Introduction

In 1968, negative index material (NIM) was first introduced by Veselago1. NIM can have negative permittivity and permeability simultaneously. Pendry et al.2,3,4,5,6 gave a deep discussion and pointed out that a configuration which was called split-ring resonator (SRR)3 could put negative refraction into practice, apart from some particular configurations with non-trivial symmetry breaking7. From then on, negative refraction became a focus in scientific research, for example NIM can be used to fabricate perfect lenses to enhance local field and detection sensitivity8,9,10,11. Two years later, Shelby et al.12 realized NIM experimentally. Metamaterial, made up of SRRs or molecules which consist of SRRs, e.g. extended metal atom chains, becomes a new branch of study13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37. Many new directions are developed, such as electromagnetic cloaking38,39,40,41,42,43, toroidal moment44, liquid crystal magnetic control45, etc. On the other hand, in forensic science, on highly reflective surface, the latent fingermarks are difficult to be observed. The traditional method of visualizing the invisible fingermarks is using fluorescent tag. In Boddis and Russell’s paper46, they made use of antibody-magnetic particle conjugates to visualize them. During this procedure, they find the latent fingermarks of smokers and non-smokers are quite different. As a result, the latent fingermarks of these two kinds of donors are observed to be inverse and thus they can be used to identify the smokers. The research of distinguishing drug users by metabolites becomes a new focus in forensic science field47,48. In this paper, we first introduce negative refraction phenomenon to forensic science. We point out that when we put those latent fingermarks of drug addicts and non-drug users in the light field, they can also be identified. Furthermore, our method is physical and non-damaged, because the latent fingermarks will not be destroyed. More importantly, due to quantum effect, a small volume of molecules could sufficiently respond negatively to the applied electromagnetic fields49. We give the theoretical derivation and calculation of this phenomenon. Our result is not only suitable for smokers but also for drug addicts. In other words, except for cotinine, the metabolite of nicotine, benzoylecgonine and morphine can also be detected using our method.

Results

Tight-binding Approximation and Hückel Model

Many molecules of drug metabolites have a broken ring configuration, i.e. SRR. This structure gives them special optical properties. Without loss of generality, we calculate cotinine, i.e. the metabolite of nicotine, as an example.

Figure 1(a) demonstrates the structure of cotinine molecule. The main part of cotinine is the hexagon part which is called pyridine as shown in Fig. 1(b). In this part, one carbon atom of the ring is substituted with one nitrogen atom. For the sake of simplicity, the remaining part of cotinine molecule is simplified as a methyl in the same plane. This simplification is reasonable because the main contribution to the optical property comes from the π electrons in the conjugate part of cotinine molecule, i.e. single-nitrogen-substituted heterocyclic annulene50 or 3-methylpyridine in Fig. 1(c). Hückel calculation is justifiable and sufficient for the 3-methylpyridine model, though it is a simplified model. Recently, by using a small set of empirical parameters, the Hückel method has been successfully applied to calculate the energy of highest-occupied π orbital and the first π − π* transition energy for a large set of organic molecules with less than 13% deviation. As a consequence, we utilize the Hückel model with the set of empirical parameters to simulate the optical properties of cotinine molecules in our paper. Alternatively, we also notice that by taking σ orbital into account, extended Hückel theory51,52,53,54 may provide better result with much less consuming time as compared to other more accurate method, e.g. time-dependent density function theory.

Figure 1
figure 1

(a) The chemical structure of cotinine molecule (C10H12N2O). Cotinine has two rings, one is pyridyl and the other is pyrrolidine. (b) The simplified model. The molecule is simplified into a pyridyl and a methyl. (c) The spatial distribution of atoms in the simplified model. The origin is set at the center of the hexagon. Seven sites are labeled sequentially. Site 1 is a nitrogen atom and others are carbon atoms.

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π electrons of the pyridyl interact with the electromagnetic fields and therefore result in special optical properties of cotinine. The structure of 3-methylpyridine consists of one pyridyl and one methyl with sites labeled as in Fig. 1(c). Although there are seven electrons in 3-methylpyridine, it has been discovered that one of the electrons is mainly located at the methyl and does not contribute to the current in the ring. Because the magnetic response originates from the circular current, hereafter we shall restrict our calculation to the pyridyl. However, due to the presence of methyl, two carbon-carbon bond lengths have been slightly modified. The problem starts from toluene (methylbenzene) which adds a methyl on a benzene. The quantum dynamics of six π electrons of benzene are described by the Hückel model as55

where j is the site label, denotes the state with a π electron at site j, αj is the site energy and αC = −6.7 eV56. The coupling strength between jth and (j + 1)th sites is given by the resonant integral . Here we use Harrison expression57,58

where is the Planck constant, me is the mass of electron, dij is the bond length and can be attained from NIST56,59. Benzene has four energy levels ε1, ε2, ε3, ε4, which are labeled sequentially from the lowest eigen energy. Both the degeneracies of ε2 and ε3 are two, while ε1 and ε4 are non-degenerate.

For toluene, the resonant integrals of benzene have been modified, because the bond lengths are revised by the presence of the methyl. The bond length of benzene is d = 1.397 A, while toluene have two values, i.e. d1 = 1.394 A and d2 = 1.395 A56,59. The smaller one is the bond length between carbon which is connected to the methyl and its adjacent atoms. The revised resonant integrals are β1 = −2.469 eV and β2 = −2.473 eV. As labeled in Fig. 1(c), the Hückel Hamiltonian of toluene reads

In this case, the energy spectrum of toluene can be solved exactly as

which are schematically shown in Fig. 2. And we do not list the eigen vectors here for simplicity. Because of the methyl, the degeneracy is lifted.

Figure 2
figure 2

The energy spectra of (left) benzene, (middle) toluene and (right) simplified cotinine molecule.

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For simplified cotinine, the Hamiltonian is described as

The nitrogen atom is located at site 1, cf. Fig. 1(c). In this configuration, the site energies and coupling constants are explicitly given as

By diagonalization, the Hückel Hamiltonian (5) can be reexpressed as

where is kth single-electron molecular orbital, εk is the eigen energy.

In order to obtain and εk, we use the perturbation theory in quantum mechanics. We assume that the unperturbed system is a toluene. Then, the perturbation originates from

where is the creation operator on jth site and aj is the annihilation operator. As a consequence, the simplified cotinine molecule has also six energy levels (see Fig. 2) and all of them are non-degenerate. Hereafter, for the sake of simplicity, we further assume αC = 0.

According to the perturbation theory, the energy spectrum of the simplified cotinine molecule reads

Following the degenerate perturbation theory, we can obtain the wave function to the first order. Since their explicit expressions are very complicated, we do not list the analytical expressions of energy and wave function here.

The cotinine molecule has six non-interacting π-electrons. On account of the spin degree (see Fig. 3), the ground state can be expressed in the second-quantization form as

Figure 3
figure 3

Schematic for the single-electron excitation of cotinine molecule (red solid up-arrow) from the ground state (red dashed up-arrow), i.e. with transition energy ΔE3 = E3 − E0 = ε5 − ε3.

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and E0 represents the ground-state energy of the whole cotinine system, i.e.

where is the creation operator of the orbital k with spin σ .

The system has eighteen single-excitation states, for example, the 2th and 3th excited states are

with corresponding eigen energies

respectively. In the first case, the electron with energy ε3 and spin down is excited to energy level ε4 with spin conserved. In the second case, the electron with energy ε3 and spin up is excited to energy level ε5 with spin conserved. Here the flip of electronic spin is not taken into consideration. To sum up, the single-excitation states read

where p = 1, 2, 3, q = 4, 5, 6, and the eigen energies are

In the subspaces spanned by the ground state and single-excitation states, the Hamiltonian without electro-magnetic field reads

Perturbation Theory in Electromagnetic Field

When there is a time-dependent electromagnetic field applied on the molecule, based on the dipole approximation, the total Hamiltonian including the interaction between the molecule and the electromagnetic field can be written as

where and denote the electric and magnetic dipole moments respectively. By assuming the spatial scale of the molecule is much smaller than the wave length of the field since the coordinate is chosen as Fig. 1(c), we have

By a unitary transformation

the Hamiltonian becomes time-independent

where

In other words, we change the system from the static frame into a rotating frame. In the rotating frame, the state and operator become and , respectively. Moreover, due to the interaction with electromagnetic field, the molecular ground state becomes

Permittivity

The electric dipole moment in the rotating frame reads

For the ground state, the expectation value for the dipole operator in the rotating frame is

In the electromagnetic field, the electric displacement field in a volume V with N identical molecules

reads

Thus, the total permittivity in different direction is

The relative dielectric constant of the system, i.e. the permittivity, gives

where is the unit vector of the lab coordinate system.

Because we choose the symmetric center of pyridyl as the origin of coordinate, the electric dipole moment reads

where is the vector of jth electron and −e is the electric charge of an electron. Because ’s are single-electron operators, the matrix elements of electric dipole operators are given by

where is the overlap of between two single-electron wave functions, i.e.

Permeability

To account for the magnetic response of cotinine molecule, we start from the Heisenberg equations of motion,

where we assume ħ = 1.

The magnetic dipole moment is related to the angular momentum of the system. The angular momentum operators read

Obviously, only the response in z direction is present as all atoms in the cotinine molecule are restricted in the xy plane. Therefore, the magnetic dipole moment is

where

Similar to the electric response, the expectation value for the magnetic dipole operator in the rotating frame is

The magnetic induction in a volume V with N identical molecules

is explicitly given by

Notice that μ is the permeability of medium, different from the electric dipole moment above.

The relative permeability of cotinine medium is simplified as

Analysis

Equations (29) and (43) present the analytical results for the relative permittivity and permeability of cotinine molecules in electromagnetic field. According to the expressions of these two quantities, they can be negative simultaneously when the second parts of the expressions are greater than unity. In order to fulfill this requirement, the denominators of the second parts should be small enough. In other words, needs to be much smaller than numerator which means . For a given initial energy of the electron before transition E0, we can observe simultaneous negative permittivity and permeability of cotinine molecules in electromagnetic field when the driving frequency ω is tuned approximately equal to the transition frequency .

Numerical Simulation of Permittivity and Permeability

In the above section, the analytical derivation suggests that relative permittivity and permeability of cotinine molecules might be negative simultaneously in certain frequency regime. Here we show and analyze the numerical result. In the investigated model, the cotinine molecule is simplified as a pyridine and a methyl, c.f. Fig. 1(b). The simplified cotinine model is of two dimension. Thus, we only need to analyze the electromagnetic responses of the molecules in two directions. Figure 4 shows the numerical simulation of relative dielectric constants in the xy plane and relative magnetic permittivity in z direction of the system. Here we assume the site energies αC = 0, αN = αC − 1.2 eV and the coupling strengths βCC = −2.462 eV, revised coupling strengths β1 = −2.469 eV, β2 = −2.473 eV, βCN = −2.676 eV55,57,58,59. The excited-state life time τ = 10 ns is within the range of experimentally observation, e.g. 90 μs60. For a transition to the first excited state, e.g. a spin-up electron is excited from ε3 to ε4, the contributions from transition dipoles μ01 and m01 are much larger than others i.e., . In Fig. 4(a), both relative dielectric constants in the two main axes and are different from unity in the vacuum case, as the presence of nitrogen atom breaks the reflection symmetry along the axis connecting site 3 and the origin. Furthermore, Fig. 4 clearly shows the negative permittivity and permeability at the same time. This result suggests that cotinine molecules can be detected by negative refraction.

Figure 4
figure 4

The numerical results of (a) permittivity and (b) permeability of cotinine molecules vs the light frequency ω. The permittivity (red solid line) along one main axis in the xy plane is negative near the resonance frequency, while (blue dashed line) along the other main axis is always constant. In the magnetic response, the permeability along z direction is shown.

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Discussion

In this paper we research the optical properties of drug metabolites in latent fingermarks. All of these drug metabolites have a structure in common, i.e. SRR which could realize negative refraction. And negative refraction makes the optical properties of latent fingermark quite different between drug addicts and non-drug users and thus can be used to distinguish them. Illuminated by the same incident field, the latent fingermarks of these two kinds of donors may be observed in the different directions with respect to the normal of the interface. The method is to print the donor’s fingermarks on the transparent media and to observe them in the light transmission direction on the opposite side with respect to the side for the normal refraction. In the ordinary case, the refracted light and incident light are on the opposite sides of the normal. However, if the donor is a drug addict, we can detect the refracted light on the same side of incident light with respect to the normal. Although the concentration of drug metabolites may not be evenly distributed in the fingermark, some parts of the fingermark can be detected by negative refraction once the concentrations of drug metabolites in these parts are sufficiently large. Because of negative refraction, the fingermarks of drug addicts can be distinguished from those of non-drug users. Without loss of generality, we take cotinine as an example to calculate electromagnetic response of metabolites in latent fingermarks of smokers. According to our analytic derivation and numerical simulation, we demonstrate the presence of negative refraction in cotinine molecules. The advantage of this method is that it is physical and non-damaged. Our method is suitable for all drug metabolites which have the SRR structure. And this method can also be conveniently applied to distinguish drug addicts and non-drug users. For example, except for cotinine, benzoylecgonine and morphine can also be detected using our method.

Additional Information

How to cite this article: Shen, Y. and Ai, Q. Optical properties of drug metabolites in latent fingermarks. Sci. Rep. 6, 20336; doi: 10.1038/srep20336 (2016).