Probing bianisotropic biomolecules via a surface plasmon resonance sensor

Maoyan Wang; Hailong Li; Tong Xu; Hu Zheng; Mengxia Yu; Guiping Li; Jun Xu; Jian Wu

doi:10.1364/OE.26.028277

1. Introduction

Surface plasmon resonance (SPR) [1–13], which is extremely sensitive to the change in the refractive index of the surrounding medium, has been widely exploited in the fields of real-time and label-free monitoring biomolecular interactions and detecting biological analytes. Compared with optical waveguide and optical fiber SPR sensors [3], the Kretschmann configuration surface plasmon resonance sensor [5–7] is simple, easy for fabrication, and the most popular. Liu et al. [6] put forward a SPR phase detection method for measuring the thickness of thin metal films. Luo et al. [7] described a dual-angle technique for simultaneous measurement of refractive index and temperature based on a SPR sensor. However, the surface plasmon resonance conditions of chiral biomolecules are frequently underestimated due to the neglected chirality parameter, which plays an important role in its propagation and biological characteristics.

The chirality (enantiomerism) of biomolecules, such as DNA and proteins, strongly influences their functionalities [14–23]. The detection of the chirality parameter [15–19] is necessary for the enantiomeric purity test, concentration of a chiral sample, biological processes, and so on. Chen et al. [11] have experimentally investigated the enantiospecific detection of the leucine derivative by using the SPR technique. Phan et al. [9] proposed a method for enhanced circular dichroism detection based on a SPR prism coupler and an ellipsometry technique. The cutoff condition and propagation length of surface plasmon polaritons at a chiral-metal interface was analyzed [12,13]. Moreover, Berthod A [15]. indicated that enantiomers have exactly the same properties in isotropic conditions and behave differently only in anisotropic conditions. Nee et al. [17] theoretically investigated the optical scattering depolarization property of a biomedium with anisotropic biomolecules. Whereas, few have studied the surface plasmon resonance affected by the optical activity and bianisotropy of biomolecules so far.

Furthermore, the frequency-domain transfer matrix method (TMM) [24–32] is one of the convenient and fast methods [24–37] to compute electromagnetic characteristics of continuous layered structures [27], even if the structures are dispersive, high-index-contrast, nonlinear, anisotropic and chiral. The Fresnel equations [5–7], which do not incorporate the chirality parameter, are generally used to simulate the surface plasmon resonance biosensor. He S. L [28]. presented an approximation based on eigenvalues and Zarifi et al. [29] proposed the state transition matrix method by introducing the Cayley-Hamilton theorem to study the scattering problem for a stratified bi-anisotropic slab. Ning et al. [30] proposed a hybrid matrix method by applying a thin-layer asymptotic approximation. The technique of the generalized spectral-domain exponential matrix was employed in [31]. The exact solution [24] for a monolayer uniaxial bianisotropic slab has been given. Nevertheless, the previous transfer matrix methods calculating the interaction of waves with multilayered bianisotropic slabs are generally approximate solutions.

Thus, the exact transfer matrix method is developed to probe bianisotropic biomolecules by using a Kretschmann configuration prism surface plasmon resonance sensor. Section II presents a theoretical model of the SPR sensor and gives how to obtain wave vectors, as well as tangential field components expressed in four eigenstates for a bianisotropic medium. Then, the co- and cross-polarized reflectivity of the stratified bianisotropic slab for linearly or circularly polarized incident plane waves are derived. In Section III, the cross-polarized transmission coefficient of a monolayer biisotropic chiral slab is computed to validate the TMM by comparing with the available numerical result. The effects of the bianisotropy, refractive index, chirality parameter, thickness, wavelength, and polarization states of incident waves on the reflectivity of the surface plasmon resonance sensor are investigated. Finally, conclusions are made in Section IV.

2. Theory

2.1 Theoretical model

A surface plasmon resonance based biosensor with chalcogenide (Ge₂₀Ga₅Sb₁₀S₆₅, 2S2G) prism, graphene, and gold multilayer is proposed in this paper to study the behavior of bianisotropic biomolecules. Figure 1 shows the schematic diagram of an experimental setup for the proposed biosensor.

Fig. 1 Schematic diagram of a Kretschmann configuration surface plasmon resonance sensor.

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The 2S2G glass is used as a coupling prism in this paper, because of its high refractive index with large operating window. The dispersive refractive index of the chalcogenide over a broad wavelength can be expressed by [38]

n_{2S2G} (λ) = 2.24047 + 2.693 \times 10^{- 2} / λ^{2} + 8.08 \times 10^{- 3} / λ^{4},

where the incident wavelength λ in this paper is in micrometers.

To improve the absorption of biomolecules and prevent the oxidizable metal from oxidized, the graphene layer is introduced to functionalize the gold (Au) film. The wavelength dependent refractive index of the graphene is given by the following relation [39],

n_{G} = 3.0 + i C_{1} λ / 3,

in which C₁ = 5.446 μm⁻¹. The thickness of a monolayer graphene is 0.34 nm.

Due to high stability and good resistance to corrosion in different environmental conditions, the gold is used. The complex refractive index of the Au layer can be given by the following Drude–Lorentz model [40]

n_{m} = \sqrt{1 - λ^{2} λ_{c} / [λ_{p}^{2} (λ_{c} + i λ)]},

where λ_c = 8.9342 × 10⁻⁶ m and λ_p = 1.6826 × 10⁻⁷ m denote the collision and plasma wavelengths, respectively.

The refractive index and chirality parameter of chiral biomolecules absorbed by the graphene layer are n_s and γ₄, respectively. The refractive index [1] of the water solution is n_t = 1.33.

2.2 Wave vectors for a biaxial bianisotropic medium

We consider a linearly biaxial bianisotropic medium characterized with magnetoelectric coupling constitutive relations [41]

D = \bar{\bar{ε}} E + i \bar{\bar{γ}} H, B = \bar{\bar{μ}} H - i \bar{\bar{γ}} E .

where the permittivity tensor

\bar{\bar{ε}}

, permeability tensor

\bar{\bar{μ}}

, and chirality tensor

\bar{\bar{γ}}

can be represented by 3 × 3 complex-valued matrices in the rectangular coordinate system,

\bar{\bar{ε}} = ε_{0} [\begin{array}{l} ε_{x} 0 0 \\ 0 ε_{y} 0 \\ 0 0 ε_{z} \end{array}], \bar{\bar{μ}} = μ_{0} [\begin{array}{l} μ_{x} 0 0 \\ 0 μ_{y} 0 \\ 0 0 μ_{z} \end{array}], \bar{\bar{γ}} = \sqrt{ε_{0} μ_{0}} [\begin{array}{l} γ_{x} 0 0 \\ 0 γ_{y} 0 \\ 0 0 γ_{z} \end{array}] .

The other two of the Maxwell equations in the frequency domain are expressed as

\nabla \times E = i ω B, \nabla \times H = - i ω D .

We reorganize Eqs. (4) and (6) as

B = (1 / i ω) \nabla \times E, H = (B + i \bar{\bar{γ}} E) / \bar{\bar{μ}},

D = (i / ω) \nabla \times H = \bar{\bar{ε}} E + i \bar{\bar{γ}} H .

By dispersing Eq. (7), one can find:

\begin{array}{l} B_{x} = (\partial E_{z} / \partial y - \partial E_{y} / \partial z) / i ω, H_{x} = (B_{x} + i γ_{x} \sqrt{ε_{0} μ_{0}} E_{x}) / (μ_{x} μ_{0}), \\ B_{y} = (\partial E_{x} / \partial z - \partial E_{z} / \partial x) / i ω, H_{y} = (B_{y} + i γ_{y} \sqrt{ε_{0} μ_{0}} E_{y}) / (μ_{y} μ_{0}), \\ B_{z} = (\partial E_{y} / \partial x - \partial E_{x} / \partial y) / i ω, H_{z} = (B_{z} + i γ_{z} \sqrt{ε_{0} μ_{0}} E_{z}) / (μ_{z} μ_{0}) . \end{array}

If the bianisotropic media is z-stratified and a plane wave propagates in the zx plane, one can get:

\partial / \partial y = 0, \partial / \partial x = i k_{x}, \partial / \partial z = i q,

k_x and q are the x- and z-components of the wavevector, respectively. E_x, E_y, and E_z satisfy the following equations by substituting Eqs. (5), (9), and (10) into dispersive Eq. (8),

\begin{array}{l} [(ε_{x} μ_{y} - γ_{x}^{2} μ_{y} / μ_{x}) ω^{2} / c^{2} - q^{2}] E_{x} - i q (γ_{x} μ_{y} / μ_{x} + γ_{y}) E_{y} ω / c + q k_{x} E_{z} = 0, \\ i q (γ_{y} \frac{μ_{x}}{μ_{y}} + γ_{x}) \frac{ω}{c} E_{x} - i k_{x} (γ_{y} \frac{μ_{x}}{μ_{y}} + γ_{z} \frac{μ_{x}}{μ_{z}}) \frac{ω}{c} E_{z} \\ + [(ε_{y} μ_{x} - γ_{y}^{2} \frac{μ_{x}}{μ_{y}}) \frac{ω^{2}}{c^{2}} - \frac{μ_{x}}{μ_{z}} k_{x}^{2} - q^{2}] E_{y} = 0, \\ q k_{x} E_{x} + i k_{x} (γ_{z} μ_{y} / μ_{z} + γ_{y}) E_{y} ω / c + [(ε_{z} μ_{y} - γ_{z}^{2} μ_{y} / μ_{z}) ω^{2} / c^{2} - k_{x}^{2}] E_{z} = 0. \end{array}

To obtain the non-zero solution of the electric field E, the coefficients of E_x, E_y, and E_z in Eq. (11) have to meet the condition

| \begin{array}{l} (ε_{x} μ_{y} - γ_{x}^{2} \frac{μ_{y}}{μ_{x}}) \frac{ω^{2}}{c^{2}} - q^{2} - i q \frac{ω}{c} (γ_{x} \frac{μ_{y}}{μ_{x}} + γ_{y}) q k_{x} \\ i q \frac{ω}{c} (γ_{y} \frac{μ_{x}}{μ_{y}} + γ_{x}) (ε_{y} μ_{x} - γ_{y}^{2} \frac{μ_{x}}{μ_{y}}) \frac{ω^{2}}{c^{2}} - k_{x}^{2} \frac{μ_{x}}{μ_{z}} - q^{2} - i k_{x} (γ_{y} μ_{x} / μ_{y} + γ_{z} μ_{x} / μ_{z}) ω / c \\ q k_{x} i k_{x} \frac{ω}{c} (γ_{z} \frac{μ_{y}}{μ_{z}} + γ_{y}) (ε_{z} μ_{y} - γ_{z}^{2} μ_{y} / μ_{z}) ω^{2} / c^{2} - k_{x}^{2} \end{array} | = 0,

solutions q₁, q₂, q₃, and q₄ of the quartic Eq. (12) can be obtained.

Four sets of E_x/E_y and E_z/E_y can be further got by solving any two formulas in Eq. (11). The tangential field components E_x, H_x, and H_y expressed in E_y are thus

\begin{array}{l} E_{x} = E_{x y} E_{y}, E_{x y} = E_{x} / E_{y}, H_{x} = \sqrt{ε_{0} / μ_{0}} [i γ_{x} E_{x y} / μ_{x} - c q / (ω μ_{x})] E_{y}, \\ E_{z y} = E_{z} / E_{y}, H_{y} = \sqrt{ε_{0} / μ_{0}} [i γ_{y} / μ_{y} + c (q E_{x y} - k_{x} E_{z y}) / (ω μ_{y})] E_{y} . \end{array}

2.3 TMM for stratified bianisotropic media

As shown in Fig. 2, an s linearly polarized plane wave with a time harmonic variation e⁽⁻ⁱ^ωt⁾ is incident on a stratified bianisotropic chiral slab between two isotropic half-spaces (z<d₀ and z>d_n). The incident angle is θ₀. The material parameters in different regions are illustrated in Fig. 2. The total tangential field components in region 0 and region t are

\begin{array}{l} E_{0 y} = (A_{0} e^{- i q_{0} z} + B_{0} e^{+ i q_{0} z}) e^{i k_{x} x}, H_{0 x} = q_{0} (A_{0} e^{- i q_{0} z} - B_{0} e^{+i q_{0} z}) e^{i k_{x} x} / (ω μ_{s} μ_{0}), \\ E_{0 x} = c_{0} (C_{0} e^{- i q_{0} z} + D_{0} e^{+ i q_{0} z}) e^{i k_{x} x}, H_{0 y} = \sqrt{ε_{s} ε_{0} / (μ_{s} μ_{0})} (- C_{0} e^{- i q_{0} z} + D_{0} e^{+i q_{0} z}) e^{i k_{x} x}, \\ E_{t y} = (A_{t} e^{- i q_{t} z} + B_{t} e^{+ i q_{t} z}) e^{i k_{x} x}, H_{t x} = q_{t} (A_{t} e^{- i q_{t} z} - B_{t} e^{+ i q_{t} z}) e^{i k_{x} x} / (ω μ_{t} μ_{0}), \\ E_{t x} = c_{t} (C_{t} e^{- i q_{t} z} + D_{t} e^{+ i q_{t} z}) e^{i k_{x} x}, H_{t y} = \sqrt{ε_{t} ε_{0} / (μ_{t} μ_{0})} (- C_{t} e^{- i q_{t} z} + D_{t} e^{+ i q_{t} z}) e^{i k_{x} x} . \end{array}

where

\begin{array}{l} n_{0} = \sqrt{ε_{s} μ_{s}}, c_{0} = \cos θ_{0}, n_{t} = \sqrt{ε_{t} μ_{t}}, c_{t} = \cos θ_{t} = \sqrt{1 - {(n_{0} \sin θ_{0} / n_{t})}^{2}}, \\ q_{0}^{2} + k_{x}^{2} = ε_{s} μ_{s} ω^{2} / c^{2}, k_{x} = \sqrt{ε_{s} μ_{s}} \sin θ_{0} ω / c, q_{t} = \sqrt{ε_{t} μ_{t}} \cos θ_{t} ω / c, \end{array}

n₀, n_t, q₀, and q_t are indices of refraction and z components of wavevectors in region 0 and region t, respectively. A₀, B₀, C₀, D₀, A_t, B_t, C_t, and D_t are coefficients of incident, reflected, and transmitted fields, respectively.

Fig. 2 Structure of a planar stratified bianisotropic media.

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Because of multiple reflections, electromagnetic fields inside the bianisotropic medium are made up of four eigenstates [42]. The tangential fields in the lth region are

\begin{array}{l} E_{l y} = i {f^{'}}_{l +} - i {f^{'}}_{l -} + i {b^{'}}_{l +} - i {b^{'}}_{l -}, E_{l x} = E_{x y 1 l} i {f^{'}}_{l +} - E_{x y 2 l} i {f^{'}}_{l -} + E_{x y 3 l} i {b^{'}}_{l +} - E_{x y 4 l} i {b^{'}}_{l -}, \\ H_{l y} = \sqrt{\frac{ε_{0}}{μ_{0}}} {\begin{cases} i \frac{c}{ω μ_{y l}} \times [\begin{array}{l} (q_{1 l} E_{x y 1 l} - k_{x} E_{z y 1 l}) {f^{'}}_{l +} + (k_{x} E_{z y 2 l} - q_{2 l} E_{x y 2 l}) {f^{'}}_{l -} \\ + (q_{3 l} E_{x y 3 l} - k_{x} E_{z y 3 l}) {b^{'}}_{l +} + (k_{x} E_{z y 4 l} - q_{4 l} E_{x y 4 l}) {b^{'}}_{l -} \end{array}] \\ + \frac{γ_{y l}}{μ_{y l}} ({f^{'}}_{l -} - {f^{'}}_{l +} + {b^{'}}_{l -} - {b^{'}}_{l +}) \end{cases}}, \\ H_{l x} = \sqrt{\frac{ε_{0}}{μ_{0}}} [\begin{array}{l} \frac{γ_{x l}}{μ_{x l}} (E_{x y 2 l} {f^{'}}_{l -} - E_{x y 1 l} {f^{'}}_{l+} + E_{x y 4 l} {b^{'}}_{l -} - E_{x y 3 l} {b^{'}}_{l +}) \\ + i \frac{c}{ω μ_{x l}} (q_{2 l} {f^{'}}_{l -} - q_{1 l} {f^{'}}_{l+} + q_{4 l} {b^{'}}_{l -} - q_{3 l} i {b^{'}}_{l +}) \end{array}], \\ {f^{'}}_{l +} = f_{l +} e^{[i(k_{x} x + q_{1 l} z)]}, {f^{'}}_{l -} = f_{l -} e^{[i(k_{x} x + q_{2 l} z)]}, {b^{'}}_{l +} = b_{l +} e^{[i(k_{x} x + q_{3 l} z)]}, {b^{'}}_{l -} = b_{l -} e^{[i(k_{x} x + q_{4 l} z)]} . \end{array}

In Eq. (16), ${f^{'}}_{l \pm}$ and ${b^{'}}_{l \pm}$ denote eigenstates propagating in the positive and negative z directions, respectively. E_xyjl and E_zyjl (j = 1, 2, 3, 4) can be computed with the corresponding q_jl. q₁_l and q₂_l are wavevectors for waves propagating in the + z-direction, whereas q₃_l ( = −q₁_l) and q₄_l ( = −q₂_l) are for waves propagating in the –z-direction.

The transfer matrices can be determined via the application of boundary conditions at z = d₀, z = d_l, and z = d_n as follows,

\begin{array}{l} [\begin{array}{l} A_{0} \\ B_{0} \\ C_{0} \\ D_{0} \end{array}] = V_{10} [\begin{array}{l} f_{1 +} e^{i q_{1}_{1} d_{0}} \\ f_{1 -} e^{i q_{2}_{1} d_{0}} \\ b_{1 +} e^{i q_{3}_{1} d_{0}} \\ b_{1 -} e^{i q_{4}_{1} d_{0}} \end{array}], V_{10} = i {[\begin{array}{l} 1 1 0 0 \\ 0 0 c_{0} c_{0} \\ c_{0} Y_{s} - c_{0} Y_{s} 0 0 \\ 0 0 - Y_{s} Y_{s} \end{array}]}^{- 1} [\begin{array}{l} u_{11}^{1} u_{12}^{1} u_{13}^{1} u_{14}^{1} \\ u_{21}^{1} u_{22}^{1} u_{23}^{1} u_{24}^{1} \\ u_{31}^{1} u_{32}^{1} u_{33}^{1} u_{34}^{1} \\ u_{41}^{1} u_{42}^{1} u_{43}^{1} u_{44}^{1} \end{array}], Y_{s} = \sqrt{\frac{ε_{s}}{μ_{s}}}, \\ u_{1 j}^{l} = (^{- 1) j + 1}, u_{2 j}^{l} = (^{- 1) j + 1} E_{x y j l}, u_{3 j}^{l} = (^{- 1) j + 1} [i γ_{x l} E_{x y j l} / μ_{x l} - c q_{j l} / (ω μ_{x l})], \\ u_{4 j}^{l} = (^{- 1) j + 1} [(c q_{j l} E_{x y j l} - c k_{x} E_{z y j l}) / (ω μ_{y l}) + i γ_{y l} / μ_{y l}], (j = 1, 2, 3, 4) . \end{array}

[\begin{array}{l} f_{l +} e^{i q_{1}_{l} d_{l}} \\ f_{l -} e^{i q_{2}_{l} d_{l}} \\ b_{l +} e^{i q_{3}_{l} d_{l}} \\ b_{l -} e^{i q_{4}_{l} d_{l}} \end{array}] = V_{(l + 1) l} [\begin{array}{l} f_{(l + 1) +} e^{i q_{1}_{(l + 1)} d_{l}} \\ f_{(l + 1) -} e^{i q_{2}_{(l + 1)} d_{l}} \\ b_{(l + 1) +} e^{i q_{3}_{(l + 1)} d_{l}} \\ b_{(l + 1) -} e^{i q_{4}_{(l + 1)} d_{l}} \end{array}], V_{(l + 1) l} = {[\begin{array}{l} u_{11}^{l} u_{12}^{l} u_{13}^{l} u_{14}^{l} \\ u_{21}^{l} u_{22}^{l} u_{23}^{l} u_{24}^{l} \\ u_{31}^{l} u_{32}^{l} u_{33}^{l} u_{34}^{l} \\ u_{41}^{l} u_{42}^{l} u_{43}^{l} u_{44}^{l} \end{array}]}^{- 1} [\begin{array}{l} u_{11}^{l + 1} u_{12}^{l + 1} u_{13}^{l + 1} u_{14}^{l + 1} \\ u_{21}^{l + 1} u_{22}^{l + 1} u_{23}^{l + 1} u_{24}^{l + 1} \\ u_{31}^{l + 1} u_{32}^{l + 1} u_{33}^{l + 1} u_{34}^{l + 1} \\ u_{41}^{l + 1} u_{42}^{l + 1} u_{43}^{l + 1} u_{44}^{l + 1} \end{array}],

\begin{array}{l} [\begin{array}{l} f_{n +} e^{i q_{1 n} d_{n}} \\ f_{n -} e^{i q_{2 n} d_{n}} \\ b_{n +} e^{i q_{3 n} d_{n}} \\ b_{n -} e^{i q_{4 n} d_{n}} \end{array}] = V_{t n} [\begin{array}{l} A_{t} \\ B_{t} \\ C_{t} \\ D_{t} \end{array}], V_{t n} = - i {[\begin{array}{l} u_{11}^{n} u_{12}^{n} u_{13}^{n} u_{14}^{n} \\ u_{21}^{n} u_{22}^{n} u_{23}^{n} u_{24}^{n} \\ u_{31}^{n} u_{32}^{n} u_{33}^{n} u_{34}^{n} \\ u_{41}^{n} u_{42}^{n} u_{43}^{n} u_{44}^{n} \end{array}]}^{- 1} [\begin{array}{l} Q P 0 0 \\ 0 0 c_{t} Q c_{t} P \\ c_{t} Y_{t} Q - c_{t} Y_{t} P 0 0 \\ 0 0 - Y_{t} Q Y_{t} P \end{array}], \\ Y_{t} = \sqrt{ε_{t} / μ_{t}}, P = e^{i q_{t} d_{n}}, Q = e^{- i q_{t} d_{n}} . \end{array}

Then, the relation between column matrices in regions 0 and t for the stratified bianisotropic chiral slab can be simplified as

[\begin{array}{l} A_{0} \\ B_{0} \\ C_{0} \\ D_{0} \end{array}] = V_{t 0} [\begin{array}{l} A_{t} \\ B_{t} \\ C_{t} \\ D_{t} \end{array}] = [\begin{array}{l} v_{11} v_{12} v_{13} v_{14} \\ v_{21} v_{22} v_{23} v_{24} \\ v_{31} v_{32} v_{33} v_{34} \\ v_{41} v_{42} v_{43} v_{44} \end{array}] [\begin{array}{l} A_{t} \\ B_{t} \\ C_{t} \\ D_{t} \end{array}] = V_{10} \cdot V_{21} \cdot \dots V_{(l + 1) l} \cdot \dots V_{n (n - 1)} \cdot V_{t n} [\begin{array}{l} A_{t} \\ B_{t} \\ C_{t} \\ D_{t} \end{array}],

where A₀ = r_ss, B₀ = 1, C₀ = r_sp, D₀ = 0, A_t = 0, B_t = t_ss, C_t = 0, and D_t = t_sp. By solving Eq. (20), we can write the co- and cross-polarized reflection and transmission coefficients for the stratified slab as

r_{s s} = \frac{v_{12} v_{44} - v_{14} v_{42}}{v_{22} v_{44} - v_{24} v_{42}}, r_{s p} = \frac{v_{32} v_{44} - v_{34} v_{42}}{v_{22} v_{44} - v_{24} v_{42}}, t_{s s} = \frac{v_{44}}{v_{22} v_{44} - v_{24} v_{42}}, t_{s p} = \frac{- v_{42}}{v_{22} v_{44} - v_{24} v_{42}} .

If the incident wave is p linear polarization, the coefficients of electromagnetic waves in Eq. (14) become A₀ = r_ps, B₀ = 0, C₀ = r_pp, D₀ = 1, A_t = 0, B_t = t_ps, C_t = 0, and D_t = t_pp, where

r_{p p} = \frac{v_{22} v_{34} - v_{32} v_{24}}{v_{22} v_{44} - v_{24} v_{42}}, r_{p s} = \frac{v_{14} v_{22} - v_{12} v_{24}}{v_{22} v_{44} - v_{24} v_{42}}, t_{p p} = \frac{v_{22}}{v_{22} v_{44} - v_{24} v_{42}}, t_{p s} = \frac{- v_{24}}{v_{22} v_{44} - v_{24} v_{42}} .

The corresponding reflection and transmission coefficients of the positive helicity (p + is)/2^1/2 and negative helicity (p−is)/2^1/2 circularly polarized waves are

\begin{array}{l} r_{+ \pm} = \pm (r_{s s} \mp r_{p p}) /2 - i (r_{s p} \pm r_{p s}) /2, t_{+ \pm} = (t_{p p} \pm t_{s s}) /2 + i (t_{s p} \mp t_{p s}) /2, \\ r_{- \mp} = \pm (r_{s s} \mp r_{p p}) /2 + i (r_{s p} \pm r_{p s}) /2, t_{- \mp} = (t_{p p} \pm t_{s s}) /2 - i (t_{s p} \mp t_{p s}) /2 . \end{array}

The reflectivity R for various polarized light in the structure can be given by [1,5,38]

R_{p p} = | r_{p p} |^{2}, R_{s s} = | r_{s s} |^{2}, R_{+ +} = | r_{+ +} |^{2}, R_{- -} = | r_{- -} |^{2}, R_{p s} = | r_{p s} |^{2} .

3. Numerical results

3.1 Validation example

In order to validate the accuracy of the transfer matrix method developed in this paper, the cross-polarized transmission coefficient t_sp of a monolayer chiral slab with ε_x = ε_y = ε_z = 5, μ_x = μ_y = μ_z = 1, and γ = 0.01 is plotted in Fig. 3. The slab is suspended in free space, that is, ε_s = μ_s = ε_t = μ_t = 1 for the two isotropic half-spaces are chosen. The incident wave is the s polarization and the normalized thickness of the slab is k₀(d₁−d₀) = 10. The ο and dashed line denote results of the vector circuit theory in [20] and the transfer matrix method in this paper, respectively. The cross-polarized transmission coefficients computed with the two methods are in excellent agreement.

Fig. 3 Cross-polarized transmission coefficients of a chiral slab versus the incident angle.

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3.2 Co- and cross-polarized electric field distributions

If there is no special instructions, material parameters considered for the SPR sensor in the following paper are ε_s = n_2s2G × n_2s2G, ε_t = n_t × n_t, ε₁ = n_G × n_G, ε₂ = n_m × n_m, ε₃ = n_G × n_G, ε_x₄ = ε_y₄ = ε_z₄ = n_s × n_s, n_s = 1.33, ${\bar{\bar{μ}}}_{l} =$ I, d₀ = 0, d₁−d₀ = 0.34 nm, d₂−d₁ = 50 nm, d₃−d₂ = 0.34 nm, d₄−d₃ = 100 nm [1], and λ = 632.8 nm. The incident plane wave is the p linear polarization. The smaller the number of graphene layers, the higher the sensitivity of the sensor due to the loss within the graphene layers. Thus, the monolayer graphene is considered in the paper to increase biomolecules adsorption and prevent oxidation.

Figure 4 compares the total electric field distributions of the proposed Kretschmann prism for sensing achiral biomolecules with γ_x₄ = γ_y₄ = γ_z₄ = 0 and bianisotropic biomolecules with γ_x₄ = γ_z₄ = 0.05 and γ_y₄ = 0.1. The wavelength and angle of the incident wave are λ = 632.8 nm and θ = 37.8125° respectively. This electric field enhancement at the gold-graphene interface originates from the surface plasmon resonance. In Fig. 4, the magnitude of the standing wave E_x composed of the incident wave and refletion wave changes varying from −50 nm to 0 nm. Whereas the transmitted |E_x| and |E_y| away from the biomolecules, as well as the reflected |E_y| in the 2S2G prism keep unchanged. By comparing the black solid line and magenta dash line in Fig. 4, one can find that the introduction of the chirality parameter makes the reflected |E_x| decrease and transmitted |E_x| increase. Due to the magnetoelectric coupling effect of bianisotropic biomolecules and the plasmon absorption of the gold thin film, the magnitude of cross-polarized electric field first increases, then decreases varying from 0 nm to 151.02 nm.

Fig. 4 Total electric field distributions of the SPR based biosensor along the z direction perpendicular to the 2S2G prism base for achiral and bianisotropic biomolecules.

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3.3 Effects of bianisotropy and polarization states

Figure 5 shows co- and cross-polarized surface plasmon resonance curves for various biomolecules with different chirality parameters. The solid line, dash line, dotted line, ◦, and • represent the reflectivity for isotropic (γ_x₄ = γ_y₄ = γ_z₄ = 0), biisotropic (γ_x₄ = γ_y₄ = γ_z₄ = 0.15), bianisotropic case I (γ_y₄ = γ_z₄ = 0.15, γ_x₄ = 0.2), case II (γ_x₄ = γ_z₄ = 0.15, γ_y₄ = 0.2), and case III (γ_x₄ = γ_y₄ = 0.15, γ_z₄ = 0.2) biomolecules respectively.

Fig. 5 Co- and cross-polarized reflectivity of the surface plasmon resonance based biosensor versus the angle of incidence for various biomolecules with different chirality parameters. (a) co-polarized, (b) cross-polarized.

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In Fig. 5(a), the inset figure is the zoomed out view with the angle. The resonance angles are 37.5°, 37.6875°, 37.625°, 37.8125°, and 37.6875°, and minimum reflectance are 0.0094, 0.0148, 0.0144, 0.0158, and 0.0157 for isotropic, biisotropic, bianisotropic case I, case II, and case III biomolecules, respectively. It is the relatively large impedance mismatch between graphene and chiral biomolecules, as well as impedance mismatch between chiral biomolecules and water that cause the increase of the co-polarized minimum reflectance.

In Fig. 5(b), resonance angles and maximum reflectance of the cross-polarized reflectivity are 37.5625° and 0.0176 for biisotropic, 37.5° and 0.0185 for bianisotropic case I, 37.6875° and 0.0224 for bianisotropic case II, 37.5625° and 0.0236 for bianisotropic case III biomolecules, respectively. The maximum reflectance of the cross-polarized reflectivity is more easily affected by the z component and less easily affected by the x component of the chirality parameter.

The resonance angles of co-polarized reflectivity are larger than those of cross-polarized reflectivity for the same kind of biomolecules due to the complicated mechanism of the chirality parameter acting on the surface plasmon resonance. The increase of γ_x₄ and γ_y₄ shift the surface plasmon resonance towards smaller and larger resonance angles, respectively. The resonance angles of co- and cross-polarized reflectivity are insensitive to the z component of the chirality parameter.

Figure 6 plots the reflectivity of the SPR sensor for various polarized normally incident plane waves. The chirality parameter of biomolecules is γ_x₄ = γ_z₄ = 0.1 and γ_y₄ = 0.2. There is no surface plasmon resonance phenomenon as the s linearly, positive or negative helicity circularly polarized wave is incident. Both R₊₊ and R_{− −} display a small resonant peak. The difference between R₊₊ and R_{− −} is enhanced around the surface plasmon resonance angle. Most of s and p linearly polarized waves are reflected, whereas less of positive and negative helicity circularly polarized waves are reflected below the SPR angle.

Fig. 6 Reflectivity as a function of the angle for the surface plasmon resonance based biosensor illuminated by different polarized incident waves.

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3.4 Effects of refractive index and chirality parameter

Figure 7 illustrates effects of the refractive index n_s, real and imaginary parts of chirality parameter of bianisotropic biomolecules on the resonance angle of the sensor. The anisotropic chirality parameter are γ_x₄ = γ_z₄ = Im(γ_y₄) = 0 in Fig. 7(a) and γ_x₄ = γ_z₄ = Re(γ_y₄) = 0 in Fig. 7(b), respectively.

Fig. 7 Resonance angles of the SPR sensor versus real and imaginary parts of chirality parameter for various refractive indices. (a) versus Re(γ_y₄), (b) versus Im(γ_y₄).

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The resonance angles for the refractive index n_s from 1.33 to 1.335, the y components of the real and imaginary part of the chirality parameter ranging from 0.01 to 0.2 are shown in Fig. 7. One can find that the resonance angle increases with the increase of Re(γ_y₄) and n_s, or the decrease of Im(γ_y₄).

3.5 Effects of thickness and wavelength

Figure 8 presents the resonance angle and the minimum reflectivity of the SPR biosensor for wavelengths of the incident wave λ = 632.8 nm, 694.3 nm, and 850 nm. The anisotropic chirality parameter for the biomolecules are γ_x₄ = γ_z₄ = 0.05 and γ_y₄ = 0.1. The resonance angle increases and the minimum reflectivity decreases as the wavelength of the incident light decreases. As the thickness of the bianisotropic biomolecular layer increases, the resonance angle and the minimum reflectivity first obviously increase, and then almost keep fixed. The smaller the thickness of biomolecules is, the larger the changes in the resonance angle and the minimum reflectivity of the SPR sensor.

Fig. 8 Resonance angle and minimum reflectivity of the SPR sensor versus the thickness of bianisotropic biomolecules for different wavelengths. (a) resonance angle, (b) minimum reflectivity.

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4. Conclusion

The transfer matrix method is expanded to model effects of the anisotropic chirality parameter of bianisotropic biomolecules on a 2S2G/graphene/gold/graphene sandwich-like sensor in this paper. The transfer matrices connecting four eigenstates in adjacent bianisotropic chiral media are derived. In comparison with the transmitted field through a homogeneous biisotropic slab computed by the vector circuit theory, the validation of the transfer matrix method is confirmed. Numerical results demonstrate that the reflectivity of the biosensor for circularly polarized incident waves are obviously different to those for linearly polarized incident waves and enhanced around the SPR angle due to the magnetoelectric coupling constitute relations of bianisotropic biomolecules. The resonance angle of the SPR biosensor increases with the increase of refractive index, real part of chirality parameter, thickness of bianisotropic biomolecules, and the decrease of imaginary part of chirality parameter and working wavelength. To further distinguish a bianisotropic and an isotropic chiral biomolecules, the genetic algorithm and the transfer matrix method can be employed to obtain the chirality parameter based on experimental data by fabricating and using the proposed Kretschmann SPR sensor. The theoretical work implies that prism surface plasmon resonance sensors may find applications in detecting of chirality parameter of molecules in the solution state, as well as in the solid state.

Funding

Fundamental Research Funds for the Central Universities (ZYGX2015J041, ZYGX2015J039, ZYGX2016J026); National Key Laboratory of Electromagnetic Environment; National Natural Science Foundation of China (41304119, 41104097).

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Probing bianisotropic biomolecules via a surface plasmon resonance sensor

Abstract

1. Introduction

2. Theory

2.1 Theoretical model

2.2 Wave vectors for a biaxial bianisotropic medium

2.3 TMM for stratified bianisotropic media

3. Numerical results

3.1 Validation example

3.2 Co- and cross-polarized electric field distributions

3.3 Effects of bianisotropy and polarization states

3.4 Effects of refractive index and chirality parameter

3.5 Effects of thickness and wavelength

4. Conclusion

Funding

References

Cited By

Figures (8)

Equations (24)

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