Ultrahigh refractive index sensitivity of TE-polarized electromagnetic waves in graphene at the interface between two dielectric media

The behavior of the TE and TM electromagnetic waves in graphene at the interface between two semi-infinite dielectric media is studied. The dramatic influence on the TE waves propagation even at very small changes in the optical contrast between the two dielectric media is predicted. Frequencies of the TE waves are found to lie only in the window determined by the contrast. We consider this effect in connection with the design of graphene-based optical gas sensor. Near the frequency, where the imaginary part of the conductivity of graphene becomes zero, ultrahigh refractive index sensitivity and very low detection limit are revealed. The considered graphene-based optical gas sensor outperforms characteristics of modern volume refractive index sensors by several orders of magnitude.

The dispersion relation ( ) q ω of electromagnetic waves with TE (transverse electric) and TM (transverse magnetic) polarization, propagating in the 2D electron system, surrounded by the homogeneous dielectric medium with permittivity ε , and decaying exponentially in the transverse directions, is given by [61,62]: where ( ) σ ω is the local dynamic conductivity of the 2D electron system and c is the velocity of light in the free space. The dispersion relation (1) for TM waves (also known as surface plasmon polaritons) in the nonretarded limit ( q c ω ε ) reduced to the plasmon dispersion. In fact, collective oscillations of 2D charge density ρ described by 2D plasmons are excited by the in-plane electric field E of incident light: is the in-plane electric current as a δresponse of 2D electron system to the E . The resulting pattern of the charge density oscillations for TM waves can be represented in terms of electric dipole wave (see Fig. 1(a)). However, TE waves cannot be reduced to the common plasmons.
Since their in-plane electric field oscillations are perpendicular to the propagation vector q ( 4 0 divE πρ ε = = ), the electric current is also perpendicular to q ( 0 div j = ) and 2D charge density ρ is zero. The resulting pattern of self-sustained oscillations of the current in the case of TE waves can be described in terms of magnetic dipole wave. Figure 1(b) shows schematic representation of this wave: induced currents provide local magnetic dipoles with corresponding magnetic field; electric field is always directed opposite to the current (that follows from the condition of TE wave existence Im ( ) 0 σ ω < ).
The previous works related to the study of TE waves in graphene [27,31,41,59,60,[63][64][65][66][67][68][69][70] are mainly focused on the case when graphene sheet is embedded into a homogeneous medium or devoted to the investigation of quasi-TE waves. The case of different dielectrics above and below graphene, though mentioned in Refs [60,70], was not under a detailed consideration. In our work we make consistent calculations of the behavior of TE and TM waves in graphene at the interface between two semi-infinite dielectric media. We show that unlike TM waves, the behavior of TE waves strongly depends on the small changes in the optical contrast between the two dielectric media. We argue that TE waves do not exist in some frequency range depending on the contrast even at Im ( ) 0 σ ω < . Solving the electrostatic problem it is easy to show that optical contrast has no dramatic influence on the common plasmon dispersion where dielectric constants of surrounding media are included as the half-sum. We obtain that the same situation will be for plasmon polaritons (TM waves). Here for the first time we estimate TE waves refractive index sensitivity and detection limit in connection with the design of graphene-based optical gas sensor. We propose a novel approach for volume optical sensing employing surface TE waves (STE) in graphene which incorporates some features of the surface plasmon resonance (SPR) sensing [71] and volume optical sensing [72,73].

II. GRAPHENE BETWEEN TWO DIELECTRIC MEDIA
Let us consider graphene at the interface between two semiinfinite dielectric media. Usually (see [71]), the sensitivity to the changes in the optical contrast is expressed in terms of refractive index. Hereinafter we will operate with normalized quantities: ) can be expressed as [74,75]: and τ are temperature in units of energy and a finite carrier relaxation time in graphene, respectively. Imaginary part of conductivity becomes zero at Fig. 2). That is TE waves exist only at Due to the Landau damping at 2 Ω > (at finite temperatures a little less) TE waves should be excited only in the range 1.667 2 < Ω < . To reduce the role of the temperature here and below we will consider high doped graphene: 1eV For high frequencies (above the phonon frequency 0.2eV ∼ ) the carrier relaxation time in graphene is mainly determined by electron-phonon scattering mechanism and can be incorporated through an effective ) [28]. In case of TE waves it is easy to find complex analytical solution of the Eq. (4) by solving the system of complex equations: Solving system (7) we obtain the real and imaginary parts of normalized transverse wave vectors: Setting 2 1 n n > , we obtain that for ( ) (9)) 2 Re z K becomes negative, which leads to an exponential growth of the wave field with the distance from graphene into the medium with refractive index 2 n . Such solutions should be rejected as unphysical, because they do not satisfy the boundary conditions at infinity. Therefore, when the relative permittivity of dielectrics above and below graphene differs more than the 2 ( ) f Ω , TE wave cannot propagate along graphene layer.
Further we will consider this effect applied to the design of graphene-based optical gas sensor. Its possible registration system is shown in Fig. 3. Incident light excites TE wave in graphene, e.g., by means of grating substrate, then, after passing the suspended part of graphene, TE wave decouples to light by another grating substrate (see Fig. 3(a)). We assume that after the appearance of the investigated gas refractive index below the graphene layer ( 2 n ) is increased by x n while refractive index above the graphene layer ( 1 n ) remains the same (i.e. If the concentration of the investigated gas exceeds critical value corresponding to x n , TE wave will no longer exist in suspended part of graphene and, ideally, there will be no output signal (see Fig. 3(b)). Using Eqs. (5) and (6)  πσ Ω Ω = at zero and room temperatures ( Fig. 4(a)).
The inset from Fig. 4(a) shows that the smallest detectable refractive index change ( ) min Re   The refractive index sensitivity (sensitivity to the refractive index changes) depending on the normalized frequency can be written as: Figure 4(c) shows that near 0 Ω = Ω the sensitivity do not depend on temperature and tends to infinity. Further we will refer to the frequency at which the sensitivity reaches its maximum value as the sensitivity point. But in fact it is limited by the value corresponding to the required confinement (see below) and by the accuracy of setting the necessary frequency of the wave. The last is defined by the charge inhomogeneity in graphene [76] which leads to the accuracy of the Fermi level 3 10 eV ( ) where ( )  . It always lies to the right of the most inclined light line and, hence, cannot exist as leaky modes. Taking into account the damping, the dispersion relation of TE waves, which is generally defined by the Eq. (11), is also very close to the dispersion of light. At 300 T K = for 5 10 RIU x n − = TE waves exist at 1.78 Ω > and for 6 10 RIU x n − = at 1.69 Ω > (see Fig. 5(a)), which is in agreement with the condition (9). The Eq. (3) for TM waves has to be solved numerically. As we have expected the optical contrast has no dramatic influence on the TM waves dispersion (see Fig. 5(b)).  6 10 RIU x n − = (red line), 5 10 RIU

III. TE WAVES CONFINEMENT
For the case of common plasmons in the nonretarded limit  8) and (9) shown in Fig. 6(a). At  (2)).
When the refractive index change begins to exceed ( ) min x n (e.g., 5 10 RIU x n − = ) the behavior of the TE wave confinement changes significantly (see Fig. 6(b) red line (3)). The confinement 1 Re z K continues to increase and 2 Re z K continues to decrease, but in such a way that at the frequency 1.78 Ω = it becomes zero. This is in agreement with the results for TE waves dispersion represented by the Fig. 5(a). Thus, the absence of TE waves at frequencies less that those which satisfy the Eq. (10) is caused by the delocalization of the wave at the side with the highest refractive index (the side filled with the investigated gas). In the sensitivity point (where the sensitivity S goes to infinity) the confinement 2 Re z K goes to zero. With the increase of x n the sensitivity point shifts towards the damping region (see Fig.  6(b)). (green (2)), 5 10 RIU The above calculations were carried out under the assumption that graphene is surrounded by two semi-infinite media. In reality one deals with finite volume filled by the investigated gas. In order that the upper and lower boundaries of the medium above and below the graphene layer, respectively, do not affect the TE wave refractive index sensitivity, the wave decay length z L (see above) should be less than the required distances above and below the graphene layer. In this case the wave will not bound with upper or lower boundaries, and so their influence can be neglected. The quantity 1 z z L k = determines the 1 e field decay, where ( ) For the proper comparison of our results with the characteristics of modern refractive index sensors let us express the decay length and the sensitivity depending on the wavelength in non-normalized variables: z L in the units of length and 1 ( ) in nm RIU . The sensitivity reaches its maximum value near the Fig. 4(c)) which corresponds On the other hand, the decay length defining the transverse size of the investigated volume grows with the increase of the wavelength. Depending on the required measurements one should find the optimum balance between the decay length and the sensitivity.

IV. DISCUSSION
Let us discuss some details which can be important in experiment. Our calculations were held for high Fermi level of graphene ( 1 ) in order to reduce the role of the temperature. Such Fermi level can be achieved in graphene only by means of strong chemical doping, which will lead to a very high suppression of the carrier mobility in graphene. However, as can be seen from Fig. 4(b) at 300 T K = the refractive index resolution is almost independent from the carrier mobility and, hence, strong doping does not impair the performance of the sensor. At low temperatures (for helium temperature all results will be almost the same as for 0 T K = ) it is not necessary to use high doped graphene. It is possible to work at low Fermi levels (less than 0.2eV ) and to achieve very high carrier mobility in graphene, which can improve the refractive index resolution by several orders of magnitude (see Fig. 4(b)).
Some difficulties may occur in searching the optimal length of the suspended part of graphene in the direction of the wave propagation. On the one hand, in order to avoid a mechanical sagging of graphene the length should be several or a few tens of microns. In this case, the registration system suggested here (see Fig. 3) may give the suppression of the output signal depending on the length of the suspended part of graphene rather than its absence. Possibly, for sufficiently small suspended graphene length the considered effect can become unobservable. On the other hand, at small distances between left and right grating couplers it will be difficult to provide low-background measurement with independent illumination of the left coupler and collection signal light from the right coupler. The possible solution to this is placing the coupler and decoupler far enough from the suspended part of graphene. But the distance between each of gratings and the suspended part should not exceed the propagation length of TE waves. Since the propagation length is of the order of several hundred microns it seems possible to distance the coupler and decoupler by considerable measure, while the length of the suspended part of graphene remains several microns.
The main problem of the experimental observation of TE waves in graphene is their very small field confinement. It can be not so easy to distinguish TE wave propagating along the graphene layer from the total electromagnetic field of incident light. In fact, there are different ways to improve TE waves confinement in graphene. Combining graphene with waveguide [41,78] or making multilayer graphene system [63,79] it is possible to get high-confined quasi-TE waves containing waveguide component. Perhaps, it will be possible to increase the confinement by the usage of strained graphene sheets [64], or by the applying of quantizing magnetic field leading to the hybrid TM-TE waves [66][67][68]. Also, TE wave confinement may become higher if we take into account the spatial dispersion effect in graphene [69]. But the most possible solution to the problem can be the usage of another atom-thick systems with richer electron band structure instead of monolayer graphene. An example of such a system can be a Bernal-type bilayer graphene where TE wave confinement can be improved in comparison with monolayer graphene by two orders of magnitude [60]. However, it is important to emphasize that TE wave confinement described by is proportional to the absolute value of Im ( ) σ ω (see Eqs. (8), (9)), whereas their sensitivity, discussed here, is inversely proportional to the gradient of Im ( ) σ ω . Due to the monotonic behavior of Im ( ) σ ω near the sensitivity point, the increase of the absolute value, which causes the improvement of the confinement, inevitably leads to the increase of the gradient, which results in the decrease of sensitivity. In other words high refractive index sensitivity of TE waves in graphene, predicted in this work, is a reverse side of their small field confinement.
In connection with the comparison of our results with the characteristics of modern refractive index sensors it should be noted that STE sensing proposed here incorporates some features of SPR sensing [71] and volume optical sensing [72,73]. Both SPR and STE sensing are based on the interaction between a sample and an evanescent electromagnetic wave. But SPR sensing uses common plasmons with typically small field confinement (10-300 nm) and, hence, operates with thin layers of analyte. On the other hand, based on the beam deviation technique volume optical sensing similarly to STE sensing operates with tens of micrometers of the investigated gas or liquid. Thus, due to the similarity of application areas, it will be more correct to compare our results with the characteristics of volume optical sensors.
Finally, we would like to mention that our calculations were held under the assumption of the homogeneous density of the investigated gas. Otherwise (particularly when we do not try to investigate gas but very thin layers on the surface of graphene) it is necessary to calculate the scattering of TE waves by a finitesize barrier located on the graphene surface. This problem will be the subject of the future investigation.

V. CONCLUSION
To conclude, we have shown that, unlike TM electromagnetic waves, TE waves in graphene at the interface between two semiinfinite dielectric media can exist only in some frequency range depending on the optical contrast between these media. We have obtained the analytical expressions describing the TE waves frequency range and their sensitivity to the changes in the optical contrast. The effect was considered in connection with the design of graphene-based optical gas sensor. We have found that near the frequency, where the imaginary part of the conductivity of graphene becomes zero, this sensor may have very high refractive index sensitivity and very low detection limit. At zero temperature we found the minimal detection limit to be ( )  3 10 nm RIU S λ ∼ ) by three orders of magnitude. By changing input signal frequency or Fermi level in graphene one can find the optimum balance between required field confinement and refractive index sensitivity. Unlike SPR sensors, TE waves graphene-based sensor poroposed here, as well as any volume optical sensor, is suitable for applications requiring thick surface functionalization or measurements through bigger biological samples, such as living cells.