Precision improvement of surface plasmon resonance sensors based on weak-value amplification

Lan Luo; Xiaodong Qiu; Linguo Xie; Xiong Liu; Zhaoxue Li; Zhiyou Zhang; Jinglei Du

doi:10.1364/OE.25.021107

1. Introduction

Since its first proof-of-concept demonstration by Nylander and Liedberg in 1982 [1, 2], the surface plasmon resonance (SPR) sensor technology has been widely applied to food safety, pharmaceutics and immunoassay due to its advantages of label-free and real-time detection. Over the past three decades, researchers have made a great effort to improve the performance of the SPR sensors, and several types of commercial SPR sensors have been developed for practical applications. However, how to improve the precision is always of concern, especially for the detection of the objects with extreme low concentrations or low affinity. Several methods used to improve the precision of SPR sensors have been proposed, such as long range SPR [3], differential measurements [4], nano-structure-enhanced SPR [5]. Nevertheless, these methods require accurately control of the thickness of metal film, complex instruments, and advanced nano-fabrication technology.

Recently, weak-value amplification (WVA), originally proposed by Aharonov, Albert, and Vaidman in 1988 [6], has stepped into a public spotlight. With appropriate pre- and post-selections, WVA can improve the precision of the measurement limited by the technical noise, since the detected signal is enhanced while the technical noise is retained or even suppressed [7–11]. Using the WVA technique, Hosten and Kwiat demonstrated the first observation of the spin Hall effect of light with an incredible sensitivity of 0.1nm [12]. Since the WVA has been widely used for estimation of small parameters of interest, such as beam shift [13–17], nanometal film thickness [18], graphene layers [19], optical phase [20–23], velocity [24], magneto-optical constant [25] and even nonlinear effect of a single photon [26]. More recently, the bio-sensors based on WVA has also been proposed, e.g., glucose concentration sensor [27], phase-sensitive label-free sensor [28], and chiral sensor [29]. The precision of the WVA techniques met and even surpassed that of the standard techniques in most of these experiments.

In this work, we investigate the precision of prism coupler-based SPR sensors with intensity measurement when the post-selection comes into play. A propagation model to describe the post-selected intensity is established. The technical noise depending on the light intensity is suppressed by the post-selection, while the intensity variation signal is enhanced by WVA. In addition, the intensity contrast ratio (ICR) is introduced to reveal the mechanism of the precision improvement based on WVA. Our results show that the precision can be improved by nearly one order of magnitude compared to the conventional SPR sensor.

2. Theoretical analysis

We first establish a theory model in the WVA formalism to describe the post-selected SPR sensing system. Consider a Gaussian light beam prepared in a linear polarization state,

| ψ_{i} 〉 = \cos α | H 〉 + \sin α | V 〉,

here, |H〉 and |V〉 denote horizontal and vertical polarization states, respectively (the stokes polarization operator

\hat{A} = | H 〉 〈 H | - | V 〉 〈 V |

is referred to as the observable of the system). α is a experimental controlled parameter corresponding to the different incident polarization states. Upon reflection, the polarization state evolves into

| ψ_{p r e} 〉 = | R_{p} | \cos α \exp (i ϕ_{p}) | H 〉 + | R_{s} | \sin α \exp (i ϕ_{s}) | V 〉,

where

R_{p, s} = \frac{r_{p, s} + r_{p, s}^{'} \exp (2 i k_{0} \sqrt{n_{2}^{2} - n_{1}^{2} \sin^{2} θ} d)}{1 + r_{p, s} r_{p, s}^{'} \exp (2 i k_{0} \sqrt{n_{2}^{2} - n_{1}^{2} \sin^{2} θ} d)},

denotes the generalized Fresnel reflection coefficient of a three-layers structure for the horizontal and vertical polarizations, respectively (here is a glass-film-sample structure). ϕ_p,_s is the phase angle of the generalized Fresnel reflection. r_p,_s and

r_{p, s}^{'}

are the Fresnel reflection coefficients at the glass-film and film-sample interfaces, respectively. θ and k₀ are respectively the incident angle and central wave vector. n₂ and n₁ represent the refractive index of the metal film and glass, respectively (the refractive index of sample of interest is included in

r_{p, s}^{'}

). d denotes the thickness of film. From the angular spectrum theory, a light beam involves different spectrum components corresponding to different incident angles. The Fresnel reflection coefficients are sensitive to the incident angle near the resonance angle. Therefore, we consider the first order approximation of their Taylor expansions of the in-plane wave vector k_ix,

R_{p, s} (k_{i x}) = R_{p, s (k_{i x} = 0)} + k_{i x} {[\frac{\partial R_{p, s} (k_{i x})}{\partial k_{i x}}]}_{k_{i x} = 0} .

In our case, the polarization state |ψ_pre〉 [Eq. (2)] without refractive index change of sample plays the role of the pre-selection. In the presence of a small refractive index change, the horizontally polarized component will undergoes a significantly amplitude (i.e., intensity) and phase variations as a consequence of SPR. However, it is quite different for the vertical polarization. Therefore, two orthogonal polarization components undergo different amplitude and phase evolutions (i.e., polarization-dependent amplitude and phase evolutions corresponding to the weak interaction in the WVA formalism). Consider a small refractive index change, the polarization state is given as

\begin{matrix} | ψ^{'} 〉 = (| R_{p} | + Δ | R_{p} |) \cos α \exp [i (ϕ_{p} + Δ ϕ_{p})] | H 〉 \\ + (| R_{s} | + Δ | R_{s} |) \sin α \exp [i (ϕ_{s} + Δ ϕ_{s})] | V 〉, \end{matrix}

here, Δ|R_p,_s | and Δϕ_p,_s, corresponding to the refractive index change of concern, denote the amplitude and phase variations for horizontally and vertically polarized components, respectively. The above eqnarray can be recast in the WVA formalism

\begin{matrix} | ψ^{'} 〉 ≃ \exp (i \bar{ϕ} + \bar{R}) [| R_{p} | \cos α \exp (i ϕ_{p}) \exp (\frac{i Δ ϕ + Δ R}{2}) | H 〉 + | R_{s} | \sin α \exp (i ϕ_{s}) \exp (\frac{i Δ ϕ + Δ R}{2}) | V 〉] \\ = \exp (i \bar{ϕ} + \bar{R}) \exp (\hat{A} \frac{i Δ ϕ + Δ R}{2}) | ψ_{p r e} 〉, \end{matrix}

where

\bar{ϕ} = \frac{Δ ϕ_{p} + Δ ϕ_{s}}{2},

\bar{R} = \frac{1}{2} (\frac{Δ | R_{p} |}{| R_{p} |} + \frac{Δ | R_{s} |}{| R_{s} |}),

Δ ϕ = Δ ϕ_{p} - Δ ϕ_{s},

Δ R = \frac{Δ | R_{p} |}{| R_{p} |} - \frac{Δ | R_{s} |}{| R_{s} |} .

In the WVA strategy, a strong interference between two eigen-states (H and V polarizations) induced by the post-selection leads to a amplified detected signal. Here, the light beam is post-selected in

| ψ_{p o s t} 〉 = \cos β \exp (i \frac{ϕ}{2}) | H 〉 + \sin β \exp (- i \frac{ϕ}{2}) | V 〉

here, β is the post-selected angle. ϕ denotes the post-selected phase which is used to control the phase difference [ϕ_p −ϕ_s, Eq. (2)] between the horizontally and vertically polarized components due to reflection. The post-selected SPR sensor is based on the intensity measurement. The intensity of the post-selected light is given by

\begin{array}{l} I = {| 〈 ψ_{p o s t} | \exp (i \bar{ϕ} + \bar{R}) \exp (\hat{A} \frac{i Δ ϕ + Δ R}{2}) | ψ_{p r e} 〉 |}^{2} \\ = {| 〈 ψ_{p o s t} | \exp (i \bar{ϕ} + \bar{R}) (\cos \frac{Δ ϕ}{2} + i \hat{A} \sin \frac{Δ ϕ}{2}) (\cosh \frac{Δ R}{2} + \hat{A} \sinh \frac{Δ R}{2}) | ψ_{p r e} 〉 |}^{2} \\ = {| 〈 ψ_{p o s t} | ψ_{p r e} |}^{2} {| \exp (i \bar{ϕ} + \bar{R}) |}^{2} {{[1 - I m (A_{ω}) \frac{Δ ϕ}{2} + R e (A_{ω}) \frac{Δ R}{2}]}^{2} + {[R e (A_{ω}) \frac{Δ ϕ}{2} + I m (A_{ω}) \frac{Δ R}{2}]}^{2}} \\ = I_{0} \exp (2 \bar{R}) [1 + \frac{1}{4} (Δ ϕ^{2} + Δ R^{2}) {| A_{ω} |}^{2} + R e (A_{ω}) Δ R - I m (A_{ω}) Δ ϕ], \end{array}

where

\begin{array}{l} I_{0} = {| 〈 ψ_{p o s t} | ψ_{p r e} 〉 |}^{2} \\ = {| R_{p} |}^{2} \cos α^{2} \cos β^{2} + {| R_{s} |}^{2} \sin α^{2} \sin β^{2} + \frac{1}{2} | R_{p} | | R_{s} | \sin 2 α \sin 2 β \cos (ϕ_{p} - ϕ_{s} - ϕ) \end{array}

denotes the initial light intensity in the absence of refractive index change and

\begin{array}{l} A_{ω} = \frac{〈 ψ_{p o s t} | \hat{A} | ψ_{p r e} 〉}{〈 ψ_{p o s t} | ψ_{p r e} 〉} \\ = \frac{| R_{p} | \cos α \cos β \exp [i (ϕ_{p} - ϕ_{s} - ϕ)] - | R_{s} | \sin α \sin β}{| R_{p} | \cos α \cos β \exp [i (ϕ_{p} - ϕ_{s} - ϕ)] + | R_{s} | \sin α \sin β} . \end{array}

is the weak value of the observable

\hat{A}

. The approximations in Eq. (12) are feasible with Δϕ/2 ≪ 1 and ΔR/2 ≪ 1.

From Eq. (12), the intensity of the post-selected light is associated with the weak-value. Furthermore, the larger weak-value is, the lower initial light intensity I₀ is [Eqs. (13) and (14)]. However, it is not clear what role WVA plays in our scheme. Here, we introduce the intensity contrast ratio (ICR) to reveal the mechanism of the precision improvement based on WVA. The ICR is given by

\begin{array}{l} η = \frac{Δ I}{I_{0}} \\ ≃ \exp (2 \bar{R}) [1 + \frac{1}{4} (Δ ϕ^{2} + Δ R^{2}) {| A_{ω} |}^{2} + R e (A_{ω}) Δ R - I m (A_{ω}) Δ ϕ] - 1, \end{array}

here, ΔI = I − I₀ denotes the intensity variation of the post-selected light (detected signal) corresponding to the refractive index change of sample. As we all know, the practical measurements always suffer from various types of technical noise. In our experiment, the main technical noise is induced by the power fluctuation of light source and imperfection of optical element which are proportional to the light intensity I₀, and the electronic noise of detector which is independent of I₀. Therefore, to some extent, the ICR can be interpreted as the ratio between the magnitude of detected signal and the magnitude of noise, i.e., signal-to-noise ratio (SNR). From Eq. (15), one can obtain a large ICR by constructing a large weak-value with post-selection, that is to say, a high measured precision can be obtained based on WVA. However, we should note that the ICR is not completely equal to the SNR. In addition, due to the significant reduction of light intensity induced by the post-selection process, a low-saturation detector with low electronic noise can be used for intensity measurement.

3. Experimental observation

The experimental setup is shown in Fig. 1. A monochromatic Gaussian light beam at a wavelength of 632.8nm is generated by the He-Ne laser, and the half-wave plate (HWP) is used to adjust the light intensity to avoid the saturation of the detector. The first Glan polarizer (P1) is rotated by an angle of α with respect to the horizontal direction to prepare the incident polarization state, see Fig. 1(b), α is chosen as 0.084rad. Then the light beam is reflected at the prism-gold film interface. The concentration of sample (sodium chloride solution) is changed from 0% to 0.23%, and the corresponding refractive index change is 0 ~ 4.1 × 10⁻⁴RIU. After reflection, the phase difference between two polarization components is controlled using the Soleil-Babinet compensator [SBC, see Fig. 1(c)]. The second Glan polarizer (P2) is used for polarization post-selection [Fig. 1(d)]. Finally, the intensity variation is recorded by the CCD.

Fig. 1 (a) Experimental setup of SPR sensor based on WVA. Light source, He-Ne laser; HWP, half-wave plate; P1 and P2, Glan laser polarizers; SBC, Soleil-Babinet compensator; CCD, Charge-coupled device. (b), (c), and (d) show the different orientations of the P1, SBC, and P2, respectively.

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In our experiment, the gold films (d = 42.9nm and 57.2nm) deposited on the surface of BK7 prisms were prepared by using the vacuum thermal evaporation technique. We note that the optimal thickness of the gold film for SPR is 46nm, however, the thickness of the gold film is difficult to control accurately under our experimental conditions.

The experimental results, along with the theoretical predictions obtained from Eqs. (12) and (15), are shown in Fig. 2. We have measured the intensity variation of the post-selected light at the incident angles of 73.0° and 73.8° for two SPR sensing systems (d = 42.9nm and 57.2nm), respectively [Figs. 2(a) and 2(b)]. The post-selected phases ϕ are −1.275rad and −1.327rad, respectively. The post-selected angles β are chosen as 1.033rad and 1.802rad, respectively. As we can see, the intensity of the post-selected light is highly sensitive to the refractive index change of the sample, especially for the case of d = 42.9nm. We have calculated the ICR by using the measured values of intensity. As shown in Figs. 2(c) and 2(d), the ICR for the case of d = A2.9nm is always larger than that for the case of d = 57.2nm. From our above theoretical analysis, a larger ICR implies a higher measured precision δn. By using δn = δI/(ΔI/Δn) (δI denotes the standard deviation of light intensity obtained from the statistic of measurements), we can estimate the measured precision. In these measurements, δI is always less than 1 nw, and thus the corresponding measured precision of two SPR sensing systems is 2.9 × 10⁻⁷ RIU and 8.2 × 10⁻⁷ RIU, respectively.

Fig. 2 Experimental results with post-selection. (a) and (b) show the light intensity changing with the refractive index of sample for the SPR sensing systems with d = 42.9nm and 57.2nm, respectively. (c) and (d) show the corresponding ICR for different refractive index. The dots and curves denote the experimental measured values and theoretical predictions, respectively.

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In addition, we have performed the SPR refractive index sensing without post-selection. The incident light is prepared in the horizontal polarization state (i.e., α = 0). As shown in Figs. 3(a) and 3(b), the incident angles are chosen as the same as that for the post-selected SPR sensing systems (73.0° and 73.8°). The intensity of light without post-selection is much larger than that in the case of with post-selection. By calculating the ICR [see Figs. 3(c) and 3(d)], we found that the light intensity shows lower sensitivity to refractive index change. In the same way, we obtained the corresponding measured precision (5.5 × 10⁻⁶ RIU and 2.6 × 10⁻⁵ RIU) of the SPR sensing systems without post-selection.

Fig. 3 Experimental results without post-selection. The upper row shows the light intensity changing with the refractive index of sample for two SPR sensing systems (d = 42.9nm and 57.2nm) at different incident angles. The lower row shows the corresponding ICR for different refractive index. The dots and curves denote the experimental measured values and theoretical predictions, respectively.

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In order to further confirm the advantage of our method, both two SPR sensing systems without post-selection have been demonstrated at the other incident angles (72.0° and 73.0°). As shown in Figs. 3(e)–3(h), the light intensity increases significantly and the ICR is also improved with respect to the case of Figs. 3(a)–3(d). The corresponding measured precision is 2.5 × 10⁻⁶ RIU and 6.3 × 10⁻⁶ RIU, respectively. Therefore, the measured precision of the refractive index is roughly proportional to the ICR in our experiment.

By comparison of the performances of two SPR sensing systems in the case of both with and without post-selection, we found that, the ICR increases significantly with post-selection and the measured precision can be improved by nearly one order of magnitude. However, the measured precision is not completely proportional to the ICR. This is because the electronic noise of CCD independent of the light intensity cannot be neglected when the intensity of the post-selected light is significantly reduced. In addition, the measured precision of the SPR sensing system with d = 42.9nm is higher. Actually, the low reflectivity of the horizontal polarization plays a role of self-post-selection which can reduce the influence of stray light. Therefore, we infer that the SPR sensing system likely performs better when the thickness of the gold film is closer to the optimal thickness (~ 46nm).

4. Conclusion

In conclusion, we have theoretically and experimentally demonstrated a high precision SPR refractive index sensing based on the WVA. This method have the potential to outperform the conventional SPR sensor by nearly one order of magnitude. We have found that, the intensity variation signal is significantly enhanced with respect to the main technical noise which is dependent on the intensity of the post-selected light. In addition, our method is robust against thickness deviation of the metal film. These researches may be applied to high sensitive chemical and bio-sensing.

Funding

National Natural Science Foundation of China (Grant Nos. 11674234); Sichuan Provincial Funds for Distinguished Young Scientists (Grant Nos. 2017JQ0021).

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Precision improvement of surface plasmon resonance sensors based on weak-value amplification

Abstract

1. Introduction

2. Theoretical analysis

3. Experimental observation

4. Conclusion

Funding

References and links

Cited By

Figures (3)

Equations (15)

Optics Express