Eigenmodes of a chiral sphere placed in a dielectric medium were investigated in details. Excitation of these eigenmodes by a plane wave and a chiral molecule radiation was studied both analytically and numerically. It was found that decay rates of “right” and “left” enantiomers are different in the presence of the chiral sphere. Strong dependence of radiation pattern of the chiral molecule placed in the vicinity of the chiral sphere on chirality strength was also demonstrated. An interesting correlation between chirality of sphere and spatial spirality (helicity, vorticity ...) of the electromagnetic fields in the presence of chiral sphere was observed for the first time.
© 2014 Optical Society of America
Chirality is a geometric property of a three-dimensional body not to coincide with its mirror reflection for any shifts and turns . It is important in many areas including molecular biology, analytical chemistry, and pharmaceutics since a variety of organic compounds (such as proteins and amino acids) have chiral properties.
Now chiral media  which can rotate the plane of light polarization, are of great interest in physics. It is due to a rapid advance in creation of metamaterials based on chiral objects [3–6], and due to importance of studying of biomaterials which also have chiral properties.
There are plenty of works considering interaction of electromagnetic waves with chiral objects [7–14]. At present, analytical solutions for a range of different geometries and parameters are available. For example, scattering of a plane wave by homogeneous and heterogeneous chiral spheres is studied in  and , correspondently. General solutions are also obtained for one  or several [10,11] spherical chiral shells. Plane wave interaction with a chiral cylinder and a chiral cylindrical shell are discussed in [12,13], respectively. Scattering of a Hermitian-Gaussian beam by a chiral sphere was considered in .
The influence of a chiral structure on molecules radiation seems also to be very important. Due to molecules’ small size, they can be considered as oscillating with optical frequency (that corresponds to the transition frequency) point electrical dipoles for a nonchiral molecule. A chiral (optically active) molecule can be considered as a combination of electric and magnetic dipoles oscillating with optical frequency [15,16]. For more general type of emitters, one should also take into account quadrupole transition .
Radiation of a point electric dipole situated in the center of a chiral sphere was studied in , where expressions for a field were obtained and it was pointed out that the field outside the sphere can be presented as a superposition of fields radiated by electric and magnetic point dipoles placed in vacuum. An arbitrary position of an electromagnetic source was considered in , where a formal expression for fields was also obtained for source situated both inside and outside the sphere. A case of an electric dipole placed at the center of the sphere was analyzed then and it was stressed that for a particular set of parameters purely circular polarization of the far field radiation can be achieved. Besides, in both articles [18,19] a dependence of the dipole decay rate on the radius of a chiral sphere was examined and it was shown that the decay rate can be increased in some cases. Unfortunately, all specific investigations were focused on the simplest case – an electric dipole situated at the sphere center. In , it was shown that there are two different points outside the chiral sphere, that have very interesting properties: if an electric dipole is placed at one of these points, there will be a right- or left-handed polarized plane wave in the far field. Eigenmodes of a chiral spherical particle with conducting walls are discussed in .
Interaction of biomolecules with chiral structures is of particular interest, because it is natural to expect that it will be very effective. Indeed, it leads to a number of new effects such as dramatic changes of circular dichroism , pure optical separation of enantiomers of biological molecules [23–25], creation of media with negative refractive index [3,26], focusing of radiation of chiral molecules [27,28].
However, despite the existence of formal analytical solutions, optical properties of a chiral sphere were not investigated in details until now due to its complexity. We will try to fill the gap in this work where we present results of a detailed study of optical properties of a chiral sphere excited by a plane wave as well as a chiral molecule. Analysis of eigenmodes will be an important issue because they determine response of a chiral sphere on the external field. Special attention will be paid to investigation of spatial properties of electromagnetic fields in the presence of chiral particles, because interesting interplay between internal chirality of an object and the spatial structure of electromagnetic fields exists.
Despite a large number of works considering the interaction of chiral structures with electromagnetic field, a number of geometries having an exact analytical solution is limited severely. Therefore, it is very important to have a tool for a numerical investigation of electromagnetic fields in the presence of chiral objects of an arbitrary geometry. To solve such a task it was suggested in  to use a modified method of T-matrix. However, to check accuracy of numerical results one should compare them with analytical test models, but we do not know such studies. One of the standards of a numerical simulator is the finite element method (FEM) realized by COMSOL Multiphysics. We have used our own modification of the RF module of COMSOL allowing us to work with chiral objects of arbitrary shape. We have used 4.3b version of COMSOL for our simulation.
In real life, a chiral sphere can be implemented by covering of a sphere made from a usual material with a layer of chiral organic molecules (for example, sugar) or by embedding helical structures to a homogeneous matrix – see Fig. 1. However, for simplicity in this work we will use the description of the sphere medium through effective parameters, that is through permittivity, permeability, and chirality (see Eq. (1)). It should be noted that despite the fact that a sugar shell itself consist of chiral molecules we can consider it as a uniform medium with effective optical properties.
The article has a following structure. In Section 2, it is described what is needed to work with chiral materials in COMSOL Multiphysics. An analytical expressions for eigenmodes (fields and eigenvalues) and a detailed study including their spatial properties are presented in Section 3. In Section 4, interaction of a plane wave with a chiral sphere is considered and results of numerical simulations are compared with analytical solutions. It is also shown that interaction of a plane wave with different types of polarization with a chiral sphere differs significantly. Moreover, existence of a spatial spiral pattern of the electromagnetic field in the near field is observed for the first time. This pattern is similar to one of vortex light and it is possible that in the far field a chiral sphere will produce beam carrying an orbital momentum .
Radiation of optically active molecules (approximated as a combination of oscillating electric and magnetic dipoles) placed near a chiral dielectric sphere is considered in Section 5, where a good agreement between numerical and analytical results is demonstrated. It is very important, that the decay rates of molecules with different type of chirality (“left” and “right” enantiomers) are strongly influenced by the chiral particle and this effect can be used, in principle, for pure optical separation of enantiomers of biological molecules which is very important for current pharmaceutics.
In Section 5, it is also demonstrated that molecule radiation pattern changes substantially in the presence of a chiral sphere and there are several regions of parameters where radiation patterns are qualitatively different. Throughout the paper, all fields are assumed to be monochromatic with time dependence , where is the angular frequency of the electromagnetic wave oscillation and .
2. Modeling chiral particles in COMSOL Multiphysics
As it was already mentioned, a number of analytical solutions for chiral particles is very limited and an effective numerical approach should be elaborated. We use Comsol Multiphysics software for numerical simulation of chiral particles. To describe a chiral medium, in this work we use constitutive equations in the Drude-Born-Fedorov form [31–33]:
The choice of constitutive equations in the form (1) is arbitrary to some extent. Another possibility is to use the so called Boys-Post form:34,35] (see also discussion on constitutive equations in chiral media in [2,36]).
The RF module of the present version of COMSOL Multiphysics , is based on solution of second order differential equations for electric field instead of first order system of Maxwell’s equations. So, Maxwell’s equations for media with constitutive Eq. (1) are reduced to a wave equation for vector:Eq. (3) are not trivial, although they follow from the usual condition of continuity of tangential components of vectors. To take them into consideration, one needs to add surface current on the interface between media:
As a result, one needs to perform following modification to work with chiral objects in COMSOL:
- 1. Replace the wave equation for usual media by that for chiral media Eq. (3).
- 3. Add current (4) to all boundaries.
The proposed method works for chiral objects of an arbitrary shape. In particular, it is confirmed by comparison of numerical and analytical results of present study (see sections 4 and 5).
3. Eigenmodes of a chiral sphere
It is well known that eigenmodes define the physics of phenomena. As far as we know the eigenmodes of a chiral sphere have never been analyzed before, except a simple case of a chiral sphere covered by perfectly conducting walls . In this section, we will find the eigenmodes for a chiral sphere situated in vacuum. The generalization to the case of an arbitrary medium is straightforward.
In (8), is the radius vector, – a scalar spherical harmonic (J = L, R):39], is the associated Legendre function , is the wavenumber in chiral medium. Wavenumbers of left- and right-handed polarized waves have the following form:
In (12), are scalar spherical harmonics:39].
To find the expansion coefficients, one should use the conditions of continuity of tangential components of the electric and magnetic fields on the surface of a chiral spherical particle:39], the prime of a function in (15) and below means the derivative with respect to its argument, J = L, R. Let us note, that the expression for does not depend on the azimuthal quantum number m, but only on the orbital quantum number n. From physical point of view, the condition corresponds to excitation of electromagnetic oscillation (eigenmodes). Due to mathematical complexity of it we study eigenmodes numerically.
The expression for depends on a number of parameters: the sphere radius a, dielectric permittivity and magnetic permeability, the dimensionless parameter of chirality, and the wavenumber which values for different modes we will call. Fixing the parameters, one can find a set of wavenumbers satisfying the Eq. (15). It is important that the solution exist only for a complex value of with a negative imaginary part. With selected dependence on time, it corresponds to a constant pumping of energy into the system and the exponential growth of the mode amplitude at infinity. This paradoxical aspect of open resonators is discussed in detail in . Further, we will consider a nonmagnetic chiral sphere with everywhere.
In Fig. 2, the dependence on and the imaginary part of the wavenumber is shown for parameters, and the fixed real part of the wavenumber determined by the wavelength. One can easily see a set of peaks corresponding to the eigenmode. In order to distinguish them, one should introduce the third quantum number (its physical meaning will be clarified below). Nine peaks in Fig. 2 correspond to nine modes with, .
Let us note that even for a lossless material of the sphere, , the imaginary part of the wavenumber is nonrezo. This is due to presence of radiative losses.
Solving of the eigenmodes problem for fixed in respect to parameters,, is more representative from the physical point of view. The projection of these curves onto plane is shown in Fig. 3 for a chiral spherical particle with and n = 1.
When the chirality parameter approach a critical value (one of denominators in Eq. (10) tends to zero), the real part of also tends to zero. That means that corresponding eigenmodes exist only in quasistatic limit, i.e. when the wavelength of oscillation is much bigger than the sphere radius (further increasing of chirality leads to negative refractive index for one of the circularly polarized waves).
In this area of chirality, Q-factor of modes decreases together with increasing of the radial quantum number. For each point in Fig. 3 (i.e. for each eigenvalue), one can find eigenmodes of a chiral sphere by solving Eq. (14) relative to and substituting found coefficients into Eq. (6) and Eq. (11). In doing so, one should remember that when equations in the Eq. (14) are linearly dependent and one can consider only three of them and choose. As a result, we have:Eq. (15) J = L, R.
In Fig. 4, the spatial structure of the of eigenmodes (see Eq. (11)) on the surface of a chiral sphere with , is shown for quantum numbers , , . This distribution is weakly dependent on the absorption in the sphere. . From this figure, one can clearly see that has a spiral spatial structure where a number of turns is proportional to the orbital number.
4. Plane wave scattering by a chiral sphere
Let us now consider a chiral sphere with the radius, dielectric permittivity, magnetic permeability, and the dimensionless parameter of chirality , surrounded by a vacuum and illuminated by a plane wave. The geometry of the problem is shown in the inset in Fig. 5.
Extinction efficiency was calculated using our modification of COMSOL Multiphysics® (see Section 2) and analytically. Results of calculations of extinction efficiency as a function of the dimensionless parameter of chirality are shown in Fig. 5 for a sphere with and the radius , for different type of polarization – linear (black), right-handed (red), and for left-handed (blue) and the wavelength . First of all, one can see from the figure that numerical results (circles) are in a fine agreement with analytical ones (solid lines).
As one can see from Fig. 5, the difference between incident waves with different type of polarization is significant. Indeed, for the selected constitutive equations Eq. (1), a sphere with a positive value of interacts poorly with a right-hanged wave, while the interaction is strong for a left-handed wave . For negative values of , situation is opposite. A plane wave in turn can be presented as a superposition of left- and right-handed waves, so a corresponding extinction efficiency will be an average between them. Left- and right-handed circularly polarized (counterclockwise and clockwise) waves are defined from the point of view of the receiver.
Main resonances in Fig. 5 correspond to eigen-oscillations with quantum numbers: orbital , azimuthal and radial = 1,2,3,4. As approaches the critical value , each subsequent resonance becomes weaker. As it is known the plane wave can excite only the mode with .
Let us note that a weak resonance corresponding to can also be seen in Fig. 5, and it is marked with a red arrow. If one reduces the imaginary part of dielectric permittivity from 0.04 as in Fig. 5 to 0.0004, the resonance broadening will disappear due to nonradiative losses and it will become strong and narrow.
The electric field component corresponding to main resonances in Fig. 5, is shown in Fig. 6. First of all, this figures is remarkable because it shows the spiral spatial structure of the electric fields. Here, the electric field distribution is determined completely by corresponding eigenmodes.
As it can be seen from Fig. 6, the z component of an electric field in xy-plane is equal to zero for . As the dimensionless parameter of chirality increases, eigenmodes corresponding to peaks in Fig. 5 are excited. Most remarkable features of this figure is that the field structure changes substantially with increasing of chirality: the higher chirality is the higher spiralling of the field is.
It is important that this spiral structure of the electric field still survives outside the sphere. Distribution of corresponding to the main maximum ( in Fig. 5) for a chiral sphere with, , illuminated by a plane wave is shown in Fig. 7 in different planes with . It can be seen that the beam outside sphere has a helical structure again.
Thus, in this section we have analyzed the interaction of a plane wave with a chiral sphere. It was demonstrated that efficiency of extinction increases significantly as the dimensionless parameter of chirality does. Herewith, a right-handed circularly polarized wave has very weak interaction with a chiral sphere with , while a left-handed wave excites eigenmodes with the azimuthal quantum wave number efficiently The strongest excited mode is a chiral dipole mode with . As the mode order increases, the extinction efficiency peak intensity gets down. For a very low imaginary part of , high-Q modes , appear. As the imaginary part increases, modes with disappear.
5. Radiation of a chiral molecule placed near a chiral sphere
Radiation of a chiral molecule in the vicinity of a chiral nonabsorbing sphere was studied analytically in  within nonrelativistic QED. In this case, it is difficult to use the notion of cross sections and usually interaction is described by the molecule decay rate. In the case of a chiral molecule, the normalized (to the decay rate of the same molecule in vacuum) total decay rate can be described by the following form :
It should be noted that the expression (18) corresponds to the transition rate of a molecule from the excited state to the ground one. In this case, and should be considered as matrix elements of the electric and magnetic dipole moments of the selected transition. To take into account the possibility of transition into several states, one should sum corresponding partial radiation rates.
It is important that the equation forthe total decay rate (18) can be obtained in both QED and classical electrodynamics with classical electric and magnetic dipoles instead of corresponding matrix elements [24,42–44].
Geometry of the problem is shown in the inset of Fig. 8. The chiral sphere has dielectric permittivity and the radius. The wavelength is fixed,. The molecule is placed on the z axis at. Having in mind spiral structures of a chiral molecule we will consider “left” and “right” molecules which have different relative orientations of electric and magnetic dipoles: for the “right” molecule they are collinear while for the “left” molecule they have opposite directions . Magnetic dipole amplitude is assumed to be .
Results of calculation of the relative total decay rate versus the dimensionless parameter of chirality χ is shown in Fig. 8 for molecules with different type of chirality (“left” and “right”) and different orientation relative to the sphere (radial and tangential).
It can be seen again that there is a very good agreement between numerical and analytical results. Another important issue is that addition of chirality to a sphere changes drastically total decay rate of the chiral molecule. The spectrum of eigenmodes excited by the molecule is richer than exited by a plane wave. Indeed, it can be shown that a radially oriented molecule excites modes with azimuthal quantum number while a tangentially oriented molecule excites modes with . Moreover, in contrast to a plane wave exciting only modes with orbital quantum number , the chiral molecule can excite modes of a higher order. In the case of rather big absorption, , resonances with different overlap and only the first three of them can be distinguished. So, the first three peaks on each line in Fig. 8 correspond to modes with (), () and ().
As the distance between the chiral molecule and the sphere increases, the interaction gets stronger for modes with and weaker for modes with larger. In the extreme case of the molecule situated in the infinity, its radiation corresponds to a plane wave.
As it can also be seen from Fig. 8, the total decay rate for a radially oriented molecule (where ) is higher than for a tangentially oriented one (where ). It is even more important that there is a significant difference between the decay rates of molecules with different chirality (“left” or “right” molecules) which is placed near the chiral particle. This is a very significant issue and it can be used for detection and separation of enantiomers in the analogous manner as it was proposed for nanoparticles with metamaterial properties [23–25] (see also the end of the section).
It is interesting that increasing chirality of the sphere can also change far-field pattern which is defined as:Eq. (19)) is a total power radiated by the molecule to the far field and is angular distribution of this quantity. For an isotropic emitter, and the radiation pattern is a sphere. For electric dipole situated in a homogeneous medium, and the radiation pattern has a doughnut shape. To find the radiation pattern of a chiral molecule placed in the vicinity of a chiral sphere, we have used far field asymptotes of expressions for fields outside the sphere from .
The radiation pattern for radially oriented molecule (which excite modes with ) is shown in Fig. 9 for three different values of the dimensionless parameter of chirality in the xz plane. The black line corresponds to the absence of chirality () and the far-field pattern is the same as for a molecule in free-space, that is doughnut-shaped. This is due to a small effective size of the sphere. The radiation pattern of the first resonance ( see Fig. 8) is very similar to a nonchiral case. However, the radiation patterns for 2nd (, , red dashed line in Fig. 9) and 3rd (, , blue dash-dot line in Fig. 9) resonances suffer strong modification. Due to axial symmetry, in these cases the radiation pattern has the shape of a doughnut directed to upper or lower half-space depending on . It should be mentioned that difference in the radiation pattern for molecules with different chirality is negligible.
The case of a tangentially oriented molecule (where modes with are excited) is much more interesting. The far-field pattern in this case is shown in Fig. 10 for four values of the dimensionless parameter of chirality : 0, 0.495, 0.563, 0.5925 in two different projections (for better understanding of the shape). The last three values of correspond to the first three peaks in Fig. 8. The colored planes are: z = 0 – red, y = 0 – blue, x = 0 – green. The lower set of projections corresponds to the bottom view of z = 0 plane (from negative z).
From Fig. 10, it can be seen that for , the sphere does not affect much the radiation pattern as it does for a radially oriented molecule. However, as one adds chirality, a few significant changes occur. First, as it follows from the bottom sets of pictures, the radiation pattern rotates as a whole and twists. And second, redistribution of the energy between the upper and lower half-spaces (relative to OZ axis) occurs. For () and (), the most part of the energy goes up (in direction), while for () it goes down (in direction). The maximal difference of the energies flowing up and down corresponds to the third, strongest resonance in Fig. 8 (). In fact, in this case we have a unidirectional radiator, which can be naturally called a chiral Huygens element.
Thus, in this section it was demonstrated that adding the chirality to a dielectric sphere enriches significantly its interaction with a chiral molecule. It was shown that the total decay rate is increasing substantially with the chirality growth. A radially oriented molecule, exciting only eigenmodes with the azimuthal quantum number , interacts with a chiral sphere more effectively than a tangentially oriented molecule, exciting modes with from the point of view of total decay rate (it is higher for radially oriented one). Besides, there is a difference in total decay rate of molecules with different type of chirality (different mutual orientation of electric and magnetic dipole). This leads to a brand new possible applications, such as pure optical separation of enantiomers of chiral molecules.
The key element in this scheme of separation is a reaction chamber, which contains chiral particles. The racemic mixture of enantiomers is placed in this chamber and then excited by one or another way (e.g., photoexcitation). Due to presence of chiral particles one type of optically active enantiomers radiates efficiently and goes to the ground state quickly, while remaining excited enantiomers can be ionized by a resonant field, and then removed from the chamber. Other methods of removal of excited molecules or their decay products are also possible. As a result, the desired pure enantiomer will be accumulated in the chamber. A more detailed description of such procedure in the case of chiral nanoparticles with special (metamaterial) properties can be found in [23–25].
Furthermore, it was shown that not only integral value of radiated power changes (total decay rate) but also its radiation pattern. The effect is higher for tangentially oriented molecules. Besides, the direction of a main lobe of radiation pattern changes from the upper half-sphere (the first and third resonances in Fig. 8) to the lower one (second resonance in Fig. 8).
Thus, in this work eigenmodes of a chiral sphere were investigated in details both analytically and numerically, and a good agreement was found between these approaches. For numerical analyses modifications of commercial software Comsol Multiphysics were developed. It is important that these modifications can be used not only for spherical objects but for an object of an arbitrary shape.
It was shown that increasing chirality of a dielectric sphere enhance Q-factor of eigenmodes significantly. Excitation of these eigenmodes by a plane wave and a chiral molecule is studied. It was demonstrated that increasing of the sphere chirality leads to dramatic enhancement of efficiency of extinction (for the plane wave) and total decay rate of the chiral molecule. Difference in lifetimes of the excited state of chiral molecules with different chirality (left and right enantiomers) was observed. Besides, it was demonstrated that the radiation pattern of the chiral molecule in the vicinity of a chiral sphere changes drastically for different sets of parameter. Finally, an interesting correlation between chirality of sphere and spatial spirality of the fields induced by either radiating molecule or a plane wave was found.
The authors would like to thank the Russian Foundation for Basic Research (grants 11-02-91065, 11-02-92002, 11-02-01272, 12-02-90014 and 14-02-00290), the Presidium of the Russian Academy of Sciences, the Russian Quantum Center and the Skolkovo Foundation for financial supports of the present work.
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