum in the scattering cross section.
RESULTS AND DISCUSSION
The dependences of the normalized scattering cross-section on the wavelength for various refractive indices were determined and their graphical illustrations were prepared (Fig. 1). In the Mie scattering theory, the normalized scattering cross-section is a dimensionless value specifying the efficiency of light scattering by a spherical particle in relation to its geometric size. Fig. 1 shows that with an increased refractive index of the particle at a constant refractive index of the medium, the Mie resonances relevant to various harmonics are shifted to the region of longer waves. The air with a refractive index of nenv = 1 was taken as the surrounding medium. The peaks designated by MD correspond to the magnetic-dipole harmonics, and the peaks designated by ED correspond to the electric-dipole harmonics. The weaker peaks in the region of 1.5–2 μm are associated with the resonances of higher harmonics: quadrupole and octupole.
Next, the relations between the particle refractive index and the volume fraction of available pores were calculated. The particles were made of silicon, and the pores were filled with air. The refractive index values of the particle np were found numerically using the formula 1. The porosity was varied in the range from 0 to 50%. The dependency graph of the refractive index np and the particle porosity is shown in Fig. 2.
The graph shows that with the increasing porosity (volume fraction of pores), the effective refractive index of the particle is decreased linearly. Then, the wavelength values relevant to the maxima of light scattering were calculated depending on the particle porosity.
Figure 3 shows the process of linear decrease in the Mie resonance wavelength for various components (magnetic dipole, electric dipole, magnetic quadrupole, and electric quadrupole) when the particle porosity is increasing.
This conclusion is very important for the practical use of particles in the enhanced light scattering method. The availability of pores will deviate the Mie resonances from the target values. When the particle porosity is changed by only 5%, the magnetic dipole peak shifts by 50 nm. For example, for alkanes, in particular, asymmetric and symmetric vibrations of C−H bonds, the absorption peak is at 3 360–3 390 nm and 3 470–3 500 nm, respectively, i. e. the interval for absorption waves is 30 nm [34]. Thus, any change in porosity by only 5% in a particle placed in a matrix of the target substance of interest leads to the fact that the enhanced light scattering will be observed at the wavelengths that do not correspond to the typical absorption bands of the target substances. That is, the absorption region does not fall into the spectral range to identify concentration of the substance of interest.
CONCLUSION
Detection of low substance concentrations is one of the urgent tasks in photonics. Each substance has some typical absorption bands. It is necessary to enhance absorption not in the entire spectrum of wavelengths, but at the certain wavelengths relevant to the absorption bands of a specific substance. The method is required that allows selective light amplification at the wavelengths of interest. This type of selected enhancement of the optical response is needed in many applications, for example, in the development of IR spectroscopic sensors for rapid detection of low hydrocarbon concentrations. One solution to this problem is the use of dielectric particles with a high refractive index. These particles enhance the light scattering at certain wavelengths. The position of Mie resonance depends on many properties of both the particle and its environment. Most particles have pores that affect the overall refractive index of the particle. In this paper, the dependences of refractive index of a dielectric particle and the wavelength of Mie resonances on the dielectric particle porosity are calculated and plotted. It is shown that an increase in the particle porosity leads to a linear decrease in the particle refractive index and to a linear shift of the Mie resonance to the shorter wavelengths. It is indicated that even small changes in the particle porosity lead the Mie resonance shift. This shift results in the enhanced light scattering being observed at the wavelengths that do not correspond to the typical absorption bands of the target substances.
ACKNOWLEDGEMENTS
The author expresses his gratitude to M. Kh. Salakhov, Doctor of Physical and Mathematical Sciences, A. R. Gazizov, Ph.D. in Physical and Mathematical Sciences, and A. I. Garifullin, Ph.D. in Physical and Mathematical Sciences for consultations during the work.
Author
Gainutdinov Azat Radikovich, postgraduate student, Kazan (Volga Region) Federal University; e-mail: azat794@mail.ru; Kazan, Russia.
ORCID: 0009-0000-8711-8711
CONFLICT OF INTEREST
The author confirms that there is no conflict of interest.
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