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Abstract
The spatial and angular Goos-Hänchen shifts (GHSs) and Imbert-Fedorov shifts (IFSs) are theoretically investigated in a modified Kretschmann-Raether configuration consisting of glass prism, J-aggregate cyanine dye, and air. With the excitation of surface excitation polaritons (SEPs), the spatial and angular GHSs and IFSs for the transverse magnetic polarized light are strongly enhanced around the resonant angle of SEP. A highly sensitive gas sensor based on the SEP enhanced GHS (or IFS) is proposed, which exhibits the refractive index sensitivity on the order of 106λ/RIU (or 105λ/RIU) (λ: illumination wavelength; RIU: refractive index unit) for the GHS- (or IFS-) based gas sensor, respectively.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
A light beam reflected from the interface between two different media will experience a lateral shift with respect to the position predicted by the geometrical optics. This lateral shift is known as Goos-Hänchen shift (GHS), which was first surmised by Newton in the 17th century [1], and experimentally demonstrated in 1947 by Goos and Hänchen [2]. In addition to the spatial shift, theoretical predictions and experimental validations have shown that the reflected light beam also exhibits a small angular deviation from the law of reflection [3–10], i.e., the angular GHS. The spatial and angular GHSs occur in the plane of incidence. On the other hand, the optical shift occurring in the transverse direction (i.e., perpendicular to the plane of incidence), is known as Imbert-Fedorov shift (IFS) [10–17].
The GHS and IFS for light beams reflected from the interface between two isotropic homogeneous media are typically on the order of illuminated wavelength or less, which limits the applications of GHS and IFS. Various structures [e.g., Kretschmann-Raether (K-R) configuration] and novel materials (e.g., absorbing media, chiral metamaterial, and negatively refractive media) have been proposed and demonstrated to enhance the GHS and IFS [18–31]. In particular, a larger GHS of ${\sim }50\lambda$ ($\lambda$: the wavelength of incident light) is experimentally achieved in the K-R configuration with the excitation of surface plasmon polaritons (SPPs) [18]. A theoretical investigation has demonstrated that both the spatial and angular GHSs for the transverse magnetic (TM) polarized light are significantly enhanced at the incident angle close to the resonant angle [27]. The SPPs also exhibit enhancement effects on the spatial and angular IFSs for both TM and transverse electric (TE) polarized incident light, but less pronounced than that of GHS for TM polarized light.
In the last few years, various sensors based on GHS have been explored, such as temperature sensors [31–33], biosensors [34], and refractive index (RI) sensors [28–30,35,36]. For example, Yin $et$ $al.$ proposed a SPP-GHS sensor by monitoring the variations of the GHS with the excitation of SPP [29]. The sensitivity of ${\sim }5.5\times 10^{7}$ nm/RIU (RIU: refractive index unit) was achieved with the SPP-GHS sensor by changing the concentration of sodium chlorite solution. The superior RI sensitivity of the SPP-GHS sensor is a result of the SPP enhanced GHS. In addition, sensing applications based on the IFS have also been explored in recent years [31,37,38].
Similar to exciting SPPs in the K-R configuration [39,40], another type of polaritons, surface exciton polaritons (SEPs), can be excited in a modified K-R configuration at room temperature [41–43]. In the modified K-R configuration, the metal film is replaced by an organic film, 5,6-dichloro-2-[[5,6-dichloro-1-ethyl-3-(4-sulfobutyl)-benzimidazol-2-ylidene]-propenyl]-1-ethyl-3-(4-sulfobutyl)-benzimidazolium hydroxide (TDBC). The SEPs in the modified K-R configuration share some similar properties with the SPPs, for instance, the strongest electric field at the TDBC-air (or metal-air) interface, and exponentially decays into air [42]. Based on this, a SEP-based sensor is proposed, which shows higher RI sensitivity than the conventional SPP sensor in gas sensing and biosensing [42]. The enhancement effects of SPP on the GHS and IFS are well-known [18,27,31], then it is expected to achieve SEP enhanced GHS and IFS due to the similarity between SEP and SPP.
At this juncture, it is timely to investigate the effect of SEP on the spatial and angular GHSs and IFSs. Specifically, the GHS and IFS in the modified K-R configuration consisting of glass prism, TDBC film, and air will be investigated. As the SEP is excited, enhanced spatial and angular GHSs and IFSs for the TM polarized light near to the resonant angle are observed. The SEP enhanced GHS and IFS are found to be sensitive to the ambient RI, and a high sensitivity gas sensor based on the GHS or IFS is proposed. The superior RI sensitivity of the proposed gas sensor is attributed to the excitation of SEPs.
2. Theoretical model
A modified K-R configuration [44] was employed for the SEP excitation, in which the metal film was replaced by a TDBC film, as shown in Fig. 1. In the modified K-R configuration, the TDBC film was deposited on a BK7 glass substrate, and the substrate was attached to the base of BK7 glass prism. The dielectric constant of TDBC can be calculated from the following equation [41]:
Fig. 1. Schematic diagram of the light beam reflection at the BK7 prism/TDBC interface in the modified K-R configuration consists of BK7 prism, TDBC film, and air. The reflected light beam undergoes both the spatial and angular GHSs and IFSs. Here, the spatial and angular GHSs are shown as an example.
In geometrical optics, there are no angular or spatial shifts during the light reflection. In fact, a spatial GHS (IFS) occurs accompanying the angular GHS (IFS), an example of spatial and angular GHSs is shown in Fig. 1. Consider a monochromatic beam of $\alpha$ ($\alpha$=TM, TE) polarized light with wavelength $\lambda$ and finite waist $w_0$ as the incident light, the spatial shift $\Delta _\alpha$ and angular shift $\Theta _\alpha$ of GHS and IFS can be described with a unified representation as suggested by Aiello $et$ $al.$ [45]:
The total beam beam displacement $\delta ^{GH(IF)}_\alpha$ is a liner combination of the spatial shift $\Delta _\alpha$ and angular shift $\Theta _\alpha$, which is also known as the composite GHS (composite IFS) [31,50–53]. In this manuscript, the reflectance, spatial and angular Goos-Hänchen and Imbert-Fedorov shifts for TM and TE polarized incident light are obtained by using MATHEMATICA software package.
3. Spatial and angular Goos-Hänchen and Imbert-Fedorov shifts
Takatori $et.$ $al.$ first experimentally observed SEPs at room temperature with the modified K-R configuration (see Fig. 1) under 532 nm illumination wavelength [41]. Therefore, in this work, we will consider the incident light with wavelength $\lambda =532$ nm only. The dielectric constant of TDBC under 532 nm illumination wavelength is $-2.59+0.375i$, and the RI of BK7 prism is 1.52 [49]. The reflectance $R_\alpha =|r_\alpha |^2$ for $\alpha$ polarized light as a function of the incident angle $\theta$ is shown in Fig. 2(a). The reflectance curve for TM polarized light exhibits a dip at the resonant angle of $54.65^\circ$, which is a result of the excitation of SEP. In contrast, the reflectance $R_{TE}$ increases monotonically with the incident angle [see Fig. 2(a)]. It should be noted that both the reflectance $R_{TM}$ and the resonant angle depend on the TDBC film thickness (see Fig. S1). Here, the thickness of TDBC film in Fig. 2 is 60 nm, which gives the deepest reflectance dip for the TM polarized incident light (see Fig. S1). The phase $\phi _\alpha$ of the reflection coefficient $r_\alpha$ is shown in Fig. 2(b). In contrast to the phase $\phi _{TE}$ of TE polarized incident light, the phase $\phi _{TM}$ exhibits a significant change near the resonant angle of SEP. Therefore, it is expected to obtain a large spatial GHS for the TM polarized light near the resonant angle of SEP according to Eqs. (2)–(3).
Fig. 2. (a) Reflectance $R_\alpha$ and (b) phase $\phi _\alpha$ as a function of the incident angle for TM polarized ($\alpha$=TM) and TE polarized ($\alpha$=TE) light incident into the BK7 prism/TDBC/air structure. The thickness of TDBC film is 60 nm, and the illumination wavelength is $\lambda =532$ nm.
The spatial GHS $\Delta _\alpha ^{GH}$ as a function of incident angle for the TM and TE polarized light are shown in Fig. 3(a). The TE polarized light exhibits negligible spatial GHS ($|\Delta _{TE}^{GH}|<\lambda$). In contrast, an extremely large negative spatial GHS for the TM polarized light ($\Delta _{TM}^{GH} \approx -51\lambda$, $\lambda =532$ nm in this work) was found at the resonant angle, where SEPs were excited. This enhanced spatial GHS for the TM polarized light at the resonant angle of SEP is similar to the SPP enhanced spatial GHS, which has been theoretically and experimentally demonstrated [18,27,29,31]. For angular GHS $\Theta _\alpha ^{GH}$, it is also extremely small for the TE polarized light, as shown in Fig. 3(b). The angular GHS of TM polarized light was significantly enhanced with the excitation of SEPs. A positive peak ($\sim 79.4 \theta _0^2$) and a negative peak ($\sim -76.8 \theta _0^2$) are observed just below and above the resonant angle. Here, the waist is $w_0=40\lambda$, and the angular spread of the incident beam is $\theta _0=0.025\pi$. The sudden change of the sign of angular GHS $\Theta _{TM}^{GH}$ was also experimentally observed near the Brewster angle at an air-glass interface [3], and theoretically predicted near the resonant angle in a conventional prism-coupled SPP structure [27].
Fig. 3. (a) Spatial GHS $\Delta ^{GH}_\alpha$ and (b) angular GHS $\Theta ^{GH}_\alpha$ as a function of incident angle for TM polarized ($\alpha =$TM) and TE polarized ($\alpha$=TE) light.
The spatial IFS $\Delta _\alpha ^{IF}$ exhibits different behaviors from that of spatial GHS $\Delta _\alpha ^{GH}$. As shown in Fig. 4(a), the $\Delta _{TE}^{IF}$ is positive in the incident angle range from $45^\circ$ to $75^\circ$, whereas the spatial GHS for TE polarized light $\Delta _{TE}^{GH}$ is negative. The spatial IFS of TM polarized light $\Delta _{TM}^{IF}$ is enhanced with the SEP excitation around the resonant angle, but with a much smaller magnitude as compared with that of spatial GHS $\Delta _{TM}^{GH}$. For example, the spatial IFS of TM polarized light $\Delta _{TM}^{IF}$ shows a negative peak ($\sim -2.1\lambda$) and a positive peak ($\sim 4.86\lambda$) just below and above the resonant angle [see Fig. 4(a)]. In Fig. 4(b), a sudden change of the sign of angular IFS was observed for the TM polarized light: a large negative peak ($\sim -20.5\theta _0^2$) just below the resonant angle, and a small positive peak ($\sim 1.21\theta _0^2$) above the resonant angle. The TE polarized light also exhibits negative and positive angular IFSs around the resonant angle, but with a much smaller magnitude.
Fig. 4. (a) Spatial IFS $\Delta ^{IF}_\alpha$ and (b) angular IFS $\Theta ^{IF}_\alpha$ as a function of the incident angle for TM polarized ($\alpha$=TM) and TE polarized ($\alpha$=TE) light.
The study above discussed only one particular TDBC film thickness (i.e., 60 nm). The TDBC thickness dependent spatial and angular GHSs and IFSs are shown in Figs. S2-S3. The TM polarized light exhibits positive and negative peaks for the spatial GHS $\Delta _{TM}^{GH}$ [see Fig. S2(a)] at the incident angle near the resonant angle, which depends on the thickness of TDBC film. The change of the sign of angular GHS $\Theta _{TM}^{GH}$ around the resonant angle was also observed with other thicknesses of TDBC film [see Fig. S2(b)]. It should be noted that the magnitude of spatial (or angular) GHS peak of TM polarized light for the BK7 prism/TDBC/air structure with 60 nm TDBC film is much larger than those for the SEP structure with 40 nm, 50 nm, 70 nm, 80 nm, and 90 nm TDBC films. This significantly enhanced spatial (or angular) GHS with 60 nm TDBC film can be attributed to the strongest excitation of SEPs (see Fig. S1). The TDBC thickness dependent spatial and angular GHSs of TE polarized light share the similar behavior, as shown in Figs. S2(c) and (d), respectively. Both the magnitudes of $\Delta ^{GH}_{TE}$ and $\Theta ^{GH}_{TE}$ decrease with the TDBC film thickness in the incident angle range from 45$^\circ$ to 75$^\circ$. For the spatial and angular IFSs of the TM and TE polarized light, their magnitudes and signs (positive or negative shifts) also can be controlled by the TDBC film thickness, as shown in Fig. S3. Overall, for the spatial (angular) GHS of the TM polarized light, its magnitude is higher than that of spatial (angular) IFS of TM polarized light. In contrast, for TE polarized light, the magnitude of spatial IFS is comparable to that of spatial GHS, while the magnitude of angular GHS is smaller than that of angular IFS.
The composite GHS $\delta ^{GH}_\alpha$ and composite IFS $\delta ^{IF}_\alpha$ are shown in Fig. 5(a) and Fig. 5(b), respectively. Here, the thickness of TDBC film is 60 nm, and the distance is $l=10$ cm. The composite GHS $\delta ^{GH}_{\text {TM}}$ and composite IFS $\delta ^{IF}_{\text {TM}}$ of TM polarized light are significantly enhanced around the resonant angle of SEP. Both the composite GHS $\delta ^{GH}_{\text {TM}}$ and composite IFS $\delta ^{IF}_{\text {TM}}$ of the TM polarized light show a sudden change of its sign around the resonant angle. The positive (negative) peak of $\delta ^{GH}_{\text {TM}}$ is ${\sim }919\lambda$ (${\sim }-942\lambda$), while it is ${\sim }15.9\lambda$ (${\sim }-243\lambda$) for the $\delta ^{IF}_{\text {TM}}$. Compared with the composite GHS (IFS) of the TM polarized light around the resonant angle, the composite GHS (IFS) of the TE polarized light can be neglected (see Fig. 5), thus can be served as a reference signal in the sensing applications.
Fig. 5. (a) Composite GHS $\delta ^{GH}_\alpha$ and (b) composite IFS $\delta ^{IF}_\alpha$ as a function of incident angle for TM polarized ($\alpha$=TM) and TE polarized ($\alpha$=TE) light.
4. Refractive index sensing
The SEP enhanced composite GHS and composite IFS can be used for sensing applications. Here, gas sensing with the RI range from 1.0 to 1.001 is investigated by monitoring variations of the composite GHS (IFS) difference $d\delta ^{GH(IF)}$ between the TM and TE polarized light. The RI sensitivities for composite GHS ($S^{GH}$) and composite IFS ($S^{IF}$) as a function of incident angle around the resonant angle are shown in Figs. 6(a) and (b), respectively. Significant RI sensitivities are obtained around the resonant angle with composite GHS (or composite IFS). Both the composite GHS and composite IFS exhibit positive and negative RI sensitivities. For example, the maximum positive RI sensitivity of composite GHS is about $1.178\times 10^6\lambda /\text {RIU}\approx 6.27\times 10^5 \ \mu \text {m/RIU}$ at the incident angle of $54.71^\circ$. The fitting line shown in the inset of Fig. 6(a) indicates the good linearity between the composite GHS difference $d\delta ^{GH}$ and the gas RI. The negative RI sensitivities of $-1.275\times 10^5\lambda /\text {RIU}\approx -6.78\times 10^4 \ \mu \text {m/RIU}$ and $-1.942\times 10^5\lambda /\text {RIU}\approx -1.03\times 10^5 \ \mu \text {m/RIU}$ are obtained for composite GHS at $54.37^\circ$ and $55^\circ$, respectively. The composite IFS exhibits the maximum positive RI sensitivity of $7.884\times 10^4\lambda /\text {RIU}\approx 4.19\times 10^4 \ \mu \text {m/RIU}$ at $54.56^\circ$ [see Fig. 6(b)], which is about one-fifteenth that of the positive peak of composite GHS RI sensitivity. The negative peak of $-1.368\times 10^5\lambda /\text {RIU}\approx -7.28\times 10^4 \ \mu \text {m/RIU}$ is obtained at the incident angle of $54.79^\circ$, whose magnitude is comparable to that of the negative peak of composite GHS RI sensitivity.
Fig. 6. Sensitivities for (a) composite GHS- and (b) composite IFS-based gas sensors. The insets show the linear fitting lines of the composite GHS difference $d\delta ^{GH}/\lambda$ and composite difference IFS $d\delta ^{IF}/\lambda$ versus the gas refractive index at the incident angle where the maximum magnitude of composite GHS or IFS are obtained.
To investigate the contributions of spatial and angular GHSs to the RI sensitivity of the GHS-based gas sensor, the RI sensitivities of spatial GHS ($S_{\Delta }^{GH}$) and angular GHS ($S_{\Theta }^{GH}$) are shown in Figs. 7(a) and (b), respectively, by monitoring the variations of the difference of $\Delta _\alpha ^{GH}$ and $l\Theta _\alpha ^{GH}$ between the TM and TE polarized light. It is found that the RI sensitivity for the composite GHS $S^{GH}$ (composite IFS $S^{IF}$) is mainly contributed by the angular GHS (angular IFS). The RI sensitivity for the spatial GHS has a positive peak ($1.971\times 10^4\lambda /\text {RIU}\approx 1.05\times 10^4 \ \mu \text {m/RIU}$) and a negative peak ($-2.22\times 10^4\lambda /\text {RIU}\approx -1.18\times 10^4\mu \text {m/RIU}$) around the resonant angle [see Fig. 7(a)]. For the spatial IFS, two positive peaks ($224.4\lambda /\text {RIU}\approx 1.19\times 10^2\ \mu \text {m/RIU}$ and $1127\lambda /\text {RIU}\approx 6.0\times 10^3\ \mu \text {m/RIU}$) and a negative peak ($-4259\lambda /\text {RIU}\approx 2.27\times 10^3\ \mu \text {m/RIU}$) are observed around the resonant angle [see Fig. 7(c)]. The ultra-high sensitivities of spatial GHS and IFS are the result of the excitation of SEPs.
Fig. 7. The sensitivities for GHS-based gas sensor contributed by (a) spatial GHS $S_\Delta ^{GH}$, and (b) angular GHS $S_\Theta ^{GH}$. The sensitivities for IFS-based gas sensor contributed by (c) spatial IFS $S_\Delta ^{IF}$, and (d) angular IFS $S_\Theta ^{IF}$.
The TDBC thickness-dependent RI sensitivities for the composite GHS $S^{GH}$ and composite IFS $S^{IF}$ are shown in Fig. S4. The maximum RI sensitivity is obtained with a 60 nm-thick TDBC film, where the excitation of SEPs is strongest. It should be noted that the spatial and angular shifts are two different manifestations of the optical displacement, and occur simultaneously when lossy media (TDBC in this model) is included [45]. Therefore, the measured GHS and IFS in experiments are the total beam displacement or composite shift $\delta ^{GH(IF)}$. Thus for real practical gas sensing, the obtained RI sensitivity is the combination of the RI sensitivity for the spatial and angular shifts, i.e., $S^{GH(IF)}$.
5. Conclusions and discussions
In this work, we have theoretically investigated the effect of SEPs on the spatial and angular GHSs and IFSs for the TM and TE polarized light in a modified K-R configuration consists of BK7 prism, TDBC film, and air. Both the spatial and angular GHSs and IFSs for the TM polarized light are remarkably enhanced around the resonant angle of SEP. In contrast, the magnitudes of spatial and angular GHSs and IFSs for TE polarized light are much smaller. Based on the SEP enhanced GHS and IFS, a highly sensitive gas sensor is proposed. Both positive and negative RI sensitivities are achieved, which depend on the incident angle. The maximum magnitudes of the RI sensitivities for the GHS- and IFS-based gas sensors are $6.27\times 10^5 \ \mu \text {m/RIU}$ and $7.28\times 10^4 \ \mu \text {m/RIU}$, respectively. The RI sensitivities of the GHS-based (or IFS-based) gas sensors are mainly contributed by the angular shift effect. However, due to the simultaneous occurrence of spatial and angular shifts in the lossy system, the measured optical shift is the combination of spatial and angular shifts. Our results provide the potential applications of SEPs at room temperature and open up new opportunities for high sensitivity gas sensing.
As mentioned in Section 1, SPPs also show the enhancement effect on the spatial and angular GHSs and IFSs. L. Salasnich theoretically investigated the SPP enhanced spatial and angular GHSs and IFSs with an optimized K-R configuration [27]. The maximum magnitude of the calculated SPP enhanced spatial (angular) GHS is ${\sim }80\lambda$ (${\sim }90\theta ^2_0$), which is higher than that of the SEP enhanced spatial (angular) GHS with the value of $51\lambda$ ($79\theta ^2_0$). Although the SEPs exhibit less pronounced enhancement effect on the spatial and angular GHSs as compared to the SPPs, the angular and spatial IFSs are significantly enhanced with SEPs. Particularly, the maximum magnitude of ${\sim }1.1\lambda$ is obtained for the SPP enhanced spatial IFS of the TM polarized light, which is a quarter of that achieved with SEPs ($4.86\lambda$). The SPP enhanced angular IFS of TM polarized light has a maximum magnitude of ${\sim }2\theta ^2_0$. In contrast, a significantly enhanced angular IFS with the magnitude of $20.5\theta ^2_0$ is obtained with SEPs.
Since the SPPs exhibit more pronounced enhancement effect on the spatial and angular GHSs than the SEP, it is expected to obtain higher RI sensitivity with the SPP-GHS than the SEP-GHS. For example, a sensitivity of $\sim 5.5\times 10^{4}\ \mu$m/RIU was obtained with the SPP enhanced spatial GHS sensor for the RI around 1.33 [29], which is higher than that of SEP enhanced spatial GHS ($1.18\times 10^{4}\ \mu$m/RIU) in the RI range from 1.0 to 1.001. However, it should be noted that the magnitudes of spatial and angular IFSs with SEPs are greater than those with SPPs, higher RI sensitivity should be achieved with SEP-IFS.
Funding
National Natural Science Foundation of China (62205078); China Postdoctoral Science Foundation (2021M690235); National Research Foundation Singapore and A*STAR under its Quantum Engineering Programme (NRF2021-QEP2-02-P03); Singapore University of Technology and Design for the Start-Up Research (SRG SMT 2021 169); National Research Foundation Singapore (NRF2021-QEP2-03-P09).
Acknowledgments
Y. Xu acknowledges the project funded by National Natural Science Foundation of China (Grant No. 62205078) and China Postdoctoral Science Foundation (Grant No. 2021M690235). L. K. Ang acknowledges the National Research Foundation Singapore and A*STAR under its Quantum Engineering Programme (NRF2021-QEP2-02-P03). L. Wu acknowledges the Singapore University of Technology and Design for the Start-Up Research Grant SRG SMT 2021 169 and National Research Foundation Singapore via Grant No. NRF2021-QEP2-03-P09.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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