Refractive index sensing and imaging based on polarization-sensitive graphene

Lixun Sun; Yuquan Zhang; Chonglei Zhang; Yanmeng Dai; Ziqiang Xin; Siwei Zhu; Xiaocong Yuan; Changjun Min; Changjun Min; Yong Yang; Yong Yang

doi:10.1364/OE.27.029273

1. Introduction

Graphene, as a novel two-dimensional optoelectronic material, has raised enormous research interest in the fields of electronics, photonics and plasmonics during the past decade [1–5]. The propagation and manipulation of light on graphene’s surface is strongly influenced by its dynamic conductivity, which is extremely sensitive to exterior electromagnetic field, chemical doping, gate bias voltage, and the material of substrate and upper cladding [6–8]. Owing to the adjustable dynamic conductivity, surface plasmons of both transverse electric (TE) and transverse magnetic (TM) propagation modes with low loss could be realized on graphene from infrared to terahertz (THz) regimes [9–12]. Promising applications, such as graphene opto-electric detector, THz modulator, perfect absorber, as well as graphene/meta-surface structures for super-resolution imaging, have been investigated recently, manifesting huge potential of graphene in electric, optical, and photonic researches [13–20].

Besides the TE and TM propagation modes of graphene, there also exists a distinctive transverse electric absorption (TEA) mode of graphene in high-frequency regime of the electromagnetic spectrum from near infrared to ultraviolet wavelengths [21], where graphene shows stronger confinement and absorption to TE-polarized light as compared to TM-polarized light. The absorption is very sensitive to the variations of ambient refractive index, thus could be used to design high-performance refractive index sensors. The theory of refractive index sensing has been explained based on the assumption that graphene is characterized by an effective complex refractive index with a certain thickness [22,23]. Based on this theory, cell identification and sorting has been reported in a graphene-prism structure at visible frequency with ultrahigh sensitivity [24]. In our previous studies [25,26], label-free refractive index imaging and real-time microscopy of living cells with high spatial resolution are realized in a graphene-based tight focusing system. Moreover, observation of the dynamic evolution of cell nucleolus has been achieved, which could hardly be detected by surface plasmon resonance (SPR) microscopy due to its limited longitudinal detecting range. For graphene-based sensing theory with effective refractive index approach, precise determination of the refractive index values for different layers of graphene would be critical to obtain accurate theoretical prediction. But the measuring value of the refractive index of graphene varies with different experimental conditions, such as temperature, incident wavelength, or substrate type [27–30]. Hence, the measurement difference will bring corresponding deviations for the theoretical calculation. Therefore, the mechanism of graphene-based sensing and imaging needs to be established with systematic analysis and modeling. The limit of such sensing and imaging ability of graphene also needs to be quantitatively investigated for future applications.

In this paper, we theoretically investigate the mechanism of optical refractive index sensing and imaging based on the polarization-sensitive property of graphene using a surface dynamic conductivity approach. The dispersion relations of graphene for TE and TM modes are obtained by solving the Maxwell’s equations under specific boundary conditions, where the atom-thick graphene is characterized by a surface dynamic conductivity with negligible thickness. Then the expanded Fresnel’s formulas considering the surface dynamic conductivity of graphene is derived to quantitatively calculate the absorptances of TE-and TM-polarized incident lights for sensing. Finite difference time domain (FDTD) approaches are used to visualize the polarization-sensitive discrepancy between TE-and TM-polarized focal spots for imaging. Through the numerical results, we demonstrate that the graphene-based sensing method can provide an ultrahigh refractive index resolution of 2.09×10⁻⁸ refractive index unit (RIU) and a sensitivity of 1.09×10⁷ mV/RIU, and the graphene-based imaging method can achieve a high lateral spatial resolution of ∼200 nm and a large longitudinal detecting range of ∼750 nm, manifesting good resolving power for subwavelength structures. This work could contribute to future researches on high performance refractive index sensing and imaging, with potential applications including cell monitoring, subcellular investigation, and other biomedical researches.

2. Electromagnetic modes on graphene surface

Graphene is monolayer of carbon atoms arranged in honeycomb lattices. In electromagnetic calculation, the electromagnetic property of graphene is usually characterized by the surface dynamic conductivity ${\sigma _g}$, and its thickness (∼0.34 nm for monolayer) can be ignored as compared with that of the surrounding dielectrics [21]. Dispersion relations of graphene for incident TE-/TM-polarized modes can be derived with the Maxwell’s equations under boundary conditions including ${\sigma _g}$. The Cartesian coordinate system is established as shown in Figs. 1(a)–1(b), with a graphene layer at z = 0 plane.

Fig. 1. Analysis of the electromagnetic modes on graphene surface. Diagram of dielectric/graphene/dielectric layers under (a) TE and (b) TM mode incidences from the bottom at y = 0 plane, the graphene lies at z = 0 plane. Plots of the dynamic conductivity of monolayer graphene varied with different excitation (c) frequencies and (d) wavelengths.

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For TE-polarized incident light in Fig. 1(a), the electromagnetic field oscillations could be expressed as:

(1)$$\left\{ \begin{array}{l} {\boldsymbol{E}_{y}}(r,t) = {E_y}{e^{ - i\omega t}}\\ {\boldsymbol{H}_{x}}(r,t) = {H_x}{e^{ - i\omega t}}\\ {\boldsymbol{H}_{z}}(r,t) = {H_z}{e^{ - i\omega t}} \end{array} \right.$$

Considering a propagating surface wave that is confined on the surface of graphene and decays rapidly along z-direction, the electric field components ${E_{1y}}$ and ${E_{2y}}$ in nonmagnetic Dielectric 1 (z < 0, ${\varepsilon _1}$) and Dielectric 2 (z > 0, ${\varepsilon _2}$), as well as the corresponding magnetic field components ${H_{1x}}$, ${H_{1z}}$, ${H_{2x}}$, ${H_{2z}}$, are written as:

(2)$$\left\{ \begin{array}{l} {E_{1y}} = {A_1}{e^{i{k_\rho }x}}{e^{{k_{1z}}z}}\\ {H_{1x}} = \frac{{i{k_{1z}}}}{{\omega {\mu_0}}}{E_{1y}}\\ {H_{1z}} = \frac{{{k_\rho }}}{{\omega {\mu_0}}}{E_{1y}} \end{array} \right.$$

and

(3)$$\left\{ \begin{array}{l} {E_{2y}} = {A_2}{e^{i{k_\rho }x}}{e^{ - {k_{2z}}z}}\\ {H_{2x}} = - \frac{{i{k_{2z}}}}{{\omega {\mu_0}}}{E_{2y}}\\ {H_{2z}} = \frac{{{k_\rho }}}{{\omega {\mu_0}}}{E_{2y}} \end{array} \right.$$

respectively. Here, ${A_1}$ and ${A_2}$ are the amplitude factors, ${k_\rho }$ represents the wavevector of the surface wave on graphene surface along x-direction, ${k_{1z}}$ and ${k_{2z}}$ are the wavevectors along z-direction in Dielectric 1 and Dielectric 2, respectively, $\omega$ is the incident angular frequency and ${\mu _0}$ is the vacuum permeability. The magnetic field components are obtained by applying the Maxwell’s equation $\nabla \times \boldsymbol{E} = - \frac{{\partial \boldsymbol{B}}}{{\partial t}}$ and the matter equation $\boldsymbol{B} = \mu \boldsymbol{H}$, where $\boldsymbol{B}$ is the magnetic induction, and $\mu = {\mu _0}$ for nonmagnetic material.

With the introduction of graphene, the boundary condition between two dielectrics at z = 0 turns into:

(4)$$\left\{ \begin{array}{l} \hat{\boldsymbol{n}} \times ({{\boldsymbol{E}_{2}} - {\boldsymbol{E}_{1}}} )= 0\\ \hat{\boldsymbol{n}} \times ({{\boldsymbol{H}_{2}} - {\boldsymbol{H}_{1}}} )= \boldsymbol{j} \end{array} \right.$$

Where, $\boldsymbol{j} = {\boldsymbol {\sigma} _g}{\boldsymbol{E}_{y}}$ is the surface current density introduced by graphene and $\hat{\boldsymbol{n}}$ is the unit vector along z-direction. According to Eq. (4), the boundary condition for TE mode is written as:

(5)$$\left\{ \begin{array}{l} {E_{2y}} - {E_{1y}} = 0\\ {H_{2x}} - {H_{1x}} = {\sigma_g}{E_{1y}} \end{array} \right.$$

On substituting the expressions of ${E_{1y}}$, ${E_{2y}}$, ${H_{1x}}$, ${H_{2x}}$ in Eq. (2) and Eq. (3) into Eq. (5), we can get the relation as follows:

(6)$${k_{1z}} + {k_{2z}} = i\omega {\mu _0}{\sigma _g}$$

By connecting ${\boldsymbol{E}_{y}}$ in Eq. (1) with the wave equation ${\nabla ^2}\boldsymbol{E} - \frac{{{\varepsilon _r}}}{{{c^2}}}\frac{{{\partial ^2}\boldsymbol{E}}}{{\partial {t^2}}} = 0$, the expressions of ${k_{1z}}$ and ${k_{2z}}$ are derived as:

(7)$$\left\{ \begin{array}{l} {k_{1z}} = \sqrt {k_\rho^2 - k_0^2{\varepsilon_1}} \\ {k_{2z}} = \sqrt {k_\rho^2 - k_0^2{\varepsilon_2}} \end{array} \right.$$

where ${k_0} = \frac{\omega }{c}$ is the wavevector of light in vacuum.

Finally, the dispersion equation for TE mode incidence on graphene is obtained with Eq. (6) and Eq. (7) as:

(8)$$\sqrt {k_\rho ^2 - k_0^2{\varepsilon _1}} + \sqrt {k_\rho ^2 - k_0^2{\varepsilon _2}} = i\omega {\mu _0}{\sigma _g}$$

Similarly, for TM-polarized incident light in Fig. 1(b), the electromagnetic field oscillations of ${\boldsymbol{H}_{y}}$, ${\boldsymbol{E}_{x}}$, and ${\boldsymbol{E}_{z}}$ need to be considered with boundary conditions ${E_{2x}} - {E_{1x}} = 0$ and ${H_{1y}} - {H_{2y}} = {\sigma _g}{E_x}$. Then the dispersion equation for TM mode can be derived as:

(9)$$\frac{{{\varepsilon _1}}}{{\sqrt {k_\rho ^2 - k_0^2{\varepsilon _1}} }} + \frac{{{\varepsilon _2}}}{{\sqrt {k_\rho ^2 - k_0^2{\varepsilon _2}} }} = - \frac{{i{\sigma _g}}}{{\omega {\varepsilon _0}}}$$

From Eq. (8) and Eq. (9), we find that the complex dynamic conductivity ${\sigma _g}$ plays a key role in the behavior of light on graphene surface. The expression of ${\sigma _g}$ for monolayer graphene could be determined from the Kubo formula, which consists of both intraband and interband contributions [6,9,21,31–33]:

(10)$$\begin{aligned} {\sigma _g} & =\sigma _g^{\prime} + i\sigma _g^{{\prime}{\prime}} = {\sigma _{intra}} + {\sigma _{inter}}\\ & =\frac{{ - i{e^2}}}{{\pi {\hbar ^2}(\omega + i2\Gamma )}}\int\limits_0^\infty {\xi \left( {\frac{{\partial {f_d}(\xi )}}{{\partial \xi }} - \frac{{\partial {f_d}( - \xi )}}{{\partial \xi }}} \right)} d\xi + \frac{{i{e^2}(\omega + i2\Gamma )}}{{\pi {\hbar ^2}}}\int\limits_0^\infty {\frac{{{f_d}( - \xi ) - {f_d}(\xi )}}{{{{(\omega + i2\Gamma )}^2} - 4{{(\xi /\hbar )}^2}}}d\xi } \end{aligned}$$

where e, $\hbar$ and $\Gamma $ denote the electron charge, reduced Planck’s constant and scattering rate, respectively, $\omega = 2\pi f = 2\pi c/\lambda$ is the angular frequency, ${f_d}(\xi ) = {[{{e^{(\xi - {\mu_c})/{k_B}T}} + 1} ]^{ - 1}}$ is the Fermi-Dirac distribution, with ${\mu _c}$ the chemical potential, ${k_B}$ the Boltzmann constant, and T the Kelvin temperature. For most simulation and experimental conditions, it satisfies ${k_B}T \ll |{{\mu_c}} |$ [6,10,21,34], so Eq. (10) could be simplified to:

(11)$${\sigma _g} = {\sigma _{intra}} + {\sigma _{inter}} = \frac{{i{e^2}{\mu _c}}}{{\pi {\hbar ^2}(\omega + i2\Gamma )}} + \frac{{i{e^2}}}{{4\pi \hbar }}\ln \left[ {\frac{{2|{{\mu_c}} |- (\omega + i\Gamma )\hbar }}{{2|{{\mu_c}} |+ (\omega + i\Gamma )\hbar }}} \right]$$

Based on Eq. (11), the relation curves of the dynamic conductivity ${\sigma _g}$ for monolayer graphene varied with different incident frequencies and wavelengths are plotted by setting $\Gamma = 0.2 \times {10^{11}}{\raise0.5ex\hbox{$\scriptstyle \textrm{1}$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle \textrm{s}$}}$, $T = 300\textrm{K}$, ${\mu _c} = 0.2\textrm{eV}$ (values of these parameters are chosen for the graphene/glass-substrate structure at room temperature in the experiment [6,21]), as shown in Fig. 1(c) and Fig. 1(d). For the analysis of multilayer graphene in semi-classical model, the band effects due to interlayer coupling are usually ignored and graphene of a few layers could be considered as a stack of monolayer graphene [21,35–37]. Therefore, the dynamic conductivity of N-layer graphene $\sigma _g^N$ can be expressed as the product of ${\sigma _g}$ and a conductivity scaling N that indicates the layer numbers:

(12)$$\sigma _g^N = N{\sigma _g}$$

From Eq. (8), Eq. (9) and the results in Figs. 1(c)–1(d), we find that graphene could support three kinds of electromagnetic modes through the frequency spectrum of the electromagnetic waves: transverse magnetic propagation (TMP) mode, transverse electric propagation (TEP) mode, and transverse electric absorption (TEA) mode. At lower frequency range (green region, $\hbar \omega /{\mu _c} < 1.667$), the real part of ${\sigma _g}$ is zero and the imaginary part is positive. According to Eq. (9), the wavevector ${k_\rho }$ has real solutions, so graphene works in the TMP mode. At middle frequency range (red region, $1.667 < \hbar \omega /{\mu _c} < 2$), the real part of ${\sigma _g}$ is still zero, but the imaginary part becomes negative, hence ${k_\rho }$ has real solutions in Eq. (8), and supports the TEP mode. The other TEA mode exists at the higher frequency range (blue region, $\hbar \omega /{\mu _c} > 2$), where the imaginary part of ${\sigma _g}$ is negative, indicating it works in TE mode. While the real part of ${\sigma _g}$ is positive, which means ${k_\rho }$ would be complex and have an imaginary part according to Eq. (8), thus it corresponds to the TEA mode. Note that the increasement of chemical potential ${\mu _c}$ lead to blueshift of the angular frequency $\omega $, thus the frequency regions of TMP, TEP, and TEA modes will change accordingly. Considering that 532 nm laser is both commercially available and biocompatible, we choose this wavelength to perform the following calculation and simulation. The dynamic conductivity of graphene at 532 nm wavelength turns out to be ${\sigma _g} = 60.853 - 0.078i$ in Fig. 1(d). We will show that high incident frequency with the TEA mode is suitable for both refractive index sensing with high sensitivity and imaging with high spatial resolution.

3. Absorption of graphene with oblique incidence of light for refractive index sensing

Next, we use the above surface dynamic conductivity of graphene to calculate the reflectance, transmittance and absorptance of light in the dielectric/graphene/dielectric structure shown in Fig. 1. The incident wavelength $\lambda $ is 532 nm and the corresponding dynamic conductivity for monolayer graphene in the TEA mode is ${\sigma _g} = 60.853 - 0.078i$. The refractive indices of lower and upper dielectrics (${n_1}$ and ${n_2}$) are 1.53 and 1.33 respectively, where the aquatic environment is considered in upper layer on a glass prism substrate for potential biomedical applications. The dispersion and loss of upper and lower media are assumed to be negligible. Let us consider the incident and reflected electric fields (${E_1}$ and $E_1^{\prime}$) of light propagating in y = 0 plane in lower dielectric (${n_1}$) and the transmitted electric field (${E_2}$) propagating in upper dielectric (${n_2}$). For TE-polarized light, the electric fields oscillate in y-direction, ${E_1}$, $E_1^{\prime}$, and ${E_2}$ are expressed as:

(13)$$\left\{ \begin{array}{l} {E_{1y}} = {A_1}{e^{i({k_{1x}}x + {k_{1z}}z)}}\\ E_{1y}^{\prime} = A_1^{\prime}{e^{i(k_{1x}^{\prime}x + k_{1z}^{\prime}z)}}\\ {E_{2y}} = {A_2}{e^{i({k_{2x}}x + {k_{2z}}z)}} \end{array} \right.$$

Where the temporal term ${e^{ - i\omega t}}$ has been omitted. ${A_1}$, $A_1^{\prime}$, and ${A_2}$ are the amplitude factors of the incident, reflected and transmitted light. ${k_{1x}} = k_{1x}^{\prime} = {k_0}{n_1}\sin {\theta _1}$ and ${k_{2x}} = {k_0}{n_2}\sin {\theta _2}$ are the wavevectors along x-direction with ${\theta _1}$ and ${\theta _2}$ the incident and refracted angles, respectively. ${k_{1z}} = - k_{1z}^{\prime} = {k_0}{n_1}\cos {\theta _1}$, ${k_{2z}} = {k_0}{n_2}\cos {\theta _2}$ are the wavevectors along z-direction. The corresponding magnetic field oscillations in x-direction are expressed as:

(14)$$\left\{ \begin{array}{l} {H_{1x}} = - \frac{{{k_{1z}}}}{{\omega {\mu_0}}}{A_1}{e^{i({k_{1x}}x + {k_{1z}}z)}}\\ H_{1x}^{\prime} = - \frac{{k_{1z}^{\prime}}}{{\omega {\mu_0}}}A_1^{\prime}{e^{i(k_{1x}^{\prime}x + k_{1z}^{\prime}z)}}\\ {H_{2x}} = - \frac{{{k_{2z}}}}{{\omega {\mu_0}}}{A_2}{e^{i({k_{2x}}x + {k_{2z}}z)}} \end{array} \right.$$

Based on Eq. (13) and Eq. (14) together with the boundary condition ${E_{1y}} + E_{1y}^{\prime} = {E_{2y}}$, and ${H_{2x}} - ({H_{1x}} + H_{1x}^{\prime}) = {\sigma _g}{E_{2y}}$ at z = 0, we get the following relations as:

(15)$$\left\{ \begin{array}{l} {A_1} + A_1^{\prime} = {A_2}\\ {n_1}\cos {\theta_1}({A_1} - A_1^{\prime}) = ({n_2}\cos {\theta_2} + c{\mu_0}{\sigma_g}){A_2} \end{array} \right.$$

with c the light velocity in vacuum. Therefore, the reflectance ${r_s}$, transmittance ${t_s}$ and absorptance ${a_s}$ for TE-polarized light at the dielectric/graphene/dielectric interface could be obtained as:

(16)$$\left\{ \begin{array}{l} {r_s} = {\left|{\frac{{A_1^{\prime}}}{{{A_1}}}} \right|^2} = {\left|{\frac{{{n_1}\cos {\theta_1} - {n_2}\cos {\theta_2} - c{\mu_0}{\sigma_g}}}{{{n_1}\cos {\theta_1} + {n_2}\cos {\theta_2} + c{\mu_0}{\sigma_g}}}} \right|^2}\\ {t_s} = \frac{{{n_2}\cos {\theta_2}}}{{{n_1}\cos {\theta_1}}}{\left|{\frac{{{A_2}}}{{{A_1}}}} \right|^2} = \frac{{{n_2}\cos {\theta_2}}}{{{n_1}\cos {\theta_1}}}{\left|{\frac{{2{n_1}\cos {\theta_1}}}{{{n_1}\cos {\theta_1} + {n_2}\cos {\theta_2} + c{\mu_0}{\sigma_g}}}} \right|^2}\\ {a_s} = 1 - {r_s} - {t_s} \end{array} \right.$$

In a similar way, the reflectance ${r_p}$, transmittance ${t_p}$ and absorptance ${a_p}$ for TM-polarized light can be derived as:

(17)$$\left\{ \begin{array}{l} {r_p} = {\left|{\frac{{A_1^{\prime}}}{{{A_1}}}} \right|^2} = {\left|{\frac{{{n_2}\cos {\theta_1} - {n_1}\cos {\theta_2} + c{\mu_0}{\sigma_g}\cos {\theta_1}\cos {\theta_2}}}{{{n_2}\cos {\theta_1} + {n_1}\cos {\theta_2} + c{\mu_0}{\sigma_g}\cos {\theta_1}\cos {\theta_2}}}} \right|^2}\\ {t_p} = \frac{{{n_2}\cos {\theta_2}}}{{{n_1}\cos {\theta_1}}}{\left|{\frac{{{A_2}}}{{{A_1}}}} \right|^2} = \frac{{{n_2}\cos {\theta_2}}}{{{n_1}\cos {\theta_1}}}{\left|{\frac{{2{n_1}\cos {\theta_1}}}{{{n_2}\cos {\theta_1} + {n_1}\cos {\theta_2} + c{\mu_0}{\sigma_g}\cos {\theta_1}\cos {\theta_2}}}} \right|^2}\\ {a_p} = 1 - {r_p} - {t_p} \end{array} \right.$$

Note that Eq. (16) and Eq. (17) could reduce to the well-known Fresnel’s formulas when graphene is absent at the dielectric interface (${\sigma _g} = 0$).

The results of reflectance, transmittance and absorptance of TE- and TM-polarized incident lights for different layers of graphene (from 0- to 5-layers, and 10-layers) are plotted in Figs. 2(a)–2(d) with different incident angles. The 0-layer indicates that graphene is absent at the interface, and 10 layers approaches the 2D-limit of graphene [1,38]. It clearly shows in Figs. 2(a) and 2(b) that, as the graphene layer number increases, the reflectance of TE-polarized light ${r_s}$ are significantly reduced in contrast to those of TM-polarized light ${r_p}$, due to the polarization-sensitive absorption property of graphene. As shown in Figs. 2(c) and 2(d), both the absorptance ${a_s}$ and ${a_p}$ become larger with the increasing graphene layer numbers, but ${a_s}$ is an order of magnitude larger than ${a_p}$ for all layers of graphene due to the TEA mode, and ${a_s}$ meets the maximum absorption value around the total reflection angle (${\sin^{ - 1}}(1.33/1.53) = 60.4^\circ $). When the graphene is absent (0-layer), there is no absorbing medium existing at the dielectric interface, so the absorptance curves of ${a_s}$ and ${a_p}$ will be zero according to Eq. (16) and Eq. (17), thus has been omitted in 2(c)–2(d).

Fig. 2. Quantitative analysis of the polarization-sensitive property of graphene with oblique incidence. (a)–(d) Reflectance, transmittance and absorptance of TE- and TM-polarized lights for graphene of different layers (from monolayer to 5-layers, and 10-layers) with different incident angles, the results when graphene is absent (0-layer) are presented for comparison; Absorptance variations of (e) TE- and (f) TM- polarized light for 10-layer graphene at incident angles from 60° to 66° as upper dielectric refractive index changes; (g) Normalized variation rates of the TE-polarized absorptance for different incident angles; (h) Linear relation between the incident angles and the corresponding most sensitive region for refractive index sensing; (i) Reflectance comparison between 0-layer and 10-layer graphene varied with n₂ at 61°, and calculation of the refractive index resolution limit. Inset: Reflectance difference for 10-layer graphene around the total reflection angle.

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To explain the refractive index sensing mechanism, we calculate the variations of ${a_s}$ and ${a_p}$ for 10-layer graphene, as a function of the refractive index of upper dielectric ${n_2}$ (from 1.33 to 1.40) at different incident angles (from 60° to 66°) near the total reflection angle. As shown in Figs. 2(e)–2(f), ${a_s}$ varies much more rapidly than ${a_p}$ as ${n_2}$ changes, and the most sensitive region (gray area in Fig. 2(e)) for refractive index sensing locates just before the corresponding total reflection angle for different incident angles, where the variation of ${a_s}$ is approximately linear in such small refractive index range. It is noted that the most sensitive region varies with different incident angles, so we calculate the normalized variation rates (absolute value) V of the absorptance curves in Fig. 2(e) based on the following Eq. (18), to demonstrate the most sensitive region of refractive index for different incident angles.

(18)$$V = \mathop {\lim }\limits_{n^{\prime} \to n} \left|{\frac{{{a_s}(n^{\prime}) - {a_s}(n)}}{{n^{\prime} - n}}} \right|$$

where $n^{\prime}$ and n indicate the refractive indices, ${a_s}(n^{\prime})$ and ${a_s}(n)$ are the corresponding absorptances of TE-polarized light. As shown in Fig. 2(g), there exists a variation rate peak for each incident angle that is corresponding to the most sensitive region for refractive index detecting. The relation between the incident angles ${\theta _{in}}$ and the refractive indices ${n_{sens}}$ of the peaks are given in Fig. 2(h), and it turns out to be linear as:

(19)$${n_{sens}} = 0.012{\theta _{in}} + 0.606$$

Therefore, depending on the refractive index range of the sample, choosing an appropriate incident angle according to Eq. (19) will give rise to ultrahigh sensitivity for refractive index detection.

In experiment, ${a_s}$ and ${a_p}$ are relatively difficult to measure, so the reflectance difference between ${r_s}$ and ${r_p}$ is actually detected in a common-path optical configuration with significant reduction of the systematic and environmental noises [24,26]. Measuring the reflectance difference between TE- and TM-polarized reflected light experimentally brings the excellent refractive index sensitivity, and the sensing essence is owing to the different absorption responses of graphene to TE- and TM-polarized light as upper refractive index ${n_2}$ changes. Figure 2(i) shows the comparison of ${r_s}$ and ${r_p}$ at the incident angle of 61° for 0- and 10-layer graphene, respectively. The reflectance difference ${r_{diff}} = |{{r_p} - {r_s}} |$ with 10-layer graphene (black solid line) varies much more rapidly than that without graphene (black dash line) due to the polarization-sensitive absorption of graphene. As shown in the inset of Fig. 2(i), the reflectance difference for 10-layer graphene presents approximate linear variation with ${n_2}$ just before the total reflection angle:

(20)$${r_{diff}} = - 493.13{n_2} + 660.59$$

The theoretical limit of the refractive index resolution can be calculated based on the following equation [39]:

(21)$$\Delta {n_{\min }} = \frac{1}{{|k |}}\Delta {r_{\min }} = \frac{1}{{|k |}}\sqrt {\frac{{2h\upsilon \Delta f}}{{\eta {P_0}}}} $$

Where, $\Delta {n_{\min }}$ and $\Delta {r_{\min }}$ denote the minimum refractive index and reflectance difference that could be detected, $k = - 493.13$ is the gradient of Eq. (20), $h\upsilon $ is the photon energy, $\Delta f$ is the bandwidth of the photodetector, $\eta $ is the quantum efficiency, and ${P_0}$ is incident laser power. We choose the parameters of a common balanced detector (Thorlabs, PDB210A), with $\Delta f = 1{\mathop{\textrm {MHz}}\nolimits} $, $\eta = 0.35$ for 532 nm laser, ${P_0} = 20{\mathop{\textrm{ mW}}\nolimits} $ of the threshold power, to calculate the refractive index detecting limit according to Eq. (21). It turns out to be $\Delta {n_{\min }} = 2.09 \times {10^{ - 8}}$ RIU, which is defined as the refractive index resolution of such graphene-based sensing method. Then, the theoretical limit of the refractive index sensitivity could be calculated with the following Eq. (22) [24]:

(22)$$S = |k |\frac{P}{\alpha }$$

Here, we adopt the same values of the incident power $P = 80\mu \textrm{W}$ and the response of balanced detector $\alpha = 0.00361\mu \textrm{W/mV}$ in Ref. [24], so the refractive index sensitivity can be obtained to be 1.09×10⁷ mV/RIU.

4. Absorption of graphene with tight focusing of light for refractive index imaging

Since higher frequency of light could generate smaller focusing spots under the same condition, the graphene working in the TEA mode with high frequency could be applied to generate tightly focused spot for refractive index imaging with high spatial resolution. As explained in Fig. 2(a), the absorptance of TE-polarized light for graphene is extremely sensitive to the variation of incident angles. Therefore, to control the focusing angle for graphene in microscopic system, a ring-shaped perfect optical vortex (POV) beam with very thin ring width and tunable beam radius is adopted [40–42], whose focusing angle $\theta $ could be adjusted by changing the the beam radius r based on the Abbe’s sine condition $r = f\sin \theta $, where f is the effective focal length of the objective [43,44], as illustrated in Fig. 3(a). Compared with a Gaussian beam, the POV beam has the advantages of smaller transverse focusing spot size and larger longitudinal propagating length [45–47].

Fig. 3. Visualization of the polarization-sensitive property of grahene within tight focusing system. (a) Diagram of the focusing process with the ring-shaped POV beam and three kinds of polarization/phase sates: (1) radial polarization, (2) azimuthal polarization, and (3) azimuthal polarization with [0, 2π] spiral phase; Total electric field intensity distributions ${|E |^2}$ of (b)–(c) focused radial POV beam (TM-polarization) in propagating direction (y = 0 plane) when graphene is (b) absent and (c) present at z = 0; Focusing electric field distribution of (d)–(e) azimuthal POV beam (TE-polarization) when graphene is (d) absent and (e) present at z = 0 plane; Focusing electric field distribution of (f)–(g) azimuthal POV beam with spiral phase (compressed TE-polarization) when graphene is (f) absent and (g) present at z = 0 plane. The simulation region of Figs. 3(b)–3(g) is 5 µm× 2µm. Inset: (b₁)–(g₁): total electric field intensity comparison at z = 0 plane, (b₂)–(g₂): intensity profiles across the spot centre along z-axis (white dot lines in Figs. 3(b)–3(g)); (h) Reflectance vaired with the thickness $\tilde{l}$ of the upper refractive index ${n_2}$. (FWHM: Full width at half maximum; SP: spiral phase.)

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To study the polarization-sensitive absorption of graphene in tight focusing imaging system, focused electric field intensity distributions ${|E |^2}$ of three different polarized POV beams at a sensitive focusing angle of 60° are calculated with calculation region of 5µm×2µm based on the FDTD method, as shown in Figs. 3(b)–3(g). A 3D FDTD model with boundary condition of perfectly matched layers (PML) and minimum mesh accuracy of 0.034 nm is built to observe both the electric field distributions at z = 0 plane and y = 0 plane. Graphene of ten layers lies at the interface (z = 0 plane) between the cover glass substrate (${n_1}$=1.53) and upper refractive index medium (${n_2}$=1.33). The thicknesses of ${n_1}$ and ${n_2}$ can be regarded as semi-infinite based on the PML boundary condition. The light source imported into FDTD model is generated based on the vector diffraction method [48–52]. The coordinate system is established in Fig. 3(a) with the origin located at the focusing spot centre, and z-axis indicates the propagation direction. The incident wavelength is $\lambda $=532 nm and the objective numerical aperture is NA = 1.49. Figures 3(b) and 3(c) show the focused electric field intensities of a radially-polarized POV beam (TM-polarization in all ring-shaped region) without and with 10-layer graphene at the interface, respectively. Intensities in the two figures are normalized with same scale, and show very similar distribution due to the low absorptance of graphene to the TM-polarized focal spot. Figures 3(d)–3(e) are the focal results for an azimuthally-polarized POV beam (TE-polarization in all ring-shaped region). It is clear that graphene has obviously larger absorption to the focal spot with azimuthal polarization than radial polarization due to the polarization-sensitive absorption. However, the size of the donut-shaped TE-polarized focusing spot (FWHM: 0.78$\lambda $) is more than twice larger than that of TM-polarization (FWHM: 0.36$\lambda $), which strongly limits its spatial resolution in optical imaging. To obtain both sensitive absorption and high spatial resolution in the focal spot, we consider tight focusing of an azimuthally-polarized POV beam with first-order spiral phase [0, 2π], which forms a bright focal spot with compressed subwavelength size (FWHM = 0.32$\lambda $) [45], as shown in Figs. 3(f) and 3(g). In this case, the main electric field component of focal spot is still TE-polarized, so it is adopted to realized both high-sensitive sensing and high-resolution imaging with graphene. To measure the longitudinal detecting length of such TE-polarized focusing spot with graphene, we change the thickness of the sample $\tilde{l}$ in the range from 0 µm to 2 µm, and the background refractive index above ${n_2}$ is set as ${n_3} = 1$. The reflectance variation is recorded in Fig. 3(h), and the reflectance reducing to 1/e is considered as the longitudinal detecting length, which turns out to be 750 nm and much larger than the ∼200 nm longitudinal detecting length of SPR imaging [53–56].

Similar to the analysis with oblique incidence of light in Fig. 2, here the reflectance, transmittance and absorptance of TM-polarized POV beam (Fig. 3(c)) and TE-polarized POV beam with spiral phase (Fig. 3(g)) under tight focusing incidence are calculated with different focusing angles, as shown in Fig. 4(a). The absorptance of focused TE-polarized light ${a_s}$ (red solid line) is much larger than TM-polarization (blue solid line) when it approaches the maximum output angle of the objective (${\sin^{ - 1}}(1.49/1.53) = 76.9^\circ $). Compared with the results for oblique incidence in Figs. 2(a) and 2(b), the results in Fig. 4(a) shows no obvious inflection point indicating the total reflection angle, due to the fact that the ring width of generated POV beam covers not a single focusing angle but a small angle range in FDTD simulation around the centre angle. Thus, the relation curve in Fig. 4(a) is smoothed by the focusing angle range in contrast with that in Figs. 2(a) and 2(b) with single incident angle. To illustrate the sensing mechanism in tight focusing configuration, the relation curves varied with upper refractive index (${n_2}$) are calculated in Fig. 4(b) at the focusing angle of 70°. ${a_s}$ (red solid line) appears to be approximate linear variation with ${n_2}$ while ${a_p}$ (blue solid line) varies barely with ${n_2}$. Similar to the case in Section 3, here the refractive index resolution limit can also be estimated with the calculated ${r_{diff}}$ in Fig. 4(b) (black solid line), which satisfies the following equation by linear fitting:

(23)$${r_{diff}} = - 3.24{n_2} + 4.64$$

According to Eq. (21) and Eq. (23), the refractive index resolution limit of the graphene-based imaging method turns out to be 3.18×10⁻⁶ RIU. In our previous study [26], the corresponding experimental refractive index resolution has been measured to be 4.40×10⁻⁵ RIU. It shows that the resolution could be further improved about an order of magnitude in the experiment in the future.

Fig. 4. Graphene in tight focusing configuration. Refectance, transmittance and absorptance of the focused TE- and TM-polarized lights (a) with different incident angles, and (b) with different upper refractive indices at 70°; (c) Determination of spatial resolution by scanning across a slit with different spacing $\tilde{d}$, and (d) corresponding line scanning curves when $\tilde{d}$ is 500 nm, 400 nm, 300 nm, and 200 nm respectively; (e) Diagram of a simulated grating sample with decreasing slit widths on the surface of graphene; (f) Computational imaging of the grating structure based on the polarization absorption property of graphene.

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Then the theoretical spatial resolution of the graphene-based imaging is calculated by scanning the compressed TE-polarized focusing spot (Fig. 3(g)) across a slit with different spacing $\tilde{d}$, as shown in Fig. 4(c). The refractive indices of the structure and background are chosen as 1.40 and 1.33 respectively, which cover most biological samples such as cells and tissues, and the culture medium. The refractive index curves are calculated in Fig. 4(d) with 4 µm total scanning distance as $\tilde{d}$ varies from 500 nm to 200 nm with a 100 nm step size, respectively. According to the Rayleigh’s criterion, the saddle-to-peak intensity ratio of 81.1% is usually taken as the spatial resolution limit [57]. We find that in the case of slit width of 200 nm, 20.5% intensity difference between the saddle and the peak can be just separated, thus 200 nm can be considered as the spatial resolution of the graphene-based tightly-focused imaging system, which is close to the FWHM of focus in Fig. 3(g). With the advantages of high spatial resolution, an example of computational refractive index imaging is carried out, as illustrated in Fig. 4(e). A grating with different slit widths (from 400 nm to 100 nm) is constructed as the imaging sample. The refractive indices of the grating structure and the slits are chosen as 1.40 and 1.33. The imaging result in Fig. 4(f) shows good resolving power as the slit width reduces from 400 nm to 200 nm. Note that although the 100-nm-width slit can still be observed in Fig. 4(f), the measured slit width and refractive index are inaccurate because it has been beyond the resolution limit of the system. The graphene-based microscopic imaging method would be applied to researches of many fields such as cell pathology and hematology.

5. Summary

In this paper, we systematically investigate the refractive index sensing mechanism with a surface dynamic conductivity approach, based on the polarization-sensitive property of graphene in TEA mode. It works at the high frequency regime of the electromagnetic spectrum (e.g. visible light), where relatively less attention has been received as compared with the extensive studies on the graphene plasmonics from infrared frequencies to radio waves. We theoretically calculate the surface dynamic conductivity of graphene, and use it in the finite difference time domain (FDTD) approaches to quantitatively study the limit of sensing and imaging ability of such graphene-based methods. By choosing an appropriate incident angle, we prove that graphene with oblique incidence of light offers an ultrahigh refractive index resolution of ∼2.09×10⁻⁸ RIU and sensitivity of 1.09×10⁷ mV/RIU. With the tight focusing of an azimuthally polarized beam with first-order spiral phase [0, 2π], the graphene-based refractive index microscopy is examined with ultrahigh lateral spatial resolution (∼200 nm) and large longitudinal detecting length (∼750 nm). Label-free detection would be new trends for future research on cell sensing and imaging, so we hope the proposed method could be useful in the study of living cell metabolism, kinetics, and pathology, as well as other biomedical researches.

Funding

National Natural Science Foundation of China (11604219, 11774256, 61427819, 91750205, 61490712, U1701661, 61605117, 61805157); National Basic Research Program of China (973 Program) (2015CB352004); Leading Talents Program of Guangdong Province (00201505); Natural Science Foundation of Guangdong Province (2016A030310063, 2016A030312010, 2017A030313351, 2018A030310559); Shenzhen Science and Technology Innovation Commission (JCYJ20180507182035270, KQJSCX20170727100838364, KQTD2017033011044403, ZDSYS201703031605029); China Postdoctoral Science Foundation (2018M643160).

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Refractive index sensing and imaging based on polarization-sensitive graphene

Abstract

1. Introduction

2. Electromagnetic modes on graphene surface

3. Absorption of graphene with oblique incidence of light for refractive index sensing

4. Absorption of graphene with tight focusing of light for refractive index imaging

5. Summary

Funding

References

Cited By

Figures (4)

Equations (23)

Optics Express