† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant Nos. 61505111 and 11604216), the China Postdoctoral Science Foundation (Grant No. 2016M600667), the Science and Technology Planning Project of Guangdong Province, China (Grant No. 2016B050501005), the Fund from the Educational Commission of Guangdong Province, China (Grant No. 2016KCXTD006), and the Natural Science Foundation of Guangdong Province, China (Grant No. 2015A030313549).
A hybrid structure based on a planar waveguide (PWG) mode coupling a long-range surface plasmon resonance (LRSPR) mode is proposed to enhance the GH shift. Both the PWG mode and LRSPR mode can be in strong resonance, and these two modes can be coupled together due to the normal-mode splitting. The largest GH shift of PWG-coupled LRSPR structure is 4156 times that of the incident beam, which is 23 times and 3.6 times that of the surface plasmon resonance (SPR) structure and the LRSPR structure, respectively. As a GH shift sensor, the highest sensitivity of 4.68 × 107λ is realized in the coupled structure. Compared with the sensitivity of the traditional SPR structure, the sensitivity of our structure is increased by more than 2 orders, which theoretically indicates that the proposed configuration can be applied to the field of high-sensitivity sensors in the future.
The Goos–Hänchen (GH) effect refers to a lateral spatial shift that an electromagnetic wavepacket experiences in its incident plane when it is reflected from a surface. It can be traced back as early as the era of Newton, but it was not until 1947 that Goos and Hänchen first observed it.[1,2] A theoretical explanation of the effect was given by Artmann who used the stationary phase method.[3] In the past few decades, the lateral shift has been widely studied in optical,[4,5] chemical,[6] and biomedical fields.[7] Usually, GH shift was studied at the interface of the structure containing two homogeneous materials with different optical characteristics, and the GH shift is very small in this case, almost comparable to the wavelength of the incident beam.[8–11] For better research, measurement, and application, thus, how to obtain a giant GH shift has always been the goal pursued by researchers.
Currently, researchers are working on the enhancing of the GH shift via various methods, and one of the effective ways is to excite the surface plasmon polaritons (SPPs). As described by Yin et al.,[12] a lateral spatial displacement that is greater than 50 wavelengths for the reflected beam due to the surface plasmon polaritons (SPPs), was observed experimentally. The SPP is a perpendicularly confined evanescent electromagnetic wave that occurs at the metal-dielectric interface,[13–15] and it can be excited by attenuated total reflection (ATR) configuration, where the wave vector mismatch between vacuum and SPP is compensated for by using high-index prisms. When SPPs are excited at the metal–dielectric interface, there will be a reflectance dip in the SPR reflectivity curve, and the electromagnetic fields near the interface will become very strong, which can lead to a giant GH shift.
After that, a new mode called long-range surface plasmon resonance (LRSPR) has been proposed to enhance the GH shift.[16] When the metal in the ATR structure is thin enough and clamped by two kinds of dielectrics (ε1, ε2) with similar refractive indices (|ε1 – ε2| ≪ ε1, ε2), the SPRs on the upper and lower surfaces of the metal can be coupled together to form an LRSPR. The lateral beam shift at the resonance of LRSPR can be two orders of magnitude greater than the wavelength. Similarly, a dielectric planar waveguide (PWG) coupling system consisting of a high refractive index core layer and two low refractive index cladding layers is another effective way to enhance the GH shift.[17–20] Although the amplified shift has been widely studied and applied, the larger GH shift is always pursued by researchers.
In recent years, hybrid configurations with two different electromagnetic modes have received a lot of attention.[21–23] When the strong resonance formed by the two modes can be coupled together, a stronger resonance is formed. This phenomenon is known as the normal-mode splitting.[24] Therefore, inspired by the above, a coupled structure of the PWG mode and the long-range surface plasmon resonance (LRSPR) mode is proposed to enhance the GH shift in this paper. Both PWG and LRSPPs modes can form a strong resonance, and once the two modes are coupled together, a new strong resonance based on PWG–LRSPP will occur.
We find that the GH shift in the coupled configuration can be enhanced obviously, which is much larger than the GH shift in the conventional SPR and LRSPR configuration. The GH shift generated by the coupled PWG–LRSPP mode is nearly 3.7 times more than the GH shift produced by the LRSPP mode. We can adjust the thickness of the cytop to improve the GH shift. In the application, the GH shift in our structure is very sensitive to the refractive index of the substrate medium; through the further improvement of the structure, it can be used as a high sensitivity sensor. We believe that this scheme could be potentially valuable for the high-sensitivity sensors, the measurement methodology, etc.
The structure analyzed here is shown in Fig.
In this structure, the refractive index of 2S2G prism is given by the following expression: np = 2.24047 + 2.693 × 10−2/λ2 + 8.08 × 10−3/λ4.[28,29] The refractive index of the cytop layer is n1 = n3 = ns = 1.34 at λ = 633 nm.[26] The third layer is silicon (Si) film and its refractive index is calculated from the expression n2 = A+A1 e−λ/t1 + A2 e−λ/t2,[30,31] where A = 3.44904, A1 = 2271.88813, A2 = 3.39538, t1 = 0.058304, and t2 = 0.30384. The fifth layer is Au film, and its dielectric constant follows the Drude–Lorentz model,[32]
The structure of PWG consists of cytop–Si–cytop, and the dispersion relation for PWG is calculated from the following formula:[17]
![]() |
![](cpb_27_8_087302/cpb_27_8_087302_ieqn2.gif)
![]() |
![](cpb_27_8_087302/cpb_27_8_087302_ieqn3.gif)
![](cpb_27_8_087302/cpb_27_8_087302_ieqn4.gif)
For a multilayer system, we can use the transfer matrix method (TMM) and Fresnel equations based on an N-layer model to analyse the phase (ϕp) and reflectivity (Rp).[35,36] The layers are stacked parallel to the Z direction, perpendicular to the dielectric interface. The total transfer matrix is the multiple multiplication of the transfer matrix for each layer,
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
Figure
![]() | Fig. 2. (color online) Plot of effective refractive indices of PWG mode (solid red line) and LRSPP mode (solid blue line) versus thickness. |
Using the TMM method, we can obtain the curve of reflectance (solid red line) and phase (blue dotted line) as a function of the incident angle in Fig.
![]() | Fig. 3. (color online) Variation of reflectance and phase with incident angle when d2 = 14.1 nm, d4 = 11.4 nm. |
In our theoretical research, we find that the GH shift will present a great red shift when we change the refractive index of the bottom cytop layer (ns). Therefore, we can use the proposed structure as a high sensitivity sensor based on the changing shift. The underlying cytop (ns) is replaced by the sensing medium, and in order to ensure that the configuration of the waveguide can be coupled with the LRSPR, it is better for the refractive index of the sensing layer to be about 1.34 (pure water). Here, we define ΔGH as the maximum value of the changed GH shift. From Fig.
![]() | Fig. 6. (color online) Variation of GH shift with incident angle when changing the refractive index of sensing medium in (a) SPR structure, (b) LRSPR structure, and (c) PWG–LRSPR structure. |
In this study, we have coupled PWG structure and LRSPR structure together to obtain a new structure, where the GH shift based on the PWG-coupled LRSPR structure can be greatly enhanced. By optimizing the thickness of dielectric layers, the largest lateral shift we can theoretically obtain almost attains to the millimeter magnitude. As a GH shift sensor is based on the changing shift, our structure can obtain sensitivity as high as 4.68 × 107λ. Compared with the sensitivity of a traditional SPR structure, the sensitivity of our structure is increased by more than 2 orders. It should be noted that the refractive index of the dielectric layer is 1.34, so it is applicable to the sensing medium with a refractive index of about 1.34.
[1] | |
[2] | |
[3] | |
[4] | |
[5] | |
[6] | |
[7] | |
[8] | |
[9] | |
[10] | |
[11] | |
[12] | |
[13] | |
[14] | |
[15] | |
[16] | |
[17] | |
[18] | |
[19] | |
[20] | |
[21] | |
[22] | |
[23] | |
[24] | |
[25] | |
[26] | |
[27] | |
[28] | |
[29] | |
[30] | |
[31] | |
[32] | |
[33] | |
[34] | |
[35] | |
[36] |